New Frontiers in Practical Risk Management

English edition Issue n. 7 - Summer 2015

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New Frontiers in Practical Risk ManagementYear 2 - Issue Number 7 - Summer 2015

Published in September 2015First published in October 2013

Last published issues are available online:www.iasonltd.comwww.energisk.org

Summer 2015

NEW FRONTIERS IN PRACTICAL RISK MANAGEMENT

Editors:Antonio CASTAGNA (Co-founder of Iason ltd and CEO of Iason Italia srl)Andrea RONCORONI (ESSEC Business School, Paris)

Executive Editor:Luca OLIVO (Iason ltd)

Scientific Editorial Board:Fred Espen BENTH (University of Oslo)Alvaro CARTEA (University College London)Antonio CASTAGNA (Co-founder of Iason ltd and CEO of Iason Italia srl)Mark CUMMINS (Dublin City University Business School)Gianluca FUSAI (Cass Business School, London)Sebastian JAIMUNGAL (University of Toronto)Fabio MERCURIO (Bloomberg LP)Andrea RONCORONI (ESSEC Business School, Paris)Rafal WERON (Wroclaw University of Technology)

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Articles submission guidelinesArgo welcomes the submission of articles on topical subjects related to the risk management. Thetwo core sections are Banking and Finance and Energy and Commodity Finance. Within these twomacro areas, articles can be indicatively, but not exhaustively, related to models and methodologiesfor market, credit, liquidity risk management, valuation of derivatives, asset management, tradingstrategies, statistical analysis of market data and technology in the financial industry. All articlesshould contain references to previous literature. The primary criteria for publishing a paper are itsquality and importance to the field of finance, without undue regard to its technical difficulty. Argois a single blind refereed magazine: articles are sent with author details to the Scientific Committeefor peer review. The first editorial decision is rendered at the latest within 60 days after receipt of thesubmission. The author(s) may be requested to revise the article. The editors decide to reject or acceptthe submitted article. Submissions should be sent to the technical team (argo@iasonltd.com). LaTex orWord are the preferred format, but PDFs are accepted if submitted with LaTeX code or a Word file ofthe text. There is no maximum limit, but recommended length is about 4,000 words. If needed, forediting considerations, the technical team may ask the author(s) to cut the article.

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3

NEW FRONTIERS IN PRACTICAL RISK MANAGEMENT

Table of Contents

Editorial pag. 05

Antonio Castagna, Andrea Roncoroni and Luca Olivo introduce the new topics of this n. 7 Argo edition.

energy & commodity finance

Off-shore Wind Investmentunder Uncertainty: Part II pag. 07

James Savage, Mark Cummins and Jean Charpin

OW Bunker: from IPO to Bankruptcy pag. 15

Alessandro Mauro

banking & finance

Off-shore Wind Investmentunder Uncertainty: Part II pag. 17

James Savage, Mark Cummins and Jean Charpin

Modelling of Libor-Ois Basis pag. 27

Antonio Castagna, Andrea Cova, Matteo Camelia

Front Cover: Giacomo Balla Pessimismo e ottimismo, 1923.

EDITORIAL

Dear Readers,not a really relaxed Summer the one we just went through,

characterised by great movements in financial markets. Grexitscenario before and (especially) China slowdown after causeda sharp fall of prices in the main equity indeces, revivingthe fear of crisis and recession that seemed to be far away.

Fall of asset prices may be expected in financial markets, but mayalso be the trigger for cases of sudden turmoil, especially in thefield of Energy & Commodities finance. The Summer 2015 issue isopened by the second part of the article written by James Savage,Mark Cummins, and Jean Charpin: they report the results of thenew real option valuation model introduced in the previous Argoissue. The section continues with the interesting contribution ofAlessandro Mauro, Energy Risk professional, that illustrates the caseof OW Bunker: this Danish company was one of the biggest tradersof bunker oil in the world before its end in November 2014, due toa mixture of bad economic trends, fraud and risk management loss.

The Banking & Finance section carries on with the contribution ofAntonio Castagna, Andrea Cova and Matteo Camelia, that introducea set of models aimed at explaining the market phenomenologyof Libor forward fixings implied in swap prices. The models theypresent are all based on the idea that the Libor fixings refer to apanel of primary banks whose composition may change over time.

We conclude as usual by encouraging the submissionof contributions for the next issues of Argo in orderto improve each time this newsletter. Detailed informa-tion about the process is indicated at the beginning.

Enjoy your reading!

Antonio CastagnaAndrea Roncoroni

Luca Olivo

Summer 20155

NEW FRONTIERS IN PRACTICAL RISK MANAGEMENT

AAAA

Energy & CommodityFinance

Real Option Model

Business Story

6

Offshore Wind Investmentunder UncertaintyPart II

The authors present a new real op-tions model, different from the al-ready known ones, aimed at provid-ing in a simultaneously way optimisedoutcomes with respect to the timing,the size and the financing leverageof investments. In this second partresults of the previously describedanalysis are shared and discussed.

James SAVAGEMark CUMMINSJean CHARPIN

In this two-part article series 1, we apply a novelreal options model, recently proposed in theliterature, which provides simultaneously op-timised outcomes with respect to (i) when to

invest, (ii) how much to invest, and (iii) by howmuch to lever the financing of the investment. Themodel is superior to previous models of this typein its joint optimisation of investment timing, sizeand leverage. We value an assumed offshore windinvestment located in Swedish territorial waters,giving a practical assessment of the investment de-cisions. We define a base case scenario and presenta comprehensive scenario analysis, which exhibitshow the model responds to variations in its parame-ter inputs, i.e. its key investment drivers of capacitycapital cost, operating cost, net capacity factor, dis-count rate, taxation rate and bankruptcy cost, inaddition to expected growth rate and volatility of

Nordic electricity returns. The investment entryprice in the base case is shown to be approximatelyEUR 48, with optimal installed capacity being ap-proximately 2.5MW of the proposed 3MWs andoptimal financing comprising approximately 89%debt. We identify an economic limitation of themodel for offshore wind investment. Counterin-tuitively, the higher the net capacity factor of thewind farm, and hence production elasticity, thenthe later the decision is made to invest under themodel. Because wind farms have no fuel costs thenthe higher the net energy yield, the greater the re-turn on investment is for the same capital costs. Thenormal commercial expectation here would be forearlier, rather than later, investment. Despite thislimitation, and some others discussed, the modelshows the importance of considering and under-standing how investment drivers to offshore windinvestment impact on the investment decisions oftiming, size and leverage jointly.

Results and Discussion

For the scenario analysis of the SVM applied to ourwind farm case study, we consider a range of pa-rameter scenarios for contrast against the base casescenario set out in the previous section. This servesto provide insights into the investment decisions,i.e. optimal investment timing, investment size andinvestment leverage, that would be made aroundthe wind farm project and how the investment deci-sions would change with parameter variations. Forthe parameter scenarios, each input parameter istaken in turn, variations of this parameter are thenconsidered within reasonable bounds while the re-

1Part I of the article has been published in Argo Issue n. 06 - Spring 2015, available online at: http://www.energisk.org/research/

Summer 20157

REAL OPTIONS MODEL

Scenario Base 2 3 4 5 6 7 8NCF 42% 36% 38% 40% 44% 46% 48% 50%

a 0.21 0.07 0.17 0.21 0.25 0.29 0.33 0.37β 4.75 14.275 8.385 4.754 3.957 3.411 3.013 2.710P* 48.77 37.39 40.44 44.09 54.92 63.37 75.73 95.58Q* 1.03 0.89 0.89 0.94 1.20 1.46 1.93 2.80c* 43.47 25.23 28.62 34.19 59.25 87.53 144.29 277.82Pb 32.49 25.91 27.67 29.77 36.07 41.01 48.24 59.90Pe 12.26 12.26 12.26 12.26 12.26 12.26 12.26 12.26K* 2.46 2.47 2.35 2.35 2.72 3.18 4.02 5.61

c* (%) 9.75% 10.19% 10.03% 9.89% 9.61% 9.48% 9.36% 9.25%% Debt 89.17% 88.41% 88.66% 88.91% 89.46% 89.76% 90.08% 90.42%

TABLE 1: Production Elasticity (a) Scenarios

maining parameters are maintained at the base casechoices. Tables 1-7 present the detailed results fromthis parameter scenario analysis.

Taking Table 1, which examines how the SVMmodel responds to changes in the NCF of the windfarm and hence the production capacity a, it canbe seen that in the base case scenario the entryprice trigger is estimated at EUR 48.77. This isabove the average price for the rolling 3-year fu-tures contract in the sample period but well withinthe maximum-minimum range. In contrast to thisbase case scenario, it is notable that in the caseof the lower NCFs of 38% and 36%, the optimalP∗ falls to EUR 40.44 and EUR 37.39 respectively,close to and below the average power price. Forthe higher NCFs, the optimal P∗can be seen to in-crease quickly with increasing production elasticityparameter a, to levels that ultimately (for the 48%and 50% NCFs) exceed that maximum 3-year fu-tures power price observed over the sample period.A question mark is therefore raised here aroundthe sensitivity of the SVM to the choice of a. Amore fundamental economic issue emerges, whichcalls into question the suitability of the SVM for thewind farm case study and renewable energy basedpower projects more generally. What changes in thelevel of production elasticity demonstrate is thatthe higher a becomes, the later the decision is madeto invest. For wind farms, though, this is a coun-terintuitive finding. In other words, because windfarms have no fuel costs, the higher the net energyyield is, the greater the return on investment is forthe same level of capital costs, kK. In addition, themore energy that is produced in UT per unit of K,the less sensitive investors should be to volatilityin the state variable P. As such, the normal com-mercial expectation here would be that the higherthe value of any production output parameter, theearlier the investment should be made. This is dueto both the project and the debt being valued underthe SVM as functions of a perpetual stream of net

operating cash flows, and not as functions of thecost of building the project, kK.

Converting Q∗ back into megawatts of installedcapacity K∗, according to K∗ = Q∗/NCF,2 it can beseen that this increases with increasing NCF andhence production elasticity. The base case NCFof 42% suggests an optimal installed capacity ofapproximately 2.5MW of the proposed 3MW. Incontrast, for an NCF of 46% and higher, the K∗islarger than the proposed 3MW and in fact for the50% NCF it is twice that at approximately 6MW.On the one hand, it is reasonably commonplacefor large offshore wind farms to be developed inphases over time. However, the concept that thedevelopment of a single wind farm needs to havethe flexibility to be up to multiple times some ini-tial size gives rise to a number of major practicalchallenges. For example, to be able to implementsuch an investment strategy properly, it would benecessary to procure a site, a set of planning ap-proval conditions, and a grid connection agreementthat all provide the investor with enough flexibility,thereafter, to significantly vary the ultimate sizeof the wind farm. Moreover, even if such arrange-ments could be entered into, the right to hold suchoptions would not be cost-free and certainly wouldnot be for any significant duration. To take an ex-treme example, if every wind farm developer hadthe unfettered right to connect large amounts ofwind power to the grid at a time of their choosing,then this would result in large amounts of gener-ation looking to connect when P is at an optimallevel and very little generation looking to connectwhen P is sub-optimal. In other words, in the hy-pothetical situation where this could happen, whatwould begin as an optimal course of action thatwas being followed by each individual developerwould soon culminate in the collapse of the systemas a whole (i.e. either too much or too little newcapacity being added).

Finally, in terms of the optimal debt c∗, it can2As we use a unit time of one hour, this allows for consistent conversion between MWh production and MW capacity.

8energisk.org

Scenario Base 2 3 4 5k (euro/MWh) 104 83 97 110 117

P* 48.77 48.77 48.77 48.77 48.77Q* 1.03 1.10 1.05 1.02 1.00c* 43.47 46.17 44.29 42.83 42.13Pb 32.49 32.49 32.49 32.49 32.49Pe 12.26 12.26 12.26 12.26 12.26K* 2.46 2.61 2.51 2.42 2.38

c* (%) 9.75% 9.75% 9.75% 9.75% 9.75%% Debt 89.17% 89.17% 89.17% 89.17% 89.17%

TABLE 2: Unit Capital Cost (k) Scenarios.

Scenario Base 2 3 4 5w (euro/MWh) 25.50 15.50 20.50 30.50 35.50

P* 48.77 29.65 39.21 58.34 67.90Q* 1.03 0.90 0.97 1.08 1.13c* 43.47 23.14 32.98 54.54 66.10Pb 32.49 19.75 26.12 38.86 45.24Pe 12.26 7.45 9.86 14.66 17.07K* 2.46 2.15 2.32 2.58 2.69

c* (%) 9.75% 9.75% 9.75% 9.75% 9.75%% Debt 89.17% 89.17% 89.17% 89.17% 89.17%

TABLE 3: Unit Operational Cost (w) Scenarios

be seen that with increasing NCF and hence pro-duction elasticity, the percentage coupon c∗/D (P∗)remains relatively stable in the approximate rangeof 9-10%, with the optimal debt level being approxi-mately 89% (in most cases). Hence, according to theSVM, offshore investment would require an optimalfinancial structure dominated by debt financing.

Although the above discussion raises some con-cerns about the suitability of the Sarkar [5] modelin appropriately capturing the production elastic-ity associated with offshore wind, the discussion tofollow shows the importance of considering and un-derstanding how the other key investment driversimpact on investment decisions jointly. We turnfirst to Tables 2 and 3, which presents scenario anal-ysis results for the unit capital cost k (in euro perMWh terms) and the unit operating cost w respec-tively. Variations in k do not have an effect on theentry trigger price P∗, with a consistent price ofEUR 48.77 reported for all scenarios in line with thebase case. This characteristic of the SVM model isconsistent with the dynamics discussed by Sarkar[5]. In contrast to this, what does change is theoptimal production capacity Q∗ determined by theSVM. As might be expected, as the unit capital costincreases, the incentive to put in place more in-stalled capacity for increased production decreases.With varying unit operating cost w, increases in thiscost are accompanied by higher price entry triggersP∗. Indeed, for the operating costs of EUR 30.50and EUR 35.50 it can be seen that the P∗ exceedsthe base case by approximately EUR 10 and EUR20 per megawatt hour respectively. Of note also is

that the higher the unit operating cost w then thehigher the optimal production capacity Q∗ returnedby the SVM. The percentage coupon c∗/D (P∗) isunchanged across all of the k and w scenarios atalmost 9.75%, while the optimal debt level matchesthe base case at 89.17%.

