Events 2000-2001

Mr Martin Gambrell

Royal Bank of Scotland

Risk management: Meeting the e-commerce challenge

● Defining e-commerce and e-risk
● Decomposing the risk (identifying the distinctive incremental risk characteristics of e-commerce)
● Adapting the risk management process
● Enabling the business

Dr Rodney Coleman

Imperial College

Quantifying extreme operational risk

The US banking regulators are set to impose a capital allocation to reserves to cover extreme loss due to operational risk (losses from activities other than market or credit risk, and from external factors). The allocation will be at the same rate for all banks, regardless of the quality of management and supervision. This is because there is no agreed formula for quantifying the risk. In this talk we will look at the problems of fitting long-tailed probability distributions to loss data. In particular we will look at the generalised extreme value distribution when the data sample is small. With no very large losses in the sample, the large losses are under-represented; with one or more large losses, they are over-represented. In neither case will the fitted distribution model the true situation.

Konstantina Armata

Imperial College, London

Closed form solutions for the Asian option pricing model

Asian Options have been proved extremely difficult contracts to price. No closed form solution for the value of such an option has been produced so far and research has been evolving around the development of different types of analytical or numerical approximations. By changing some fundamental assumptions of the problem, we derive closed form solutions for the price of both fixed and floating strike Asian options.

Professor Michael Dempster

Cambridge Systems Associates and Centre for Financial Research

Wavelet-based PDE valuation of derivatives Wavelet-based discretisation techniques for parabolic PDEs in 1,2, 3 and higher dimensions

Wavelet-based PDE valuation of derivatives Wavelet-based discretisation techniques for parabolic PDE's in 1,2, 3 and higher dimensions Suitability of wavelet-based techniques to highly nonlinear and discontinuous boundary conditions relevant to European barrier and digital option payoffs Progress toward using these techniques for Bermudan and American style derivatives including fast Monte Carlo benchmarks Numerical results for European vanilla, cash-or-nothing and supershare calls in 1 space dimension and for domestic and cross-currency Bermudan swaptions in 3 space dimensions relative to finite difference methods.

Dr Abigail James

Weather Risk Advisory, Cambridge

An Introduction to Weather Derivatives

Weather derivatives, developed in the late 1990's, enable exposure due to abnormal weather conditions to be hedged. The market for trading weather derivatives is growing rapidly and now encompasses many industries. This talk will give a general overview of how weather derivatives are used, how they are structured and the development of the European market. An example will be shown of how weather can be correlated to business indices and how this can be used to hedge weather risk. Weather derivatives are most commonly valued against derived weather indices. These weather indices will be described along with an overview of common problems encountered when dealing with weather data.

Professor Mark Davis

Imperial College, London

Pricing Weather Derivatives by Marginal Value

Weather derivatives are a classic incomplete market. This paper gives a preliminary exploration of weather derivative pricing using the >marginal substitution value or shadow price approach of mathematical economics. Accumulated heating degree days (HDD) and commodity prices are modelled as geometric Brownian motion, leading to explicit expressions for swap rates and option values.

Professor Nicos Christofides

CQF, Imperial College, London

Valuation in incomplete markets

The classical approach to the pricing and hedging, of derivative instruments involves the construction and trading of a portfolio of basic assets so as to replicate the possible derivative payoffs. The whole approach is based on the no-arbitrage principle. In many markets, however, such replication is not always possible (the market is incomplete) either because of jumps in the underlying price process (as is the case with pricing credit derivatives), or because the underlying cannot be traded in the quantities needed (because of liquidity restrictions), or for a variety of other reasons. In such cases, the arbitrage considerations alone can only provide upper and lower bounds on the option price - not an exact value. The talk will develop the "pseudo-arbitrage" and "near-arbitrage" arguments which can form a sufficient basis for an exact pricing methodology. Computational pricing comparisons for some credit derivatives will be given.

