Option Prices as Probabilities: A New Look at Generalized Black–Scholes Formulae, by C. Profeta, B. Roynette and M. Yor

Option Prices as Probabilities: A New Look at Generalized Black–Scholes Formulae, by C. Profeta, B. Roynette and M. Yor, Springer (2010), Hardcover. ISBN: 978-3-642-10394-0.

A fundamental problem in mathematical finance is the computation of the European put and call prices

and

for strike K and maturity t, where the asset price M is a continuous non-negative local martingale on some filtered probability space . The celebrated Black–Scholes formulae solve this problem when M is the geometric Brownian motion, that is , where B is a standard Brownian motion.

The monograph under review begins with the fascinating observation that if

is the last passage time of Brownian motion with drift ν to the level a, then the put price for is

and the call price is

This is first shown by an elementary direct computation using the fact that

Alternatively, one can check that the partial derivatives in t of the claimed formulae agree with the corresponding derivatives of the Black–Scholes formulae and that the values at t = 0 agree. These two identities are quite unexpected in the way that they connect together expected values of a priori unrelated functionals of ℰ.

Moreover, these observations extend readily to more general M. If M is a continuous non-negative local martingale such that M ₀ is constant and almost surely, then, setting

there is the identity

and, if, moreover, and M is a true martingale, then

where is the probability measure defined by the ‘change of numéraire’

The proofs in chapter 2 of both of these identities hinge on Doob's identity that has the same distribution as , where U is a uniform random variable on that is independent of .

As remarkable as these identities are, their practical utility depends on the extent to which it is possible to come to grips with the distribution of , and so examples are given in chapter 2 of processes for which the last passage time has a closed-form distribution. For example, it is possible to write the distribution of the last passage time of a transient diffusion in terms of its scale function and its transition densities with respect to the speed measure.

One aspect of the identity is not so surprising: the function is a priori non-decreasing with limits 0 and 1 at t = 0 and and so, for a fixed K, there should be some random variable such that . This remark is extended much further in chapter 6 by an investigation into whether there is a probability measure γ on such that

This problem is related to the existence of so-called pseudo-inverses, and the final two chapters, chapters 7 and 8, contain a detailed study of pseudo-inverses for Bessel processes and more general diffusions.

The monograph ends with two appendices. The first appendix provides a number of complements to the results in the main body. For example, it presents an exposition of work by Yen and Yor on the European call associated with a local martingale that is not a true martingale. The second appendix contains a very useful summary of facts about Bessel and related special functions and a compendium of results about Bessel and squared Bessel processes.

It may seem from the description given above that the monograph focuses on a somewhat narrow range of problems. However, it is difficult to convey in the space of a brief review the manner in which the authors attack those problems from numerous perspectives using a remarkable array of methods. They often provide several proofs of the same result using quite different techniques and points of view. The monograph is certainly not ‘self-contained’ (and the authors do not purport that it is), but results that are used from the literature are stated clearly and accompanied by adequate references. The reader will be rewarded by seeing many of the tools from stochastic analysis and diffusion theory that have been most useful in modern mathematical finance deployed in a wonderfully imaginative and insightful way to study a particular set of closely related questions.

Steven Evans

Department of Statistics and Mathematics

University of California at Berkeley