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

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): Bruno Dupire

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 Bruno Dupire

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. Bruno Dupire

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 Bruno Dupire

Solution of Skorokhod Calibrated Martingale Then satisfies If , then is a solution of Skorokhod as Bruno Dupire

ROOT Solution Possibly simplest solution : hitting time of a barrier Bruno Dupire

Barrier Density Density of PDE: BUT: How about Density Barrier? Bruno Dupire

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

PDE computation of ROOT (2) Define as the hitting time of Then Thus , and B is the ROOT barrier Bruno Dupire

PDE computation of ROOT (3) Interpretation within Potential Theory Bruno Dupire

ROOT Examples Bruno Dupire

Minimize one expectation amounts to maximize the other one Realized Variance Call on RV: Ito: taking expectation, Minimize one expectation amounts to maximize the other one Bruno Dupire

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 Bruno Dupire

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

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

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 Bruno Dupire

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. Bruno Dupire

Uniform case 1  : associated Root barrier -1 Bruno Dupire

Uniform case Scaling by : Bruno Dupire

Example 1. Homogeneous scheme: Bruno Dupire

Example Adaptive scheme: 2a. With a barrier: L L Case 1 Case 2 Bruno Dupire

Example 2. Adaptive scheme: 2b. Close to maturity: Bruno Dupire

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

Approximation of can be very well approximated by a simple function Bruno Dupire

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

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. Bruno Dupire