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Functional Ito Calculus and PDE for Path-Dependent Options

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

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Functional Ito Calculus and PDE for Path-Dependent Options

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  1. Functional Ito Calculusand PDE for Path-Dependent Options Bruno Dupire Bloomberg L.P. PDE and Mathematical Finance KTH, Stockholm, August 19, 2009

  2. Outline • 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

  3. 1) Functional Ito Calculus

  4. Why?

  5. Review of Ito Calculus current value • 1D • nD • infiniteD • Malliavin Calculus • Functional Ito Calculus possible evolutions

  6. Functionals of running paths 12.87 6.34 6.32 T 0

  7. Examples of Functionals

  8. Derivatives

  9. Examples

  10. Topology and Continuity t s Y X

  11. Functional Ito Formula

  12. Fragment of proof

  13. Functional Feynman-Kac Formula

  14. Delta Hedge/Clark-Ocone

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

  16. Functional PDE for Exotics

  17. Classical PDE for Asian

  18. Better Asian PDE

  19. 2) Robust Volatility Hedge

  20. Local Volatility Model • Simplest model to fit a full surface • Forward volatilities that can be locked

  21. 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

  22. S&P500 implied and local vols

  23. 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

  24. 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

  25. Implied and Local Volatility Bumps implied to local volatility

  26. P&L from Delta hedging

  27. Model Impact

  28. Comparing calibrated models

  29. Volatility Expansion in LVM

  30. Frechet Derivative in LVM

  31. One Touch Option - Price Black-Scholes model S0=100, H=110, σ=0.25, T=0.25

  32. One Touch Option - Γ

  33. Up-Out Call - Price Black-Scholes model S0=100, H=110, K=90, σ=0.25, T=0.25

  34. Up-Out Call - Γ

  35. Black-Scholes/LVM comparison

  36. Vanilla hedging portfolio I

  37. Vanilla hedging portfolios II

  38. T T K K Example : Asian option

  39. T T K K Asian Option Superbuckets

  40. 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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