Given the assumption of the GBM process forthe rolling three-year futures price, Table 4 presentsresults for varying volatility coefficient σ. Higherlevels of market volatility would have led to modestincreases in the level of the price entry P∗, hittingapproximately EUR 54 per MWh for the highestassumed volatility case of 19.6%. The optimal pro-duction capacity Q∗ in all cases is approximately1MWh. The percentage coupon c∗/D (P∗) variesacross the scenarios, increasing marginally withincreased volatility, while the optimal debt leveldecreases with increasing volatility, from approxi-mately 91% for the lowest volatility case down tounder 88% for the highest volatility case. For thediscount rate ρ, some interesting findings emergefrom Table 5. With increasing discount rate, theoptimal installed capacity suggested by the modelcan be seen to reduce, with a K∗ equal to approxi-mately 2MW at the 10 discount rate, a full MW lessthan that proposed. The optimal entry price trig-ger P∗ can also be seen to decline with increasingdiscount rate, with the lowest 5% rate command-ing a price entry level well above the maximumobserved price historically. The reason for this pat-tern in entry prices is that as ρ increases in value,there is a decline in the net value gained from wait-ing for further information. This results in a lower

Summer 20159

REAL OPTIONS MODEL

Scenario Base 2 3 4 5 6 7 8 9 10 11σ 0.171 0.146 0.151 0.156 0.161 0.166 0.176 0.181 0.186 0.191 0.196P* 48.77 44.08 44.96 45.87 46.81 47.77 49.80 50.87 51.97 53.11 54.29Q* 1.03 0.99 1.00 1.01 1.02 1.02 1.04 1.05 1.06 1.07 1.08c* 43.47 35.02 36.56 38.17 39.85 41.62 45.42 47.45 49.58 51.82 54.16Pb 32.49 31.06 31.32 31.60 31.88 32.18 32.82 33.16 33.51 33.88 34.27Pe 12.26 13.03 12.87 12.72 12.56 12.41 12.11 11.96 11.81 11.67 11.53K* 2.46 2.36 2.38 2.40 2.42 2.44 2.48 2.50 2.52 2.54 2.56

c* (%) 9.75% 9.39% 9.46% 9.53% 9.60% 9.67% 9.82% 9.89% 9.96% 10.03% 10.11%% Debt 89.17% 91.01% 90.61% 90.23% 89.86% 89.51% 88.85% 88.54% 88.24% 87.95% 87.67%

TABLE 4: Volatility Coefficient (σ) Scenarios.

Scenario Base 2 3 4 5 6ρ 0.08 0.05 0.06 0.07 0.09 0.10

k(euro/MWh) 104 81 89 96 111 119P* 48.77 91.53 63.35 53.76 45.65 43.47Q* 1.03 1.77 1.38 1.17 0.93 0.86c* 43.47 264.24 106.51 63.09 32.66 25.80Pb 32.49 54.02 39.37 34.73 31.18 30.32Pe 12.26 7.88 9.75 11.16 13.15 13.90K* 2.46 4.22 3.27 2.79 2.22 2.04

c* (%) 9.75% 5.89% 7.16% 8.45% 11.05% 12.35%% Debt 89.17% 89.02% 89.08% 89.13% 89.21% 89.25%

TABLE 5: Discount Rate (ρ) Scenarios.

entry-trigger price P∗.

For the final pair of parameters, that is the taxa-tion rate τ and bankruptcy cost α, the results of thescenario analysis are given respectively in Tables 6and 7. Interestingly, increases in the taxation ratecan be seen to have negligible impact on both theprice entry trigger P∗ and the optimal productioncapacity Q∗ relative to the base case; although thedirection of these changes are in line with thosenoted by Sarkar (2011). Although the optimal debtcoupon c∗ increases modestly with the increasingtaxation rate τ, the optimal debt level increasessubstantially with increased taxes; reflecting the in-creasing attractiveness of taxation benefits. Indeed,for an assumed taxation rate of 32%, the SVM sug-gests that the extreme case of optimally financingthe project through debt alone. So from a policyperspective, taxation serves to have minimal im-pact on investment timing and size but influenceshow the project is financed. Similar to taxation,increasing bankruptcy costs can be seen to haveminimal impact on both the price entry trigger P∗

and the optimal production capacity Q∗ relative tothe base case. However, the level of debt financ-ing can be seen to vary notably from 89% underthe base case down to approximately 69% underthe extreme bankruptcy cost assumption of 46.4%.This reflects that fact that the higher are bankruptcycosts for a company considering a large scale invest-ment project, the more discouraged that companywill be to use debt financing.

A final comment on the SVM worth noting is

that the P∗ is usually significantly higher than theexit-trigger price Pe. For the base case, P∗ is almost4 times higher than the exit price. Variations inthis ratio are seen across the parameter scenariosbut the most noticeable differences are observed inthe GBM volatility coefficient and the discount ratecases. Indeed, for the base case, the exit price Pe isestimated to be EUR 12.26/MWh, which is well be-low the minimum three-year futures price observedover the sample period. What this broadly suggestsis that once the irreversible investment decision hasbeen made, there is little prospect thereafter of thewind farm ever being shut down. Intuitively, giventhe fact that wind farms have zero exposure to fuelcosts, as well as the assumption made by Sarkar[5] that debt is treated as equity post-bankruptcy(i.e. no further interest payments are made), such afinding is economically plausible. In contrast, P∗ isonly 1.5 times the bankruptcy-trigger price Pb in theSVM. The fact that optimal leverage in the projectis as much as 89% in the base case may explainsuch modest headroom. At the same time, giventhat price volatility in the SVM is 17.1% per annum,this remains a finding that both debt and equityinvestors would need to consider with care.

Conclusion

Wind energy investment is confronted with the is-sues of irreversibility, uncertainty, and when to in-vest, all issues that are magnified when consideringoffshore wind energy investment. We contribute to

10energisk.org

Scenario Base 2 3 4 5τ 0.22 0.12 0.17 0.27 0.32P* 48.77 48.55 48.66 48.89 49.02Q* 1.03 1.04 1.04 1.03 1.02c* 43.47 40.99 42.56 44.06 44.47Pb 32.49 31.14 31.97 32.87 33.17Pe 12.26 12.26 12.26 12.26 12.26K* 2.46 2.48 2.47 2.45 2.43

c* (%) 9.75% 9.38% 9.58% 9.90% 10.06%% Debt 89.17% 77.15% 83.37% 95.16% 101.64%

TABLE 6: Taxation Rate (τ) Scenarios.

Scenario Base 2 3 4 5α 0.064 0.164 0.264 0.364 0.464P* 48.77 48.89 48.96 49.00 49.03Q* 1.03 1.03 1.03 1.03 1.03c* 43.47 39.80 36.90 34.52 32.52Pb 32.49 30.83 29.51 28.42 27.51Pe 12.26 12.26 12.26 12.26 12.26K* 2.46 2.45 2.45 2.45 2.44

c* (%) 9.75% 9.61% 9.52% 9.45% 9.40%% Debt 89.17% 82.68% 77.37% 72.93% 69.12%

TABLE 7: Bankruptcy Cost (α) Scenarios.

the renewable energy investment literature throughthe application of the novel, flexible real optionsmodel proposed by Sarkar [5], which provides forsimultaneously optimised outcomes with respectto (i) when to invest, (ii) how much to invest, and(iii) by how much to lever the financing of the in-vestment; all three of which are fundamental toany investment decision. The Sarkar [5] modellies within the seminal real options framework ofDixit and Pindyck [6] and extends related works[7], [8], [9], [10], that previously have not been ableto consider jointly investment timing, investmentsize and investment leverage. We value offshorewind investment in the jurisdiction of Sweden, ap-plying the Sarkar [5] model to practically assessthe optimal investment decisions for such a largescale renewable energy investment. Performing ascenario analysis, insights are given into the opti-mal price entry trigger for this investment, alongwith the optimal installed capacity suggested bythe model for the wind project and the optimal mixof debt and equity, where in the latter case debtis seen to dominate the financing landscape, beingapproximately 89% in the base case. This scenarioanalysis shows the importance of understandinghow the investment decisions, in terms of invest-ment timing, size and leverage, jointly interplaywith the factors that drive the investment, i.e. unitcapacity capital cost, unit operating cost, net capac-ity factor (i.e. production elasticity), discount rate,taxation rate and bankruptcy cost, in addition tothe estimated expected growth rate and volatility

of the driving Nordic power price series (underassumed Geometric Brownian Motion price dynam-ics). Such interplay is lost in previous real optionsmethodologies.

However, despite the heightened flexibility ofthe Sarkar [5] real options model, an importantlimitation is identified in this study. In particular,it is shown that the higher the production elastic-ity, driven by a higher net capacity factor, the laterthe decision is made to invest. However, for windfarms, this is a counterintuitive finding because aswind farms have no fuel costs, the higher the net en-ergy yield is, the greater the return on investment isfor the same level of capital costs. The normal com-mercial expectation here would be that the higherthe value of any production output parameter, theearlier the investment should be made. Additionalto this, the model has other more practical limita-tions; it does not allow for construction lag, finiteoperating lifetimes and the inclusion of additionalstate variables (such as green certificate prices thatadd value within the offshore wind investment ap-praisal). In terms of offshore wind investment, oncethe irreversible decision has been made to invest,it takes at least two years before this investmentbegins to generate a net operating cash flow. Theinclusion of such a construction lag in a real optionsmodel has been found to promote earlier invest-ment [43], [44]. Wind farms have finite operatinglifetimes, with a wind turbine having an expectedtechnical lifespan of only 20-25 years. Of course, alater decision could be made to extend the lifetime

Summer 201511

REAL OPTIONS MODEL

of these machines. However, this would be to as-sume that there has been insufficient technologicalinnovation during the interim period for manage-ment not to have better options by that stage. Thismeans that the full investment return (includingthe repayment of all debt) must be achieved withinthis timeframe. When such a restriction is addedto a real options model, the effect is to reduce theoptimal entry-trigger price [45], [46]. In terms ofconsidering additional state variables, it is unlikelythat capital costs will remain constant while elec-tricity prices drift upwards. Moreover, Boomsmaet al. [11] use steel prices as an additional statevariable and a market-observable proxy for capitalcosts. In doing so, they find that option value isnegatively related to changes in steel prices. Theprice of green certificates could also be used as asecond state variable in our assessment of offshorewind investment in Sweden. In this instance, [11]find that the option value is higher for projects thatbenefit from the support of such certificates thanit is for projects with either a fixed feed-in tariff or

without the benefit of any such support (i.e. theyare reliant upon the value of electricity alone). In-troducing the price of green certificates as a secondstate variable would significantly add to the off-shore wind investment analysis presented in thispaper. In addressing these various limitations, onewould need to examine of course the effect thatthese would have on the tractable semi-analytic re-sult of the Sarkar [5] model. Such considerationsare deferred for future research.

ABOUT THE AUTHORS

James Savage: Senior Equity Analyst, State Street GlobalAdvisors, Dublin, Ireland.

Mark Cummins: Mark Cummins, DCU Business School,Dublin City University, Dublin 9, Ireland.

Email address: Mark.Cummins@dcu.ie

Jean Charpin: University of Limerick, Limerick, Ireland.

ABOUT THE ARTICLE

Submitted: May 2015.Accepted: June 2015.

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[10] Lyandres, E. and A. Zhdanov. Accelerated investment ef-fect of risky debt. Journal of Banking & Finance, n. 34,pp. 2587-2599. 2010.

[11] Boomsma, T. K., N. Meade and S. E. Fleten. Renewableenergy investments under different support schemes: Areal options approach. European Journal of OperationalResearch, n. 220, pp. 225-237. 2012.

[12] Fleten, S.E., K. M. Maribu, I. Wangensteen. Optimalinvestment strategies in decentralized renewable powergeneration under uncertainty. Energy, n. 32, pp. 803-815.2007.

[13] Kjærland, F. A real option analysis of investments in hy-dropower - the case of Norway. Energy Policy, n. 35, pp.5901-5908. 2007.

[14] Bockman, T., S. E. Fleten, E. Juliussen, H. G. Langham-mer and I. Revdal. Investment timing and optimal capac-ity choice for small hydropower projects. European Jour-nal of Operational Research, n. 190, pp. 255-267. 2008.

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Fleten, S. E. and G. Ringen. New renewable electricitycapacity under uncertainty: The potential in Norway.The Journal of Energy Markets, n. 2, pp. 71-88. 2009.

[15][16] Yang M., F. Nguyen, P. De T’Serclaes and B. Buchner.Wind farm investment risks under uncertain CDM bene-fit in china. Energy Policy, 38, pp. 1436-1447. 2010.

[17] Munoz, J. I., J. Contreras, J. Caamano and P.F. Correia .A decision-making tool for project investments based onreal options: The case of wind power generation. Annalsof Operations Research, n. 186, pp. 465-490. 2011.

[18] Siddiqui, A. and S. E. Fleten. How to proceed with com-peting alternative energy technologies: A real optionsanalysis. Energy Economics, n. 32, pp. 817-830. 2010.

[19] Lee, S. C. Using real option analysis for highly uncertaintechnology investments: The case of wind energy technol-ogy. Renewable and Sustainable Energy Reviews, n. 15,pp. 4443-4450. 2011.

[20] Levitt, A.C., W. Kempton, A.P. Smith, W. Musial and J.Firestone. Pricing offshore wind power. Energy Policy, n.39, pp. 6408-6421. 2011.

[21] Venetsanos, K., P. Angelopoulou and T. Tsoutsos. Re-newable energy sources project appraisal under uncer-tainty: The case of wind energy exploitation within achanging energy market environment. Energy Policy, n.30, pp. 293-307.2002.

[22] Davis, G. A. and B. Owens. Optimizing the level ofrenewable electric R&D expenditures using real optionsanalysis. Energy Policy, n. 31, pp. 1589-1608. 2003.

[23] Siddiqui, A. S., C. Marnay, R. H. Wiser. Real optionsvaluation of US federal renewable energy research, devel-opment, demonstration, and deployment. Energy Policy,n. 35, pp. 265-279. 2007.

[24] Menegaki, A. Valuation for renewable energy: A com-parative review. Renewable and Sustainable Energy Re-views, n. 12, pp. 2422-2437. 2008.

[25] Martinez-Cesena, E. A. and J. Mutale. Application of anadvanced real options approach for renewable energy gen-eration projects planning. Renewable and SustainableEnergy Reviews, n.15, pp. 2087-2094. 2011.

[26] Bilgili, M., A. Yasar and E. Simsek. Offshore wind powerdevelopment in Europe and its comparison with on-shore counterpart. Renewable and Sustainable EnergyReviews, n. 15, pp. 905-915. 2011.

[27] Mauer, D. C. and S. H. Ott. Agency costs, underinvest-ment, and optimal capital structure. In: Brennan MJ,Trigeorgis L, editors. Project flexibility, agency and com-petition: New developments in the theory and appli-cations of real options, New York: Oxford UniversityPress, Inc., p.151-179. 2000.

[28] Leland, H. E. Corporate debt value, bond covenants, andoptimal capital structure. The Journal of Finance, n. 49,pp. 1213-1252. 1994.

[29] Dicorato, M., G. Forte, M. Pisani and M. Trovato. Guide-lines for assessment of investment cost for offshore windgeneration. Renewable Energy, n. 36, pp. 2043-2051. 2011.

[30] Esteban, M. D., J.J. Diez, J.S. Lopez and V. Negro. Whyoffshore wind energy?. Renewable Energy, n. 36, pp. 444-450. 2011.

[31] Manwell, J. F., J. G. McGowan and A.L. Rogers. Windenergy explained: Theory, design and application. Chich-ester: John Wiley & Sons, Ltd. 2002.

[32] Krohn, S., P.E. Morthorst and S. Awerbuch. The eco-nomics of wind energy: A report by the European WindEnergy Association. Brussels: European Wind EnergyAssociation. 2009.

[33] KPMG. Offshore wind in Europe: 2010 market report.KPMG AG. 2010.

[34] Weaver, T. Financial appraisal of operational offshorewind energy projects. Renewable and Sustainable En-ergy Reviews, n. 16, pp. 5110-5120. 2012.

[35] Kaiser, M.J. and B.F. Snyder.Offshore wind energy costmodelling: Installation and decommissioning. London:Springer-Verlag. 2012.

[36] Hassan, G.UK offshore wind: Charting the right course.London: British Wind Energy Association. 2009.

[37] Green, R. and N. Vasilakos.The economics of offshorewind. Energy Policy, n.39, pp. 496-502. 2011.

[38] Koch, C. Contested overruns and performance of offshorewind power plants. Construction Management and Eco-nomics, n.30, pp. 609-622. 2012.

