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Wilmott Associates Technical Reports
Technical papers downloadable in compressed Adobe portable document format (pdf).
Hedge with an Edge ( 20k) by Elizabeth Whalley and Paul Wilmott. We show how to optimally hedge an option when there is a cost associated with trade in the underlying. We find that there is an optimal hedging bandwidth around the Black-Scholes delta; if this bandwidth is breached then the option must be rehedged immediately. We also find the size of the optimal rehedge. We consider an arbitrary cost structure. Published in Risk Magazine.
Untitled ( 32k) by Jeff Dewynne and Paul Wilmott. This paper addresses advanced issues in pricing path-dependent exotic options in a unified partial differential equation framework.
A Model for the Value of a Business, Some Optimisation Problems in its Operating Procedures and the Valuation of its Debt (163k) by M.Z.Apabhai, N. Georgikopoulos, D.Hasnip, R.Jamie, M.Kim and Paul Wilmott. We model the earnings of a firm and derive differential equations for its value which depend on its legal status and operating procedures.
An Asymptotic Analysis of the Davis, Panas and Zariphopoulou Model for Option Pricing with Transaction Costs ( 78k) by Elizabeth Whalley and Paul Wilmott. This is an asymptotic analysis of an optimal hedging strategy of Davis, Panas and Zariphopoulou. The optimal hedging bandwidth is found.
Optimal Hedging of Options with Small but Arbitrary Transaction Cost Structure ( 139k) by Elizabeth Whalley and Paul Wilmott. This paper examines the asymptotic analysis of an optimal hedging strategy. The optimal rebalance point and hedging bandwidth are found.
Option Prices and Subjective Beliefs ( 36k) by Ralf Korn and Paul Wilmott. We consider the differences between the risk-neutral value of an option and the subjective view of a trader towards the asset drift. We examine also the risk inherent in buying options for speculation and not for hedging. We present new models for the drift of an asset and show how to determine the optimal time at which to close an option position if it becomes unfavorable.
Expect the Worst! ( 253k) by David Epstein and Paul Wilmott. We find an equation for the lower bound for a portfolio of cashflows. We show how to optimally statically hedge this portfolio. Published in Net Exposure.
A Review of Key Results in the Modeling of Discrete Hedging and Transaction Costs ( 43k) by Elizabeth Whalley and Paul Wilmott. We review the most important models and results for pricing derivatives in the presence of transaction costs.
Crash Course (28k) by Philip Hua and Paul Wilmott. In this paper we present a new model for pricing and hedging a portfolio of derivatives that takes into account the effect of an extreme movement in the underlying. We make no assumptions about the timing of this crash or the probability distribution of its size, except that we put an upper bound on the latter. The pricing and hedging follow from the assumption that the worst scenario actually happens i.e. the size and time of the crash are such as to give the option its worst value. The optimal static hedge follows from the desire to make the best of this worst value. There are many applications for this crash modelling, we shall focus on using the model to evaluate the Value at Risk for a portfolio of options. Published in Risk Magazine.
Uncertain Parameters, an Empirical Stochastic Volatility Model and Confidence Limits (69k) by Asli Oztukel and Paul Wilmott. The focus of this paper is to build upon the uncertain parameter framework for valuing derivatives in a worst-case scenario. We improve its applicability to practical option valuation. We start by deriving an empirical stochastic volatility model. We use this stochastic model to examine the time evolution of volatility from an initial known value to a steady-state distribution. This empirical model is then incorporated into the uncertain parameter option valuation framework. Published in International Journal of Theoretical and Applied Finance.
The Valuation of a Firm Advertising Optimally ( 48k) by David Epstein, Nick Mayor, Philipp Schonbucher, Elizabeth Whalley and Paul Wilmott. In this paper we model the value of a firm based on its current earnings and cash balances. The value is modelled on the assumption that earnings follow a mean-reverting process. The effect of advertising on earnings is modelled, and the condition for optimal advertising derived. The way in which the value depends on the legal structure and banking arrangements of the firm is discussed.
Risk of Default in Latin American Brady Bonds ( 64k) by Ingrid Blauer and Paul Wilmott. In this paper, we describe a stochastic model for the instantaneous risk of default, applicable to many fixed-income instruments and Brady bonds in particular. We make some simplifying assumptions about this model and a model for the riskless short-term interest rate. These assumptions allow us to find explicit solutions for the prices of risky zero-coupon bonds and floating rate coupons. We apply the model to Latin American Brady bonds, deducing the risk of default implied by market prices. Published in Net Exposure.
The Value of Market Research when a Firm is Learning: Option Pricing and Optimal Filtering ( 66k) by David Epstein, Nick Mayor, Philipp Schonbucher, Elizabeth Whalley and Paul Wilmott . In this paper we model the value to a firm of undertaking market research into a particular product opportunity. The way in which information about the potential of the project arrives and knowledge evolves during the life of the project is modelled using the theory of optimal filtering. The value of the project and optimal entrance decision rule is then derived as the solution to a partial differential equation, using boundary conditions which reflect the structure of the project.
Room for a View (18k) by Ralf Korn and Paul Wilmott. The market view of an investor plays little role in the pricing of options. We make some theoretical and practical suggestions for how the view on the direction of the underlying may affect investment decisions.
Parisian Options (383k) by Rich Haber, Philipp Schonbucher and Paul Wilmott. Parisian options are barrier options for which the knock-in/knock-out feature is only activated after the price process has spent a certain prescribed, consecutive time beyond the barrier. This specification is motivated by the need to make the option more robust against short-term movements of the share price, a single outlier cannot trigger the barrier. We present a flexible approach to valuing such options using the numerical solution of a partial differential equation.
