Advanced Option Models
This chapter extends the discussion of option pricing to more advanced models that attempt to deal with the observed non-constancy of volatility in the foreign exchange market. As I documented in Chapter 5, quoted volatility rises and falls with market conditions. More perplexing yet are the phenomena of skew and smile volatility patterns. Derman (2007) remarked:
The Black-Scholes model has no simple way to obtain different implied volatilities for different strikes. The size of a building cannot depend on the angle from which you photograph it, except perhaps in quantum mechanics. Similarly, the volatility of a stock itself cannot depend upon the option with which you choose to view it. Therefore the stock's volatility should be independent of the option strike or time to expiration, because the option is a derivative that “sits above” the stock. (p. 3)
Derman was writing about options on common stocks, but his remarks are germane to options on foreign exchange. Hence the interest in new models, ones that are designed to be less restrictive than BSM is in the treatment of volatility. Such is the case for the models that I will discuss in this chapter. They incorporate stochastic processes that are more complex than the BSM diffusion process. These models are of special importance to the pricing of barrier options.
However, there are costs associated with transitioning to new models. Many of these models are mathematically complex and relatively ...