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

2. 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

3. 1) Functional Ito Calculus

4. Why?

5. Review of Ito Calculus 1D nD infiniteD Malliavin Calculus Functional Ito 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 Ito 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. Frechet 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 Superbuckets K T K T

43. 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