In October the SA Branch was pleased to welcome Professor Robert Elliott, Research Professor at the University of South Australia, to give the ninth E.A. Cornish Memorial Lecture. Robert is known for his contributions to control theory, game theory, stochastic processes and mathematical finance. Robert has published the paper “Malliavin Calculus in a Binomial Framework” written with Samuel Cohen, his former University of Adelaide student and Statistical Society member, who is now based in Oxford. The paper describes some advanced concepts in the binomial model used in financial pricing.
Martingale representation, backward stochastic differential equations and the Malliavin calculus are difficult concepts in a continuous time setting. In order to describe these advanced concepts Robert presented these ideas using the simple, discrete time binomial model. This is based on the discrete time random walk and is a standard framework used to introduce the risk neutral pricing of financial assets. In this binomial framework it is assumed there are two assets, a bond B and a risky asset, or share S, which can go either up or down at each time step. Risk neutral pricing is defined in this binomial model and provides the analog of the Nobel Prize winning Black Scholes formula for option prices. Indeed, in this simple model the price of any asset is the discounted expected value of its final value, where the expectation is calculated using the risk neutral probability.
Robert then defined the Malliavin derivative of a stochastic process in the simple binomial model. These can determine the integrands in stochastic integrals and, in finance, the sensitivity of prices to changes in the underlying parameters. In the binomial model the only randomness comes from the martingale representation of the random walk. In continuous time modeling, where the noise is represented by Brownian motion, the martingale representation theorem states that any martingale is given by a stochastic integral with respect to the Brownian motion. The martingale representation theorem can be used to establish the existence of a hedging strategy. A similar result holds in the simple binomial framework.
Robert then described backward stochastic diﬀerence equations (BSDEs) as dynamics which “look at a process going backwards in time”. In Financial/Insurance modelling a fundamental question is: what value should be assigned today for a quantity to be delivered (or lost) in the future. An answer can be given in terms of a risk measure. Robert described a characterization of risk measures as the solution of a BSDE.
The binomial framework provides an introduction to financial modelling and Robert’s talk gave a glimpse of concepts developed in modern, stochastic modeling as used in quantitative finance.
By Paul Sutcliffe