Mr Andrew Smith

Bacon & Woodrow

Pricing beyond the curve

Wherever possible, practitioners seek to price derivatives in a way which is consistent with an observed yield curve. But some products extend beyond the observable yield curve. Simplistic extrapolation methods may be unstable and lead to predictable convexity losses on revaluation. This paper considers extrapolation methods which arise from Markovian models and are consistent with absence of arbitrage. We identify the long spot rate as the leading eigenvalue of an integral operator, and obtain asymptotic long bond formulas which extend the result of Dybvig, Ingersoll and Ross. We illustrate the techniques by pricing various long pension liabilities, expressed in currency terms, or relative to inflation, or defined by compounded inflation subject to caps and floors applied annually. Andrew Smith is an associate at Bacon & Woodrow, actuaries and consultants. He is responsible for the development and implementation of Monte Carlo worldwide capital market models. He applies these to pricing and risk management problems for financial institutions.

Professor Anthony Neuberger

London Business School

Rational bounds on the prices of exotic options

The paper explores the bounds which can be placed on the price of exotic options while making minimal assumptions about the price process. In particular, we identify the bounds on the price of a general barrier option given the price of a set of European call options. We show the hedging strategies which enforce those bounds. The hedging strategies are robust in that, after inception, trading occurs only when the barrier is breached. The hedge strategies put a floor on the maximum loss. The distribution of hedge errors under the strategy is compared with that under alternative strategies.

Professor Jean-Philippe Bouchaud

Science et Finance & CEA-Saclay

Historical option pricing: Smiles, skews and hedged Monte Carlo

In this seminar we will present recent advances in option pricing beyond Black-Scholes, in particular we will discuss the relationship between smile and skew/kurtosis in the terminal distribution. The main point is to understand why the smile survives to long maturity. From an empirical analysis of the statistics of stock prices, we will illustrate the impact of long-range volatility correlations on kurtosis and that of price-volatility correlations on skewness. In the light of this analysis, we will comment on recent multifractal models.

Professor Stuart Hodges

FORC, Warwick Business School

The sampling properties of volatility cones

We extend the original work on volatility cones by Burghardt and Lane (1990) to consider of the sampling properties of the variance of variance (and the standard deviation of volatility) under a rich class of models that includes stochastic volatility and conditionally fat-tailed distributions. Because the volatility cone examines volatility at quite long horizons, the estimation requires the use of overlapping data. Our principal contribution is to identify the downward bias when estimation is done on an overlapping basis what this bias is and to derive an adjustment factor that approximates an unbiased estimate of the true variance of variance. Using the theory presented we test the bias adjustments to the standard deviation of volatility using simulations for both a GBM i.i.d. process and a non-i.i.d. process associated with the stochastic volatility model suggested by Heston (1993) and show that the bias becomes insignificant after making the theoretical adjustments. These results are relevant to those who must sell options and understand the nature of quadratic variation in asset prices and they provide clearer insights into the nature of hedging errors when options are dynamically hedged.

Professor M A H Dempster

Cambridge Systems Associates & Judge Institute

Pricing spread options with the Fast Fourier Transform

We investigate a method for pricing the generic spread option beyond the classical two-factor Black-Scholes framework by extending the fast Fourier Transform technique introduced by Carr & Madan (1999) to a multi-factor setting. The method is applicable to models in which the joint characteristic function of the underlying assets forming the spread is known analytically. This enables us to incorporate stochasticity in the volatility and correlation structure - a focus of concern for energy option traders - by introducing additional factors within an affine jump-diffusion framework. Furthermore, computational time does not increase significantly as additional random factors are introduced, since the fast Fourier Transform remains two dimensional in terms of the two entities defining the spread. This yields considerable advantage over Monte Carlo and PDE methods and numerical results are presented to this effect.

Dr William Shadwick

The Risk Partnership

Pricing and hedging options: Some cautionary notes on the Emperor's tailoring

The options business is a remarkable example of the power of financial modelling. The path from the results of Black Scholes and Merton to the current vast and still growing market is a complex interplay of theory and implementation. An essential component of the latter is the computation of option prices and sensitivities. It is an unfortunate fact that industry standards for these computations are far from optimal even in simple cases. Significant unnecessary exposures are undoubtedly arising as a result. I will illustrate some of the difficulties and discuss their significance for the finance industry now and in the near future. As well as offering some speculations as to how we came to this position, I will suggest some remedies.

Dr Robert Barnes

UBS Warburg, London

Doing the Splits

Split capital invest trust instruments are listed on the stock exchange as shares, but perhaps they are more accurately valued as equity derivatives.