DECIPHERING THE BLACK-SCHOLES MODEL: INSIGHTS INTO OPTIONS AND PROFIT-LOSS ATTRIBUTION

Abstract

The Black-Scholes formula is a cornerstone in pricing European options, widely used in financial markets. However, its reliance on the constant volatility assumption often falls short of capturing real-world complexities. To address this, the concept of a volatility surface, encompassing two dimensions of maturity and strike level, is introduced, allowing the formula to better align with market prices. This paper delves into the theoretical underpinnings of option trading, shedding light on profit and loss attribution throughout the trade's lifecycle. In practice, various entities, such as oil producers, airlines, and insurers, engage in option transactions with investment banks to manage their business-related risks. Investment banks, in turn, must adeptly handle these client demands while prudently managing their own risk exposure. One prevalent strategy employed by investment banks is delta hedging, which aims to maintain a risk-neutral portfolio. This paper offers a novel mathematical derivation, grounded in the assumption of a flat and sideways market, demonstrating that the gains from delta hedging precisely match the option's price. While this concept is well-known, the mathematical framework presented here provides a fresh perspective and a reimagined formulation of the Black-Scholes formula, enriching our understanding of option pricing and risk management.