# Black Scholes Model To calculate the Premium Mark Price for options contracts and their margin requirements, Pascal employs an extended version of the Black Scholes model (BSM) ’76. The BSM and its 1976 extension constitute a mathematical model for estimating the prices of European-style options. At its core, it features a partial differential equation representing the market as a stochastic process. This equation governs the price evolution of options, accounting for various market dynamics and uncertainties: $$ \frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} - rV = 0 $$ In this equation, $$V$$is value of the option (fair premium price), $$T$$ is time to expiration, $$\sigma$$ is implied volatility of underlying asset, $$S$$ is price of underlying asset, and $$r$$ is the risk-free interest rate. According to '76 extension to the model, spot price $$S$$of underlying asset is replaced by a discounted futures price $$F$$: $$ S = e^{-rT}F $$ While the futures price, time to maturity, and the risk-free rate are known parameters, solving the Black-Scholes-Merton equation establishes a relationship between the option's premium price and its implied volatility. --- # Agent Instructions: Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://docs.pascalprotocol.com/black-scholes-model.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.