Embed Size (px)
Citation preview
Electronic copy available at: http://ssrn.com/abstract=1281743Electronic copy available at: http://ssrn.com/abstract=1281743
Stepping Through Fourier Space
Sebastian Jaimungal
Department of Statistics and
Mathematical Finance Program,
University of Toronto, Toronto, ON, Canada
Vladimir Surkov
Department of Computer Science,
University of Toronto, Toronto, ON, Canada
October 9, 2008
ABSTRACTDiverse finite-difference schemes for solving pricingproblems with Levy underliers have been used inthe literature. Invariably, the integral and diffu-sive terms are treated asymmetrically, large jumpsare truncated, the methods are difficult to extendto higher dimensions and cannot easily incorporateregime switching or stochastic volatility. We presenta new efficient approach which switches betweenFourier and real space as time propagates backwards.We dub this method Fourier Space Time-Stepping(FST). The FST method applies to regime switch-ing Levy models and is applicable to a wide class ofpath-dependent options (such as Bermudan, barrier,shout and catastrophe linked options) and options onmultiple assets.
KEY WORDSOption Pricing, Levy Processes, Regime Switching,Fourier Methods, American Options, CatastropheOptions.
Jump-diffusion, more generally Levy models, havebeen extensively applied in practice to partially cor-rect the defects in the Black and Scholes (1973) andMerton (1973) (BSM) model and explain the im-plied volatility surface’s shape and dynamics. Popu-lar models include the variance gamma (VG) model(Madan, Carr, and Chang 1998) and normal inverse
Gaussian (NIG) model (Barndorff-Nielson 1997).Under such models, the pricing PDE transforms intoa partial-integro differential equation (PIDE) witha non-local integral term. By writing the price asconvolution and performing a transform on log-strike(Carr and Madan 1999) introduced an FFT methodwhich prices European calls/puts with a series ofstrikes in “one shot”.
For path dependent options or options with earlyexercise clauses, the full PIDE must be solved. Aplethora of finite difference methods for solving thesePIDEs have been proposed in the recent literature(see e.g. Andersen and Andreasen (2000), Briani,Natalini, and Russo (2004), Cont and Tankov (2004),and d’Halluin, Forsyth, and Vetzal (2005)), yet theyall have many points in common: Firstly, the inte-gral and diffusion terms of the PIDE are often treatedseparately – invariably, the integral term is evaluatedexplicitly in order to avoid solving a dense systemof linear equations. Secondly, within infinite activ-ity processes, the small jumps are approximated bya diffusion and incorporated into the diffusion term.Thirdly, the integral term is localized to the boundeddomain of the diffusion term, i.e. large jumps aretruncated. Finally, the separate treatment of diffu-sion and integral components requires that functionvalues are interpolated and extrapolated between the
1
Electronic copy available at: http://ssrn.com/abstract=1281743Electronic copy available at: http://ssrn.com/abstract=1281743
diffusion and integral grids.These factors together make finite difference meth-
ods for option pricing under jump models quite com-plex, and potentially prone to accuracy and stabil-ity problems, especially for path dependent claims orclaims with early exercise clauses. In this work, wepresent a new Fourier Space Time-stepping (FST)algorithm which avoids the problems associated withfinite difference methods by switching back and forthbetween Fourier and real space. Working in Fourierspace transforms the PIDE into a system of eas-ily solvable ordinary differential equations (ODE).The characteristic exponent of the underlying Levyprocess naturally appears and since it is available,through the Levy-Khintchine formula, in closed formfor all independent increment processes, makes theFST method quite flexible and generic. The FSTalgorithm naturally leads to a symmetric treatmentof the diffusion and jump terms, avoids steppingthrough time between observation dates, easily gen-eralizes to higher dimensions and can incorporateregime switching rather simply.
The method, in its first incarnation together withextensive numerical experiments and further exam-ples can be found in Jackson, Jaimungal, and Surkov(2007). Recently, Lord, Fang, Bervoets, and Ooster-lee (2008) have independently worked along similarlines of thought without making the connection tothe PIDE or extending to regime switching modelsand instead carried out an interesting analysis of themethod’s convergence.
