King's College London Financial Mathematics |

Wednesday, 12 July 2000, Lecture Theatre 2C, King's College London

Abstracts

**Levy Processes in Finance**

*Dr. Philippe Balland, Merrill Lynch, London*

ballaphi@mle.co.uk

Abstract:

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*

nick.bingham@brunel.ac.uk

Abstract:

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*

D.C.Brody@damtp.cam.ac.uk

(Joint work with L.P. Hughston, King's College London.)

Abstract:

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*

hurdt@mcmail.cis.mcmaster.ca

Abstract:

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*

iorig@essex.ac.uk

Abstract:

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*

Steven.Leppard@enron.com

Abstract:

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*

william.shaw@nomura.co.uk

Abstract:

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*

nick.webber@warwick.ac.uk

Abstract:

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*

gwoo@eqe.co.uk

Abstract:

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*

mihail.zervos@kcl.ac.uk

Abstract:

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.