Consistent Upper Price Bounds for Exotic Options

Abstract

We consider the problem of finding a consistent upper price bound for exotic options whose payoff depends on the stock price at two different predetermined time points (e.g. Asian option), given a finite number of observed call prices for these maturities. A model-free approach is used, only taking into account that the (discounted) stock price process is a martingale under the no-arbitrage condition. In case the payoff is directionally convex we obtain the worst case marginal pricing measures. The speed of convergence of the upper price bound is determined when the number of observed stock prices increases. We illustrate our findings with some numerical computations.

On Intermediate Marginals in Martingale Optimal Transportation

“…. , m i , i = 1, 2, 3, compare also [10] and [28]. Since by construction the marginals (µ * i ) i=1,2,3 possess the same mean S 0 , Assumption 2.1 is for the marginals (µ * i ) i=1,2,3 fulfilled if and only if…”

“…First note that in this situation Proposition 3.7 is not applicable since, as shown in [42] or [36], conditional on S t 1 the law of S t 3 is supported on three and not only on two values, under the optimal martingale measure for the minimization problem 10 . We show that in this setting including the marginal µ 2 nevertheless improves the price bounds.…”

“…We show that in this setting including the marginal µ 2 nevertheless improves the price bounds. In [42] model-independent price bounds for this specific derivative were extensively studied, and it was shown that the solution of inf Q∈M(µ 1 ,µ 3 ) E Q [c] is determined via the dual strategy given by 10 As shown in [43], under the unique optimal measure for the maximization problem, the conditional law of St 3 is supported only on two values, and therefore Proposition 3.7 is applicable to the analogue maximization problem.…”