Functional Itô Calculusand Volatility Risk Management

Bruno DupireBloomberg L.P/NYU

AIMS Day 1

Cape Town, February 17, 2011

Outline1) Functional Itô Calculus

• Functional Itô formula• Functional Feynman-Kac• PDE for path dependent options

2) Volatility Hedge

• Local Volatility Model• Volatility expansion• Vega decomposition• Robust hedge with Vanillas• Examples

1) Functional Itô Calculus

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2) Robust Volatility Hedge

Local Volatility Model• Simplest model to fit a full surface• Forward volatilities that can be locked

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Summary of LVM

• Simplest model that fits vanillas

• In Europe, second most used model (after Black-Scholes) in Equity Derivatives

• Local volatilities: fwd vols that can be locked by a vanilla PF

• Stoch vol model calibrated

• If no jumps, deterministic implied vols => LVM

),(][ 22 tSSSE loctt

S&P500 implied and local vols

S&P 500 FitCumulative variance as a function of strike. One curve per maturity.Dotted line: Heston, Red line: Heston + residuals, bubbles: market

RMS in bpsBS: 305Heston: 47H+residuals: 7

Hedge within/outside LVM

• 1 Brownian driver => complete model

• Within the model, perfect replication by Delta hedge

• Hedge outside of (or against) the model: hedge against volatility perturbations

• Leads to a decomposition of Vega across strikes and maturities

Implied and Local Volatility Bumps

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local volatility

P&L from Delta hedging

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One Touch Option - Price

Black-Scholes model S0=100, H=110, σ=0.25, T=0.25

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Up-Out Call - Price

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Forward Parabola Hedge

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= 1, = 2, = 100, = 101T 2T 0S σ

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Γ/VegaLink

Robustness

• Superbucket hedge protects today from first order moves in implied volatilities; what about robustness in the future?

• After delta hedge, the tracking error is

• In the case of discrete hedging in the model or fast mean reversion of vol, variance of TE is proportional to

TE 1

2(v t v0(y t , t)) xx f (Yt ) dt

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T

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0

T

Hedgeability

• The optimal vanilla hedge PF minimizes

and thus coincides with the superbucket hedge as

• The residual risk is

• The hedgeability of an option is linked to the dependency of the functional gamma on the path

xxPF (y, t) E[ xx f (Yt ) y t y]

Var[ xx f (Yt )) y t y] (y, t) dy dt0

T

E[( xx f (Yt ) xxPF (y, t))2 y t y] (y, t) dy dt0

T

Bruno Dupire 53

The Geometry of Hedging

• In practice, determining the best hedge is not sufficient

Need to measure of residual risk

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Skewness Products

• Negative Skewness: fat tail to the left• Linked to UP Dispersion < Down Dispersion

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Some products are based onUp Dispersion – Down Dispersion

Hedge with Components

• Decorrelated Case

• Good hedge with risk reversals on the components

Hedge with Components

• Correlated Case

• No hedge with components

BMW

Sample Mean of Best third

+

Sample Mean of Worst third

-2 x

Sample Mean of Middle third

Perfect Hedge for BMW, Correlation = 0% Short 6 Shares, Short 9 Puts at –K*, Long 9 Calls at K*

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-4

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0

2

4

6

Level

Pa

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ffComponent Hedge

Norm Cdf = 2/ 3

Norm Cdf = 1/ 3

K* = 0.4307

BMW with 30 stocks , Correlation = 0 Hedge with Puts and Calls using 120 strikes

R-squared = 80.7%, MC runs = 10,000

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0

0.5

1

1.5

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t

BMW with 30 stocks , Correlation = 40% Hedge with Puts and Calls using 120 skrikes

R-squared = 5.8%, MC runs = 10,000

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Analysis

• Correlation shifts the returns, which affects the components hedge but not the target

• In the absence of hedge, it is a mistake to price the product with a calibrated model

• Implied individual skewness is irrelevant !

• It is a wrong way to play implied versus realized skewness

Conclusion

• Itô calculus can be extended to functionals of price paths

• Price difference between 2 models can be computed

• We get a variational calculus on volatility surfaces

• It leads to a strike/maturity decomposition of the volatility risk of the full portfolio