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COMPARISON OF DIFFERENT METHODS TO COMPUTE THE GREEKS
Date
2013-10-29
Author
Yılmaz, Bilgi
İnkaya, Bülent Alper
Yolcu Okur, Yeliz
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The Greeks in finance are the partial derivatives of a financial quantity with respect to any of the model parameters. These derivatives could serve to measure the stability of the financial quantity under study (e.g. delta is the derivative of an option price with respect to the initial price) or to hedging a certain payoff (Higa and Montero, 2004). The Greeks are useful tools in finance which help to understand how the option reacts to a change in the parameters. The information gained from calculated Greeks of options help the investors in their portfolio management decisions. However the calculation of Greeks are not always straightforward, because of some technical difficulties. For example, the structure of the payoff function of some contingent claims are very complex. For this reason computing their Greeks could be cumbersome. On the other hand, sometimes the payoff function does not have a closed form and one may have to estimate the Greeks numerically, which could be a time consuming and fraught workout. There are essentially three methods used to compute the Greeks: The Finite Difference Method, the Malliavin Calculus and the Finite Element Method. The most widely used method to compute the Greeks is the finite difference method. This method requires to compute the financial quantity of interest at two nearby points and approximate the differential of the payoff function at that point. The problem here is, the identification of “two nearby points" is not clear. L'Ecuyer and Perron (1994) attempt a solution to resolve this issue asymptotically. Since this method is deeply related with the Kernel Density Estimation method in Statistics, it is also called the Kernel Density Estimation Method. The Malliavin calculus, also known as Stochastic Calculus of Variations or Calculus in infinite dimensions, was introduced by Paul Malliavin in 1976 (Henao, 2005). It is first used as a tool to prove results in Calculus through the use of probabilistic theory and it is an an area of research which for many years has been considered highly theoretical and technical from the mathematical point of view. In recent years, it played a major role in applications of Mathematical and Computational Finance. It can also be applied to Monte Carlo simulations, and therefore it is used in computation of the Greeks. The finite element method is a numerical method to solve partial differential equations (PDE). This method is entirely deterministic. Although the system of difference equations can be usually solved quickly in low dimensions, the method is not suitable to compute the Greeks that are not directly related to the derivatives computed in the PDE. Cases where it can be applied successfully are the calculation of delta and gamma. In other cases it involves increasing amounts of recalculations which can be cleverly reduced in certain cases. The main aim of this study is give the details of the three methods given above and compute the Greeks of European options in Black-Sholes environment via Malliavin Calculus. The secondary aim is to compare the methods using the Monte Carlo Simulation Method.
URI
https://hdl.handle.net/11511/82524
Conference Name
Uluslararası 8. İstatistik Kongresi, (27 - 29 Ekim 2013)
Collections
Graduate School of Applied Mathematics, Conference / Seminar
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B. Yılmaz, B. A. İnkaya, and Y. Yolcu Okur, “COMPARISON OF DIFFERENT METHODS TO COMPUTE THE GREEKS,” presented at the Uluslararası 8. İstatistik Kongresi, (27 - 29 Ekim 2013), Antalya, Türkiye, 2013, Accessed: 00, 2021. [Online]. Available: https://hdl.handle.net/11511/82524.