Python for Finance: Analyze Big Financial Data

Implied Volatilities

Given an option pricing formula like the seminal one of Black-Scholes-Merton (1973),

implied volatilities are those volatility values that, ceteris paribus, when put into the

formula, give observed market quotes for different option strikes and maturities. In this

case, the volatility is not an input parameter for the model/formula, but the result of a

(numerical) optimization procedure given that formula.

The example we consider in the following discussion is about a new generation of options,

namely volatility options on the VSTOXX volatility index. Eurex, the derivatives

exchange that provides these options on the VSTOXX and respective futures contracts,

established a comprehensive Python-based tutorial called “VSTOXX Advanced Services”

in June 2013 about the index and its derivatives contracts.

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However, before proceeding with the VSTOXX options themselves, let us first reproduce

in Equation 3-1 the famous Black-Scholes-Merton formula for the pricing of European

call options on an underlying without dividends.

Equation 3-1. Black-Scholes-Merton (1973) option pricing formula