# Black Scholes Model

To calculate the Premium Mark Price for options contracts and their margin requirements, Pascal[^1] 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. 

[^1]: