Joint work with B.A. Shadwick, The Finance Development Centre Limited

In spite of the growing sophistication of option pricing technology, relatively primitive approaches to the computation of prices and hedging parameters are still in widespread use. This appears to be an inevitable side effect of the growth of derivatives markets. As one practitioner noted, "The problem with the Black Scholes formula is that it makes every idiot think he can price an option." Increasingly, the options he thinks he can price are what used to be known as exotic.

One of the most common examples of this problem is the variety of ad hoc methods which have been devised to deal with volatility smile or skew. There is a straightforward extension of the original Black Scholes Merton 1-factor model which good engineering practice would suggest should be exhausted prior to moving to more complex remedies. However, there is no shortage of examples of trading desks which purport to be using more 'advanced' approaches, many of which prove to be self contradictory under a discouragingly low level of scrutiny.

It is not uncommon to find that traders and quants who espouse these advances are, in their view, routinely making large 'profits' buying or selling long dated options to large sophisticated counterparties. Needless to say, these P&L effects often use mark to model and proprietary risk exposure analysis in an essential way.

We review the maximal 'local volatility' extension of the 1-factor Black Scholes Merton (BSM) model and illustrate one good reason for this situation. We show that the computation of prices and sensitivities in this framework requires numerical expertise sufficient to solve the forced, variable coefficient BSM equation and that as a result, one really does have all or nothing.

The low level of the pde solvers in common use in the finance industry then explains the prevalence and longevity of the ad hoc approaches to variable volatility. We provide some examples which show just how dangerous these approaches can be in the process of pricing or hedging even simple derivative positions.