functional ito calculus and pde for path-dependent options
DESCRIPTION
Functional Ito Calculus and PDE for Path-Dependent Options. Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009. Outline. Functional Ito Calculus Functional Ito formula Functional Feynman-Kac PDE for path dependent options 2)Volatility Hedge - PowerPoint PPT PresentationTRANSCRIPT
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
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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.
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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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Up-Out Call - PriceBlack-Scholes model S0=100, H=110, K=90, σ=0.25, T=0.25
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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