functional itô calculus and pdes for path-dependent options bruno dupire bloomberg l.p. rutgers...
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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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Hedge within/outside LVM
• 1 Brownian driver => complete model
• Within the model, perfect replication by Delta hedge
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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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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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