Elices 2009 Bespoke model validation - 1er Seminaire parisien de …aelices/seminars/Elices_2009_Bespoke... · 2012. 3. 19. · 1er Seminaire parisien de validation de mod èles financiers

ModelModel ValidationValidation GroupGroupRisk Risk MethodologyMethodologyGrupo SantanderGrupo Santander

BespokeBespoke modelmodel validationvalidation

1er 1er SeminaireSeminaire parisienparisien de de validationvalidation de de modmod èèlesles financiersfinanciers ..InstitutInstitut Telecom, Telecom, amphiteatreamphiteatre TheveninThevenin ..Paris, Paris, MarchMarch 15th, 201015th, 2010

Alberto ElicesAlberto Elices

2

Outline

� Introduction.� Model validation philosophy.� Reconcile FO and Risk interests: provisions.� Examples of bespoke validation:

� Double no touch, single barrier and portfolio of faders.� FX self-quanto options.� Effect of stochastic interest rates on callable products.

� Conclusions.

3

Introduction

� After the crisis in the 2nd half of 2008, a big concernabout pricing models has been raised.� Risk management and model validation raise nowconsiderably more attention.� Model validation:

� Validation of model implementation is not enough.� Estimation of model risk.� Periodic reviews to identify adjustments and reflect changes

in market conditions.

� Risk management:� Apply reserves: provisions.� Limit model risk exposure (reduce volume of operations).

4

Introduction

� Models are limited: they just approximate real prices.� Risk of prices out of market.� Risk of assumptions far from market practice.

� Taxonomy of model risk:� Not appropriate or incomplete theoretical framework.� Implementation errors.� Calibration errors.� Stability problems.� Market data inconsistencies.� User errors.

5

Model Validation Philosophy

� Tests that could and sometimes should be performed:� Consistency with simpler products already validated.� Implementation of premium and greeks.� Stability of premium and greeks.� Convergence of premium and greeks.� Model robustness against inconsistent or illiquid market data

-> back testing vs real market data in fluctuating conditions.� Tests to estimate model risk.

� Model implementation:� Premium: compare with an independently developed model

for a representative set of deal inputs and market data.� Greeks: compare with simpler products already validated.

6

Model Validation Philosophy

� Model risk estimation (premium based):� Premium sensitivity to non-calibrated (unobserved) or

innaccurately calibrated parameters: e.g. correlations, dividends.

� Comparison with other models with more accurate or simplydifferent hypothesis:

– Compare same product with different models available in FO.– Development of “toy” models:

Get sets of model parameters calibrated manually to market.Generate market and stressed market scenarios.Compare model under validation with “toy” model valuation.

� Simulation of hedging strategies: either back test with real or“toy” model market data.

7

How to reconcile Front Office and Risk department inter ests?: provisions

� The provision should cover the expected hedging loss:� When hedging is done using a model with aggressive

prices, the expected hedging loss is around the fair priceminus the aggresive price: that difference is the provision.

� Provisions as a means to approve campaigns usinglimited models with controllable risk:

� A provision allows approving campaigns which would not be possible with a more sophisticated model (performance).

� Provisions to foster improvement of FO models:� Models with limitations should be given provisions which

should be released the more the model is improved.

8


� Provision calculation philosophy:� They should be transparent, easy to compute.� They should be dynamic stable, with smooth evolution

through time (they might ↑ or ↓ approaching expiry).� They should balance risk limitation and trading mitigation.� Front Office should be able to reproduce them.

– Use provision tables calculated from studies.– Use FO pricing models to calculate provisions:

Changing market or deal parameters (move correlations, barriers, coupons, leverages, volatilities etc).

Compare prices of deals valued with different FO models(better models might take too long on a daily basis).

� Other possible but more complicated approaches:– Simulate or back test portfolio hedging: sometimes impractical.– Compare FO versus Risk models.

9


� Market provisions.� Liquidity: inflation, credit spreads, volatility.� Movement of unobserved market parameters: correlations,

dividends.

� Model provisions: applied to specific models.� Digital jumps: varying call spread width depending on digital

jump and gamma sensitivity through tables.� Sensitivity to calibration set selection.� Movement of unobserved model parameters: mean

reversion, vol of vol, etc.� Model risk: tables from studies comparing different models.