1 The Model
Levy models on their own do not fully capture thestylized features observed in stock returns nor arethey capable of fitting the entire implied volatilitysurface. It is well known that stochastic volatilityeffects are necessary to model the mid/long term dy-namics while jump components assist in fitting theshort term dynamics. Stochastic time changes alaCarr and Wu (2002) is one approach for integrat-ing Levy processes and stochastic volatility models.They demonstrate that the Laplace transform of thetime-changed process can still be obtained in closedform via a clever complex valued measure change.
Regime switching models provide an enticing alter-native for combining stochastic volatility and Levymodels. Elliott and Osakwe (2006) employ a Markovmodulated pure jump process, determine its Laplacetransform and proceed to value European styled op-tions on a single underlier.
Motivated by these works, we are interested inmodeling a vector of returns when the world switchesbetween a finite number of regimes. Within eachregime, returns are driven by a vector valued Levyprocess incorporating both diffusive and jump com-ponents, while the regime changes may induce:
• a new covariance structure in the underlying dif-fusions – preserving variances but changing cor-relations, or changing variances while preservingcorrelations, or changing both variances and cor-relations;
• a new jump measure to incorporate regimes ofhigh uncertain and a series of small marketcrashes – perhaps in one state the jump mea-sure may be absent all together and in the otherstate the jump process is turned on and has in-finite activity with large negative jumps;
• a new co-dependence in the jump structure –Levy copulas have recently been introduced byKallsen and Tankov (2006) which allow non-trivial co-dependence between jump componentsof two or more processes;
• new interest rate levels – to mimic stochastic in-terest rates;
• any combination of the above.
A typical sample path using a two-state regimeswitching VG model together with the underlyingLevy paths are depicted in Figure 1. In regime 1(panel (a)) the jumps are of moderate size and sym-metric, while in regime 2 (panel (b)) jumps are biaseddownwards and are larger. In the regime switchingcase (panel (c)) volatility and downward jumps clus-ter – non-stationary features that cannot be capturedby Levy models alone.
In our framework, the state of the world, enumer-ated 1, . . . ,K, is driven by a Markov chain denotedZt with generator matrix A. Conditional on the
2
8.0%
4.0%
0.0%
‐4 0%0 2 4
4.0%
‐8.0%
‐12.0%
(a) Low volatility and moderate symmetric jumps
8.0%
4.0%
0.0%
‐4 0%0 2 4
4.0%
‐8.0%
‐12.0%
(b) High volatility and large downward biased jumps
8.0%
4.0%
0.0%
‐4 0%0 2 4
4.0%
‐8.0%
‐12.0%
(c) Regime Switching between models (a) and (b)
Figure 1: Sample returns from VG models both withand without regime switching. The top and mid-dle panels are produced by the underlying VG mod-els, while the bottom panel is produced by a Markovchain inducing switching between the two models.
world state, the d-dimensional vector of stock pricelog returns accumulate according to a d-dimensionalLevy process X(k)
t (when state k prevails). Let-ting (Xt)i = ln(St)i denote the logarithm of theith stock price, the model can succinctly be writtendXt = dX(Zt)
t . In this manner, the logarithmic re-turns, rather than the prices themselves, are modu-lated by the Markov chain.
In the remainder of this article, the Levy triple(which characterizes the diffusive and jump compo-nents) for state k is denoted (γ(k),Σ(k),ν(k)). Re-call that for a Levy process, γ is the uncompensateddrift vector, Σ is the covariance matrix of the Brow-nian pieces of the Levy process, and ν(dy × ds) isthe Levy density representing the rate of arrival ofjumps of size (y1, y1 + dy1] × · · · × (yd, yd + dyd]. Itis sometimes useful to incorporate a time-dependent
Levy density – to mimic seasonality effects for exam-ple – for brevity, such simple generalizations are leftfor the interested reader to explore.
