The second partial derivative of option price with respect to the strike: A historical reminiscence

An option’s market price reflects the risk-neutral probability that it will end up in the money. Research has been increasing in recent years that shows how, given a set of market prices for options covering a range of strikes, an estimate of the entire risk-neutral probability distribution can be obtained. The technique is based on the fact that the second partial derivative of the option pricing function with respect to the strike price is the risk-neutral density (discounted from option expiration). This idea is generally attributed to Breeden and Litzenberger’s 1978 paper. In this article, Zimmermann shows that the connection between the second partial derivative of the option price with respect to the exercise price and risk-neutral probabilities has a much longer history, including a little-known 1974 note by Fischer Black, and going all the way back to Bachelier in 1900.