[39] Yescombe, E. R. Principles of project finance. San Diego.Academic Press. 2002.

[40] Kost, C., J.N. Mayer, J. Thomsen, N. Hartmann, C.Senkpiel, S. Philipps, S. Nold, S. Lude, N. Saad andT. Schlegl. Levelised cost of electricity: Renewable en-ergy technologies. Fraunhofer Institut for Solar EnergySystems ISE. 2013. Available online.

[41] Guillet, J. An overview of the deals closed in 2012 andthe current bank market approach to offshore wind. Pa-per presented at the 11th Hamburg Offshore Wind Con-ference, Hamburg. 2013.

[42] Thorburn, K. S. Bankruptcy Auctions: Costs, Debt Re-covery, and Firm Survival. Journal of Financial Eco-nomics, n.58, pp. 337-368. 2000.

[43] Majd, S. and R.S. Pindyck. Time to build, option value,and investment decisions. Journal of Financial Eco-nomics, n. 18, pp. 7-27. 1987.

[44] Bar-Ilan, A. and W.C. Strange. Investment lags. TheAmerican Economic Review, n.86, pp. 610-622. 1996.

[45] McDonald, R. and D. Siegel. The value of waiting to in-vest. Quarterly Journal of Economics, n.101, pp.707-728.1986.

[46] Gryglewicz S., K.J.M. Huisman and P.M. Kort. Finite projectlife and uncertainty effects on investment. Journal of EconomicDynamics & Control, n.32, pp. 2191-2213. 2008.

Summer 201513

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OW BunkerHow the World’s largest Marine Fuel Traderwent from IPO to Bankruptcy

OW Bunker was a Danish company thatbecame one of the largest traders ofbunker oil in the world before fallinginto bankruptcy after IPO in 2014. Theauthor distinctly describes all the phasesof this fall, from the IPO in March2014 to the ruinous end in November2014. The story is an important caseof study for all the companies operat-ing in bunker fuel trading, a bad exam-ple of risk management to be aware of.

Alessandro MAURO

OW Bunker was a company founded in1980, at the end of a decade marked bythe two oils shocks that changed theoil market forever and saw the birth

of modern oil trading. OW Bunker was one ofthe World’s largest traders of bunker oil. It hadoperations in 29 countries and claimed to controlaround 7% of worldwide bunker trade[1]. Marineor bunker fuel is a residual product of crude oil re-fining process and it is burned in seagoing vesselsto navigate around the World. Similarly to manyother commodity traders, apart from basic goodstransformations (blending), OW Bunker businessmodel was about buying in order to sell at a laterstage. It bought the fuel from suppliers, mainlyrefiners or other traders, and later sold it to ship-owners and distributors or stored these goods fora period of time. To run the business, they used togive financing to their customers and use to receivefinancing from their suppliers and banks.

Marine fuel trading, as many segments of oiltrading in general, is a low margins business due tofierce competition among players. Higher revenuesand profits can be made only by intermediating ahigher volume of goods. However, low margins donot imply low risk in the oil trading business. Com-panies such as OW Bunker face severe market andcredit risk, but at the same time well establishedand widely known risk mitigation techniques caneffectively remove most of those risks.

In this low margin and highly competitive en-vironment, OW Bunker had at least one appeal:it was big. In a market crowded by a plethora ofsmall and even minuscule shops, it was large andorganized. Its dimension and profitability justifiedchoosing a path normally avoided by the majorityof traders. OW Bunker went public with an InitialPublic Offering (“IPO”) which took place in March2014 on the NASDAQ OMX Copenhagen exchange.The IPO was what people called a success. On the28th of March OW Bunker CEO proudly welcomed

“the more than 20,000 new shareholders”[2]. The sharesprice went up about 20% on the first day of tradingand the value of the company got close to one bil-lion US dollars. Not bad for a company engagedexclusively in an old fashion and low margin busi-ness.

During the following months no major eventcame to disturb the honeymoon between OWBunker and its happy shareholders. Unfortunately,after spring and summer, the first days of fallstarted delivering bad news. On the 7th of Octoberthe company released a profit warning, mainly dueto “unrealized accounting loss before tax of approx.USD 22 million in Q3 2014”[3], triggered by theslide in oil prices. On the 23rd of October the com-

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pany further restated the loss at USD 24.5 millionand, in an Investor presentation, gave some furtherdetails about the drivers of this loss.

No more details were to follow. Abruptly onthe 5th of November the shares were suspendedfrom trading on NASDAQ OMX. On the same dayOW Bunker management declared a loss of USD275 million[4]. Two separate issues were behindthis drama. A fraud had been discovered, put inplace by senior employees in a previously unknownSingapore-based subsidiary named Dynamic OilTrading. This fraud resulted in a USD 125 millionloss. The second cause was a “risk management loss”in addition to the USD 24.5 million already com-municated. Apparently the loss was found after areview of “OW Bunker’s risk management expo-sure” and the total loss was now estimated at USD150 million.

On the 7th of November, after no other viable so-lution was found, the company filed for bankruptcyin Denmark[5]. Further bankruptcy filings came inthe following days in other jurisdictions.

The IPO prospectus

The rest of this article is centred on what OWBunker management described as the “risk man-agement loss”. The information released by thecompany, in occasion of the IPO and later till thebankruptcy, allows to go in further interesting de-tails. That is not possible for the fraud maturedin the Singapore subsidiary, as it was made publicjust two days before bankruptcy. Unless otherwisestated, this article is based on public informationrealised by the Company, the same information thathad been made available to its shareholders, coun-terparties and creditors. Sentences sourced by OWBunker’s official documents are reported in Italics.

The prospectus produced at the time of the IPOis the crucial source that helps understanding moreabout OW Bunker, its business and, among others,its risk management processes. This document, asit is habitual for every IPO, is full with warningsabout several risks potential investors will be facingby buying shares in the Company. At the same time,some crucial topics are left in the dark. One cannotfind much information about the real functioningof OW Bunker’s market risk management process.The few reported data are repeated several times,in order to convey the idea of solid operations inthis domain, as in many others the Company wasengaging in. A long list of theoretical reasons whythings could go wrong is given. If we concentratethe attention on the practical information and leaveapart theory, what potential investors were specifi-cally told at this point can be summarized in these

sentences:

1. “The primary goal of our marine fuel and ma-rine fuel component price risk management policy,which is approved by the Board of Directors, is toensure that our business generates a stable grossprofit per tonne by limiting the effects of marinefuel price fluctuations.”[6]

2. “The overall risk limit set in our marine fuel andmarine fuel component price risk management pol-icy is defined by a maximum net open (unhedged)position for the Group. Currently, the maximumnet open position approved by the Board of Di-rectors is 200,000 tonnes. However, we operatewith a lower internal risk management guidelinewith a maximum net open position of 100,000tonnes, which is set by the Company’s Chief Ex-ecutive Officer (the “CEO”) and applied in ouroperations.”[6]

3. “The Executive Vice President for our physicaldistribution operations is responsible for marinefuel price risk management and reports directly tothe CEO.”[6]

The first point essentially tells that financialderivatives are used in order to hedge Company re-sults against the volatility in the prices of goods theCompany buys and sells. An important goal thatin recent years could be easily achieved by tradingexclusively plain vanilla derivatives, considering oilmarkets showed risibly low volatility.

With the second point the Company commu-nicates the internal self-imposed rules in place tolimit the risk it faces due to oil market prices. Inthe IPO prospectus it is further clarified that OWBunker’s “net open position from marine fuel transac-tions and derivative instruments can be either long orshort and is at any time below our policy limits.”[6]Insummary, this information is meant to communi-cate the market risk appetite of the Company.

However, the benefit of this piece of informationis limited. How a potential shareholder, or most ofthe stakeholders in general, could assess the dimen-sion and the severity of market risk the Companywill be facing? How big is the risk of being longor short 100,000 metric tonnes of anything? Whatis the probability of a negative event generated bythis volumetric exposure? Income statements, bal-ance sheets and financial ratios are expressed in USdollars, i.e. money, not quantities. There exist well-established risk evaluation techniques that answerto these questions. The IPO prospectus could haveexpressed the market risk in terms of Value-at-Risk(“VaR”), a statistically based measure of the maxi-mum possible monetary loss. VaR is widely used by

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FIGURE 1: OW Bunker chronology from IPO to bankruptcy

commodity trading firms’ risk management. In fact,many public companies frequently report their VaRfigures, often comparing it with their shareholders’equity.

Among other benefits, VaR takes in to accountphysical and financial exposure simultaneously.VaR also allows to communicate market risk currentlevels and limits without the need to disclose sensi-tive information about company business3. Unfor-tunately nowhere, in the OW Bunker’s IPO prospec-tus or in other documents, is VaR mentioned. Thatraises some reasonable doubts about the real sophis-tication of OW Bunker’s market risk managementvaluation process and related IT systems. To makethings clear, however, it is not 100,000 or 200,000metric tons an exposure sufficient to cause the largefinancial loss that materialized just months later.

The third point above opens questions abouthow risk management governance in OW Bunker,a public company, was shaped. In the Annual re-port 2013 there was no mention to an employeespecifically responsible for risk management. It wasreported, as being part of the management team,the existence of an employee which job title was“Executive Vice President – Physical Distribution”.In the IPO prospectus this employee becomes also

“responsible for marine fuel price risk management”. Hadthis employee sufficient experience in market riskmanagement? Was a sole employee in charge oftrading operations and risk management? Was thisa self-controlling employee, without anyone elsebalancing this power?

We will come back to some other specific pointsof the IPO prospectus while analysing the otherdocuments that were realised some months later.

Fall brings bad news

With the 7th of October announcement the Com-pany, apart from the estimated loss of USD 22 mil-lion, started disclosing new important features ofthe pursued risk management strategy. Informationis however ambiguous and in some important partseven contradicting.

“As part of its risk management policy, OW Bunkerhedges its commercial inventory and marine fuel trans-actions within an expected oil price range. Conse-quently, price fluctuations within such range only havea marginal effect on OW Bunker’s results. Conversely,when the oil price breaks the expected range, it may af-fect a given quarter by changes in the valuation (markto market) of the derivatives contracts used for hedgingof inventory and marine fuel transactions”[3]. It is dif-ficult to interpret this statement. The starting pointshould be the exposure generated by the physicalbusiness, i.e. “its commercial inventory and marine fueltransactions”, before hedging with financial deriva-tives is put in place. Any exposure can be either“long” or “short”. A long exposure will gain moneyif price increases and will lose money if price re-duces. For a short exposure, the opposite wouldhappen. In the IPO prospectus it is stated that “Ourtypical open position before hedging varies from a longposition of 250,000 tonnes to a short position of 100,000tonnes.”[6] However the company never clarified ifthe physical exposure, in the months preceding thebankruptcy, was actually short or long.

Let’s assume that the physical exposure waslong, but the reasoning would be still valid in theopposite case. In order to hedge this long physicalexposure, the company needed to be short on finan-

3 For a practical introduction to Value-at-Risk in energy markets the reader can refer to A. Mauro, “Price Risk Management in theEnergy Industry: The Value at Risk Approach”, available at https://amaurorisk.wordpress.com/articles/

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BUSINESS STORY

cial derivatives, by selling Futures, Swaps, Optionsor combinations of them. In this way, apart fromproblems related to the efficacy of the derivativeshedging instruments (“marginal effect”), the com-bined physical plus derivatives transactions shoulddeliver rather predictable financial results. In thestatement reported above, the company clarifiedthat this was the case, but the global hedging strat-egy was more complex. In fact, at that point OWBunker’s hedging strategy was active “within anexpected oil price range”, i.e. within a high and a lowprice boundary, normally identified as “cap” and“floor”, constituting overall a “collar”.

A collar can be built exclusively with deriva-tives instruments, not with physical deals, and itconsists of a combination of long and short options.If we believe in what the company managementwas announcing, then the derivatives collar wascounterbalancing the physical exposure only insidea price range. If prices would move outside therange, then the derivatives will become inactiveand the company will be simply un-hedged on thephysical business.

If the physical exposure was long, a reduction inoil prices under the collar floor would have finallydetermined realised losses on the physical businesswithout any benefit on the financial derivatives side.

The company further announced that “The re-cent slide in the oil price, in particular during September,is outside the range expected and OW Bunker will asa consequence of its risk management policy report anunrealized accounting loss before tax of approx. USD 22million in Q3 2014. This is based on a mark to marketvaluation of OW Bunker’s derivatives contract as at endSeptember 2014.”[3] At this point it seems that thecompany is not giving a complete picture, becauseit is discussing exclusively the derivative contracts.It is true that the value of the collar derivative con-tract should change even when prices are outsidethe collar range. However, at the same time, alsothe physical exposure would change in value. If, forevery metric ton of physical exposure, one metricton of collar was executed in the financial market,then the change in value of the physical exposureshould be more important than the change in valueof the collar contract. In fact the physical exposureis linear while the collar one is not.

Issues related to accountancy rules do not seemto bring an explanation to the incongruence of theannouncement. In the IPO prospectus the com-pany had stated that its derivatives did not qualifyfor hedging accounting treatment. Consequently,

“Changes in the fair value of these derivative instrumentsare recognised immediately in the income statement”[7],while changes in value in the physical transactions

would be recognised at a later stage. This is quitecommon for commodity trading firms and it doesnot bring any surprise. Why, then, in the 7th ofOctober Company announcement nothing is saidabout the value of the physical transactions and thefact that the loss recognised on derivatives in Q32014 will be balanced later by the physical part ofthe business? The obvious explanation of the omis-sion about the physical business can be only one:the volume of traded derivatives was much biggerthan the physical exposure. At least at this point intime, the main driver of company results was con-stituted by the derivative contracts. Profits or losseson the physical business could not counterbalancethe derivatives results. Amusing conclusion from astarting point in which derivatives were meant tocounterbalance the results of the physical commer-cial activity!

Furthermore, the 7th of October Company an-nouncement informs that the loss of USD 22 million“includes a substantial element of protection taken upagainst further falls in the oil price” and that “We havetaken action to minimize risks against further oil pricefalls [. . . ]”[3]. This information is not sufficient tomake clear what was put in place. However the “In-vestor Presentation of the Interim Results Q3 2014”,released on the 23rd of October, would incidentallyclarify that this was a purchased Put option andit was already in place by the end of September,2014[8]. This part of the announcement seems thenrational: in order to enter in a long Put option andhedge from possible further oil prices reduction,OW Bunker had to pay a premium and this wasincluded in the communicated USD 22 million loss.As a side note, we should remark that the paidpremium was to be considered as already realised,and consequently it was not accurate to classifythe USD 22 million loss as fully unrealised. More-over, the impact of this loss on the 2014 outlookwas reduced by making the simplistic assumptionof USD 10 million of “expected regain on hedging”.This point was made clear only later in the Investorpresentation of the 23rd of October[8].

“If the oil prices rise again, we will gain on ourderivative contracts [. . . ]”. This part of the 7th ofOctober Company announcement is not simple todecipher, but can finally clarify OW Bunker’s expo-sure to oil prices. The company indirectly suggestsagain that the physical exposure is negligible in theglobal picture. Excluding the results of the addi-tional long Put option already mentioned, what weknow at this point is that:

1. OW Bunker was losing money on derivativesbecause of the reduction of oil prices.

2. OW Bunker would gain money on derivatives

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FIGURE 2: The price of Brent crude from the peak in June, 2014 to November. Source: Thomson Reuters Eikon.

if prices would increase again.

