Black Scholes Model

To calculate the Premium Mark Price for options contracts and their margin requirements, CVEX 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:

Vt+12σ2S22VS2+rSVSrV=0\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, VVis value of the option (fair premium price), TT is time to expiration, σ\sigma is implied volatility of underlying asset, SS is price of underlying asset, and rr is the risk-free interest rate. According to '76 extension to the model, spot price SSof underlying asset is replaced by a discounted futures price FF:

S=erTFS = 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.

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