King's College London
Financial Mathematics
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New Directions in Mathematical Finance
Wednesday, 12 July 2000, Lecture Theatre 2C, King's College London


Levy Processes in Finance
Dr. Philippe Balland, Merrill Lynch, London

The aim of this talk is to review some recent use of Levy processes, and more generally of processes with independent increments, for modelling the evolution of financial asset price processes. The talk will highlight the benefits and pitfalls of such a modelling approach from both a theoretical and practical point of view. In particular, the completeness or incompleteness of the resulting market will be investigated and the risk-neutral transformation will be characterised. The talk will also give some applications of Levy processes to smile models.

Hyperbolic and Related Distributions in Finance
Professor Nick Bingham, Department of Mathematics, Brunel University

We know from the Markowitz diversification theory that we should hold a diversified basket of assets. So life in mathematical finance is multi-dimensional, and we shoud be concerned with joint distributions of a basket of assets, not the distribution of one individual asset. The benchmark theory is the Black-Scholes-Merton theory, which is based on the normal/Gaussian law, i.e. on the multivariate normal distribution. This is symmetric, and has ultra-thin tails. Too bad that real financial data show asymmetry and much fatter tails. One alternative theory is that of hyperbolic distributions in place of normal/Gaussian. This has been advocated by Barndorff-Nielsen (Aarhus), Eberlein (Freiburg), and others. This can handle asymmetry, and has much fatter tails. We discuss this parametric alternative to normal/Gaussian, and semi-parametric generalizations of it.

Interest Rates and Information Geometry
Dr. Dorje Brody, Blackett Laboratory, Imperial College, and DAMTP, Cambridge University
(Joint work with L.P. Hughston, King's College London.)

The space of probability distributions on a given sample space possesses natural geometric properties. For example, in the case of a smooth parametric family of probability distributions on the real line, the parameter space has a Riemannian structure induced by the embedding of the family into the Hilbert space of square-integrable functions, and is characterised by the Fisher-Rao metric. In the nonparametric case the relevant geometry is determined by the spherical distance function of Bhattacharyya. In the context of term structure modelling, we show that minus the derivative of the discount function with respect to the maturity date gives rise to a probability distribution. This follows as a consequence of the positivity of interest rates. Therefore, by mapping the associated term structure density functions to Hilbert space, the resulting metrical geometry can be used to analyse the relationship of yield curves to one another. We show that the general arbitrage-free yield curve dynamics can be represented as a process taking values in the convex space of smooth density functions on the positive real line. As a consequence, the theory of interest rate dynamics can also be represented by a class of processes in Hilbert space.

Measures of Dependence for Multivariate Levy Distributions
Professor Tom Hurd, Department of Mathematics, McMaster University, Ontario

Recent statistical analysis of a number of financial databases suggests that the probability density functions of logarithmic returns have power law tails with constant exponent alpha approximately equal to three. This 'fat tail' asymptotic behaviour of large events is persistent across a broad range of equities and indices, and is strikingly different from the traditional Gaussian conception of financial markets. The present paper proposes a class of multivariate distributions which generalises the observed qualities of univariate time series. A consequence of the proposed class is the 'spectral measure' which completely characterises the multivariate dependences of the extreme tails of the distribution. This measure on the unit sphere in m dimensions, in principle completely general, is determined by extreme events. If it can be observed and determined empirically, it should prove to be an important new tool for scenario generation in portfolio risk management.

