1. Functional Itô Calculus and PDEs for Path-Dependent Options Bruno Dupire Bloomberg L.P. Rutgers University Conference New Brunswick, December 4, 2009

2. Outline 1)Functional It ô 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

3. 1) Functional It ô Calculus

4. Why?

5. Review of It ô Calculus 1D nD infiniteD Malliavin Calculus Functional Itô Calculus current value possible evolutions

6. Functionals of running paths 0 T 12.87 6.32 6.34

7. Examples of Functionals

8. Derivatives

9. Examples

10. Topology and Continuity ts X Y

11. Functional It ô Formula

12. Fragment of proof

13. Functional Feynman-Kac Formula

14. Delta Hedge/Clark-Ocone

15. P&L Break-even points Option Value 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. Fréchet Derivative in LVM

31. One Touch Option - Price Black-Scholes model S 0 =100, H=110, σ=0.25, T=0.25

32. One Touch Option - Γ

34. Up-Out Call - Price Black-Scholes model S 0 =100, H=110, K=90, σ=0.25, T=0.25

35. Up-Out Call - Γ

37. Black-Scholes/LVM comparison

38. Vanilla hedging portfolio I

39. Vanilla hedging portfolios II

40. Example : Asian option K T K T

41. Asian Option Hedge K T K T

42. Fwd Start Option Hedge

44. Conclusion It ô 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