This position is nothing else than a combinationof a long Call and a short Put, where the strikeprice of the Put is lower than the strike price of theCall. By adding later the long Put, the companyallegedly covered against further downside, butpreserved the upside. Not much more can be said,based on the public information that was releasedat this point.

Bigger clouds on the horizon

On the 23rd of October the Company produced anInterim Financial report for Q3 2014 and a relatedInvestor Presentation, shedding some more light onwhat was happening. In summary:

1. The losses for Q3 2014 reached USD 24.5 mil-lion, 2.5 million higher than before.

2. The forecasted “regain” of USD 10 million onderivatives was cancelled.

Using Company words, “The estimated unrealisedrisk management loss of approx. USD 22 million asannounced [. . . ] on October 7, 2014, ended at a USD24.5 million loss when final calculations were made”[9].It was just an additional 2.5 million loss, but it wasa symptom of a serious illness. First of all why

“final calculations” were necessary? The IPO prospec-tus clarified that the company traded in financialderivatives which fair value was classified, accord-ing to IFRS definition, as “level 1” and “level 2”[7],i.e. value essentially based on prices promptly avail-able from market sources. Were or were not OWBunker’s state of the art risk management systemable to calculate the value of positions at least daily(baseline in this industry)?

Additional information allows to have a betteridea regarding the derivatives position in place. Infact a “possible reduction of the unrealised risk manage-ment loss, including the additional USD 2.5 million riskmanagement loss, requires a Brent oil price of aroundUSD 92 per barrel. In case of an average Brent oil priceof USD 92 per barrel in Q4 2014, the unrealised riskmanagement loss may be reduced by around USD 12.5million [. . . ]. In case of an oil price below this level, OWBunker does not expect a reduction of the unrealised lossin 2014. However, OW Bunker is protected against fur-ther losses than the above mentioned without additionalcost to protect against further oil price falls”[9].

The described position resembles again the pay-off of a long call option. By paying a premium,the Company allegedly secured profits in case ofan increase in prices, but would not suffer from afurther reduction. However, the wording above sug-gests that the option was not a simple call option,but something similar to a “digital” call option or a“knock-in” one. These options deliver a payoff dif-ferent from zero only if the underlying price reachesa certain level. We may use simple algebra and thehypothesis that the possible USD 12.5 million profitwas based on a comparison between 92 and 84 USDper barrel, 84 being the Brent crude price in the mid-dle of October as reported in the Interim Financialreport. In this way we obtain a necessary volumeof derivatives of approximately 70,000 metric tonsfor each of the three months in Q4 2014, i.e. about210,000 metric tons in total. We need to considerthat these derivatives were options, and the expo-sure they generate is less than linear. Moreover,these options were out-of-the money, which furtherreduce the exposure.

Once again nothing is said about the exposuregenerated by the physical business. If that expo-

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BUSINESS STORY

sure was positive, then the company was long onphysical and long on derivatives, i.e. the deriva-tives were not hedging the physical business. If thephysical exposure was short, than an increase inoil prices would generate money on the derivativesbut lose money on the physical business. As saidbefore, the only way to believe what the Companymanagement was communicating at this point is tosuppose that the exposure generated by derivativesdeals was much bigger than the physical one.

Consequently we need to conclude that thisderivative position was not built for hedging pur-poses but it was instead a bet on prices going up.That would still not constitute a breach of the limitof 100,000 metric tons. In fact the IPO prospectusclarifies that, inside that limit, even pure derivativespositions could be put in place. Moreover, the CEOcould have approved a limit extension to 200,000metric tons, which was under his powers withoutapproval needed by the Board of Directors. How-ever this is not mentioned in the Interim Financialreport or in the Investor presentation.

In the latter document, some additional piece ofinformation is given. The loss of USD 24.5 millionis allocated to three main factors[9]:

1. Purchase of Put option derivative contract toprotect from further price reductions.

2. Change in forward oil price market structure,from Backwardation to Contango.

3. Change in the absolute level of oil prices.

The first two factors raise some suspicions. Re-garding the first point, this form of insurance issaid to be in place already at the end of September,and “Subsequently protection has been moved down inlight of oil price decrease."[8] It must be consideredthat, once a protection with a long Put option is inplace and the underlying price moves down sur-passing the strike price, this financial instrumentdelivers money to the holder in a linear way. Conse-quently, why OW Bunker did need to move downthe protection?

The sentence could be explained by the fact thatthe long puts, already in place at the end of Septem-ber, had a short term maturity. Later on other putoptions were bought at a lower strike price, con-sistently with the additional price reduction thatunderwent in the market in the meantime. Inciden-tally, the Investor presentation communicates thatthe underlying of the Put options is gasoil price,while it suggests a couple of times that the bench-mark price for OW Bunker business is the price ofbunker and fuel oil. This should have raised some

questions about the efficacy of these Put options ashedging instruments.

The second bullet point above mentions the im-pact of “Contango” in the market. This is a situationin which prices for prompt delivery of goods arelower than prices for forward delivery. The oppo-site is true in case of “Backwardation”. Exploitingprice time-structure in Contango when it appears inthe markets is one of the simplest, lowest risk andmost profitable way to make money for commoditytraders. It is sufficient to buy the goods, store them,and then sell the goods for forward delivery orsell derivatives maturing in some months and thegame is done. The fact that OW Bunker lost moneybecause of market prices going from Backwarda-tion in to Contango tells something more aboutthe radical payoff modification that was achievedby trading derivatives. Essentially, the Companywas long on short-dated maturities and short onlonger ones. This short time-spread position woulddeliver money in a market that moves in to Back-wardation. On the contrary, such position wouldlose money in a market where Backwardation re-duces or even changes in to Contango, and thelatter was the case for OW Bunker. However theCompany affirms that it would not be caught bysurprise again: “With the current hedge, OW Bunkerwill not be impacted by changes in market structure(i.e. a reverse of the current contango market structureto backwardation)"[8]. Even this assertion soundsstrange. In order to become insensitive to changesin the market price time structure it is necessaryto have all exposures concentrated in the promptmonth.

The third point above communicates that, inaddition to the time spread position, the Companyhad an outright long position and this lost moneydue to oil market prices reduction. Here again theCompany does not miss the opportunity to con-fuse stakeholders. In the Investor Presentation onecan read that “Typical implications from [. . . ] oil pricechanges to the business” are that “the strategy with lowprices is to increase long exposure as prices fall”, whileunder high prices “it is preferred to be long going intoan environment with rising prices”[8].

All in all, the message delivered to the mar-kets was negative for the moment being but re-assuring for the future: “Current marine fuel priceexposure: Downside risk protected and upside potentialkept”[8]. The Company had neutralized possiblefurther losses on derivatives in case prices wouldcontinue to reduce. At the same time, should pricesgo up again, either the financial profits will be sta-ble or they could even rebalance the previous loss.However, again nothing is said regarding the ex-

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posure generated by the physical business, whichis appalling for the World’s number one trader ofphysical marine fuel. At least this point, manage-ment principal or even unique matter of concernwas the financial derivatives position.

The point of no return

The catastrophe was disclosed on the 5th of Novem-ber. A fraud had been discovered in Dynamic OilTrading, an OW Bunker subsidiary in Singapore,generating a loss of USD 125 million. Moreover “areview of OW Bunker’s risk management contracts hasrevealed a significant risk management loss in additionto the loss of USD 24.5 million announced on October23, 2014 [. . . ]. As of today, the mark to market loss isaround USD 150 million”[4].

The breaking news compare with the “Down-side risk protected” picture depicted on the 23rd ofOctober, less than two weeks before. At that timethe loss was supposedly USD 24.5 million. Thatwould imply that in eight business days (marketsare closed on weekends) OW Bunker cumulatedan additional unrealised loss of around USD 125million, i.e. a daily average of USD 15.6 million.Considering the price change in the same period,i.e. approximately 4 USD per barrel down on crudeoil, a rough estimate brings to an exposure in ex-cess of 4 million metric tons. It is very improbablethat this loss was cumulated on new derivativescontracts executed after the 23rd of October. The5th of November Company announcement suggeststhat the loss was substantially there already on the23rd of October and even before, but it was notmade public. Probably on the 5th of November thepeople familiar with the outstanding derivativesposition could not conceal the catastrophe anymorebecause the counterparties issued margin calls andOW Bunker was not in the position to pay for themargin increase.

If we still believe in the information released be-fore and after the IPO, then we must infer that theloss was necessarily the result of a radical changein the amount of market risk the Company wasfacing. In the Annual Report 2013 one can readthat “If the commodity prices increase by 1% [. . . ] withall other variables being held constant, the profit for theyear will be increased by USD 0.3 million (2012: in-creased by 0.1 million 2011: lower by 0.5 million) as aresult of the changes in the oil derivative contracts asof end of the reporting period.”[10] If the risk profilein 2014 was really kept similar to the 2013 one andthe exposure was linear, then a loss of USD 150 mil-lion would request a price reduction in the order of500%, i.e. price should become negative. Anotherabsurd conclusion.

We can guess some possible explanations aboutthe lack of communication related to the change inrisk profile and the subsequent loss:

1. The top management knew about the deriva-tives position and they authorized that. Theyknew losses were cumulating, but did notcommunicate it till the 5th of November,hoping that the market prices trend wouldchange.

2. The top management did not know, but therisk management function did know aboutthat and did not communicated to top man-agement.

3. Nobody knew about the total derivatives po-sition and/or the financial loss, due to issuesin risk identification, analysis and valuation.

Whatever the truth is, it is highly probable thatthe total exposure of the company was in excess ofthe well-known 100,000 or 200,000 metric tons bysome multiples. OW Bunker built an exceptionalposition in derivatives, probably utilizing combi-nations of options, that overall resulted in a longexposure to oil prices.

A broken risk management process

This catastrophe shares many common points withother horror stories in which derivatives tradingturned sour. From now on OW Bunker will be ingood company with the likes of Metallgesellschaft,Amaranth, MotherRock and China Aviation Oil,just to name a few which got in trouble by trad-ing financial derivatives on commodities. As faras the OW Bunker case is concerned, it is actuallydifficult to find any original point or lesson to belearnt for future memory and which was not al-ready included in the horror stories gallery. Forexample, many other disasters did happen becauseof sudden changes in market conditions, after theyhad shown a stable and profitable pattern for a longperiod. Often the mechanisms and the ultimate re-sponsibilities behind these disasters have not beencompletely clarified. However, as OW Bunker wasa public company, we have here a certain amountof information delivered to the market, which hasbeen the basis for the previous pages of this article.Far to say that this information has been clear orexhaustive. Anyway, it needs to be noted that eventhis limited information should have justified somereasonable doubts in the company stakeholders.

From the narration and the analysis of the eventsand company documents, it is evident that OWBunker actively engaged in the trading of financial

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BUSINESS STORY

derivatives. By saying that “a significant risk man-agement loss [. . . ] is around USD 150 million” it wasfinally made clear that in OW Bunker “risk manage-ment” was synonymous of “derivatives trading”. Inthe IPO prospectus the company specified the oper-ational aspects of this trading activity: “Daily marinefuel and marine fuel component price risk management ishandled for the entire Group by our central risk manage-ment department. All operations hedge their exposureswith the risk management department, which, in turn,hedges the Group’s open position in the market.”[6] Inthis sentence, even the word “hedging” should beread as “derivatives trading”.

Derivatives trading activity was put in place toreach objectives that often surpassed the pure hedg-ing of exposure originated by the physical business.This is normally called speculation and it is not for-bidden by any law or any best practice or standardin risk management. Inside the general risk man-agement process, derivatives trading is an effectiveway of “risk treatment”, the sub-process that al-lows modifying the risk profile of a company. Risktreatment, and consequently also derivatives trad-ing, allows moving from a certain risk profile toanother, the latter being closer to the company riskappetite. Risk treatment does not necessarily meanrisk reduction, and it can be actioned also with theobjective of risk increasing. OW Bunker’s fault isnot in the increase of exposure to market risk byusing derivatives but in the lack of communicationof this strategy to its stakeholders, in the first placeits shareholders.

Communication of risk is a crucial part of anyrisk management process. Communication shouldbe correct and clear, but OW Bunker failed on both.From the pages before it is evident that the releasedinformation was lacunose and misleading. Ad-ditionally, OW Bunker’s risk communication wasflawed by design. As already discussed, tradinglimits expressed in metric tons do not tell muchabout the amount of risk a company is facing. Mod-ern risk communication should be based on riskmeasures of monetary loss, such as Value-at-Riskand stress testing.

An important objective of risk communicationis to make sure that the level of risk the company isbearing is aligned to the level preferred by its stake-holders. If this is not the case, either the companyshould modify its risk profile or the stakeholdersshould leave the boat. In fact, stakeholders’ riskappetite is the king, not the company managementone. Even if stakeholders, and shareholders in par-ticular, liked to bet on oil prices, this does not implythat OW Bunker was authorized to place those betsor was best placed to take those positions in the

interest of its shareholders. Nowadays there aredifferent ways to get exposure to commodity prices,for example by investing in commodity ExchangeTraded Funds (ETF). Everybody can invest a sum ofmoney directly in oil prices-linked financial instru-ments. Trading shares in public companies is farfrom being the first best if the investor is lookingexclusively for oil price exposure.

Risk treatment and risk communication do notexhaust the list of risk management sub-processes.Risk assessment is another crucial step. It shouldcome before risk treatment and should be per-formed periodically, even more frequently thandaily. In fact new deals and the modificationsof existing ones, both physical and financial, con-tinuously change the exposure to risk. We haveclarified, in the previous pages, the reasons whythere are doubts about the quality of risk assess-ment techniques in OW Bunker. While this is aserious issue in any case, it becomes of dramaticimportance whenever a company actively engagesin financial derivatives trading. There are reasonsto believe the dynamic and massive utilization offinancial derivatives, beyond the scope of hedgingthe physical business, was already there when thecompany started the IPO process. One can read inthe Investors Presentation of the Q3 2014 Interimresults that “Historically we would have moved ourhedging out in time [. . . ] ”[8], where again “hedging”is the word OW Bunker used in order to identify“derivatives trading” activity.

Other doubts should be raised around the cru-cial topic of risk governance. In the IPO prospec-tus the company proudly discuss the “Robust RiskManagement System and Culture that Underpins Sta-ble Performance”, clarifying that “Our conservativeoperating philosophy and corporate culture are reflectedin our overall governance approach, including our riskmanagement function” and that “Our risk managementdepartment [. . . ] is responsible for centrally managingour global risk exposure in line with the risk managementpolicy approved by the Board of Directors”[6]. Well, wehave not seen a copy of this risk management pol-icy. How then risk governance and controls wereshaped, if they were, in OW Bunker?

We have already discussed the controversial role,inside the OW Bunker’s organization, of the em-ployee in theory responsible for the risk manage-ment function. In the Company announcement ofthe 5th of November, it was made sure to communi-cate that the “Head of Risk Management and ExecutiveVice President” was dismissed as “a consequence ofthe risk management loss”[4]. There is no more men-tion to the fact that the same employee was firstof all in charge of physical distribution. Why the

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head of risk management was dismissed but notthe head of trading? Is not the trading function,in a trading company, the first responsible for theresults of trading activity? Has the head of phys-ical distribution/head of risk management beenanother scapegoat in the gallery of scapegoats wehave seen in the past? Was OW Bunker a shop inthe same mall described by Daniel Pennac in hisfamous novel “Au bonheur des ogres”?

