Functional Ito Calculus and PDE for PathDependent Options

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Functional Ito Calculus and PDE for Path-Dependent Options Bruno Dupire Bloomberg L. P. PDE

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

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

1) Functional Ito Calculus

Why?

Why?

Review of Ito Calculus • 1 D • n. D • infinite. D •

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

Functionals of running paths 12. 87 6. 34 6. 32 0 T

Examples of Functionals

Examples of Functionals

Derivatives

Derivatives

Examples

Examples

Topology and Continuity Y X t s

Topology and Continuity Y X t s

Functional Ito Formula

Functional Ito Formula

Fragment of proof

Fragment of proof

Functional Feynman-Kac Formula

Functional Feynman-Kac Formula

Delta Hedge/Clark-Ocone

Delta Hedge/Clark-Ocone

P&L of a delta hedged Vanilla Option Value P&L Break-even points Delta hedge

P&L of a delta hedged Vanilla Option Value P&L Break-even points Delta hedge

Functional PDE for Exotics

Functional PDE for Exotics

Classical PDE for Asian

Classical PDE for Asian

Better Asian PDE

Better Asian PDE

2) Robust Volatility Hedge

2) Robust Volatility Hedge

Local Volatility Model • Simplest model to fit a full surface • Forward volatilities

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

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 implied and local vols

S&P 500 Fit Cumulative variance as a function of strike. One curve per maturity.

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,

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

Implied and Local Volatility Bumps to ed tility pli im vola al loc

P&L from Delta hedging

P&L from Delta hedging

Model Impact

Model Impact

Comparing calibrated models

Comparing calibrated models

Volatility Expansion in LVM

Volatility Expansion in LVM

Frechet Derivative 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 - Price Black-Scholes model S 0=100, H=110, σ=0. 25, T=0. 25

One Touch Option - Γ

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 - Price Black-Scholes model S 0=100, H=110, K=90, σ=0. 25, T=0. 25

Up-Out Call - Γ

Up-Out Call - Γ

Black-Scholes/LVM comparison

Black-Scholes/LVM comparison

Vanilla hedging portfolio I

Vanilla hedging portfolio I

Vanilla hedging portfolios II

Vanilla hedging portfolios II

Example : Asian option T T K K

Example : Asian option T T K K

Asian Option Superbuckets T T K K

Asian Option Superbuckets T T K K

Conclusion • Ito calculus can be extended to functionals of price paths • Local

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