For derivative valuation, the drift vector γ must bepinned to risk-neutralize the price processes by set-ting Ψ(k)(−i1) = r(k)×1 for each state k, where r(k)
denotes the risk-free rate prevailing in world state kand Ψ(k)(ω) := EQ[exp{iω · X(k)
t }|Ft] denotes thecharacteristic exponent of the Levy process. For-tunately, the Levy-Khintchine formula provides thecharacteristic exponent for any Levy model:
Ψ(k)(ω) = iγ(k) · ω − 12 ω ·C
(k) · ω
+∫Rn/{0}
(eiω·y − 1− iy · ω 1|y|<1
)ν(k)(dy) .
This model framework is quite general and can pro-duce an extremely wide variety of effects. Here are afew examples:
• Modeling defaultable equities by associating oneof the states as a default state and making itabsorbing;
• Modeling catastrophic losses which have bothmoderate and extreme epochs;
• Modeling equity price dynamics in which volatil-ity / jumps cluster;
• Modeling currencies in which the positive andnegative jumps are emphasized in differentregimes.
It is not difficult to imagine other applications. Wenow move on to valuing derivatives based on underly-ing assets following such regime switching Levy mod-els.
2 The FST Algorithm
We now develop an algorithm for valuing variousderivatives based on the above modeling assump-tions. To this end, let v(k)(Xt, t) denote the dis-counted price of a derivative instrument given thatstate k is prevailing at time t. The discounted priceprocess is a martingale under the risk-neutral mea-sure Q; consequently, after applying Ito’s lemma for
3
Levy processes together with regime changes, theprices v(k) satisfy the system of coupled PIDEs:
∂tv(k)(x, t) + (Akk + L(k))v(k)(x, t)
+∑j 6=k
Ajkv(j)(x, t) = 0 , (1)
where L(k) is the infinitesimal generator of the multi-dimensional Levy process in state k and acts on twicedifferentiable functions f(x) as follows
L(k)f(x) =(γ(k) · ∂x + 1
2 ∂x ·Σ(k) · ∂x
)f(x)
+∫Rn/{0}
(f(x + y)− f(x)
−y · ∂xf(x) 1|y|<1
)ν(k)(dy) .
As is well known, the various derivative instru-ments are valued by augmenting the PIDE with ap-propriate boundary conditions. For example, Euro-pean options require specifying the terminal bound-ary only, Bermudan options in addition require an op-timal exercise constraint at the exercise dates, knock-out Barrier options require truncating or modifyingthe value on the barrier monitoring dates, etc. . . .
It is important to note that (1) is a coupled systemof PIDEs in multiple dimensions. In principle, anyof the usual jump-modified finite-difference schemescan be adapted to this case. However, as discussedearlier, this is quite difficult due to the non-local in-tegral terms and especially so for the present coupledmulti-dimensional problem. Instead, we use trans-form methods to simplify the problem dramatically.
Define the Fourier transform as follows:
F [h](ω) :=∫
Rd
h(x) eiω·xdx (2)
Applying this operation to both sides of (1) reducesthe coupled PIDEs to a simple coupled system ofODEs indexed by the vector of frequencies ω:
∂tF [v](ω, t) + Ψ(ω)F [v](ω, t) = 0 .
where v denotes the vector of world state prices(v(1), . . . , v(K)) and the elements of the matrix Ψ(ω)are (Ψ(ω))kl = Akl + Ψ(k)(ω) δkl. The solutions ofthese ODEs are straightforward, and taking the in-verse transform leads to the simple exact algorithm
between any two monitoring dates
v(x, tn−1) = F−1[eΨ(ω) ∆tn F [v(·, tn)](ω)
](x) .
Note that eΨ(ω) ∆tn is a K × K matrix exponentialand F [v(·, tn)](ω) is a K-dimensional vector for eachd-dimensional frequency ω.
