Coauthors: Marina Marena (SSRN page, webpage)
In this chapter, we illustrate the use of the Laplace transform in option pricing. Using the Laplace transform method we can transform a PDE into an ordinary differential equation (ODE) that in general is easier to solve. The solution of the PDE can be then obtained inverting the Laplace transform. Unfortunately when we consider interesting examples, such as pricing Asian options, usually it is difficult to find an analytical expression for the inverse Laplace transform. Then the necessity of the numerical inversion. For this reason, in this chapter we also discuss the problem of the numerical inversion, presenting the Fourier series algorithm that can be easily implemented in MATLAB or VBA. The numerical inversion is often disbelieved generically referring to its "intrinsic instability" or for "its inefficiency from a computational point of view". So the aim of this chapter is also to illustrate that the numerical inversion is feasible, is accurate and is not computational intensive. For these reasons, we believe that the Laplace transform instrument will gain greater importance in the Finance field, as already happened in engineering and physics. In Sect. 7.1 we define the Laplace transform and we give its main properties. In Sect. 7.2, we illustrate the numerical inversion problem. Section 7.3 illustrates a simple application to finance.
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