An Alternative Derivation of the Black-Sholes Model for Option Pricing

Abstract

Investment enables investors to acquire shares in capital markets or their derivative assets, with options being one of the most common instruments. An option is a contract granting the right to buy or sell an underlying asset under specific conditions. The Black–Scholes equation is widely used for option pricing, though its derivation through Backward Stochastic Differential Equations (BSDEs) is less common. BSDEs are particularly relevant in incomplete markets, where not all options can be perfectly replicated. BSDEs also avoid the need to transform probability measures into the risk-neutral, thereby simplifying the pricing procedure. This study derives the Black–Scholes equation via BSDEs, modeling investor wealth and applying the Feynman–Kac theorem. The solution for call options is obtained by transforming the variables to get a diffusion equation, while put option prices are derived using the put–call parity principle. Finally, simulations of the Black–Scholes formula are conducted to analyze option price behavior under varying volatility and risk-free interest rates.