Various Passport Options and Their Valuation (227k) by Hyungsok Ahn, Antony Penaud and Paul Wilmott. The passport option is a call option on the balance of a trading account. The option holder retains the gain from trading, while the writer is liable for the loss. We establish pricing equations for various passport options including the multi-asset passport and those with discrete trading constraints. The results are typically known as the Hamilton-Jacobi-Bellman equations with multiple layers of free boundary partial differential equations for a sequence of optimal stopping times. Also we examine the gain by selling passport options to utility maximising investors and to investors who guess the market from imperfect information.
On Trading American Options (116k) by Hyungsok Ahn and Paul Wilmott. In this paper we consider the effect that early exercise has on the profit of the writer of an American option. The American option is correctly priced in the Black-Scholes framework assuming that the holder exercises at the worst possible time for the writer. This early exercise time is unique. But why should the holder exercise at this time? If the holder of the American option wants to both eliminate all risk by delta hedging and maximize his wealth then certainly he should exercise here. But if he is following this strategy why does he bother to buy the option? Complete markets make purchasing the option unnecessary. So let's assume that he is following some other strategy, as he is free to do. Now, clearly he would always be better off selling the option than exercising it, but what if the contract is OTC, and he can only close his position by exercise? If the option holder follows some strategy that results in early exercise at a time not given by the classical optimal free boundary then the writer makes more profit than might be expected. As examples, we assume that the holder exercises according to the maximization of his own utility function. We illustrate our results by applying them to several families of utility functions, namely the CARA, the HARA, and the expected return. While the option holder maximises his utility, the issuer gains from the difference between the price maximising exercise boundary and the exercise boundary of the option holder.
The Pricing of Risky Bonds: Current Models and Future Directions (153k) Hyungsok Ahn, Varqa Khadem and Paul Wilmott. The modelling of credit risk, credit derivatives and non-hedgeable securities in general, is currently in a poor state. Ideas from equity option theory have been adopted for credit risk, but have not been adapted for the peculiarities of this more complex world. This brief paper is a review and critique of current ideas and models, and includes suggestions for a more sophisticated, realistic and ultimately more sensible approach. The bibliography at the end should prove a useful source for the current state of the art
Modelling Market Crashes: the Worst-case Scenario (139k) Philip Hua and Paul Wilmott. Jump diffusion models have two weaknesses: they don't allow you to hedge and the parameters are very hard to measure. Nobody likes a model that tells you that hedging is impossible (even though that may correspond to common sense) and in the classical jump-diffusion model of Merton the best that you can do is a kind of average hedging. It may be quite easy to estimate the impact of a rare event such as a crash, but estimating the probability of that rare event is another matter. In this paper we discuss a model for pricing and hedging a portfolio of derivatives that takes into account the effect of an extreme movement in the underlying but we will make no assumptions about the timing of this 'crash' or the probability distribution of its size except that we put an upper bound on the latter. This effectively gets around the difficulty of estimating the likelihood of the rare event. The pricing follows from the assumption that the worst scenario actually happens i.e. the size and time of the crash are such as to give the option its worst value. And hedging, delta and static hedging, will continue to play a key role.
Value-at-Risk and Market Crashes (25k) Philip Hua and Paul Wilmott. If the Black—Scholes model and its extensions were the discoveries of the 70s and 80s, then Value-at-risk (VaR) models are the darlings of the 90s. These models have many uses within an organisation; for example, a risk manager may use VaR to allocate trading limits, senior management for asset allocation and regulators set and review capital reserves for the institutions. Whatever the uses, the essence of a VaR number is to act a benchmark for measuring how ‘risky' the portfolio is across different business lines and products. This article discusses the pitfalls of traditional VaR during times of volatile market and makes some suggestions for improvements.
The Use, Misuse and Abuse of Mathematics in Finance (57k) Paul Wilmott. The once ‘gentlemanly' business of finance has become a game for ‘players.' These players are increasingly technically sophisticated, typically having Ph.D.s 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 publicised 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.
Pricing and Hedging Convertible Bonds Under Non-probabilistic Interest Rates (50k) David Epstein, Richard Haber and Paul Wilmott. Two of the authors (DE and PW) recently introduced a non-probabilistic spot interest rate model. The key concepts in this model are the non-diffusive nature of the spot rate process and the uncertainty in the parameters. The model assumes the worst possible outcome for the spot rate path when pricing a fixed-income product. The model differs in many important ways from the
traditional approaches of fully deterministic rates (as assumed when calculating yields, durations and convexities) or stochastic rates governed by a Brownian motion. In this new model, delta hedging and unique pricing play no role, nor does any market price of risk term appear. In this paper we apply the model to the pricing of convertible bonds. Later we show how to optimally hedge the interest rate risk; this hedging is not dynamic but static. We show how to solve the governing equation numerically and present results.
Exercise Class (11k) Hyungsok Ahn and Paul Wilmott. In this learning curve we explain the ideas behind the valuation of options with early exercise features, so called American options. We also aim to clarify some popular misconceptions about when an American option should be exercised. These misconceptions seem to be prevalent among both academics and practitioners.
A Nonlinear Non-probabilistic Spot Interest Rate Model (53k) David Epstein and Paul Wilmott. 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. Whether the model is useful for pricing fixed-income products is less clear. We discuss the pros and cons of the model, showing how to price and hedge various contracts, saying which are easy and which are hard.
