Endah R.M. Putri, Lutfi Mardianto, Amirul Hakam, Chairul Imron, Hadi Susanto
Black-Scholes equation as one of the most celebrated mathematical models has an explicit analytical solution known as the Black-Scholes formula. Later variations of the equation, such as fractional or nonlinear Black-Scholes equations, do not have a closed form expression for the corresponding formula. In that case, one will need asymptotic expansions, such as the homotopy perturbation method, to give an approximate analytical solution. However, the solution is non-smooth at a special point. We modify the method by first performing variable transformations that push the point to infinity. As a test bed, we apply the method to the solvable Black-Scholes equation, where excellent agreement with the exact solution is obtained. We also extend our study to multi-asset basket and quanto options by reducing the cases to single-asset ones. Additionally we provide a novel analytical solution of the single-asset quanto option that is simple and different from the existing expression. © 2021 Elsevier B.V.
Department of Mathematics, Faculty of Mathematics, Computing, and Data Sciences, Institut Teknologi Sepuluh Nopember, Jl. Raya ITS, Sukolilo, Surabaya, 60111, Indonesia; Department of Mathematics, Institut Teknologi Sumatera, Jl. Terusan Ryacudu, Way Hui, Jati Agung, Lampung Selatan, 35365, Indonesia; Department of Mathematics, Khalifa University, Abu Dhabi Campus, PO Box 127788, Abu Dhabi, United Arab Emirates; Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester, CO4 3SQ, United Kingdom
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