The model free implied volatility that we study in our CFE course is the one due to Britten-Jones and Neuberger. In a seminal work, Britten-Jones and Neuberger (2000) have shown that a model free implied volatility is the expected sum of squared returns under a risk neutral measure.

Unlike the traditional concept of implied volatility, where the implied volatility is estimated numerically from an option pricing model, the model free implied volatility (MFIV) is not dependent on any option pricing model. Britten-Jones and Neuberger demonstrate that the set of option prices (calls and puts) having the same maturity is sufficient to derive the risk neutral expected sum of the squared returns of the asset between the current date and the option maturity. The authors do not make any assumption regarding the underlying stochastic process of the asset except the fact that both the asset and the volatility exhibit no jumps.

In short, market price of options are sufficient to determine the the risk neutral variance (and hence the volatility) of the asset.

The math is fairly simple and quite elegant and its implementation in Excel is also rather straightforward. What is required is a numerical integration schedule.

Mathematically, Britten-Jones and Neuberger show that if the asset pays no dividend and if the interest rate is zero then between any two maturity dates and the risk neutral expectation of the squared returns are given by:

Where, is the market price of a call option with strike price and maturity and the domain of integration is over all the observed strike prices in the market (theoretically, from zero to infinity).

George Jiang and Yisong Tian show how to implement the Britten-Jones and Neuberger's result in the presence of dividends and when the interest rates are non-zero. For a more detailed implementation see Jiang & Tian (2003)

If we make one of the dates above as the present time, i.e. and set then the above equation can be written as:

Where, is the current value of asset price, i.e. asset price at . The model free implied volatility (MFIV) is therefore given by:

The MFIV can be easily estimated by taking a cross section of call option prices over a big range of strike prices with observable call option prices in the market and numerically integrating the above expression to get a single value for the model free implied volatility.

**References: Model Free Implied Volatility and Information Content** (March 2003), George Jiang, University of Arizona and Yisong Tian, York University;

For an excellent description of the Model Free Implied Volatility and its implementation see the book **Option Pricing Models and Volatility** by Fabrice Douglas Rouah & Gregory Vainberg

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