Computation of Greeks using the discrete Malliavin calculus and binomial tree (Q2122555)

The book contains a presentation of Malliavin calculus for discrete random processes and its usage in mathematical finance. The Malliavin calculus is a set of mathematical methods extending classical calculus of variation to stochastic setup. In particular, it allows one to define a derivative of a random variable, find a martingale representation of a random variable, and use integration by parts method for calculating expected values of stochastic integrals. Originally, Malliavin calculus was developed for Ito processes. The full understanding of these methods requires deep knowledge about stochastic calculus. In the book, the concepts of the Malliavin calculus are developed and presented for stochastic processes with discrete time. This allows for a significant simplification of the theory and allows to understand its concepts even for people who are not familiar with the nuances of infinite-dimensional mathematics. The book starts with presenting the main concepts in the simplest setup: a single-period binomial model. The author shows in this simple example how to define the main tools of Malliavin calculus, such as the Malliavin derivative or Skorohod integral. The next two chapters contain an exposition of multi-period binomial models and their usage in financial mathematics for options pricing and calculating Greek coefficients. Chapter 5 is less connected with the main topic of the book. It contains a description of a numerical algorithm for option pricing in the multiple-period binomial model. The algorithm is based on solving a system of difference equations using the spectral expansion method. The authors use this algorithm to price barrier options and to calculate their Greek coefficients. The last two chapters contain the main ideas of the book. After presenting in Chapter 6 a brief introduction to the main concepts of Malliavin calculus in continuous time, the author develops these concepts for discrete time. He presents the main theorems of the Malliavin calculus for stochastic processes with discrete time. At the end he uses these methods and tools for computation the Greek coefficients of the European call option and compares the obtained results with the coefficient obtained from the standard Black-Scholes model.