elices gimenez 2011 applying hedging strategies to...

ApplyingApplying HedgingHedging StrategiesStrategies totoEstimateEstimate ModelModel RiskRisk andandProvisionProvision CalculationCalculation

HeadHead ofof EquityEquityModelModel ValidationValidation GroupGroupRiskRisk MethodologyMethodologySantanderSantander

Alberto ElicesAlberto Elices Eduard Eduard GimGim ééneznez

ModelModel DevelopmentDevelopment GroupGroupFrontFront OfficeOfficeCaixabankCaixabank

4th 4th AnnualAnnual PricingPricing ModelModel ValidationValidationLondonLondon , 7th , 7th –– 9th 9th SeptemberSeptember , 2011, 2011

2

Outline

� Introduction.� Model Risk estimation.� Reconcile FO and Risk interests: Fair Value Adjustment.� Estimation of model risk applying hedging strategies:

� Formulation of hedging strategy.� Case study: double-no-touch option.

� Justification of findings.� Conclusions.

3

Introduction

� After the crisis in the 2nd half of 2007, a big concernabout pricing models has been raised.

� Risk management and model validation raise nowconsiderably more attention.

� Model validation:� Validation of model implementation is no longer enough.� Periodic and comprehensive review of pricing models.� Estimation of model risk.

� Risk management:� Calculate and apply provisions.� Limit model risk exposure (reduce volume of operations).

4

Model Risk Estimation

� Premium sensitivity to non-calibrated (unobserved) orinnaccurately calibrated parameters: e.g. correlation s, dividends.� Comparison with other models with more accurate orsimply different 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 withreal or “toy” model market data.

5

How to reconcile FO and Risk department interests?: FVA

� Fair Value Adjustment (FVA) should cover the expectedhedging loss and its uncertainty:� When hedging is carried out with a model with aggresive prices, the expected hedging loss is the fair minus the aggresive price: that difference plus a cushion for its uncertainty is the FVA.

� FVA as a means to approve campaigns usinglimited models with controllable risk:� FVA allows accomplishing campaigns which would not possiblewith a slower more sophisticated model.

� FVA to foster improvement of FO models:� Models with limitations should be given FVA which should be released the more the model is improved.

6

How to reconcile FO and Risk department interests?: FVA

� FVA calculation philosophy:� It should be transparent, easy to compute.� It should be dynamic, stable, with smooth evolution through time (they should decrease approaching expiry).� They should balance risk limitation and trading mitigation.� Front Office should be able to reproduce FVA.

7

How to reconcile FO and Risk department interests?: FVA

� How FVA can be estimated:� Use FVA tables calculated from studies.

� Use FO pricing models to estimate model risk:– Changing unobserved or non-calibrated model parameters (mean reversion, correlations, etc).

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

� Simulate or back test portfolio hedging: sometimes impractical.

8

Outline

� Introduction.� Model Risk estimation.� Reconcile FO and Risk interests: Fair Value Adjustment.� Estimation of model risk applying hedging strategies:

� Formulation of hedging strategy.� Case study: double-no-touch option.

� Justification of findings.� Conclusions.

9

Estimation of model risk applying hedging strategies

� Assume that market is driven by Heston’s dynamics.

� Simulate hedging strategy with different pricing models.

� Look at profit and loss (P&L) distribution of hedgingstrategy at maturity:� What is the expected hedging P&L?� What is the uncertainty (e.g. StdDev) of hedging P&L?

� Calculate Fair Value Adjustment (FVA) to account for:� Expected hedging losses.� Uncertainty of those losses: a number of StdDev of hedge loss.

� Which model is better?

10

Formulation of the hedging strategy

� Hypothesis: market is driven by Heston’s dynamics:

� Heston’s parameters are calibrated to 1y EUR/USD:

11

Formulation of the hedging strategy

� Heston’s two factors (spot and variance) are simulated:

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 p

rice

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

latilit

y

12

Formulation of the hedging strategy

� On each simulated step:� Calculate vol surf from simulated , & Heston parameters. � Sensitivities: after each factor change, vol surf is re-built.� Delta and vega are hedged with underlying and a 6m vanilla.

For Heston:- Vol surf does not change with spot.- Vol surf has only term structurechange varying variance.

13

Case study: double-no-touch option (DNT)

� A 1y double-no-touch option is considered.� Estimation and comparison of model risk for:

� VV: Volga-vanna heuristic model.� BS: Black Scholes with at-the-money volatility.