10

Example of provisions: liquidity of volatility surface.

� Liquidity provisions should allow unwinding a position.� Sensitivities are diffused through time (towards lower more

liquid maturities) and moneyness (towards closer to ATM).

Diffusion of offsetting sensitivities

for the 10 year tenor

with W=0.5

0

200

400

600

800

1,000

1,200

1 2 3 4 5 7 10 15 20 25 30 40 50 60 70

Tenor

Sensitivities (£k)

Original Sensitivity 1 tailed distribution

2 tailed distribution

Diffusion of sensitivities

for the 5% strike with W=0.05

ATM = 6%

0

200

400

600

800

1,000

1,200

2.50

%

3.00

%

3.50

%

4.00

%

4.50

%

5.00

%

5.50

%

6.00

%

6.50

%

7.00

%

7.50

%

8.00

%

Tenor

Sensitivities (£k)

2 tail diffusion 1 tailed diffusion

Original Sensitivities

11

Example of provisions: liquidity of volatility surface.

� Liquidity provisions in practice:� Movements of implied volatility surface might be mainly

explained by parallel and slope (skew) shifts.� Parallel shifts: hedged with ATM options.� Slope shifts: hedged with risk reversals sensitive to slope.

12

Example of provisions: uncertainty of volatility.

� A Heston model calibrated to market is considered:� Confidence intervals for total variance & vol are calculated.� Up and down increments of vol are calculated vs maturity.� Provision: max(|up & down movement * vega|).

0 2 4 6 8 100

0.02

0.04

0.06

0.08

0.1

0.12

Maturity in years

Tot

al v

aria

nce

mea

n &

qua

ntile

s; a

lpha

=0.

8 v0: 0.0076; Kappa: 1.9006; Theta: 0.0088; sigma: 0.1800

UpQtlMeanDownQtl

0 2 4 6 8 100.06

0.07

0.08

0.09

0.1

0.11

Maturity in years

Vol

atili

ty m

ean

& q

uant

iles;

alp

ha=

0.8

v0: 0.0076; Kappa: 1.9006; Theta: 0.0088; sigma: 0.1800

UpQtlMeanDownQtl

13

Examples of bespoke validation: Double no touch (DNT).

� Methodology: simulation of hedging strategy.� Justification of expected hedging loss as premiumdifference according to different models.� Provisions as the sum of expected hedging loss plus uncertainty of hedging loss dispersion.

14


� Double no Touch exotic options are often priced with a volga-vanna (heuristic) model.� Which model is better for hedging?

� The heuristic model.� The ATM model: Black Scholes with at-the-money volatility

� The criterion of comparison: P&L distribution at expiry.� Working hypothesis: market behaves as a Hestonmodel with certain parameters.

dtdYdW

dYdtvdv

dWvdtS

dS

tt

y

t

ρσθκ

µ

,

)(

=><

+−=

+=

15


� At each point in time:� Heston’s spot and variance are simulated according to a set

of parameters calibrated to market.� Volatility surface is calculated with those parameters.� Heston’s two risk factors (underlying and variance) are

hedged with underlying and a 6m vanilla ATM call option.

� Pricing models are analytical.� This calculation is carried out with the aid of a grid ofcomputers.

16


�Heston parameters: calibrated to 1y EUR/USD on August 2006 :� Kappa (κ) : 1.9006 Theta (θ) : 0.0088� Sigma (σ) : 0.1807 Rho (ρ) : 0.1289� Var0 (v0 ) : 0.0076

0

1

2

3

4

5

0

0.2

0.4

0.6

0.8

1

0.084

0.086

0.088

0.09

0.092

0.094

0.096

0.098

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90.086

0.087

0.088

0.089

0.09

0.091

0.092

0.093

0.094

0.095Calibrated implied volatility for maturity 1y

Vol

atili

ty in

per

uni

t

Delta Delta

Maturity in years

17


Mar06 May06 Jun06 Aug06 Sep06 Nov06 Jan07 Feb07 Apr070.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8Van1y: Spot price paths; HedgeFreq: 0.25 days

Sp

ot

price

Mar06 May06 Jun06 Aug06 Sep06 Nov06 Jan07 Feb07 Apr070

0.05

0.1

0.15

0.2

0.25

0.3

0.35Van1y: Volatility paths; HedgeFreq: 0.25 days

Vo

latility

18


� A 1 year and 2 months double no touch is considered.� Hedging is performed once a day.� Pricing:

� Heuristic model (volga-vanna): 0.0839.� ATM model: 0.0466.� Heston’s model: 0.1327.

