Advances in Finance and Stochastics

Frontmatter

Coherent Risk Measures on General Probability Spaces

Summary

We extend the definition of coherent risk measures, as introduced by Artzner, Delbaen, Eber and Heath, to general probability spaces and we show how to define such measures on the space of all random variables. We also give examples that relates the theory of coherent risk measures to game theory and to distorted probability measures. The mathematics are based on the characterisation of closed convex sets Pσ of probability measures that satisfy the property that every random variable is integrable for at least one probability measure in the set Pσ.

Freddy Delbaen

Robust Preferences and Convex Measures of Risk

Summary

We prove robust representation theorems for monetary measures of risk in a situation of uncertainty, where no probability measure is given a priori. They are closely related to a robust extension of the Savage representation of preferences on a space of financial positions which is due to Gilboa and Schmeidler. We discuss the problem of computing the monetary measure of risk induced by the subjective loss functional which appears in the robust Savage representation.

Hans Föllmer, Alexander Schied

Long Head-Runs and Long Match Patterns

Summary

The survey presents recent developments in a specific area of Extreme Value Theory that deals with long head runs and related statistics. The topic has applications in insurance, finance, reliability and computational biology (see, for instance, [6; 13]). In contrast to existing surveys, we do not restrict ourselves to any particular method but provide an overview of different approaches.

Paul Embrechts, Sergei Y. Novak

Factor Pricing in Multidate Security Markets

Abstract

The two most important static models of security markets — the Capital Asset Pricing Model (CAPM), and the Arbitrage Pricing Theory (APT) — have a common feature that expected returns of a possibly large number of securities satisfy a linear relationship with measures of sensitivity of returns to few common factors. In the CAPM, there is a single factor — the market return — and the pricing relation is exact. In the APT, there are multiple factors and the pricing relation is approximate. The nature of static (two-date) security markets models is such that trade occurs only once, followed by terminal realization of agents’ wealth and simultaneous maturity of all securities. Of course, when static models are used in empirical tests, time-series data on one-period security returns is used and various stationarity properties are assumed.

Jan Werner

Option Pricing for Co-Integrated Assets

Summary

Many financial data series are known to be co-integrated. The implications of this for option valuation are studied in this article. Since co-integration is commonly considered in a discrete time context, here we take a GARCH option pricing approach. In the course of doing so, we present new theoretical results for a discrete time price process to be co-integrated. Our option pricing results are consistent with economic principles: with deterministic volatilities the option prices do not depend on the co-integration parameters, except for the statistical effect as to the manner in which the volatilities are estimated. However, with stochastic volatilities the option prices explicitly depend upon the co-integration parameters. Our results for discrete time suggest there is considerable potential for further research on continuous time modeling of co-integrated price systems and on option models that are based on such continuous time processes.

Jin-Chuan Duan, Stanley R. Pliska

Incomplete Diversification and Asset Pricing

Summary

Investors in equilibrium are modeled as facing investor specific risks across the space of assets. Personalized asset pricing models reflect these risks. Averaging across the pool of investors we obtain a market asset pricing model that reflects market risk exposures. It is observed on invoking a law of large numbers applied to an infinite population of investors that many personally relevant risk considerations can be eliminated from the market asset pricing model. Examples illustrating the effects of undiversified labor income and taste specific price indices are provided. Suggestions for future work on asset pricing include a need to focus on identifying and explaining investor specific risk exposures.

Dilip B. Madan, Frank Milne, Robert J. Elliott

Hedging of Contingent Claims under Transaction Costs

Summary

We consider a general framework covering models of financial markets with transaction costs. Assuming that the solvency cones are proper and evolve in time continuously we prove a hedging theorem describing the set of initial endowments allowing to hedge a vector-valued contingent claim by a self-financing portfolio.

Yuri M. Kabanov, Christophe Stricker

Risk Management for Derivatives in Illiquid Markets: A Simulation Study

Summary

In this paper we study the hedging of derivatives in illiquid markets. More specifically we consider a model where the implementation of a hedging strategy affects the price of the underlying security. Following earlier work we characterize perfect hedging strategies by a nonlinear version of the Black-Scholes PDE. The core of the paper consists of a simulation study. We present numerical results on the impact of market illiquidity on hedge cost and Greeks of derivatives. We go on and offer a new explanation of the smile pattern of implied volatility related to the lack of market liquidity. Finally we present simulations on the performance of different hedging strategies in illiquid markets.