Scaling and Multi-Scaling in Financial Markets
Dr. Giulia Iori, Department of Accounting, Finance, and Management, University of Essex

Anomalous scaling laws appear in a wide class of phenomena where global dilatation invariance fails. The analysis of financial time series shows that the asymptotic behaviour of the probability distribution of stock market returns is consistent with a power law decay for relatively short time scales but already for monthly returns the shape of the Gaussian, predicted by the market efficient hypothesis, is recovered. This change of behaviour implies a non linear scaling of the moments of absolute returns. Anomalous scaling, or multiscaling, has also been detected in the autocorrelations of absolute returns for various market indices and currencies. I propose a model of heterogeneous interacting traders which can explain some of the stylised facts of stock market returns. In the model, synchronisation effects, which generate large fluctuations in returns, arise purely from communication and imitation among traders. The key element in the model is the introduction of a trade friction which, by responding to price movements, creates a feedback mechanism on future trading and generates volatility clustering. Scaling and multiscaling analysis performed on the simulated data is in good quantitative agreement with the empirical results.

Diagrammatic Approach to Real Options
Dr. Steven Leppard, Enron, London

Real option valuation requires the combination of financial option pricing methods with business-based optimisation techniques. One of the barriers to the introduction of real option valuation techniques into organisations is the need to discuss the problem formulation, and the associated valuation techniques, with non-technical management. Unfortunately the widely used dynamic programming method of valuation is far more complex than the spreadsheet-based discounted cash flow methods familiar to business school-educated practitioners. In this talk a diagrammatic representation of real option deals is introduced, which allows the full complexity of real option formulations to be discussed with non-technical practitioners. The diagrammatic notation is defined in such a way that the pricing follows automatically from the diagram. Some examples from the energy industry are presented and discussed.

Implied Volatility Instability and Smiles
Dr. William Shaw, Quantitative Analysis Group, Nomura International, London

In any modelling process the calculation of an observable effect is a mapping from the space of parameters associated with the theory to the space of observable parameters. The form of the mapping may range from an explicit formula through to an intensive numerical calculation. The inference of theoretical parameters from observations represents an inversion of such a mapping and it is necessary to be careful to establish when the inversion represents a process that is both well-defined and stable. The inverse function theorem is a critical element of the inversion process when the mapping is non-linear. In option pricing an inversion of common interest is the computation of implied volatility from market price data. This talk will explore the consequences of the failure and near-failure of the inverse function theorem as applied to volatility for some simple options of interest. I will argue that except in very limited circumstances, the implied volatility may well be meaningless.

Models of Interest Rates on Non-Linear State Spaces
Dr. Nick Webber, Department of Economics, Warwick University

A number of recent papers have investigated the pricing of barrier and double barrier options. A common assumption is that a single underlying state variable obeys a geometric Brownian motion on a linear state space. In this paper we use a general framework for the analysis of a wide range of barrier and other exotic options on non-linear state spaces. The underlying state variables are not constrained to follow geometric Brownian motion, and may include stochastic discount factors. Analysis is simplified by using a numeraire based on a rebate style option. Restricting attention to interest rate models, we discuss numerical implementation issues, formulating and comparing Monte Carlo and lattice numerical solution methods in simple non-linear state spaces. We conclude that, at least in low dimensional spaces, viable non-linear interest rate models may exist.

The Mathematics of Natural Catastrophes
Dr. Gordon Woo, Eqecat, London

Linguistic metaphors drawn from natural hazards are commonly used at times of financial crisis. A brewing storm, a seismic shock, etc., evoke the abruptness and severity of a market collapse. If the language of windstorms, earthquakes and volcanic eruptions is helpful in illustrating a financial crisis, what about the mathematics of natural catastrophes? Already, earthquake prediction methods have been applied to economic recessions, and volcanic eruption forecasting techniques have been applied to market crashes. The purpose of this contribution is to survey the mathematics of natural catastrophes, so as to convey the range of underlying principles, some of which may serve as mathematical metaphors for financial applications.

Martingale Approach to the Pricing of Real Options
Dr. Mihail Zervos, Department of Mathematics, King's College London

We formulate a general mathematical model for investments in real assets from the perspective of the real options approach. We then derive an analytic expression for its fair price under a market completeness assumption. This expression is the solution of a stochastic optimisation problem. Also, we consider certain associated control theoretic aspects and we establish the dynamic programming equation. The generality of the model is such that it can also provide a framework for the study of financial options.