Jun 29, 2010

Correct Theta for VIX options, Part 2

Using calculus we calculate total change of option price as composed of regular Black-Scholes Theta, and also Vega * change of volatility per unit of time.

dC/dT = ∂C/T + ∂C/∂σ ∂σ/T

Theta and Vega are known model parameters, while the ∂σ/T can be modeled either from time series of VIX, or fitting a model to a set of VIX derivatives, or fitting some function to the term structure of (ATM) volatility.

One should note that the first and second term have opposite signs. Theta is negative, Vega is positive, and since volatility increases as we get closer to expiration ∂σ/T is also positive. Practically this means that cēterīs paribus options will stay relatively juicy comparing to similar equity options.

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