The IPO prospectus does not use even a singletime the word “segregation”, let alone “segregationof duties”. It is probably out of fashion now, but wewere taught that best practices in risk managementincluded giving responsibility for creating valueand responsibility for controlling it to different em-ployees, in order to properly manage conflicts ofinterest. It seems there was not such a segregationin OW Bunker. In general this is bad, but it getsmuch worse when money-making objectives are as-signed to the risk management function. Nowadayswe continue to hear that the modern trend is thatrisk management should be a “business partner”,where “business” means “trading”. OW Bunkerhad probably embraced this new trend and wentone step further: risk management function waspart of the trading function. There was probablysome specialization in place: “traders” were man-aging the physical deals while “risk management”was taking care of the financial derivatives. In thissetup, who will control risk management employ-ees while they are striving to make money?

OW Bunker’s risk management function hadmonetary value creation objectives. In fact, the com-pany was selling other services to its customers,including “risk management solutions”. In the IPOprospectus we may read that “we are also able to pro-vide risk management solutions as part of our customeroffering [. . . ]. Our risk management solutions include abroad range of financial and trading instruments, suchas physical fixed price contracts, swaps, caps, collars,three-way options and other tailor-made solutions.”[6]

It is true that the IPO prospectus further speci-fies that this was a marginal driver of value creation.However marginality is not sufficient to justify lackof control. In this type of setup the incentive forcross-subsidization, i.e. using profit in one busi-ness unit to subside another, is very high. Thiscross-subsidization is normally put in place ex post,when profits or losses materialize. Probably in OWBunker there was no segregation between the hedg-ing part of the derivatives portfolio and the partthat was held to support the risk management solu-tions. Using a food analogy, this situation normallycooks a big soup in which becomes impossible todistinguish single ingredients, until one becomes

so preeminent and even disgusting that you needto throw the soup away. The incentive to conclude“discretionary”, i.e. speculative, deals grows higherbecause these deals can be easily reported as partof the “risk management solutions” portfolio. Later,if profits materialize, they will be considered intraders’ bonus compensation. On the contrary, iflosses are realized, then these deals will be ex-postconsidered as meant for hedging purposes, con-sequently dampening the result of the rest of thephysical business but not traders’ compensation.When results are just too bad, the entire companyis affected.

This could be the setup possibly used to concealthe real situation to the eyes of top managers inOW Bunker, if they were really not aware. Theywere told that the massive amount of derivativesdeals, the same ones that finally brought to thecatastrophe, were entered in order to support “riskmanagement solutions” products. However thoseproducts did not exist in that scale.

Parting thoughts

This was the story of a company that destroyed onebillion US dollars, value owned by its sharehold-ers, in a matter of months. The last question tobe answered is: could this catastrophe have beenavoided? This is the most important question if wewant to learn lessons from this case and try to avoidsimilar outcomes in the future.

The analysis above has demonstrated that therisk management process failed in every step andfell short of respecting risk management best prac-tices and standards. Differently from financial insti-tutions, commodity trading firms are not subject tolaws and regulation directly addressing their riskmanagement process. However, these firms can ap-ply risk management standards and best practiceswhich are valid in general. Their proper applicationin the commodity trading space can assure that riskmanagement is in tune with the strategic goals ofthe organization.

However, in which way stakeholders can be surethat a company is actually applying risk manage-ment best practices and standards while shapingthe risk management process? For example, a com-pany management could easily communicate thatthey are performing state of the art risk evaluation,and inform periodically about the Value-at-Riskand stress testing results. Stakeholders would feelreassured that the company risk profile and risktreatment techniques are in line with their own pref-erences. Later they could discover that this was anice staging.

Help could come from the existing and incom-

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BUSINESS STORY

ing new regulation that is reshaping the financialmarkets, with repercussions on commodity mar-kets and traders. In the plethora of rules, thereis a specific provision that could have potentiallyprevented OW Bunker masking the real dimensionof the positions taken in the commodity financialmarket. In fact, traders in financial derivatives havebeen requested to promptly report their derivativesdeals to centralized trade repositories.

This provision, together with the obligationto promptly reconcile deals with counterparties,should possibly allow to have clear and compre-hensive data related to the derivatives deals andthe net open position of companies. It is evidentthat this will not form the entire market prices ex-posure for most of the companies, as there is nota similar reporting obligation for physical deals.However critical cases, where the hypertrophy inderivatives trading is not justified by the normalcourse of physical activity, should become easier todetect.

OW Bunker was a company domiciled in the Eu-ropean Union. Consequently the “European MarketInfrastructure Regulation” (“EMIR”), was applica-ble to OW Bunker. In the IPO prospectus, OWBunker classified itself as “Non-Financial Coun-terparty” (“NFC”), which could be proven to bewrong when the true dimension of its derivativesoperations will be disclosed. The NFC classifica-tion allows to skip or postpone a number of EMIRobligations, but not all. In particular, all EU-basedcounterparties have to promptly report their deriva-

tives deals to central repositories. This requirementpotentially allows public bodies (authorities, centralbanks, etc.) to have precise knowledge of deriva-tives positions, exercise control over derivatives ac-tivity of every company and stop OW Bunker-stylebehaviour. The assumption that commodity trad-ing firms trade derivatives in order to exclusivelyhedge physical exposure should be ascertained caseby case.

Although spectacular and dramatic, OWBunker’s case does not represent the unique ex-ample of a company exiting the oil market duringthese months.More are and will come, triggeredby the relevant and sudden oil price reduction thatstarted in the middle of 2014. Much lower pricesand higher volatility, like strong winds and highwaves at sea, are showing the good and the badships, and finally force the latter to sink and disap-pear. The prodigious and efficient mechanism ofnatural selection is again at work. The only troubleis that on ships such as the OW Bunker’s one thereare passengers who would avoid the journey, if theyknew the full story. OW Bunker boarded more than20,000 once happy shareholders.

ABOUT THE AUTHOR

Alessandro Mauro: Energy Risk Professional, Geneva,Switzerland

ABOUT THE ARTICLE

Submitted: August 2015.Accepted: September 2015.

References

[1] IPO Prospectus. p.77.

[2] OW Company Announcement. No. 3/2014, p.1. 28 March2014.

[3] OW Company Announcement. No. 22/2014, p.1. 7 Octo-ber 2014.

[4] OW Company Announcement. No. 25/2014, p.1. 5November 2014.

[5] OW Company Announcement. No. 27/2014, p.1. 7November 2014.

[6] IPO Prospectus. p.101.

[7] IPO Prospectus. p.F28-F44.

[8] Investor Presentation Interim Results Q3. pp. 10-15. 23October 2014.

[9] OW Bunker Interim Financial Report Q3. pp. 8-14. 23October 2014.

[10] OW Bunker Annual Report. p 61. 2013.

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NEW FRONTIERS IN PRACTICAL RISK MANAGEMENT

AAAA

Banking & Finance

Market Models

26

Modelling ofLibor-Ois Basis

We introduce a set of models that explainthe market phenomenology of Libor for-ward fixings implied in swap prices. Themodels are all based on the idea thatthe Libor fixings refer to a panel ofprimary banks whose composition maychange over time. This effect is cru-cial to obtain the observed humped for-ward fixing curves, that could not beotherwise retrieved by a simple creditdefault model or by a forward interestrate analogy. The models differ onlyin the assumptions on how the panelcomposition will change in the future

Antonio CASTAGNAAndrea COVAMatteo CAMELIA

Since the financial crisis started in 2008 theLibor-Ois basis have been no more negligible:this implies a major change in the evaluation

of interest rate derivatives and as a consequencethe single curve interest rate models have becomeobsolete.

Ois rates can be approximately considered de-fault risk-free due to the fact that they are derivedfrom overnight deposit rates: therefore they embedthe risk of default overnight even when they arethe reference rate for longer maturities, e.g.: an Oisswap expiring in 10 years.

Libor spot and forward rates embed the riskthat the borrower (a major bank belonging to the

relevant Libor panel, depending on the currencythe debt is denominated in) may go bankrupt be-fore the expiry of the deposit. As such, a Liborrate for, say, a 6-month deposit, include a spreadover the 6-month Ois rate to remunerate the lenderfor the risk of the borrower’s default over next 6months. The Ois-Libor basis is typically increasingwith maturities (from the O/N to 1 year) for spotLibor rates: one would expect also a similar be-haviour for forward Libor rates, quoted as FRAs upto 24 months and implied in swap rates for longermaturities, but this is not what market rates exhibit.

It is now well known that the forward basiscurves, for all tenors, show a “humped” shape: thisphenomenon has been documented by some au-thors (see, for example, Morini [8] and Ametranoand Bianchetti ([1], figure 35). They find that thebasis curves are initially increasing until a certainfuture time, and then they start decreasing mono-tonically onwards, until an asymptotic value of afew basis points is reached. A confirmation of thepersistence of this feature, even in a financial envi-ronment with lower rates and Libor-Ois basis thanthe one dealing in the period 2008-2010, is givenin Figure 1, where we show the market rates inthe EUR for swaps vs 3M, vs 6M, and Eonia onNovember 1st , 2014: from these quotes we show avery basic bootstrap of the basis Euribor 3M-Eoniaand Euribor 6M-Eonia.

Although the “humped” pattern in both curvesis easily recognisable, it is worth noting that thebasis curves are very irregular even before the 10year maturity, where the market is quite liquid andactive. The weird slopes of the two curves becomeapparent around the 15 year maturity, where themarket trades less frequently. Seemingly regularswap rate curves can generate greatly inconsistentshapes of the basis curves. A general model, based

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MARKET MODELS

FIGURE 1: Swap rate rates dealing in the market on November 1st, 2014 (left hand side) and implied Euribor-Eonia basis curves(right hand side). Source: swap indicative quotes provided by major brokers.

on grounds beyond the simple smoothing criteria,can be useful also to regularise, interpolate and ex-trapolate the Libor-Ois basis curves by fitting it tomore liquid tenors.

Previous Basis Spread Modelling includes theworks by Mercurio [6] and Moreni and Pallavicini[7]. In the first one, the author derives pricingformulae for linear and volatility derivatives, as-suming stochastic dynamics for the single forwardLibor-Ois basis spread, but no connection is estab-lished amongst spreads on the different relevanttenors (i.e.: 1, 3, 6, 12 months) and different futuretimes. In the second work, the authors extend theHJM framework to account for a multi-curve en-vironment: the model establishes a link amongstthe forward Libor fixings at different future dateson the same tenor by means of their dependenceon two common stochastic state variables, whosedynamics are capable to capture the nowadays typ-ical humped term structure. The link between theLibor on different tenors is established via two de-terministic scaling functions for the rate level andthe volatility level. The framework is enough flexi-ble to fit market prices, but no financial or economicrationale lies behind the type of functions and linkschosen by the two authors.

In general, the approaches to model Libor-Oisspreads proposed so far by market practitionersand academicians aim just at matching marketprices, usually with ad hoc assumptions, withouttrying to explain the evolution of the spreads by thecredit factor that they represent. Although theseapproaches can be fully justified on the groundsof the their effectiveness to the purpose, nonethe-less they rely on the existence of a liquid marketwhere all types of main instruments (FRAs, swapson Libor with all the tenors, Caps&Floors) are ac-tively traded and quoted. When the market is notso liquid on some instruments (e.g.: swaps vs 1MLibor for maturities longer than 1 year), a generalmodel can be used to evaluate them, even if it is cal-ibrated on the traded liquid instruments. Clearly,this means that the model is able to deduce the

Libor-Ois spreads for any future date and on anytenor, which implies it is based on the common riskfactors driving all the spreads.

In what follows we introduce a unified set ofmodels that are able to reproduce the “humped”shape of forward basis, yet that are capable tomatch the upward sloping basis curve for spot start-ing deposits. The models have some nice properties:i) they are based on the default risk generating thespreads, ii) they model simultaneously basis for allthe (major) tenors (1, 3, 6 and 12 months), iii) theyall rely on the factual assumption that the panelof banks, whom the Libor refers to, may changeover time and that the any defaulting, or creditworsened, entity can be replaced within the panelitself.

THE MAIN IDEA UNDERPINNINGTHE FRAMEWORK

The Libor rates can be thought to be made of twocomponents: i) a risk-free part, generally consid-ered to be equal to the Ois rate for the correspond-ing expiry, and ii) a credit spread that remuneratesthe lender for the credit risk it bears in lendingmoney to a defaultable borrower. In our framework,the (Ois) risk-free rate is modelled as independentof credit spread; moreover, the credit spread is typi-cally referred to in the market lore as the Libor-Oisbasis. Besides, we will consider four major tenors:1M, 3M, 6M and 1Y, used in most contracts; otherspreads can be derived within the approach thatwe will outline, although they are less used as areference index in interest rate derivatives. Addi-tional factors, such as liquidity risk, are not directlyconsidered in this set of models, although theirinclusion is possible.

Classical credit spread models that consider asingle counterparty, whose default is commandedby a stochastic default intensity, generate a set ofmonotonically increasing credit spread curves, start-ing from different initial values (spot spreads) for

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FIGURE 2: Forward credit spreads for deposits with respectively 1M, 3M, 6M and 1Y maturity, derived by assuming a stochasticdefault intensity that follows a CIR process of the type in equation (29), with parameters λ0 = 0.06%, κ = 1.5, θ = 1.0%, σ = 5.0%.

our four tenors and reaching a common asymptoticvalue. An example is shown in Figure 2.

Unfortunately this is not the type of term struc-tures we observe in the market. The Libor-Ois(credit spread) basis does not simply represent therisk related to a single counterparty. Actually, theLibor rates are the interest rates, for the relevantmaturity and currency, that a panel of major banksis expected to pay when borrowing money from asimilar institutional counterparty. For reasoning’ssake, let us think of the Libor panel as identified bya single representative bank.

The representative bank of the Libor panel is anentity whose default risk may structurally changeover time. By “structural change”, we do not sim-ply mean the possibility that the default probabilitymay stochastically evolve over time; we also meanthat, since the representative bank is a sort of av-erage synthesis of the default risks of the banksincluded in the panel, if the panel changes in itscomposition, then also the default risk of represen-tative bank will change as well. This happens evenif the probabilities of default of the banks currentlyincluded in the panel, and of the banks currentlyexcluded, but which could potentially replace someof the former ones, are deterministic and known.1

To make things concrete, suppose the Liborpanel is made of 10 banks all with a probabilityto go bankrupt over next year equal to 1%: the rep-resentative bank will trivially have a 1-year defaultprobability of 1%. Assume now a bank replacesone of the current ones belonging to the panel, andlet its default probability for 1 year be 0.8%. The

representative bank should now have a 1-year de-fault probability of 0.98%. Hence, its default riskhas changed even if we did not assume any de-terministic or stochastic evolution of the defaultprobabilities of all the banks, either included oroutside the Libor panel.

In the real world, when one of the banks cur-rently belonging to the Libor panel experiences aworsening of its credit standing or even, in theextreme case, a default, then it is expected to bereplaced by a new external bank, with a good creditstanding that will likely improve the average creditquality of the panel. As a consequence, one wouldexpect the credit spread of the representative bankto be lower.

The possibility that the panel changes its com-ponent banks is crucial to account for the humpedshape of the forward Libor-Ois basis. Actually, re-stricting the observation to daily published Liborfixings, one immediately realise that Libor-Ois ba-sis are increasing with the maturity of the deposits.This means that the market expects a rising proba-bility of default over time. This is not very strange,as credit spread curves for single debtors, eithercorporates or banks, usually show the same up-ward slope. One would expect that the Libor-Oisbasis for future dates (embedded in the forwardrates applied to forward starting deposits) showan upward slope too; on the contrary, the pricesof FRAs and of swaps quoted in the market implya downward slope of the forward rates, after aninitial increase up to the maturities of 3 - 5 years.Besides, even if market forward spreads are raising,

1We do not claim we are introducing some revolutionary idea here: we are simply trying to expose what is very likely the waymarket agents (traders) think when they need to make a price for a spot starting deposit or for a FRA. Similar explanations of theLibor panel composition, and change of it, have been proposed also in older works, such as in Morini [8].

