functional itô calculus and volatility risk management bruno dupire bloomberg l.p/nyu aims day 1...
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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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• Local volatilities: fwd vols that can be locked by a vanilla PF
• Stoch vol model calibrated
• If no jumps, deterministic implied vols => LVM
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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
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Hedge within/outside LVM
• 1 Brownian driver => complete model
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• Leads to a decomposition of Vega across strikes and maturities
Implied and Local Volatility Bumps
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Up-Out Call - Price
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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
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• 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
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Var[ xx f (Yt )) y t y] (y, t) dy dt0
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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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H-Squared Parabola vs Cube
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*
-3 -2 -1 0 1 2 3-6
-4
-2
0
2
4
6
Level
Pa
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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
-1.5 -1 -0.5 0 0.5 1-1
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0
0.5
1
1.5
Hedge
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rge
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
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