The limit where Black-Scholes "works"

Abstract

The seminal (Black and Scholes, 1973) publication celebrated in this issue introduced a (stock price evolution) model and an (option value) formula�two ideas worth distinguishing. Nowadays, the formula is used to represent option prices via an implied volatility, IV (T, K), where T is a time-to-maturity and K is a strike price. The formula is very useful. At the same time, because of the K-dependence, the model is strongly rejected in statistical tests. However, with T very large at fixed K, IV (T, K) typically flattens IV(T,K)? s8imp , where s8imp is independent of K. Flattening happens both with real data and many (more complicated) models. At fixed strike, the implied volatility parameter tends to a pure constant. But a constant volatility returns us to the original BS model in a sense. As somewhat of an abuse, I say that in the limit, the BS model again works�. What is this mysterious s8imp and how do we compute it? How is the limit approached? Answering those questions is the subject of this note. While my approach is largely expository, results for non- standard cases in the Heston model are likely novel. Their analysis requires a generalized saddle point method.