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MARKET MODELS

FIGURE 3: A simplified explanatory graph depicting the gradual transition from a High forward Libor-Ois basis curve, to a Lowforward Libor-Ois basis curve (starting delayed), due to a change of the representative bank.

they do not reflect the forward spreads implied inthe Libor spot rates (see, for example, Mercurio [5]).

From this phenomena, it is possible to deducethe reasoning the lenders follow in setting spot andforward rates: if one has to lend today (spot) a givenamount of cash to a bank of the panel, she knowsexactly which is the default risk she would bear.This risk is condensed in the Libor-Ois spreads forspot starting deposits, which are increasing in time,meaning that a higher probability of default for therepresentative bank is attached to longer maturities.

If the deposit is forward starting (as the one un-derlying FRAs), then the lender should account forthe fact that on that future date the representativebank is no more the same as today, since some ofthe banks composing the panel may be replaced bynew ones. The replacement can be due to the creditworsening or by the default of one or more banks;the new banks entering in the panel to replace theexcluded ones will have very likely a better creditstanding that would improve the average creditquality of the panel and hence of the representativebank, to which the forward (FRA) Libor rates refersto. For this reason, the forward Libor fixings im-plied in the FRAs’ and swaps’ market price are notincreasing, as the spot spread curve would suggest,but decreasing, to take into account the generalexpected improvement of the credit quality of thepanel over time, originated be the possible changesof the panel of banks.

To summarise, the Libor-Ois forward basis isactually a weighted average of forward spreads ofthe single members of the panel. If we assume thatthe market expects a likely future change of thecomposition of the panel (i.e.: some panel banks

could be replaced by new banks) and / or a changein the credit worthiness of the current members,then we would not get monotonically increasingcurves anymore. We will try to give a more visualrepresentation of this concept

For simplicity, let us begin by assuming thatmarket expects only improvements, that is: changesin panel composition and / or panel members’credit standing that would result in a lower averagecredit spread. Given the current spot Libor-Ois ba-sis curve, we would now have to include the futurepossibility of a transition to a substantially lowerbasis curve when computing expected future basis.

The resulting basis curve could then be seen asa gradual transition from a high Libor-Ois basiscurve, denoted as H and corresponding to the cur-rent panel, to a low Libor-Ois basis curve, denotedas L. Assume the market will begin to monitor thenew lower credit spread curve from a certain futuretime onwards; therefore we can imagine the L basiscurve as starting with a certain time delay, as canbe seen in Figure 3. At each time in the future,there will be a certain probability of a shift fromthe Libor-Ois basis curve H to the curve L: at time0 the expected Libor-Ois basis curve will resemblethe one depicted in the figure.

The set of models we will present in this ar-ticle, all share this basic idea; they differ only inthe specific risk-neutral dynamics of the expectedfuture panel changes. To make the models ana-lytically tractable and usable in practice, we makethe simplifying assumption that only two types ofbanks exist in the market: H and L, with the formerhaving a higher credit risk than the latter. Thesetwo types of banks have the same type of dynamics

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FIGURE 4: Modularity of the modelling approach

for their respective default intensities, although adifferent set of parameters refers to the H and theL class.

To specify the risk-neutral dynamics command-ing the expected gradual transition from the H tothe future L basis curve, let us start assuming thatthe initial Libor panel is given in its compositionof H and L banks. We consider three different as-sumptions for the transition dynamics, each oneproducing a different final Libor-Ois basis mod-elling:

1. The initial Libor panel is made of a single rep-resentative bank that may be totally replacedby a new, different, representative bank, ac-cording to a continuous-time Markov processwith two states (H and L) and a well definedinstantaneous transition matrix.

2. The initial Libor panel is made of a givennumber of banks that differ in type (H or L):the replacements occur at random discrete fu-ture times, driven by continuous-time Markovprocess. At every random replacement time,each of the banks in the panel can be replacedby another bank of any type, according to atransition matrix.

3. The Libor panel is a continuous weighted av-erage of banks that differ in type (H or L) andtiming of entry in the panel: a deterministiccontinuous-time replacement process drivesthe gradual replacement dynamics towards adifferent panel composition.

All of these models assume that the replace-ment dynamics are independent from the single-counterparty default intensity dynamics and theOis instantaneous rate dynamics.

We will structure the paper as follows:

1. Specify the assumptions for Ois instantaneousrate dynamics and derive the caplet and floor-let prices.

2. Specify the assumptions for single-counterparty default intensity dynamics andderive the results for classical credit spreads

3. Specify separately for the three model vari-ants, the transition dynamics assumptionsand derive the respective results for expectedcredits spreads, spread probability densityfunctions, Libor caplet prices and impliedvolatility smiles.

Remark 1 (Modular Approach). We wish to high-light that one of the features of our approach to basismodelling is modularity (see Figure 4). Actually, weseparately specify:

1. the dynamics for the (Ois) risk-free rate;

2. the default intensity dynamics, having two dif-ferent sets of parameters for H and L, but alsosharing an identical type of dynamics for the twotypes of banks;

3. the panel reshuffling/transition dynamics, accord-ing to the chosen assumption amongst the threeproposed above;

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Moreover, we assume a mutual independence betweenall of these separate dynamics.

All this means that the user is then free to chooseher own preferred dynamics for the Ois and the defaultintensities, even if we will adopt a CIR dynamics for allof them in what follows. Our choice should be regardedas taken just for explanatory purposes.

LIBOR-OIS BASIS MODELLING

Assume that, for a given reference currency, a groupof (major) banks enter the Libor panel at time t. Tosimplify the analysis, we assume they are all equalto a representative bank that can go bankrupt withknown default probabilities for any future date.The default probabilities can be considered as anaverage of the default probabilities of the singlebanks of the panel: one may think that if she lendsmoney at time t to one of these banks, she will bearan expected default risk equal to that referring tothe representative bank.2

Consider for the moment that the representativebank is exactly similar to a specific institution withits own default risk; alternatively said, lending spot,or forward at a future date, an amount of moneyto the representative bank is no different than lend-ing to a specific bank operating in the market thatwishes to borrow (we will relax this assumptionsoon). The (risk-neutral) survival probability3 ofthe representative bank of the Libor panel, at timet up to time T, is:

SP(t, T) = EQ[

e−∫ T

t λsds∣∣∣∣Ft

](1)

where τF is the time the default occurs, λt is the(possibly stochastic) default intensity, which we as-sume independent from interest rates. We have alsothat PD(t, T) = 1− SP(t, T).

Assume for the moment we want to price adeposit starting in ts and expiring in ts + τ: themoney is lent to a defaultable counterparty witha well defined survival probability SP(ts, ts + τ).In case of default, we suffer a percentage loss ofthe notional of the deposit market value equal tothe Loss Given Default Lgd. To further simplify thenotation, assume a unit notional.

The simply compounded risk-free (Ois) rate forthe period [ts, ts + τ] is denoted by R(ts, ts + τ) andthe simply compounded credit (basis) spread isdenoted by S(ts, ts + τ). If ts > 0 these are the

simply compounded forward rates. The total Li-bor rate applied to the deposit is L(ts, ts + τ) =R(ts, ts + τ) + S(ts, ts + τ)

The credit spread represents a fair default riskpremium over the risk free rate. As such, it is cal-culated in order to equate the discounted expectedvalue of the risky deposit, under the risk neutralmeasure Q, (considering both cases of default dur-ing the contract lifetime and survival until maturity)to:

• the unit notional, if ts = 0 (i.e.: if the depositstarts today);

• the unit notional times the expected sur-vival probability of the counterparty until ts,EQ[1τF>ts+τ |Ft] if ts > 0 (i.e.: the depositstarts on a future date, and we weight thenotional by the probability that the deposit ac-tually starts, or that the counterparty survivesat the start time).

Let us start by considering a spot starting de-posit.

Spot Credit Spread

The equation to determine the spot credit spreads,assuming Recovery of Face Value (RFV), is:4

1 = EQ[

DD(0, τ) ·[1 +

(S(0, τ) + R(0, τ)

)· τ]]

=

= PD(0, τ) ·[1 +

(S(0, τ) + R(0, τ)

)· τ]· SP(0, τ)

+ (1− Lgd) · PD(0, τ) =

=1 + (S(0, τ) + R(0, τ)) · τ

1 + R(0, τ) · τ · SP(0, τ)

+(1− Lgd) · PD(0, τ)

1 + R(0, τ) · τ(2)

We have indicated with DD(t, T) the defaultrisk-free discount factor from T to t, and withPD(t, T) = EQ[DD(t, T)] = 1

1+R(t,T)(T−t) the pricein t of default risk-free zero-coupon bond expiringin T.

By some simple algebra, we get from (2):

S(0, τ) =1τ·(Lgd + R(0, τ) · τ

)· PD(0, τ)

1− PD(0, τ)(3)

If we define the adjusted default probability asPD(0, τ) = PD(0, τ)/(1− PD(0, τ)), then we can

2Clearly, once the deal is struck and the counterparty is known, the exact credit risk borne by the lender can be different from the(average) credit risk of the representative bank.

3It is likely superfluous stressing that we adopt a reduced form approach to default modelling.4This analysis is taken form Castagna and Fede [2].

32iasonltd.com

rewrite the spread as:

S(0, τ) =1τ·(Lgd + R(0, τ) · τ

)· PD(0, τ) (4)

Forward Credit Spread

In case of a forward start deposit, the forward creditspread, assuming again a Recovery of Face Value,is derived from the following equivalence:

EQ[1τF>ts]

= EQ[

DD(t,ts+τ)·[

1+(

S(ts ,ts+τ)+R(ts ,ts+τ))·τ]] (5)

Working out the expectations:

SP(0, ts) =

= PD(ts, ts + τ) ·[1 + (S(ts, ts + τ) + R(ts, ts + τ))τ

]·

· SP(0, ts + τ) + (1− Lgd) · (SP(0, ts)− SP(0, ts + τ))

=1 + (S(ts, ts + τ) + R(ts, ts + τ)) · τ

1 + R(ts, ts + τ) · τ · SP(0, ts + τ)

+(1− Lgd) · (SP(0, ts)− SP(0, ts + τ))

1 + R(ts, ts + τ) · τ(6)

In order to solve this equation, define the adjustedforward default probability as:

PD(ts, ts + τ) =EQ[τF > ts]

EQ[τF > ts + τ]− 1

=SP(0, ts)

SP(0, ts + τ)− 1

(7)

The forward credit spread S(ts, ts + τ) is retrievedwith some algebra:

SP(0, ts)

SP(0, ts + τ)(1 + R(ts, ts + τ) · τ) =

= 1 + (S(ts, ts + τ) + R(ts, ts + τ)) · τ+ (1− Lgd) · PD(ts, ts + τ)

(8)

By expressing the LHS in terms of conditionalforward default probabilities:

PD(ts, ts + τ) · R(ts, ts + τ) · τ + Lgd · PD(ts, ts + τ)

=S(ts, ts + τ) · τ

1 + R(ts, ts + τ) · τ

Finally, we get:

S(ts, ts + τ)

=1τ·(Lgd + R(ts, ts + τ) · τ

)· PD(ts, ts + τ)

(9)

For typical values of interest rates, and relevantLibor tenors (i.e.: 1,3,6 and 12 months), we can

safely assume that Lgd ≈ Lgd + R(ts, ts + τ)τ. Thespot and forward spreads can then be respectivelywritten as:

S(0, τ) ≈ Lgdτ

PD(0, τ) (10)

andS(ts, ts + τ) ≈ Lgd

τPD(ts, ts + τ) (11)

Alternatively, if R(ts, ts + τ)τ is not negligible, wecan simply replace the original Lgd with Lgd∗ =Lgd+ R(ts, ts + τ)τ and consider R(ts, ts + τ) a con-stant.

From Credit Spreads to Ois-Libor Basis

We now relax the assumption that the representa-tive bank is exactly the same as a given bank, andwe explicitly consider that it may change over time,mirroring the possible Libor panel’s changes.

We will introduce three different ways to modelthe modification of the Libor panel, that will pro-duce three different models. We will dwell moreon the first model, giving an intuitive representa-tion of the panel transition process; the other twoapproaches are a variation that can be easily un-derstood once one grasps the mechanics of the firstone.

Model 1: Stochastic Total Replacement of the Sin-gle Representative Bank

Assume we start with a given panel of banks char-acterised by a credit risk summarised in the spotLibor-Ois spread curve and referring to the repre-sentative bank at time t = 0. The credit spreadcurves are determined by the default probabilitiescommanded by an intensity process λH , as in equa-tion (1). At a future time υ > 0 a change in thepanel may occur: a new representative bank entersin the panel, replacing the bank currently enteringit. This new bank has a credit quality determinedby the default probabilities originated by anotherintensity process λL, which starts exactly when thereplacement occurs.

The representative bank may change over timedue to its credit standing change (typically a wors-ening) or default. Any transition implies a resetof the process, meaning that the new process, re-ferring to the new representative bank replacingthe old one, will start exactly when the randomtransition event occurs.

Moving from a one-time to a continuous-timereplacement process, we can generalise the ideaoutlined above in a rather straightforward fashion.Assume that we are at time t = 0, and that we areinterested in determining the Libor forward spread

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for a deposit starting in ts and expiring in ts + τ, asindicated in Figure 5. At time ts the panel will bethe one at time 0 with probability wH0(ts); duringthe period [0, ts] a continuous replacement processtakes place: at each time ξi, for i > 0, a new panelcan replace the original one, and the probabilitythat this is the panel existing at the start of the de-posit in ts is indicated by wHL(ts, ξi). To each newpanel, corresponding to a given representative bank,is associated a specific default risk, commanded bya default intensity process starting in ξi.

Hence, at time ts, loosely speaking, the represen-tative bank’s default probability will be a weightedaverage of all the default probabilities of the repre-sentative banks that can form the panel by the timeξi on.

It is important to highlight the fact that the de-posit counterparty, i.e.: the borrower bank, is a spe-cific member of the interbank population that is im-plicitly assumed to be an infinity of banks that canreplace the defaulted, or credit deteriorated, banksincluded in the current and future panels. When-ever there is a credit standing transition or a de-fault, the current representative agent changes andthe whole population changes accordingly. Thistransition in overall population characteristics isequivalent to a replacement of the representativeagent with a new kind of representative agent.

It is assumed that all banks have mutually in-dependent default intensity dynamics. Therefore,given this assumption of independence and infi-nite population, we imply that there will alwaysbe a bank (embodied in the representative bank) towhich the Libor rate can be applied when it asks toborrow money. In other words, although the Liborpanel is made of defaultable entities, the replace-ment process (jointly with the above mentionedassumptions) ensures that the process of the Libor-Ois basis never stops, and that there is always toopportunity to lend money to a Libor bank.

When we monitor the credit risk in a forwardstarting deposit with a specific representative bank,we always consider the possibility that it might gobankrupt before the contract inception in ts and thiswill be accounted for in the specific representativebank’s credit spread by using the relevant forwarddefault probabilities. In case a specific representa-tive bank defaults (i.e.: the panel stops existing anda replacement occurs), we will move our monitoringto another representative bank which is indepen-dent from the previously monitored one. Givenour assumptions we can rest assured that we willalways find a new representative bank to monitor.

