When Uncertainty is Good

As announced last week, I will look at model calibration from different angles in the posts to come.

To start with an elementary example, we want to identify a flat (Black-Scholes) volatility from prices of call options with different strikes. To be more specific, let the spot price S be 100, the flat interest rate r = 0.02 (continuous compounding), expiry T= 0.5 (years), and the true annualized volatility sigma =0.25.

The call price for K=S=100 is then C=7.517

We calculate the true call prices for a wide range of strike prices (distributed around the spot price), and perturb the true call prices with a normally distributed error with mean 0 and standard deviation 0.03. Then, we determine the implied volatilities from the perturbed prices and obtain

Image Source: A Workout in Computational Finance, Chapter 15.

We observe that we get a reasonable recovery at the money and that the quality gets worse the deeper in the money or out of the money the option ist. Why?

The reason behind this phenomenon lies in the Implicit Function Theorem from basic calculus. Assume, for a fixed strike K, we want to determine the volatilitysigma leading to the quoted (perturbed) price p, where C_K is the operator that calculates the call price for K and sigma.

Differentiating with respect to p (and obeying the chain rule), we obtain

and finally

This is also the amplifier of noise. When the call price does not change much as a function of the volatility (vega is small, the option is deep in the money or deep out of the money), then the implied volatility cannot be recovered reasonably.

How does this relate to the headline of my post? When the price contains vega risk (the price changes significantly with volatility), then we can expect to recover implied volatilities, and the implied function theorem gives us error estimates.