Functional Ito Calculus and PDE for PathDependent Options
- Slides: 43
Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L. P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009
Outline 1) 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?
Review of Ito Calculus • 1 D • n. D • infinite. D • Malliavin Calculus • Functional Ito Calculus current value possible evolutions
Functionals of running paths 12. 87 6. 34 6. 32 0 T
Examples of Functionals
Derivatives
Examples
Topology and Continuity Y X t s
Functional Ito Formula
Fragment of proof
Functional Feynman-Kac Formula
Delta Hedge/Clark-Ocone
P&L of a delta hedged Vanilla Option Value P&L Break-even points Delta hedge
Functional PDE for Exotics
Classical PDE for Asian
Better Asian PDE
2) Robust Volatility Hedge
Local Volatility Model • Simplest model to fit a full surface • Forward volatilities that can be locked
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
S&P 500 implied and local vols
S&P 500 Fit Cumulative variance as a function of strike. One curve per maturity. Dotted line: Heston, Red line: Heston + residuals, bubbles: market RMS in bps BS: 305 Heston: 47 H+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 to ed tility pli im vola al loc
P&L from Delta hedging
Model Impact
Comparing calibrated models
Volatility Expansion in LVM
Frechet Derivative in LVM
One Touch Option - Price Black-Scholes model S 0=100, H=110, σ=0. 25, T=0. 25
One Touch Option - Γ
Up-Out Call - Price Black-Scholes model S 0=100, H=110, K=90, σ=0. 25, T=0. 25
Up-Out Call - Γ
Black-Scholes/LVM comparison
Vanilla hedging portfolio I
Vanilla hedging portfolios II
Example : Asian option T T K K
Asian Option Superbuckets T T K K
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
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