Functional Itô Calculus and

PDEs for Path-Dependent Options Bruno Dupire

Bloomberg L.P.

Rutgers University Conference

New Brunswick, December 4, 2009

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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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

implied to

local volatility

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Up-Out Call - Price

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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