Price

Spot

2130.1 3622.1

2812.1=Spot

19


Aug06 Sep06 Nov06 Jan07 Feb07 Apr07 Jun07 Jul07 Sep07 Nov07−100

−50

0

50SndCDNTch−1y17usd: Hedged delta paths; HedgeFreq: 1.00 days

De

lta


0

50

100

150

200

250SndCDNTch−1y17usd: Hedged vega paths; HedgeFreq: 1.00 days

Ve

ga

� Heuristic double no touch:

20


� There is a consistent bias towards P&L losses.

Aug06 Sep06 Nov06 Jan07 Feb07 Apr07 Jun07 Jul07 Sep07 Nov07−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15SndCDNTch−1y17usd: PnL Paths; HedgeFreq: 1.00 days

Pe

r u

nit

of

no

min

al


−0.9

−0.8

−0.7

−0.6

−0.5

−0.4

−0.3

−0.2

−0.1

0SndCDNTchATM−1y17usd: Portfolio value; HedgeFreq: 1.00 days

Po

rtfo

lio p

rice

Heuristic model Black Scholes ATM model

21



−0.03

−0.025

−0.02

−0.015

−0.01

−0.005

0SndCDNTch−1y17usd: ExpVal of PnL; HedgeFreq: 1.00 days

Per

uni

t of n

omin

al

Aug06 Sep06 Nov06 Jan07 Feb07 Apr07 Jun07 Jul07 Sep07 Nov070

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05SndCDNTch−1y17usd: StdDev of PnL; HedgeFreq: 1.00 days

Per

uni

t of n

omin

al

� P&L distribution at expiry for heristic model.� ExpVal: -0.031.� StdDev: 0.045.

22



−0.08

−0.06

−0.04

−0.02

0

0.02SndCDNTchATM−1y17: ExpVal of PnL; HedgeFreq: 1.00 days

Per

uni

t of n

omin

al

Aug06 Sep06 Nov06 Jan07 Feb07 Apr07 Jun07 Jul07 Sep07 Nov070

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1SndCDNTchATM−1y17: StdDev of PnL; HedgeFreq: 1.00 days

Per

uni

t of n

omin

al

� P&L distribution at expiry for ATM model.� ExpVal: -0.09.� StdDev: 0.093.

23


Initial price of the option

Expected hedging losses0.03100.0839

0.09000.0466

0.1149 0.1366Option price with Heston model (Stdev=0.0279)0.1327

Premium + hedging costs

� The expected hedging cost plus the premium gives thefair price of the option.

0.1327

With smileSML

Without smileATM

[ ]LPEPP &modelHeston +≈

24


� To calculate the precision of Heston’s price estimator, 100 estimations with 120 paths are simulated.� Heston’s price: Exp(Pr) = 0.1327 y Std(Pr)=0.0279.

0.1366: Total hedging cost ATMExp(Pr)=0.1327

Exp(Pr)+Std(Pr)=0.1606

Exp(Pr)-Std(Pr)=0.1048

0.1149: Total hedging cost SML

� Hedging cost difference between SML and ATM can be explained because only 120 paths were used.

25


Initial price of the option

Expected hedging loss

A standard deviation of P&L is provisioned

With smile Without smile

Final price with provision

0.0450

0.0310

0.0839

0.0930

0.0900

0.0466

0.1599 0.2296

26

Examples of bespoke validation: Double no touch. Conclu sions.

� The heuristic model is better than the ATM model:� Pricing: it provides a closer price to Heston´s model (the fair

value under the assumptions taken).� Hedging: it provides a lower P&L dispersion.

� The provision chosen in this situation is approximatelyequal to the expected hedging loss + deviation of P&L.

27

Examples of bespoke validation: Simple barrier options.

� Two traded simple barriers considered:� 1y Up & In and 3 month Down & In.� Simple barrier options have signicantly less risk than DNT.