Spot

2130.1 3622.1

2812.1=Spot

Price

14

Case study: double-no-touch option (DNT)

� Qualitatively similar hedge ratios and :

BSVV

BSVV

15

Case study: double-no-touch option (DNT)

Should not the P&Lbe also very similar?

16

Case study: double-no-touch option (DNT)

� Consistent hedging losses much higher for BS:

P&L paths of hedging strategy

BSVV

17

Case study: double-no-touch option (DNT)

� Evolution through time of hedge P&L distribution:� VV has lower expected loss and lower StdDev.

Expected hedging P&L StdDev of hedging P&L

18

Case study: double-no-touch option (DNT)

19

Case study: double-no-touch option (DNT)

� Back-to-Back versus In-House.

P&L

Model Quality +_

Back-to-Back

In-House

VV MktMore

competitive

Lesscompetitive

Corporate client price: 0.19

B2B, Mkt+CVA: 0.17Margin = 0.02

With BS model: 0.1896Margin = 0.0004

With VV model: 0.1600Margin = 0.03

With Mkt model: 0.1122Margin = 0.0778

BS worse than B2BVV better than B2B

The better model themore competitive price

0.0200

0.0778

0.0300

20

Case study: double-no-touch option (DNT) : conclusion s

� Double-no-touch options have huge model risk.� Model Risk measure: accounts for risk aversion to

uncertainty of hedging loss.

� BS and VV models are compared under this measure.VV performs better than BS:� VV has less expected hedging loss.� VV has less uncertainty of hedging loss.

� The provision is equal to the model risk measure,adjusting for a given risk aversion view .

21

Justification of findings: Expected loss = market price – mo del price

� Definition of total and hedging portfolios andhedging position :

Domestic bank account.

Foreign bank account.

Amount of underlying.

Amount of vanilla option.

Price of vanilla option.

Option price to hedge.

22

Justification of findings: Expected loss = market price – mo del price

� Construction of time evolution of hedging portfolio:

Domestic zero coupon.

23

Justification of findings: Expected loss = market price – mo del price

� The expected total portfolio value at maturity does notdepend on the hedge ratios (the numeraire is ):

24

Justification of findings: why is there a consistent loss drift?

� Evolution of total portfolio when all risks are hedged:

� Evolution of any pricing model given Heston’s market:

� Evolution of option to hedge and vanilla option :

: Infinitesimal generator of Heston’s dynamics (market).

25

Justification of findings: why is there a consistent loss drift?

� Evolution of the hedge porfolio:

� Evolution of the total portfolio minus risk free return:

� Hedge ratios are chosen to eliminate randomness:

26

Justification of findings: why is there a consistent loss drift?

� Why is there a consistent loss drift?

� = 0: Market dynamics equal to model dynamics.No drift.

� > 0: Positive drift => consistent profit.

� < 0: Negative drift => consistent loss.

27

Conclusions: looking at model risk from a hedging perspective

� Two sources of model risk from a hedging perspective:� Expected hedging loss :

– It depends on model price but not on its hedge ratios.– It can be estimated by comparing with a better (usu. slower) model

or moving non-calibrated or unobserved parameters.

� Uncertainty of hedging loss (e.g. measured by its StdDev) :– It depends on hedge ratios given by the model.– More difficult to estimate: hedging simulation or back-test studies.

� Model Risk measure: measures risk aversion touncertainty of hedging loss.

28

Conclusions: looking at model risk from a hedging perspective

� No one knows market dynamics but,� There are hypothesis more plausible than others.� There are proxy models used by many participants.� A good model should monitor market prices as close as possible.

� Covering model risk with Fair Value Adjustment (FVA):� The expected hedging loss can be accounted for until expiry,

on the date of deal closing.� The uncertainty of hedging loss needs a study for each product.� Benefits of portfolio effect need simulation of the whole portfolio.� The factor allows customizing model risk aversion.

29

Conclusions: looking at model risk from a hedging perspective

For more details, look at the paper:Elices A., Giménez E.,”Applying hedging strategies to estimatemodel risk and provision calculation”, available on ArXiv at “http://arxiv.org/abs/1102.3534”.

30

Annexe: a portfolio of foreign exchange faders

� Heston hedging strategy simulation is used, faders are valued with volga-vanna model.

� 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).

� “In” faders with a lower accrual level are considered(notional increases when spot is higher than this level).

� Most faders limit client losses with a continuous lowerbarrier (when touched no more notional is accrued).

31

Annexe: a portfolio of foreign exchange faders

� 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:

32

Annexe: a portfolio of foreign exchange faders

� 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

33

Annexe: a portfolio of foreign exchange faders

� 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