Esscher transform

Abstract: The Esscher transform is a time-honored tool in actuarial science. This paper shows that the Esscher transform is also an efficient technique for valuing derivative securities if the logarithms of the prices of the primitive securities are governed by certain stochastic processes with stationary and independent increments. This family of processes includes the Wiener process, the Poisson process, the gamma process, and the inverse Gaussian process. An Esscher transform of such a stock-price process induces an equivalent probability measure on the process. The Esscher parameter or parameter vector is determined so that the discounted price of each primitive security is a martingale under the new probability measure. The price of any derivative security is simply calculated as the expectation, with respect to the equivalent martingale measure, of the discounted payoffs. Straightforward consequences of the method of Esscher transforms include, among others, the celebrated Black-Scholes optionpricing formula, the binomial option-pricing formula, and formulas for pricing options on the maximum and minimum of multiple risky assets. Tables of numerical values for the prices of certain European call options (calculated according to four different models for stock-price movements) are also provided.

Abstract: We consider the option pricing problem when the risky underlying assets are driven by Markov-modulated Geometric Brownian Motion (GBM). That is, the market parameters, for instance, the market interest rate, the appreciation rate and the volatility of the underlying risky asset, depend on unobservable states of the economy which are modelled by a continuous-time Hidden Markov process. The market described by the Markov-modulated GBM model is incomplete in general and, hence, the martingale measure is not unique. We adopt a regime switching random Esscher transform to determine an equivalent martingale pricing measure. As in Miyahara [33], we can justify our pricing result by the minimal entropy martingale measure (MEMM).

Abstract: Statistical analysis of data from the financial markets shows that generalized hyperbolic (GH) distributions allow a more realistic description of asset returns than the classical normal distribution. GH distributions contain as subclasses hyperbolic as well as normal inverse Gaussian (NIG) distributions which have recently been proposed as basic ingredients to model price processes. GH distributions generate in a canonical way Levy processes, i.e. processes with stationary and independent increments. We introduce a model for price processes which is driven by generalized hyperbolic Levy motions. This GH model is a generalization of the hyperbolic model developed by Eberlein and Keller (1995). It is incomplete. We derive an option pricing formula for GH driven models using the Esscher transform as martingale measure and compare the prices with classical Black-Scholes prices. The objective of this study is to examine the consistency of our model assumptions with the empirically observed price processes for underlyings and derivatives. Finally we present a simplified approach to the estimation of high-dimensional GH distributions and their application to measure risk in financial markets.

Abstract: We consider the problem of pricing contingent claims on a stock whose price process is modelled by a geometric Levy process, in exact analogy with the ubiquitous geometric Brownian motion model. Because the noise process has jumps of random sizes, such a market is incomplete and there is not a unique equivalent martingale measure. We study several approaches to pricing options which all make use of an equivalent martingale measure that is in different respects "closest" to the underlying canonical measure, the main ones being the Follmer-Schweizer minimal measure and the martingale measure which has minimum relative entropy with respect to the canonical measure. It is shown that the minimum relative entropy measure is that constructed via the Esscher transform, while the Follmer-Schweizer measure corresponds to another natural analogue of the classical Black-Scholes measure.