Introducing a grid in real space and Fourier space,letting v(n) denote the vector of prices at time tn
on the real grid, replacing the Fourier transform bythe discrete Fourier transform and making use of theFFT algorithm to perform the forward and inversetransforms leads to the FST method:
v(n−1) = F−1[exp{Ψ(·) ∆tn} F[v(n)(·)]
]. (3)
For Bermudan styled claims, the above price isviewed as the continuation (holding) value at timestep tn−1. Before iterating backwards, at each nodein real space, it must be compared with the intrin-sic value of the option, if the intrinsic value is larger,then that node is replaced by the intrinsic value. ForAmerican option, a penalty method can be used toaugment the system of PIDEs which results in a mod-ified version of the FST algorithm – this work is cur-rently being written up. For barrier options, a simpleapplication of the reflection principal neatly accountsfor the barrier condition. Double barrier options canbe treated by simply truncating the payoff, or alter-natively, applying several iterations of the reflectionscheme.
The FST scheme is related to the method employedby Carr and Madan (1999) for valuing European op-tions. In their work, the authors performed a trans-form in the log-strike price to obtain prices of optionsof all strikes using an FFT operation. Here, however,we employ the FFT to obtain prices for all log-spotlevels at intermediate decision dates. Another dif-ference is that the adjustment factor which pushesthe pole singularity for calls and puts off of the realaxis is absent. This issue can be treated in two ways:(i) shift the integration path appropriately; or (ii)truncate the payoff for sufficiently large spot values.We find that truncating is a very simple and stableprocedure which avoids the singularity problems; fur-thermore, we find that the results are insensitive to
4
the truncation point for sufficiently large truncation.
The FST method is far superior to any finite dif-ference inspired scheme for Bermudan or discretelymonitored barriers, shout styled options, options onmultiple asset among others. It can be applied toa wide class of Levy processes – even incorporat-ing Levy copulas in a trivial manner, can deal withregime switches efficiently and we find it to be verystable to model parameters1 and option payoffs.
It is more efficient precisely because between moni-toring dates the FST method is exact (up to the trun-cation of very large frequencies) and requires only asingle step from one date to the next. For example,a Bermudan monitored on a weekly basis will onlyrequire 50 time steps using the FST method, whileany finite-difference scheme will require at least 250time steps (assuming 1 step per day provides suffi-cient accuracy) – even without jumps. Furthermore,the jump measure is treated exactly, there is no ne-cessity to approximate small jumps with a diffusionterm, nor is it necessary to cut off large jumps.
3 Examples
Levy processes on their own have a difficult timecapturing longer termed options – allowing regimeswitches rectifies this problem quite well. In Figure2 we present the implied volatility surface obtainedby using (a) a single-sate VG model (b) a two-stateregime switching VG model which matches the shortterm smile of the single state model. The character-istic exponent within world-state i = 1, 2 is
Ψ(i)(ω) = − 1κ(i)
log(1− iµ(i)κ(i)ω +(σ(i))2κ(i)ω2
2)
Notice that the surface flattens out much more slowlyin the two-state model than in the single state model.A feature which is difficult to capture using singlestate models.
As our next application, we turn to an Americanspread option with underliers driven by two-regimeMerton jump-diffusion process. Specifically, the in-trinsic value of the option is (S2
t −S1t −K)+ and the
1Several finite difference schemes are tuned to specific mod-els and fail under certain model parameters.
(a) Single state VG Model
(b) Two state VG model
Figure 2: Implied volatility surface using (a) a singlestate VG model and (b) a two-state regime switchingVG model.
characteristic exponent is
Ψ(i)(ω1, ω2) = i(µ1 −σ2
1
2)ω1 + i(µ2 −
σ22
2)ω2
− σ21ω
21
2− ρ(i)σ1σ2ω1ω2 −
σ22ω
22
2+ λ1 (eiµ1ω1−σ2
1ω22/2 − 1)
+ λ2 (eiµ2ω2−σ22ω
22/2 − 1).