1.26850.33y1.22251.25Down & InPutBarr2

1.26851y1.421.27Up & InCallBarr1

SpotExpiryBarrier levelStrikeTypeCall/PutName

1.26851.22251.25Down & InPutBarr2

1.26851y1.421.27Up & InCallBarr1

SpotExpiryBarrier levelStrikeTypeCall/PutName

0.0025-0.00040.0112Barr2SML

0.0026-0.00050.0113Barr2ATM

0.00600.00000.0279Barr1SML

0.00850.00040.0274Barr1ATM

P&L Std DevExpected P&LPriceCase

0.0025-0.00040.0112Barr2SML

0.0026-0.00050.0113Barr2ATM

0.00600.00000.0279Barr1SML

0.00850.00040.0274Barr1ATM

P&L Std DevExpected P&LPriceCase

28

Examples of bespoke validation: FX self-quanto options.

� Self quanto options are the worst case for correlation is 1.� Four valuation methods are compared:

� Black Scholes with at-the-money (ATM) volatility (left equation).� Black Scholes with volatility selection (VolSelection) to the strike

level of the non-quanto option (left equation).� Vega-volga-vanna (VVV) method.� Estimation of probability distribution of ST using Gatheral’s

parametrization of implied volatility surface (right equation)

( )[ ]),( TtPKSprice EURQ

EUR T

+−= E

tttEUR

tUSD

tQt

Qt dWdtrr

S

dS σσ ++−= )( 2

( )[ ]0

1),(

STtPSKSprice USDTTEUR

+−= E

ttEUR

tUSD

t

t

t dWdtrrS

dS σ+−= )(

0.39500.39900.37120.3704

VVVGatheralVolSelectionATM

29


� Gatheral’s parametrization is fitted to market data:� Left plot: market fitting (maximum difference is 5.65bp).� Right plot: implied probability distribution.

−0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.80.185

0.19

0.195

0.2

0.205

0.21

0.215

0.22

0.225

0.23

Moneyness: k = log(strike/forward)

Total

varia

nce:

varia

nce x

Matu

rity

0 50 100 150 200 2500

0.2

0.4

0.6

0.8

1

1.2x 10

−3

Pro

babi

lity

dens

ity fu

nctio

nUnderlying at expiry

33.8032.2231.4731.0330.7730.5930.5531.2531.6532.1332.7933.8035.87Vol

0.950.90.850.80.750.70.50.30.250.20.150.10.05Delta

30


� A set of a hundred random scenarios changing dates, interestrates and strikes preserving shape of volatility surface is gene rated.� Call option prices are compared.

-0.0600

-0.0400

-0.0200

0.0000

0.0200

0.0400

0.0600

0.0800

0 10 20 30 40 50 60 70 80 90 100

Gatheral-VVV

Gatheral-VolSelect

Gatheral-ATM

31


� Put option prices give a lot less differences for same scenari os.� Volatility selection or ATM methods can significantly di verge fromfair value. Best method: volga-vanna.

-0.0100

-0.0050

0.0000

0.0050

0.0100

0.0150

0.0200

0 10 20 30 40 50 60 70 80 90 100

Gatheral-VVV

Gatheral-VolSelect

Gatheral-ATM

32

Examples: Effect of stochastic interest rates (IR) on cal lable products.

� Critical on long term products.� Consider a hybrid model with Hull White evolution forrates and Black Scholes for FX:

� Callable products are very related to digital options:� Products called when ST is outside a window is equivalent to

the sum of a digital OTM call and put.� Therefore, digital call options are studied.

33


� Digitals with strike around Fwd have no correction.� As Fwd paths spread around mean, more paths end at

higher or lower ST and less around mean.� Probability of calling outside a window increases.

2 4 6 8 101

1.1

1.2

1.3

1.4

1.5

1.6

1.7

Maturity in years

FX

forw

ard

Fwd Mean and quantiles with alpha=0.75

UpQtlMeanDownQtl

34


� Increasing correlation between interest rates makesdispersion lower => less correction.

35


� Higher correlation between local IR and FX:� ↑ Strikes => ↑ price; ↓ Strikes => ↓ price.� More probability to cancel at upper thresholds.� Opposite conclusion with correlation foreign IR - FX.

