Mark H. A. Davis

Portfolio selection with transaction costs

In this paper, optimal consumption and investment decisions are studied for an investor who has available a bank account paying a fixed rate of interest and a stock whose price is a log-normal diffusion. This problem was solved by Merton and others when transactions between bank and stock are costless. Here we suppose that there are charges on all transactions equal to a fixed percentage of the amount transacted. It is shown that the optimal buying and selling policies are the local times of the two-dimensional process of bank and stock holdings at the boundaries of a wedge-shaped region which is determined by the solution of a nonlinear free boundary problem. An algorithm for solving the free boundary problem is given.

European option pricing with transaction costs

The authors consider the problem of pricing European options in a market model similar to the Black–Scholes one, except that proportional transaction charges are levied on all sales and purchases o...

An Introduction to Nonlinear Filtering

In this paper we provide an introduction to nonlinear filtering from two points of view: the innovations approach and the approach based upon an unnormalized conditional density. The filtering problem concerns the estimation of an unobserved stochastic process xt given observations of a related process yt; the classic problem is to calculate, for each t, the conditional distribution of xt given ys, 0 ≤ s ≤ t. First, a brief review of key results on martingales and markov and diffusion processes is presented. Using the innovations approach, stochastic differential equations for the evolution of conditional statistics and of the conditional measure of xt given ys, 0 ≤ s ≤ t are given; these equations are the analogs for the filtering problem of the kolmogorov forward equations. Several examples are discussed. Finally, a less complicated evolution equation is derived by considering an “unnormalized” conditional measure.

THE RANGE OF TRADED OPTION PRICES

Suppose we are given a set of prices of European call options over a finite range of strike prices and exercise times, written on a financial asset with deterministic dividends which is traded in a frictionless market with no interest rate volatility. We ask: when is there an arbitrage opportunity? We give conditions for the prices to be consistent with an arbitrage-free model (in which case the model can be realized on a finite probability space). We also give conditions for there to exist an arbitrage opportunity which can be locked in at time zero. There is also a third boundary case in which prices are recognizably misspecified, but the ability to take advantage of an arbitrage opportunity depends upon knowledge of the null sets of the model.

On a multiplicative functional transformation arising in nonlinear filtering theory

SummaryThis paper concerns the nonlinear filtering problem of calculating “estimates” E[f(xt)¦y s, s≦t] where {xt} is a Markov process with infinitesimal generator A and {yt} is an observation process given by dyt=h(xt)dt +dwtwhere {wt} is a Brownian motion. If h(xt) is a semimartingale then an unnormalized version of this estimate can be expressed in terms of a semigroup Ts,tyobtained by a certain y-dependent multiplicative functional transformation of the signal process {xt}. The objective of this paper is to investigate this transformation and in particular to show that under very general conditions its extended generator is Atyf=ey(t)h(A− 1/2h2)(e−y(t)hf).

Optimal Hedging with Basis Risk

It often happens that options are written on underlying assets that cannot be traded directly, but where a ‘closely related’ asset can be traded. Rather than simply using the traded asset as a proxy for the option underlying, one should calculate some ‘best’ hedging strategy. The market is incomplete, and we address the problem using a utility maximization approach. With exponential utility the optimal hedging strategy can be computed in reasonably explicit form using the methods of convex duality. In particular, a perturbation analysis using ideas of Malliavin calculus gives the modification to the exact replication strategy that is appropriate when the option underlying and traded assets are highly, but not perfectly, correlated.

The Representation of Martingales of Jump Processes

In this paper it is shown that all local martingales of the

Nonlinear filtering with counting observations

\sigma

European option pricing with transactions costs

-fields generated by a jump process of very general type can be represented as stochastic integrals with respect to a fundamental family of martingales associated with the jump process.

Optimal investment under partial information

We apply some recent results in martingale theory and the innovations method to obtain the evolution of the conditional mean and conditional density of a process that modulates the rate of a counting process.