We will provide in next section the formulae forForward Libor-Ois basis curves referring to general

version of this model: it allows for a change in theLibor panel after a total replacement of the existingrepresentative bank with a new representative bankthat can be either of type L or H.

Forward Libor-Ois basis curves

Whenever a series of replacement events occurs be-tween t = 0 and t = ts, only the last of these eventsis relevant, since on every replacement the previousdefault intensity process stops and is replaced bythe default intensity process of the new represen-tative bank replacing the old one. Once the lastreplacement occurs, we can use the credit spreadequations defined in the beginning of this section,relating the basis to the forward default probabilityunder the last extracted λ process.

Assume the last replacement time is ξ, so thatthe new default intensity process λ starts exactlyin ξ. Consider two cases: in the first the last newbank will be of type H, in the other it will be oftype L. Given the assumption that the functionalform of the intensity process λ between switchingevents does not change, we have that the forwarddefault probability will be simply shifted in time byξ: PDz(ts − ξ, ts − ξ + τ) (with z ∈ {H, L}), where

PDz(ts − ξ, ts − ξ + τ) =SPz(0, ts − ξ)

SPz(0, ts − ξ + τ)− 1

(12)We also trivially have SPz(ts − ξ, ts − ξ + τ) =1− PDz(ts − ξ, ts − ξ + τ).

Additionally, denote the conditional creditspread, under the replacement conditions, asSH(ts − ξ, ts − ξ + τ) and SL(ts − ξ, ts − ξ + τ) re-spectively. Making use of the approximation intro-duced above, the credit spread is:

Sz(ts− ξ, ts− ξ + τ) =Lgd

τ·PDz(ts− ξ, ts− ξ + τ)

(13)where PDz(ts − ξ, ts − ξ + τ) = SPz(0, ts −ξ)/SPz(0, ts − ξ + τ)− 1.

So far we showed the calculations for a creditspread conditioned on a specific last replacementevent ξ ∈ (0, ts). We need to integrate for all thepossible ξ’s in the interval (0, ts), bearing in mindthat two last replacement time ξ’s are obviouslymutually exclusive. Therefore, to compute the un-conditional Libor-Ois basis, we have to considerthese three general possibilities:

1. the representative bank is never replaced,therefore we will calculate the basis as theone of the initial bank;

2. the representative bank is replaced at leastonce and in the last replacement the new bankis of type H;

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FIGURE 5: Visual rendering of the replacement process of the Libor panel of banks.

3. the representative bank is replaced at leastonce and in the last replaced the new bank isof type L.

The first case has probability: wH,0(0, ts), andthe spread is:

S1(ts, ts + τ) = wH,0(ts) · SH(ts, ts + τ) (14)

The weight wH,0(0, ts), as well as the other weightsin the following formulae, are derived in AppendixA5.

In the second case, the spread is calculated byintegrating over all admissible ξ’s the conditionalspread SH(ts− ξ, ts− ξ + τ) multiplied by the prob-ability density function wHH(ts, ξ).

S2(ts, ts + τ)

=∫ ts

0wHH(ts, ξ) · SH(ts − ξ, ts − ξ + τ)dξ

(15)

In the third case the spread is calculated similarlyto the second case:

S3(ts, ts + τ)

=∫ ts

0wHL(ts, ξ) · SL(ts − ξ, ts − ξ + τ)dξ

(16)

Recall that the probabilities wz1,z2(t, T) refer to thelast replacement from z1 to z2 occurring between tand T: they implicitly contain also all the possiblereplacements from the two types of representativebank occurring before the last one.

The unconditional forward Libor-Ois basis issimply the sum of the three terms above, sincethey are mutually exclusive and they are alreadyweighted for their respective probabilities:

SLibor(ts, ts + τ)

= S1(ts, ts + τ) + S2(ts, ts + τ) + S3(ts, ts + τ)

(17)

Marginal forward Libor-Ois basis p.d.f.

To derive the forward Libor-Ois basis marginalp.d.f., we need to condition it on:

• a certain state z ∈ {H, L}

• last replacement event in ξ ∈ (0, ts)

• survival until ts − ξ

We need to derive the complete density, account-ing for all possible ξ ∈ (0, ts). To this end, considerthe three cases:

1. No replacement of the representative agentoccurs in (0, ts)

2. One or more replacements occur in (0, ts),with a last replacement event in ξ ∈ (0, ts)collapsing in state H

3. One or more replacements occur in (0, ts),with a last replacement event in ξ ∈ (0, ts)collapsing in state L

5Appendices are available in the paper version published online at http://www.iasonltd.com/wp-content/uploads/2015/03/Modelling-Credit-Spreads-and-Libor-Basis-v-7.1-1.pdf.

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The respective probabilities are:

1. wH,0(ts)

2. wHH(ts, ξ)

3. wHL(ts, ξ)

The p.d.f. is:

gQ(S̃Libor, ts, τ)

= gQ1 (S̃, ts, τ) + gQ

2 (S̃, 0, ts, τ) + gQ3 (S̃, ts, τ)

(18)

where the functions gQ1 (.), gQ

2 (.) and gQ3 (.) are given

in Appendix A.

Model 2: Stochastic Partial Replacement with De-tailed Libor Panel

In the Model 2 we extend the idea of Model 1 byallowing for a detailed description of the initial Li-bor panel. In more detail, assume that the panel iscomposed by N banks6. They can be both of type Hand L, typically with a mix at the observation datecontaining more of the former if Libor-Ois curvesare humped.

Each bank in the panel can be replaced by newbanks of both types in the future; the credit spreadof these banks is commanded by an intensity pro-cess that starts at the time the replacing banks enterin the panel. The replacement process is modelledin the same way as Model 1, by a continuous timeMarkov chain.

The main difference between Model 1 andModel 2 is that in the latter we consider the ac-tual number of banks entering in the Libor panel,although they can only be of two types. The mod-ifications of the panel can occur for any of the Nbanks at random future times, contrarily to Model1, in which the (one) representative bank can bereplaced at future times by another representativebank, thus completely renewing the composition ofthe panel.

Forward Libor-Ois basis curves

If we denote the i member’s initial state as zi(0) ∈{H, L}, we have two kinds of random variablesdepending on zi(0).

Each random variable S̃i will follow dynamicsaccording to Model 1, with initial state zi(0). Wedivide the panel members in two subsets ZH ={i∣∣ zi(0) = H} and ZL = {i

∣∣ zi(0) = L}. Then

the Libor-Ois basis random variable may be ex-pressed as:

S̃Libor(ts, ts + τ)

=1N

(∑

i∈ZL

S̃i(ts, ts + τ) + ∑j∈ZH

S̃j(ts, ts + τ)) (19)

Denote with SM1,H(ts, ts + τ) and SM1,L(ts, ts +τ) the value of a forward Libor-Ois basis calculatedaccording to Model 1 with initial state H and Lrespectively. Suppose that there are m membersin ZH and N −m members in ZL. Thus the expec-tation of S̃Libor will be the weighed average of theforward Libor-Ois basis above defined:

SLibor(ts, ts + τ)

=N −m

N· SM1,L(ts, ts + τ) +

mN· SM1,H(ts, ts + τ)

(20)

Marginal forward Libor-Ois basis p.d.f.

To find out the density of S̃Libor(ts, ts + τ), we needto calculate the p.d.f. of the following random vari-able:

S̃Libor(ts, ts + τ)

=1N

(∑

i∈ZL

S̃i(ts, ts + τ) + ∑j∈ZH

S̃j(ts, ts + τ)) (21)

Suppose that the spread of the bank i is inde-pendent from the spread of the bank j for everyi 6= j. The density we are looking at is simply theconvolution of the densities S̃i∈ZL(ts, ts + τ) andS̃j∈ZH (ts, ts + τ)(ts, ts + τ).

Denote with gQM1,L(S̃, ts, ts + τ) and

gQM1,H(S̃, ts, ts + τ) the Libor-Ois basis marginal

density according to Model 1, with initial states Hand L respectively. These are the respective p.d.f.for S̃i∈ZL(ts, ts + τ) and S̃j∈ZH (ts, ts + τ).

If there are m members in ZH and N −m mem-bers in ZL, the marginal density of the Libor-Ois ba-sis is given by the convolution of two components:i) the p.d.f. of the weighted sum of the m membersin group ZH and ii) the p.d.f. of the weighted sumof N −m members in group ZL. Each member isequally weighted by 1

N .6At the time of writing, the USD Libor panel is made of 18 banks; the EUR Euribor panel is made of of 25 banks.

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gQLibor(y, ts, ts + τ) =[(

N · gQM1,H(N · S̃, ts, ts + τ)

)

· · · ∗(

N · gQM1,H(N · S̃, ts, ts + τ)

)]︸ ︷︷ ︸

m

∗ · · ·

∗[(

N · gQM1,L(N · S̃, ts, ts + τ)

)∗ · · ·

· · · ∗(

N · gQM1,L(N · S̃, ts, ts + τ)

)]︸ ︷︷ ︸

N−m

(22)

where ∗ denotes the convolution operator.

Model 3: Continuous Time Deterministic Re-placement Process of the Single RepresentativeBank

The Model 3 for the Libor-Ois spread hinges onthe assumption that new replacing representativebanks gradually replace the representative banksentering the initial Libor panel. So we have twomain differences between Model 1 and Model 3:i) the initial panel can be a combination of H andL type banks in Model 3, whereas it was a panelmade by a single type representative bank in Model1; ii) in Model 3 the replacement is not total, as inModel 1, but only a fraction of old representativebanks can be replaced by new H and L type banks;finally iii) the replacement process occurs contin-uously and in a deterministic fashion in Model 3,contrarily to the Markov chain process in Model 1.The technical details of the transition mechanics arein the Appendix B.

Forward Libor-Ois basis curves

The forward Libor-Ois basis SLibor(ts, ts + τ) in thismodel is the sum of two components:

1. the contribution of the initial panel, damp-ened by the effect of the replacement of newbanks, denoted by S1(ts, ts + τ);

2. the integral of the contributions of all the newbanks that enter the panel in (0, ts), denotedwith S2(ts, ts + τ).

Denote with S(ts, ts + τ) = SH(ts, ts + τ)SL(ts, ts +τ) the vector containing the H and L single staticcounterparty forward credit spreads. The first com-ponent is given by:

S1(ts, ts + τ) =

(B(0, ts) ·w0

)′· S(ts, ts + τ) (23)

where B(0, ts) is the decay matrix, defined in (85),Appendix B, and w0 is the initial Libor panel com-position vector.

The second component is given by:

S2(ts, ts + τ)

=∫ ts

0

(ϕ(u, ts)

)′ · S(ts − u, ts − u + τ) · du(24)

where ϕ(u, ts) is the new bank weight density vec-tor (with H and L components) defined in equation(83), Appendix B.

Finally, the complete forward Libor-Ois basis isgiven by:

SLibor(ts, ts + τ) = S1(ts, ts + τ) + S2(ts, ts + τ)(25)

The formulae for the single components are in theAppendix.

Marginal forward Libor-Ois basis p.d.f.

Let S̃H(ts, ts + τ) ∼ p̄QH(S̃H , ts, ts + τ) and S̃L(ts, ts +

τ) ∼ p̄QL (S̃L, ts, ts + τ), where p̄Q

z is the risk neutralmarginal p.d.f of a credit spread for a z-type bank,for a deposit starting in ts and maturing in ts + τ.Then:

S̃1(ts, ts + τ) ∼ gQ1 (S̃1, ts, ts + τ)

gQ1 (S̃1, ts, ts + τ)

=

[1

|bH(0, ts)wH(0)|· p̄Q

H

(S̃1(ts, ts + τ)

bH(0, ts)wH(0), ts, ts + τ

)]∗

∗[

1|bL(0, ts)wL(0)|

· p̄QL

(S̃1(ts, ts + τ)

bL(0, ts)wL(0), ts, ts + τ

)](26)

where ∗ denotes the convolution operator.Then the complete spread has the following

p.d.f.:

S̃Libor(ts, ts + τ) ∼ gQ1 (S̃Libor−S2(ts, ts + τ), ts, ts + τ)

(27)As explained in Appendix B, S2(ts, ts + τ) has in-finitesimal variance and therefore is a deterministicprocess. The detailed formula is given in AppendixB as well.

LIBOR CAPLET&FLOORLETVALUATION WITH STOCHASTIC

BASIS

The framework outlined above allows to retrievethe marginal densities for the Libor-Ois basis ineach of the three models analysed. It is then pos-sible, under the assumption of the Libor-Ois basis

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independent from the corresponding Ois rate, thesum of both being the Libor rate.7

Let us start by considering a deterministic andconstant additive spread S̄: then a Libor capletwould be equivalent to a caplet on an Ois rate withadjusted strike K̃ = K− S̄. Let R(ts, ts + τ) be theforward Ois rate between ts and ts + τ, observedat time t. A caplet on Libor rate L(ts, ts + τ) =R(ts, ts + τ) + S̄ has a pay-off at the natural expiryin ts equal to:

CapletLibor(

L(ts, ts + τ), K, ts, ts + τ)=

= max[R(ts, ts + τ) + S̄− K, 0

]= max

[R(ts, ts + τ)− (K− S(ts, ts + τ)), 0

]=

= CapletOis(

R(ts, ts + τ), K− S̄, ts, ts + τ)

At time t, the caplet is worth CapletOis(

R(ts, ts +τ), K − S̄, ts, ts + τ

)and can be computed by any

available model commonly adopted in practice, e.g.:a Black formula.

If we consider a stochastic credit spread andassume its independence from Ois rates, we cansimply evaluate the Libor caplet as the Ois capletabove conditioned on all admissible values of S̄.We may then express the value of a Libor caplet asthe convolution between the Ois caplet (as a func-tion of the strike K) and the marginal basis densitygQ(S, ts, ts + τ), whose explicit formula is given foreach of the three models:

CapletLibor(

L(ts, ts + τ), K, ts, ts + τ)=

= CapletOis(

R(ts, ts + τ), K, ts, ts + τ)∗

∗ gQ(K, ts, ts + τ) =

=∫ +∞

−∞CapletOis

(R(ts, ts + τ), K− S, ts, ts + τ

)· gQ(S, ts, ts + τ) · dS

(28)

where ∗ indicates the convolution operator, settingas the convolution domain K ∈ R.

A SPECIFICATION OF THE MODELWITH CIR INTENSITY DYNAMICS

As we have mentioned above, the framework wehave sketched is modular, in the sense that, underthe stated assumptions, we can choose any dynam-ics for the intensity processes for H- and L-typebanks, and thus specify the Models 1, 2 or 3 wehave analysed above for the Libor-Ois basis.

Besides, we can choose any dynamics for the(risk-free) Ois rate and then come up with a full

specification for the Libor rate dynamics that will al-low for the valuation of Libor derivatives, includingcaps&floors and swaptions.8

In what follows we will specify the default in-tensity dynamics as CIR processes9 and we willshow the basis curve it is possible to obtain by thethree models of Libor-Ois basis.

Forward Credit Spreads

Assume a bank of type z can go defaulted accord-ing to a jump process commanded by an intensitywhose dynamics - under the risk neutral measureQ - follows a CIR process of the type:

dλz(t) = κz (θz − λz(t)) · dt + σz

√λz(t) · dWQ

z,t(29)

where z ∈ {H, L} is a label variable indicatingwhether the counterparty is of the H or L type(high or low credit risk respectively). The initialcondition is λz(0).

