1. Applications of the Root Solution of the Skorohod Embedding Problem in Finance Bruno Dupire Bloomberg LP CRFMS, UCSB Santa Barbara, April 26, 2007

2. Variance Swaps Vanilla options are complex bets on • Variance Swaps capture volatility independently of S • Payoff: Realized Variance • Replicable from Vanilla option (if no jump):

3. Options on Realized Variance • Over the past couple of years, massive growth of - Calls on Realized Variance: - Puts on Realized Variance: • Cannot be replicated by Vanilla options

4. Classical Models Classical approach: • To price an option on X: • Model the dynamics of X, in particular its volatility • Perform dynamic hedging • For options on realized variance: • Hypothesis on the volatility of VS • Dynamic hedge with VS But Skew contains important information and we will examine how to exploit it to obtain bounds for the option prices.

5. Link with Skorokhod Problem Option prices of maturity T Risk Neutral density of : • Skorokhod problem: For a given probability density function such that find a stopping time of finite expectation such that the density of a Brownian motion W stopped at is A continuous martingale S is a time changed Brownian Motion: is a BM, and

6. Solution of Skorokhod Calibrated Martingale solution of Skorokhod: Then satisfies • If , then is a solution of Skorokhod as

7. ROOT Solution Possibly simplest solution : hitting time of a barrier

8. Barrier Density Density of PDE: BUT: How about Density Barrier?

9. PDE construction of ROOT (1) Given , define • If , satisfies with initial condition: Apply the previous equation with until Then for , Variational inequality:

10. PDE computation of ROOT (2) Define as the hitting time of • Then Thus , and B is the ROOT barrier

11. PDE computation of ROOT (3) Interpretation within Potential Theory

12. ROOT Examples

13. Realized Variance • Call on RV: Ito: taking expectation, Minimize one expectation amounts to maximize the other one

14. Link / LVM Suppose , then define satisfies Let be a stopping time. For , one has and where generates the same prices as X: for all (K,T) For our purpose, identified by

15. Optimality of ROOT As to maximize to maximize to minimize and satisfies: is maximum for ROOT time, where in and in

16. Application to Monte-Carlo simulation • Simple case: BM simulation • Classical discretization: with  N(0,1) • Time increment is fixed. • BM increment is gaussian.

17. BM increment unbounded  Hard to control the error in Euler discretization of SDE  No control of overshoot for barrier options : and  No control for time changed methods L

18. ROOT Monte-Carlo • Clear benefits to confine the (time, BM) increment to a bounded region : • Choose a centered law that is simple to simulate • Compute the associated ROOT barrier : • and, for , draw   The scheme generates a discrete BM with the additional information that in continuous time, it has not exited the bounded region.

19. Uniform case 1 •  • : associated Root barrier -1

20. Uniform case Scaling by :

21. Example 1. Homogeneous scheme:

22. Example • Adaptive scheme: 2a. With a barrier: L L Case 1 Case 2

23. Example 2. Adaptive scheme: 2b. Close to maturity:

24. Example 2. Adaptive scheme: Very close to barrier/maturity : conclude with binomial 1% 50% 50% 99% L Close to barrier Close to maturity

25. Approximation of can be very well approximated by a simple function

26. Properties Increments are controlled  better convergence No overshoot Easy to scale Very easy to implement (uniform sample) Low discrepancy sequence apply

27. CONCLUSION • Skorokhod problem is the right framework to analyze range of exotic prices constrained by Vanilla prices • Barrier solutions provide canonical mapping of densities into barriers • They give the range of prices for option on realized variance • The Root solution diffuses as much as possible until it is constrained • The Rost solution stops as soon as possible • We provide explicit construction of these barriers and generalize to the multi-period case.