Paul Wilmott

Mathematical models in finance

Influence of Mathematical Models in Finance on Practice: Past, Present and Future Applied Mathematics and Finance Stock Price Fluctuations as a Diffusion in Random Environment A Note on Super-Replicating Strategies Worldwide Security Market Anomalies Making Money from Mathematical Models Path-Dependent Options and Transaction Costs Stochastic Equality Volatility and the Capital Structure of the Firm The General Mean-Variance Portfolio Section Problem On a Free Boundary Problem That Arises in Portfolio Management Interest Rate Volatility and the Shape of the Term Structure Multi-Factor Term Structure Models Dynamic Asset Allocation: Insights from Theory Index

The use, misuse and abuse of mathematics in finance

The once ‘gentlemanly’ business of finance has become a game for ‘players’. These players are increasingly technically sophisticated, typically having PhDs in a numerate discipline. The roots of this transformation have their foundation in the 1970s. Since then the financial world has become more and more complex. Unfortunately, as the mathematics of finance reaches higher levels so the level of common sense seems to drop. There have been some well-publicized cases of large losses sustained by companies because of their lack of understanding of financial instruments. In this article we look at the history of financial modelling, the current state of the subject and possible future directions. It is clear that a major rethink is desperately required if the world is to avoid a mathematician-led market meltdown.

Uncertain Parameters, an Empirical Stochastic Volatility Model and Confidence Limits

In this paper we build upon the recently developed uncertain parameter framework for valuing derivatives in a worst-case scenario. We start by deriving a stochastic volatility model based on a simple analysis of time-series data. We use this stochastic model to examine the time evolution of volatility from an initial known value to a steady-state distribution in the long run. This empirical model is then incorporated into the uncertain parameter option valuation framework to provide confidence limits for the option value.

Path-dependent options and transaction costs

We describe research in the subjects of exotic option pricing and option pricing when trade in the underlying incurs transaction costs. These two subjects are then formally brought together to model, in terms of differential equations, problems in pricing exotic options with transaction costs. Results are presented in several cases.

THE END-OF-THE-YEAR BONUS: HOW TO OPTIMALLY REWARD A TRADER?

Traders are compensated by bonuses, in addition to their basic salary. However, little is known about how to optimally reward a trader. In this article we build a framework for the study of this problem and explore a variety of possible compensation structures.

Pricing and Hedging Convertible Bonds under Non-Probabilistic Interest Rates

Many different models for the behavior of interest rates have been proposed and investigated over the years. We have made progress in the sense that some formerly popular models have been eliminated from serious consideration, but there is still no general agreement on which model is the best. The uncertainty leads to substantial model risk in pricing and hedging interest rate derivatives. This article presents a different approach: rather than specifying a particular stochastic process for the short-term interest rate, Epstein, Haber, and Wilmott simply place bounds on its behavior, in the form of maximum and minimum feasible values and a constraint on its rate of change, while permitting any movement of the rate within the constraints. This general approach leads to an acceptable range for the interest rate and for financial instruments based upon it. Model dependence is greatly reduced in this way. The next challenge will be to develop models in this framework that produce tight enough bounds to be useful for traders and investors.

A continuum model for two-dimensional dislocation distributions

Abstract Continuum models are formulated for two-dimensional arrays of screw and edge dislocations under the assumption that the dislocations are infinitely long and parallel. The models are formulated in terms of the number density of the dislocations and the boundary of the array, both of which are unknown. Some exact steady and unsteady solutions are presented in which the applied stress is spatially linear and the boundary is an ellipse inside which the number density of dislocations is constant.

A Nonlinear Non-probabilistic Spot Interest Rate Model

We show how to use ‘uncertainty’ in place of the more traditional Brownian ‘randomness’ to model a short–term interest rate. The advantage of this model is principally that it is difficult to show statistically that it is wrong. We discuss the pros and cons of the model and show how to price and hedge various contracts.

Partial Differential Equations

Preface Here are my online notes for my differential equations course that I teach here at Lamar University. Despite the fact that these are my class notes , they should be accessible to anyone wanting to learn how to solve differential equations or needing a refresher on differential equations. Ive tried to make these notes as self contained as possible and so all the information needed to read through them is either from a Calculus or Algebra class or contained in other sections of the notes. A couple of warnings to my students who may be here to get a copy of what happened on a day that you missed. 1. Because I wanted to make this a fairly complete set of notes for anyone wanting to learn differential equations I have included some material that I do not usually have time to cover in class and because this changes from semester to semester it is not noted here. You will need to find one of your fellow class mates to see if there is something in these notes that wasnt covered in class. 2. In general I try to work problems in class that are different from my notes. However, with Differential Equation many of the problems are difficult to make up on the spur of the moment and so in this class my class work will follow these notes fairly close as far as worked problems go. With that being said I will, on occasion, work problems off the top of my head when I can to provide more examples than just those in my notes. Also, I often dont have time in class to work all of the problems in the notes and so you will find that some sections contain problems that werent worked in class due to time restrictions. 3. Sometimes questions in class will lead down paths that are not covered here. I try to anticipate as many of the questions as possible in writing these up, but the reality is that I cant anticipate all the questions. Sometimes a very good question gets asked in class that leads to insights that Ive not included here. You should always talk to someone who was in class on the day you missed and compare these notes to their notes and see what the differences are. 4. This is somewhat related to the previous …

The Mathematics of Financial Derivatives: The Black–Scholes Formulæ

Introduction In this chapter we describe some techniques for obtaining analytical solutions to diffusion equations in fixed domains, where the spatial boundaries are known in advance. Free boundary problems, in which the spatial boundaries vary with time in an unknown manner, are discussed in Chapter 7. We highlight in particular one method: we discuss similarity solutions in some detail. This method can yield important information about particular problems with special initial and boundary values, and it is especially useful for determining local behaviour in space or in time. It is also useful in the context of free boundary problems, and in Chapter 7 we see an application to the local behaviour of the free boundary for an American call option near expiry. Beyond this, though, we can also use similarity techniques to derive the fundamental solution of the diffusion equation, and from this we can deduce the general solution for the initial-value problem on an infinite interval. This in turn leads immediately to the Black–Scholes formulae for the values of European call and put options. Finally, we extend the method to some options with more general payoffs, and we discuss the risk-neutral valuation method. Similarity Solutions It may sometimes happen that the solution u ( x , τ) of a partial differential equation, together with its initial and boundary conditions, depends only on one special combination of the two independent variables.