Since the CIR process belongs to the affine ex-ponential family, forward credit spreads may beexplicitly derived. Given the survival probability:

SPz(0, t) = E[

exp(−∫ t

0λz(u)du

)]= Az(0, t) exp

(− Bz(0, t)λz(0)

) (30)

γz =√

κ2z + 2σ2

z νz =4κzθz

σ2z

Az(T, S)

=

[2γz exp [(κz + γz)(S− T)/2]

2γz + (κz + γz) (exp [(S− T)γz]− 1)

]νz/2

Bz(T, S)

=2 (exp [γz(S− T)]− 1)

2γz + (κz + γz) (exp [γz(S− T)]− 1)(31)

the forward credit spreads is:

Sz(ts, ts + τ) =Lgd

τ· PD(ts, ts + τ)

=Lgd

τ·[

Az(0,ts) exp[−Bz(0,ts)λz(0)

]Az(0,ts+τ) exp

[−Bz(0,ts+τ)λz(0)

]−1

] (32)

Practical Examples

Having specified the dynamics of the intensity λ(t)as a CIR process, we are now able show the timestructures for the Libor-Ois basis generated withinour framework. The aim of all the models is toaccurately reproduce the humped shape of the realdata (see Figure 1): it means that our model is de-signed to be flexible enough to reproduce the humpin a realistic time interval, to match the slope of the

7Similar general formulae are given also in Mercurio [5].8We have not studied the evaluation of swaptions in this work, but it is possible in the outlined framework.9See Cox, Ingersoll and Ross [3].

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Maturity 1M 3M 6M 1Y0 0.81% 0.83% 0.86% 0.90%1 0.96% 0.96% 0.97% 0.98%3 1.00% 1.00% 1.00% 1.00%5 1.00% 1.00% 1.00% 1.00%7 1.00% 1.00% 1.00% 1.00%

10 1.00% 1.00% 1.00% 1.00%15 1.00% 1.00% 1.00% 1.00%20 1.00% 1.00% 1.00% 1.00%25 1.00% 1.00% 1.00% 1.00%30 1.00% 1.00% 1.00% 1.00%

TABLE 1: Ois rates. The underlying short rate follows a CIR process with r0 = 0.80%, κOis = 1.5, θOis = 1.00% and σOis = 5.00%

CIR Parameters Replacement Parametersλ0 κ θ σ ηH ηL aHH aHL aLH aLL

Model 1 H 0.06% 1.0 1.8% 1.0% 0.2 4 60% 40% 0% 100%L 0.05% 0.5 0.5% 1.0% 0.2 4 60% 40% 0% 100%

Model 2 H 0.10% 1.0 2.5% 1.0% 0.2 4 50% 50% 0% 100%L 0.05% 0.3 0.4% 1.0% 0.2 4 50% 50% 0% 100%

Model 3 H 0.06% 1.0 1.8% 1.0% 0.2 4 60% 40% 0% 100%L 0.05% 0.5 0.5% 1.0% 0.2 4 60% 40% 0% 100%

TABLE 2: Parameters for models with constant replacement

time structure and to replicate the spot (deposits’)Libor-Ois spread.

Assume we set the parameters of the CIR inten-sity process for the H and L type banks, in each ofthe 3 models, as shown in the table 2. We can thencheck which type of shapes for the Libor-Ois basisterm structure the 3 models generate. Moreover,we test also the flexibility of the model by intro-ducing time dependent parameters in the transitionprocesses and in the exit intensity from the panel.

The starting Ois rates’ term structure is alsoneeded: we generate a curve by a CIR model for theshort rate whose parameters are chosen such thatthey fit best the market quotes dealing on Novem-ber 1st, 2014. In table 1 we show the term structuresof the forward Ois for the 1M, 3M, 6M and 1Y tenor.

The results are shown in figures 6(a,c,e), wherewe used fixed parameters for the exit intensity fromthe panel and for the transition dynamics specificto each model. Note that in the Model 2 (Stochas-tic Partial Replacement with Detailed Libor Panel)we considered a panel of 25 banks, such that 15started as H type. Moreover, in the Model 3 (Con-tinuous Time Deterministic Replacement Process ofthe Single Representative Bank) we chose a startingpanel entirely composed of H type banks, that iswH(0) = 1 and wL(0) = 0.

In figure 6(b,d,f) we show the Libor-Ois basisterm structure when allowing for time dependentparameters of the panel exit intensity and transitiondynamics. The most accurate model seems to bethe second one, that is the model with stochasticpartial replacement with detailed Libor Panel. As a

matter of fact, such a time structure does not sufferthe initial spike and moreover the hump correctlycover the interval between spot date and 10-ishyears, switching from a concave to convex slope.

Credit Spread Marginal p.d.f.

Given a CIR specification for the dynamics of λz(t),we wish to calculate the marginal p.d.f. of thecredit spread S̃z(ts, ts + τ), which is indicated asp̄Q

z (S̃z, ts, ts + τ).

Since S̃z(ts, ts + τ) = Lgdτ ·

(S̃Pz(0,ts)

S̃Pz(0,ts+τ)− 1)

, if

we define the random variable x = S̃Pz(0,ts+τ)

S̃Pz(0,ts), we

may equivalently say that the relationship betweenS̃z(ts, ts + τ) and x is:

S̃z(ts, ts + τ) =Lgd

τ·(

1x− 1)

(33)

Note that the previous relation is a deterministic,invertible and differentiable function. So, if we cal-culate the risk-neutral marginal p.d.f. of x first,which we will denote as f Q

z (x, ts, ts + τ), we areable to deduce from it p̄Q

z (S̃z, ts, ts + τ). The detailsare explained the Appendix A.

The CIR process belongs to the Affine Exponen-tial Family, therefore the random variable:

S̃Pz(ts, ts + τ) = EQ[

exp(−∫ ts+τ

tsλz(u)du

)∣∣∣∣Fts

]= A(ts, ts + τ) exp

(− B(ts, ts + τ)λz(ts)

)may be expressed in terms of the λz(ts) randomvariable.

Summer 201539

MARKET MODELS

((a)) Model 1 - constant parameters. ((b)) Model 1 - Time Variant Parameters

((c)) Model 2 - Constant Parameters ((d)) Model 2 - Time Variant Parameters

((e)) Model 3 - Constant Parameters ((f)) Model 3 - Time Variant Parameters

FIGURE 6: An example of forward basis curve with constant and time variant parameters.

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Since we wish to derive the p.d.f. of x, which isa ratio of survival probabilities, let us define Gts theprobability measure associated with the numeraireS̃P(0, ts). The associated Radon-Nikodym deriva-tive is:

dQdGts

(t) =1

S̃P(t, ts)

dQdGts

(ts) = 1

(34)

Using conditional expectations:

f Qz (x) · dx = EQ

[1SPz(ts ,ts+τ)∈(x,x+dx)

∣∣Fts

]= EGts

[1SPz(ts ,ts+τ)∈(x,x+dx) ·

dQdGts

∣∣Fts

] (35)

therefore to obtain our result we may equivalentlyswitch from Q to the Gts measure.

Let v ≤ t ≤ ts and define these auxiliary func-tions:

ψz =κz + γz

σ2z

qz(t, v) = 2 · [ρz(t− v) + ψz + Bz(t, ts)]

δz(t, v, λz(v)) =4 · (ρz(t− v))2 · λz(s) · eγz(t−v)

q(t, s)

Under the forward measure Gts , the distribution ofλz(t) conditional on λz(v) is given by:

pGts

λz(t)|λz(v)(x) = qz(t, v) · pχ2(ν,δz(t,v,λz(s)) (qz(t, v) · x)(36)

where pχ2(ν,δ)(x) denotes the marginal p.d.f. of aNon-Central Chi-Squared random variable with νdegrees of freedom and non-centrality parameter δ.

Given the Gts p.d.f. of λz(ts), by invertingthis relation we are able to derive the p.d.f. ofAz(ts, ts + τ) exp

(− Bz(ts, ts + τ)λz(ts)

)under Gts .

Set x = S̃Pz(ts, ts + τ).

λz(ts) =1

Bz(ts, ts + τ)· log

(Az(ts, ts + τ)

x

)(37)

This is an invertible and differentiable function ofx. Its first derivative is:

dλz(ts)

dx= − 1

Bz(ts, ts + τ) · x (38)

Using the classical probability result for the p.d.f.of an invertible and differentiable function of a ran-dom variable:

f Qz (x, ts, ts + τ)

=

∣∣∣∣dλz(ts)

dx

∣∣∣∣ · qz(ts, v) · pχ2(ν,δz(ts ,v))

(qz(ts, v) · λz(ts)

)(39)

We finally get:

f Qz (x, ts, ts + τ) =

=1

Bz(ts, ts + τ) · x ·

· qz(ts, v) · pχ2(ν,δz(ts ,v))

(qz(ts, v) · 1

Bz(ts, ts + τ)·

· log(

Az(ts, ts + τ)

x

))Once we have f Q

z (·), we are able to calculate thep.d.f. of the credit spread p̄Q

z according to (33)

p̄Qz (S̃, ts, ts + τ) =

τ

Lgd·

f Qz(Θ(S̃), ts, ts + τ

)(τ

Lgd S̃ + 1)2 (40)

This formula is explained in details in the Ap-pendix. We are now able to make it specific to anyof the 3 models:

1. for the first model the p.d.f. is given by theformula (76) in Appendix A;

2. once we have the p.d.f. for the first model, weeasily deduce the p.d.f. for the second modelapplying (22);

3. the p.d.f. for the third model is explicitlyshown in formulas (103) and (104), AppendixB.

Practical Examples

Given the densities for each model, the Libor Capletwill be a consequence of equation (28). We usethe same data for the Ois rates and the Libor-Oisspreads as above. For the Ois forward rates we alsoassume that they are lognormally distributed withone constant volatility set at 30%. Please note thatwe should have used a CIR model for the Ois ratesalso to evaluate caplets, to be consistent with theway we generated the Ois forward curves. Nonethe-less, the purpose of this section is to show whichis the impact on the smile shape introduced by theLibor-Ois basis models we have introduced. For thesame reason, we compare also the volatility smilesproduced by the Libor-Ois models with the smilegenerated by a simply displaced Lognormal model,with displacement set equal to the relevant forwardLibor-Ois basis, assumed to be constant.

We are then able to calculate the implied volatil-ity for each of the models as shown in Figures 7and 8 for the 6M tenor. Implied volatility smilesand implied volatility surface for other tenors (1M,3M, 1Y) are shown in Appendix C.

Summer 201541

MARKET MODELS

((a)) Model 1 - Constant parameters ((b)) Model 1 - Time Variant parameters

((c)) Model 2 - Constant parameters ((d)) Model 2 - Time Variant parameters

((e)) Model 3 - Constant parameters ((f)) Model 3 - Time Variant parameters

FIGURE 7: Volatility smiles for a caplet with expiry 10 years and tenor 6M. Ois rate is modelled by Black model with impliedvolatility equal to 30%.

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((a)) Model 1 - Constant parameters ((b)) Model 1 - Time Variant parameters

((c)) Model 2 - Constant parameters ((d)) Model 2 - Time Variant parameters

((e)) Model 3 - Constant parameters ((f)) Model 3 - Time Variant parameters

FIGURE 8: An example of volatility surfaces for a caplet with expiry 10 years and tenor 6M. Ois rate is modelled by Black modelwith implied volatility equal to 30%.

Summer 201543

MARKET MODELS

((a)) Model 1 - Constant parameters ((b)) Model 1 - Time Variant parameters

((c)) Model 2 - Constant parameters ((d)) Model 2 - Time Variant parameters

((e)) Model 3 - Constant parameters ((f)) Model 3 - Time Variant parameters

FIGURE 9: P.d.f. for Libor-Ois basis refering to 6M tenor

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We are also able to plot the p.d.f. for each modeland for each tenor. Figure 9 shows the p.d.f. for the6M tenor at different maturities for all the models.The other tenors present almost identical shapes,by differing from the 6M only for the center ofthe peaks, since each tenor’s p.d.f. is centered onits own forward rate. With the chosen set of pa-rameters, the p.d.f. of the third model implies apractically nil volatility. The other distribution forthe other two models, both in the constant and timevariant parameters, are multi-modal, due to thereplacement mechanism.

CONCLUSIONS

In the present paper we provided a framework thatis based entirely on micro-founded inference andcredit risk arguments: all of the assumptions, dy-namics and parameters derive from considerationson how market practitioners typically deal with theLibor-Ois basis.

The framework is flexible enough to capture thefeatures of the Libor-Ois basis quoted on the mar-ket. In fact it would be able to reproduce a broadvariety of complex basis term structures given theright time structure for the replacement parameters.We consider such a feature one of the strengthsof our setup. Since the replacement parameters inour models have a straightforward interpretation,they provide also a simple tool to analyze and inter-pret Libor-Ois basis expectations implied in marketquotes.

Furthermore, the present framework addresses

the issue of market illiquidity for some regions ofthe term structure. In fact, in most cases avail-able market quotes are not sufficient to cover thewhole Libor-Ois basis term structure for a giventenor: practitioners can use our framework to de-duce the illiquid parts of the curves for all tenorsfrom the quotes of actively traded instruments. Inour practical examples, the liquid 3M and 6M in-dexed instruments were sufficient to reproduce theentire set of curves.

We noticed a rather high probability of replace-ment of the Libor panel: this is likely due to the factthat not actively traded assets entered in the calibra-tion process. This factor leads to high replacementintensity, which in turn leads to small volatility forLibor-Ois basis distributions.

If we consider only actively traded instruments,we suspect that reasonable replacement parameterswill be sufficient for calibration. Our frameworkcould then be used to provide a more suitable in-terpolation and extrapolation method for the entireLibor-Ois basis term structure.

ABOUT THE AUTHORS

Antonio Castagna is Senior Consultant, co-founder ofIason ltd and CEO at Iason Italia srl.Email address: antonio.castagna@iasonltd.com

Andrea Cova worked as consultant at Iason.

Matteo Camelia worked as consultant at Iason.

ABOUT THE ARTICLE

Submitted: July 2015.

Accepted: August 2015.

References

[1] Ametrano F. M. and M. Bianchetti Everything You Al-ways Wanted to Know About Multiple Interest RateCurve Bootstrapping But Were Afraid To Ask. 2013.Available at www.ssrn.com

[2] Castagna, A. and Fede, F. Measuring and Managementof Liquidity Risk. Wiley. 2013.

[3] Cox J. C., J. E. Ingersoll and S. A. Ross. A theory of theterm structure of interest rates. Econometrica, n.53, pp.385-467. 1985.

[4] Duffie, D. and Schroder, M. and Skiadas, C. RecursiveValuation of Defaultable Securities and the Timing ofthe Resolution of Uncertainty. Annals of Applied Prob-ability, n.51. 1996.

[5] Mercurio, F. Interest Rates and The Credit Crunch:New Formulas and Market Models. 2010. Available athttp://papers.ssrn.com

[6] Mercurio, F. LIBOR Market Models with Stochastic Ba-sis. 2010. Available at http://papers.ssrn.com

[7] Moreni, N. and Pallavicini, A. Parsimonious HJM Mod-elling for Multiple Yield-Curve Dynamics. 2013. Avail-able at www.ssrn.com

[8] Morini, M. Solving the Puzzle in the Interest Rate Mar-ket. 2009. Available at www.ssrn.com

Summer 201545

in the previous issue

Spring 2015 A

banking & finance

Pricing and Risk Management withHigh-Dimensional Quasi Monte Carlo andGlobal Sensitivity Analysis

Efficient Computation of theCounterparty Credit Risk Exposure

energy & commodity finance

Off-Shore Wind Investmentunder Uncertainty: Part I

Structural Positions in Oil Futures Contracts

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