# EXPLICIT CLOSED-FORM SOLUTION OF BLACK-SCHOLES EQUATION AND ITS APPLICATION TO CASH-OR-NOTHING BINARY OPTIONS

## DOI:

https://doi.org/10.60787/jnamp-v67i1-343## Keywords:

Black-Scholes equation, Diffusion equation Fourier Transform, Option pricing## Abstract

This work deals with the explicit closed-form solution of Black-Scholes equation and its application to cash-or-nothing binary options. We first transform the Black-Scholes equation into a diffusion equation by change of variables. We then apply the Fourier Transform method to find the general solution of the diffusion equation. Finally, we establish an explicit closed-form solution for binary options. Hence, for a call (put) option, one gets the discounted risk neutral probability that the stock price is above (below) the strike price at time.

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*The Journals of the Nigerian Association of Mathematical Physics*,

*67*(1), 47-56. https://doi.org/10.60787/jnamp-v67i1-343