Abstract

Valuation of Options under Lévy Processes. An Analysis for the German Stock Index

Several empirical analyses have shown that the normal distribution curve is unsatisfactory for representing distributions of stock returns. Literature and the real world have shown that the stable non-normal distributions are more appropriate in most cases for approximating distributions.

On the other hand, this distribution class is subject to the drawback that the exponential moments necessary for computing the classical option value are seldom finite. When extending the type of distribution and when considering an aggregate of several random variables independent of one another, including distribution functions, the outcome is a stochastic process, i. e. the Lévy process, for which a number of representatives can be found possessing finite exponential moments.

It is then possible to value options on the basis of these types of processes better able to represent empirical data. But such option valuation can only in part be carried out by way of analogy with the classical Black/Scholes and Merton processes. It is true that valuation is based on arbitrage-free assumptions. But the market is incomplete in most cases so that there is no duplication and no possibility to derive an unambiguously equivalent martingale measure.

With a view to generating an unambiguous probability measure, it is necessary to make additional assumptions, e. g. to introduce a benefit function, to identify a risk-minimised strategy or to simply fix ex ante a measure conversion function. For this article it has been decided that the Esscher function shall be the measure conversion function.

The martingale measure made equivalent in this way then permits to derive call option prices. If this approach is applied to European DAX-based call options, there are visible deviations of price to be seen compared with the prices computed on the basis of Scholes and Merton. When ascertaining implied volatilities from the prices of the so computed European call options, the result is the characteristic smile effect known in connection with many studies.