Functional Ito Calculus

and PDE for Path-Dependent Options Bruno Dupire

Bloomberg L.P.

PDE and Mathematical Finance

KTH, Stockholm, August 19, 2009

Outline1) Functional Ito Calculus

• Functional Ito 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 Ito Calculus

Why?

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⎧ ⎨ ⎩

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Δ t f (X t ) + a(X t )Δx f (X t ) − r(X t ) f (X t ) + b2(X t )2

Δxx f (X t ) = 0

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df (X t ) = Δx f (X t ) dx + Δ t f (X t ) dt + 12

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The portfolio PF of option f with a short position of Δx f stocksgives :

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In the absence of arbitrage,

Δ t f (X t ) + 12

b2Δxx f (X t ) + r(X t )(Δx f (X t )x t − f (X t )) = 0

The ?/Θ trade - off for European options also holds forpath dependent options, even with an infinite number of state variables.However, in general ? and Θ will be path dependent.

Classical PDE for Asian

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2) Robust Volatility Hedge

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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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Conclusion

• Ito calculus can be extended to functionals of price paths

• Local volatilities are forward values that can be locked

• LVM crudely states these volatilities will be realised

• It is possible to hedge against this assumption

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