Chapter 13. Non-normal Distributions and Implied Volatility

The Black—Scholes formula for valuing options assumes that log share returns follow a normal distribution. First, we emphasise this assumption by showing an alternative form of the Black—Scholes formula expressed in terms of the mean and variance of the normal distribution for log share returns. The Black—Scholes formula can also be expressed in terms of the first two moments of the lognormal distribution for share prices.

In applying the Black—Scholes formula, all the input parameters are known apart from the volatility of the share returns over the life of the option. For a chosen level of volatility, we use the formula to generate an option value. This process works in the reverse direction too. Starting from an observed option price in the market, we can calculate its Black—Scholes implied volatility. The process of finding the implied volatility (or ISD for implied standard deviation) can be carried out by manual trial-and-error. An improvement is to automate the process. We discuss various methods of deciding an initial guess followed by a Newton—Raphson search to provide a good estimate of the ISD.

Practitioners are interested to know how to allow for departures from strict normality when valuing options. We look at two modifications, the first being an alternative analytic formula and the second an alternative binomial tree. The first approach suggests that share prices follow a reciprocal gamma (RG) distribution ...

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