1. Functional Itô Calculus and Volatility Risk Management
Bruno Dupire
Bloomberg L.P/NYU
AIMS Day 1
Cape Town, February 17, 2011

2. Outline Functional Itô 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 Itô Calculus

4. Why?

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

6. Functionals of running paths
12.87
6.34
6.32 T

7. Examples of Functionals

8. Derivatives

9. Examples

10. Topology and Continuitys Y X

11. Functional Itô Formula

12. Fragment of proof

13. Functional Feynman-Kac Formula

14. Delta Hedge/Clark-Ocone

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

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

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 S0=100, H=110, σ=0.25, T=0.25

32. One Touch Option - Γ

34. Up-Out Call - Price
Black-Scholes model S0=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. Forward Parabola - Γ
Payoff:
= 1, = 2, = 100, = 10 σ

43. Forward Parabola Hedge

44. Forward Cube - Γ
Payoff:
= 1, = 2, = 100, = 10 σ

45. Forward Cube Hedge

46. Forward Start - Γ
Payoff
= 1, = 2, = 100, = 10 σ

47. Forward Start Hedge

48. One Touch - Γ
Payoff
= 1, = 104 , = 100, = 5 σ

49. One Touch Hedge

51. Robustness
Superbucket hedge protects today from first order moves in implied volatilities; what about robustness in the future?
After delta hedge, the tracking error is
In the case of discrete hedging in the model or fast mean reversion of vol, variance of TE is proportional to

52. Hedgeability The optimal vanilla hedge PF minimizes
and thus coincides with the superbucket hedge as
The residual risk is
The hedgeability of an option is linked to the dependency of the functional gamma on the path

53. The Geometry of Hedging
Bruno Dupire

54. In practice, determining the best hedge is not sufficient
 Need to measure of residual risk

56. H-Squared Parabola vs Cube

57. Skewness Products Negative Skewness: fat tail to the left
Linked to UP Dispersion < Down Dispersion
Some products are based on
Up Dispersion – Down Dispersion

58. Hedge with Components Decorrelated Case
Good hedge with risk reversals on the components

59. Hedge with Components
Correlated Case
No hedge with components

60. BMW
Sample Mean of Best third + Sample Mean of Worst third -2 x Sample Mean of Middle third

61. Perfect Hedge for BMW, Correlation = 0% Short 6 Shares, Short 9 Puts at –K*, Long 9 Calls at K*

62. BMW with 30 stocks , Correlation = 0 Hedge with Puts and Calls using 120 strikes R-squared = 80.7%, MC runs = 10,000

63. BMW with 30 stocks , Correlation = 40% Hedge with Puts and Calls using 120 skrikes R-squared = 5.8%, MC runs = 10,000

64. Analysis
Correlation shifts the returns, which affects the components hedge but not the target
In the absence of hedge, it is a mistake to price the product with a calibrated model
Implied individual skewness is irrelevant !
It is a wrong way to play implied versus realized skewness

65. Conclusion Itô 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