Simulating Risk Neutral Stochastic Processes with a View to pricing Exotic Options

Abstract

Absence of arbitrage requires all claims to be priced as the expected value of cash flows under a risk neutral measure on the path space and every claim must be priced under the same measure. This motivates why we want to use the same measure to price vanilla options and path dependent products, and hence why we want to match marginal distributions.

There are many ways of matching marginal distributions. We present simulation methods for three stochastic processes that match prespecified marginal distributions at any continuous time: the Az\'{e}ma and Yor solution to the Skorohod embedding problem, inhomogeneous Markov martingale processes with independent increments using subordinated Brownian motion, and a continuous martingale using Dupire's local volatility method. Then the question is which way is a good way of matching marginal distributions.

To make a judgement, we look at the properties of the processes. Since all vanilla options are already matched, we want to use exotic options to investigate properties of the processes. One of the properties that we investigate is whether forward return distributions are close to spot return distributions as market structural features.

We price swaps associated with the first passage time to barrier levels on these processes and see which model gives the highest value of swaps, in other words, the shortest passage time to levels. Moreover, we price monthly reset arithmetic cliquets with local floors and global caps and with local caps and global floors. Then we check the model risks of these models and find how model risks change when caps or floors change. Finally, we price options on the realized quadratic variations to see how option prices move as maturity increases.