Here, µ1,2 and σ1,2 represent the mean and volatilityof the diffusive components of each asset, µ1,2 andσ1,2 represent the mean and variance of the jumpsfor each asset, and ρ(i) is the instantaneous correla-tion in regime-i (i = 1, 2). The parameter values forthe spread option are S1
0 = S20 = 100,K = 3, T = 1
and σ1 = 0.19, σ2 = 0.27, λ1 = 10.0, λ2 = 15.0, µ1 =0.01, µ2 = −0.01, σ1 = 0.017, σ2 = 0.013, q1 = q2 =0.01, r = 0.05 for the stock price process. We allowfor ρ to be in two different states, 0.3 and 0.8, wherethe regime-switching dynamics are driven by genera-
5
(a) Exercise policy
(b) Price difference in two regimes
Figure 3: Exercise slices (a) and price difference (b).
tor matrix A = [−0.2, 0.2; 0.9− 0.9].Figure 3 shows the effect of introduction of regime
switching to stock price process. The exercise pol-icy is dependent on the current state, as shown byexercise boundaries at τ = T/2 and τ = T in panel(a). Near maturity, the exercise policy is independentof the state since the payoff depends only on termi-nal stock price. In panel (b), the difference in optionprices between the low correlation state and high cor-relation state are provided. Notice that the spreadoption prices in high correlation state are lower sincethe difference between the two stock prices will typi-cally be lower.
To further illustrate the versatility of the FSTmethod, we value a non-standard path-dependent op-tion known as a double-trigger stop-loss (DTSL) con-
tract. The DTSL issues a stop-loss contract in theevent that the insurers own share value falls belowa critical level. Such options are issued as Bermu-dan contracts which can be exercised on a quar-terly, monthly, weekly or daily basis. The exer-cise value of the contract can therefore be writtenϕ = I(S < S∗) [(L− La)+ − (L− Ld)+] where S andL are the the share price and absorbed losses uponexercise.
Since large losses will drive share value down, thecodependence between loss jumps and share valuemust be incorporated. Jaimungal and Wang (2006)introduced a model which jointly models interestrates, share value and losses and value a related Euro-pean contract called a catastrophe equity put option.Here, we value the Bermudan DTSL product as wellas its European counterpart. To keep things simple,interest rates are assumed constant. The joint stockprice and loss dynamics is then
St = S0 exp{−αLt + γ t+ σWt} ,
Lt =Nt∑i=1
li .
Here, Nt is a Poisson process which drives the arrivalof losses and the random variables l1, l2 . . . are iidand gamma distributed with mean m and variance v.In this case, the characteristic exponent is
Ψ(ω1, ω2) = i γ ω1 − 12σ
2 ω21
+ λ
[1− i v
m(−αω1 + ω2)
]−m2v
− 1
.
In Figure 4, we provide the pricing surfaces for theEuropean case, together with the daily exercisableBermudan cases. The contract parameters are S∗ =100, La = 5 and Ld = 40. The joint stock price andloss parameters are σ = 0.2, r = 0.05,m = 2, v =5, λ = 1.25, α = 0.005.
Notice that the European case is a smoothed ver-sion of the payoff function. Contrastingly, the Bermu-dan price surfaces have smooth behavior when thestock price is above the trigger level of 100 whilethere is a distinct kink across this spot price levelfor losses above 5. Within this region, the DTSL
6
(a) European claim
(b) Daily exercise
Figure 4: Price surfaces for three year double triggerstop-loss contracts assuming (a) exercise at maturityonly and (b) daily exercise.
option is in-the-money. The behavior along the line(S = 100, L > 5) reflects the discontinuous behaviorof the intrinsic value of the option. These features aremost easily explained and observed by investigatingthe optimal exercise policies.
