Approximate pricing of barrier options in Lévy models

Jahncke, Giso

This thesis deals with pricing of a certain type of derivatives, namely European barrier options. We consider the question of pricing this option in geometric Lévy models, which in contrast to the famous Black-Scholes model allow jumps in the stock price. This increases the difficulty of computing an option price enormously due to the fact that the stock price does not necessarily cross the barrier continuously, but is able to jump over it from different space points. The main idea of our approach will be the interpretian of the jump model as a perturbed Black-Scholes model, where we compute a first order correction term. In the process of evaluating the approximation it will be necessary to split up our option into two, one of them paying a polynomial of the overshoot of the underlying Lévy process. The approximation for the first option will consist of moments of the stock price as well as sensitivities (so called 'greeks') of the Black-Scholes derivative price. The approximation for the overshoot option will consist of an independent result on the approximation of overshoot moments using Lévy process fluctuation theory. The correction term consists of a 2-dimensional complex integral formula depending only on the characteristic exponent of the underlying Lévy process, which may be efficiently evaluated numerically. We show in a numerical illustration for several parametric models used in practice, that our approximation yields good results if the Lévy model is reasonably close to a Black-Scholes model with same volatility in the sense that the fourth order cumulant of the Lévy process should not be too large, yet arguably being robust and simple to evaluate.

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Jahncke, Giso: Approximate pricing of barrier options in Lévy models.

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