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  1. Margin & Liquidations

Black Scholes Model

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Last updated 1 year ago

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:

∂V∂t+12σ2S2∂2V∂S2+rS∂V∂S−rV=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 ∂t∂V​+21​σ2S2∂S2∂2V​+rS∂S∂V​−rV=0

In this equation, VVVis value of the option (fair premium price), TTT is time to expiration, σ\sigmaσ is implied volatility of underlying asset, SSS is price of underlying asset, and rrr is the risk-free interest rate. According to '76 extension to the model, spot price SSSof underlying asset is replaced by a discounted futures price FFF:

S=e−rTFS = e^{-rT}FS=e−rTF

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.