Figure 5 depicts the optimal exercise behavior atdifferent points in time (exercise occurs in the areaabove the exercise curve and to the left of the lineS = 100). The exercise policy is governed by twocompeting factors. An investors is inclined to de-lay exercise to allow for the losses to accumulate andthus raise the option value at exercise. However, bywaiting the investor is running the risk that the stockprice will cross the threshold S∗ and never venture be-
Figure 5: Exercise surfaces for three year double trig-ger stop-loss contracts assuming daily exercise rights.
low before maturity, thus rendering the option worth-less. Since the latter risk is far greater than the for-mer upside potential, the exercise boundary trendsaway from the threshold S∗ as maturity approaches.Furthermore, the exercise curve flattens at aroundL = 38 for all maturities when S < 95. This canbe explained by the discrete nature of loss arrival –since the maximal payoff is achieved at L = 40, itis optimal to exercise at L = 38 rather than wait-ing and possibly exercising at the same loss level ifno additional losses arrive. In a separate experiment,as we increased the arrival rate of losses, this upperthreshold approached L = 40.
4 Conclusions
We introduced a new pricing method for regimeswitching Levy models coined Fourier Space Time-Stepping and demonstrated its usefulness in severalapplications. There are many doors still open forexploration of these techniques such as currency op-tions, interest rates, and higher dimensional options.We are also currently developing a modification ap-propriate for mean-reverting processes such as thoseappearing in Energy markets which will be usefulin valuing swing and/or storage options with mean-reversion, jumps and regime changes without resort-ing to least-square Monte-Carlo.
7
5 Acknowledgements
The authors would like to thank the participants ofthe workshop on Actuarial Science and Financial En-gineering at U. Michigan, the Computational FinanceConference at U. Waterloo, and the 2007 Daiwa In-ternational workshop on Financial Engineering inTokyo for providing interesting feedback on variousstages of this work. The Natural Sciences and Engi-neering Research Council of Canada partially fundedthis work.
References
Andersen, L. and J. Andreasen (2000). Jump-diffusion processes: Volatility smile fitting andnumerical methods for option pricing. Reviewof Derivatives Research 4, 231–262.
Barndorff-Nielson, O. (1997). Normal inverseGaussian distributions and stochastic volatil-ity modelling. Scandinavian Journal of Statis-tics 24, 1–13.
Black, F. and M. S. Scholes (1973). The pricingof options and corporate liabilities. Journal ofPolitical Economy 81 (3), 637–654.
Briani, M., R. Natalini, and G. Russo (2004).Implicit-explicit numerical schemes for jump-diffusion processes. IAC Report 38, Istituto perle Applicazioni del Calcolo IAC-CNR.
Carr, P. and D. B. Madan (1999). Option valuationusing the fast Fourier transform. The Journalof Computational Finance 2 (4), 61–73.
Carr, P. and L. Wu (2002). Time changed levy pro-cesses and option pricing. Journal of FinancialEconomics forthcoming.
Cont, R. and P. Tankov (2004). Financial Mod-elling with Jump Processes. Chapman & Hall /CRC.
d’Halluin, Y., P. A. Forsyth, and K. R. Vetzal(2005). Robust numerical methods for contin-gent claims under jump diffusion processes.IMA Journal of Numerical Analysis 25, 87–112.
Elliott, R. and C. Osakwe (2006). Option pricingfor pure jump processes with markov switch-ing compensators. Finance and Stochastics 10,250–275.
Jackson, K., S. Jaimungal, and V. Surkov (2007).Fourier space time-stepping for option pricingwith levy models. Forthcoming in Special Issueof Journal of Computational Finance.
Jaimungal, S. and T. Wang (2006). Catastropheoptions with stochastic interest rates and com-pound Poisson losses. Insurance: Mathematicsand Economics 38(3), 469–483.
Kallsen, J. and P. Tankov (2006). Characteriza-tion of dependence of multidimensional Levyprocesses using Levy copulas. Journal of Mul-tivaraite Analysis 97, 1551–1572.
Lord, R., F. Fang, F. Bervoets, and K. Oosterlee(2008). A fast and accurate FFT-based methodfor pricing early-exercise options under Levyprocesses. SIAM Journal on Scientific Comput-ing 30 (4), 1678–1705.
Madan, D., P. Carr, and E. Chang (1998). TheVariance Gamma process and option pricing.European Finance Review 2(1), 79–105.
Merton, R. C. (1973). Theory of rational optionpricing. The Bell Journal of Economics andManagement Science 4 (1), 141–183.
8