36

Conclusions

� Model validation has raised considerable more attention in the past months due to the 2008 crisis.� Validation of implementation is no longer enough. Estimation of model risk has become a major issue:

� Sensitivity to unobserved parameters.� Compare models using hedging strategies.� Compare same product with different models.

� Provisions: between perfect & reasonable:� They should cover uncertainty of hedging costs.� They may allow campaigns with limited models.� They allow to foster improvement of FO models.

37

Annexe: Examples of bespoke validation: a portfolio of F X Faders.

� Heston hedging strategy simulation is used.� This portfolio represents a real portfolio on EUR/USD.� All operations are forward contracts (long a fader calland short a fader put with same data).� All faders are “in” with a lower level (notionalincreases when spot is higher than lower level).� Most faders limit client losses with a continuous lowerbarrier (when trespassed no more notional isaccumulated).� Faders are valued with volga-vanna heuristic model.

38


� The portfolio of faders can be represented by thefollowing three positions:

175.1=DLEVEL

29.1=K

15.1=DBARR

2096.1=Spot

175.1=DLEVEL

29.1=K

2096.1=Spot

175.1=DLEVEL

29.1=K

15.1=DBARR

2096.1=Spot

yExpiry 5.1: yExpiry 8.0: yExpiry 5.0:

%54:Nominal %29:Nominal %17:Nominal

weeksfrequencyFading 4: weeksfrequencyFading 4: weeksfrequencyFading 2:

39


� Three delta periods are observed (0.5y, 1.3y y 1.5y).� Lower delta limits depend on notionals of still living options (-0.66, -0.83 y –0.54).

Jan06 May06 Aug06 Nov06 Feb07 Jun07 Sep07−1

−0.5

0

0.5

1

1.5FwdFader: Hedged delta paths; HedgeFreq: 1.00 days

Del

ta

Jan06 May06 Aug06 Nov06 Feb07 Jun07 Sep07−4

−3

−2

−1

0

1

2

3FwdFader: Hedged vega paths; HedgeFreq: 1.00 days

Veg

a

40


� Expected hedging gain at maturity is 25pb and thestandard deviation is 27bp.� No provision is needed.

Jan06 May06 Aug06 Nov06 Feb07 Jun07 Sep070

0.5

1

1.5

2

2.5

3x 10

−3FwdFader: Standard deviation of PnL; HedgeFreq: 1.00 days

Per

uni

t of n

omin

al

Jan06 May06 Aug06 Nov06 Feb07 Jun07 Sep07−0.5

0

0.5

1

1.5

2

2.5x 10

−3 FwdFader: ExpVal of PnL; HedgeFreq: 1.00 days

Per

uni

t of n

omin

al

41

Jan06 May06 Aug06 Nov06 Feb07 Jun07 Sep070

0.5

1

1.5

2

2.5

3

3.5

4x 10

−3 FwdLloss−1y5: StdDev of PnL; HedgeFreq: 1.00 days

Per

uni

t of n

omin

al

Mar06 May06 Jun06 Aug06 Sep06 Nov06 Jan07 Feb070

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

−3 FwdLloss−0y8: StdDev of PnL; HedgeFreq: 1.00 days

Per

uni

t of n

omin

al

Mar06 Apr06 May06 May06 Jun06 Jul06 Jul06 Aug06 Aug06 Sep060

0.5

1

1.5

2

2.5

3

3.5

4x 10

−3 FwdSloss−0y5: StdDev of PnL; HedgeFreq: 1.00 days

Per

uni

t of n

omin

al


� The standard deviation of each componenent is: 37pb, 42pb y 38pb.

42


� Theoretical limits of the portfolio standard deviation:� Compensación perfecta: 0 (posiciones largas y cortas del

mismo producto).� Maximum diversification (no compensation effect):

� Perfect correlation among standard deviations:

� The portfolio considered has 27pb. There exists somecompensation effect.

pbnomnomnom prodprodprodprodprodprod 39332211 =++ σσσ

pbnomnomnom prodprodprodprodprodprod 2423

23

22

22

21

21 =++ σσσ

Documents

Elices 2009 Bespoke model validation - 1er Seminaire parisien de …aelices/seminars/Elices_2009_Bespoke... · 2012. 3. 19. · 1er Seminaire parisien de validation de mod èles financiers