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
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Functional Ito Calculus
and PDE for Path-Dependent Options Bruno Dupire
Bloomberg L.P.
PDE and Mathematical Finance
KTH, Stockholm, August 19, 2009
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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
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1) Functional Ito Calculus
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Why?
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Review of Ito Calculus• 1D
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Functionals of running paths
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Examples of Functionals
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Derivatives
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Examples
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Topology and Continuity
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Functional Ito Formula
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Fragment of proof
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Functional Feynman-Kac Formula
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Delta Hedge/Clark-Ocone
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P&L
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Functional PDE for Exotics
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The portfolio PF of option f with a short position of Δx f stocksgives :
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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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Classical PDE for Asian
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Better Asian PDE
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2) Robust Volatility Hedge
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Local Volatility Model• Simplest model to fit a full surface• Forward volatilities that can be locked
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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
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S&P500 implied and local vols
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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
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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
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Implied and Local Volatility Bumps
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P&L from Delta hedging
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Comparing calibrated models
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Volatility Expansion in LVM
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One Touch Option - PriceBlack-Scholes model S0=100, H=110, σ=0.25, T=0.25
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One Touch Option - Γ
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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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Up-Out Call - Γ
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Black-Scholes/LVM comparison
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Vanilla hedging portfolio I
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Vanilla hedging portfolios II
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Example : Asian option
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Γ/VegaLink
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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