Rüdiger Frey, Pierre Patie

A Simple Model of Liquidity Effects

Summary

We consider here an agent who may invest in a riskless bank account and a share, but may only move money between the two assets at the times of a Poisson process. This models in a simplified way liquidity constraints faced in the real world. The agent is trying to maximise the expected discounted utility of consumption, where the utility is CRRA; this is the objective in the classical Merton problem. Unlike that problem, there is no closed-form solution for the situation we analyse, but certain qualitative features of the solution can be established; the agent should consume at a rate which is the product of wealth and some function of the proportion of wealth in the risky asset, and at the times of the Poisson process the agent should readjust his portfolio so as to leave a fixed proportion of wealth in the risky asset. We establish an asymptotic expansion of the solution in two slightly different formulations of the problem, which allows us to deduce that the ‘cost of liquidity’ is (to first order) inversely proportional to the intensity of the Poisson process.

L.-C.-G. Rogers, Omar Zane

Estimation in Models of the Instantaneous Short Term Interest Rate by Use of a Dynamic Bayesian Algorithm

Summary

This paper considers the estimation in models of the instantaneous short interest rate from a new perspective. Rather than using discretely compounded market rates as a proxy for the instantaneous short rate of interest, we set up the stochastic dynamics for the discretely compounded market observed rates and propose a dynamic Bayesian estimation algorithm (i.e. a filtering algorithm) for a time-discretised version of the resulting interest rate dynamics. The filter solution is computed via a further spatial discretization (quantization) and the convergence of the latter to its continuous counterpart is discussed in detail. The method is applied to simulated data and is found to give a reasonable estimate of the conditional density function and to be not too demanding computationally.

Ramaprasad Bhar, Carl Chiarella, Wolfgang J. Runggaldier

Arbitrage-Free Interpolation in Models of Market Observable Interest Rates

Summary

Models which postulate lognormal dynamics for interest rates which are compounded according to market conventions, such as forward LIBOR or forward swap rates, can be constructed initially in a discrete tenor framework. Interpolating interest rates between maturities in the discrete tenor structure is equivalent to extending the model to continuous tenor. The present paper sets forth an alternative way of performing this extension; one which preserves the Markovian properties of the discrete tenor models and guarantees the positivity of all interpolated rates.

Erik Schlögl

The Fair Premium of an Equity—Linked Life and Pension Insurance

Summary

An equity linked life and pension insurance contract consists of an nonlinear combination of a life and pension insurance with an investment strategy. In addition to the guaranteed payments the insured receives a bonus depending on the value of an investment strategy. The additional payment is similar to an Asian type option. Since the insurance contract combines mortality and financial risks in a nonlinear way, the value or premium of the contract must reflect these uncertainties. Within this context a premium sequence is called fair if the accumulated expected discounted premium is equal to the accumulated expected discounted payments of the contract. This paper shows the existence of a fair periodic premium. For two different pension policies an approximation of the fair periodic premium is derived.

J. Aase Nielsen, Klaus Sandmanne

On Bermudan Options

Summary

A Bermudan option is an American-style option with a restricted set of possible exercise dates. We show how to price and hedge such options by superreplication and use these results for a systematic analysis of the rollover option.

Martin Schweizer

A Barrier Version of the Russian Option

Summary

For geometrical Brownian motion we consider the problem of finding the optimal stopping time and the value function for a Russion (put) option, assuming that the decision about stopping should be taken before the process of prices reaches a “dangerous” barrier on the level ε > 0.

Larry A. Shepp, Albert N. Shiryaev, Agnes Sulem

Laplace Transforms and Suprema of Stochastic Processes

Summary

It is shown that moments of negative order as well as positive non-integral order of a nonnegative random variable X can be expressed by the Laplace transform of X. Applying these results to certain first passage times gives explicit formulae for moments of suprema of Bessel processes as well as strictly stable Lévy processes having no positive jumps.

Klaus Schürger

Solving the Poisson Disorder Problem

Summary

The Poisson disorder problem seeks to determine a stopping time which is as close as possible to the (unknown) time of ‘disorder’ when the intensity of an observed Poisson process changes from one value to another. Partial answers to this question are known to date only in some special cases, and the main purpose of the present paper is to describe the structure of the solution in the general case. The method of proof consists of reducing the initial (optimal stopping) problem to a free-boundary differential-difference problem. The key point in the solution is reached by specifying when the principle of smooth fit breaks down and gets superseded by the principle of continuous fit. This can be done in probabilistic terms (by describing the sample path behaviour of the a posteriori probability process) and in analytic terms (via the existence of a singularity point of the free-boundary equation).

Goran Peskir, Albert N. Shiryaev