The Skorokhod Embedding Problem and Model-Independent Bounds for Option Prices

This set of lecture notes is concerned with the following pair of ideas and concepts:

1.

The Skorokhod Embedding problem (SEP) is, given a stochastic process

X

=(

X

t

)

t

≥0

and a measure μ on the state space of

X

, to find a stopping time τ such that the stopped process

X

τ

has law μ. Most often we take the process

X

to be Brownian motion, and μ to be a centred probability measure.

2.

The standard approach for the pricing of financial options is to postulate a model and then to calculate the price of a contingent claim as the suitably discounted, risk-neutral expectation of the payoff under that model. In practice we can observe traded option prices, but know little or nothing about the model. Hence the question arises, if we know vanilla option prices, what can we infer about the underlying model?

If we know a single call price, then we can calibrate the volatility of the Black–Scholes model (but if we know the prices of more than one call then together they will typically be inconsistent with the Black–Scholes model). At the other extreme, if we know the prices of call options for all strikes and maturities, then we can find a unique martingale diffusion consistent with those prices. If we know call prices of all strikes for a single maturity, then we know the marginal distribution of the asset price, but there may be many martingales with the same marginal at a single fixed time. Any martingale with the given marginal is a candidate price process. On the other hand, after a time change it becomes a Brownian motion with a given distribution at a random time. Hence there is a 1–1 correspondence between candidate price processes which are consistent with observed prices, and solutions of the Skorokhod embedding problem. These notes are about this correspondence, and the idea that extremal solutions of the Skorokhod embedding problem lead to robust, model independent prices and hedges for exotic options.