Efficient calculation of the Greeks for exponential Lévy processes: an application of measure valued differentiation

Abstract

Monte Carlo simulation methods have become more and more important in the financial sector in the past years. In this paper, we introduce a new simulation method for the estimation of the derivatives of prices of financial contracts with respect to (w.r.t.) certain distributional parameters called the ‘Greeks’. In particular, we assume that the underlying financial process is a Lévy-type process in discrete time. Our method is based on the Measure-Valued Differentiation (MVD) approach, which allows representation of derivatives as differences of two processes, called the phantoms. We discuss the applicability of MVD for different types of option pay-offs in combination with different types of models of the underlying and provide a framework for the applicability of MVD for path-dependent pay-off functions, as Lookback Options or Asian Options.

Keywords:

• Lévy-Processes
• Estimation of the Greeks
• Measure valued differentiation
• Exotic options
• Lookback option
• Asian options

Notes

No potential conflict of interest was reported by the authors.

1 While is in most cases just the point mass at S(0), we allow here some slight generalization.

2 Notice that can be chosen in a measurable way, if (3.2(3.2) ) is the decomposition in the positive and negative part. If another decomposition is chosen, one has to require measurability.

3 Throughout the paper each estimator was simulated 300 times to get the arithmetic mean of the estimator, the arithmetic mean of the computational time and the variance.

Additional information

Funding

The author Thoma was supported by the Jubilaeumsfondsprojekt AB13377 of the Austrian National Bank.