Brownian Motion Calculus - Review

This book gives a non-rigorous treatment to Brownian motion and its applications to finance. Let me summarize a few points from various chapters.

Chapter 1 : Brownian motion

This chapter starts off by specifying Brownian motion by the properties of its increments such as independence, first and second moments, transition density etc. A discrete approximation of BM is shown via a binomial tree. Covariance of BM process is derived. A way to manufacture correlated BM is shown. Illustrations are provided to show that BM is nowhere differentiable. The most important property of BM, the quadratic variation, is shown via a few simulation runs.

Chapter 2 : Martingales

The concept of conditional expectation is dealt in this chapter. The explanation is at a 10,000 ft. view and various properties of conditional expectation are listed. In order to rigorously prove the properties of conditional expectation, measure theory is the only route. However one can always get the intuition behind the properties by working through a binomial asset pricing model. I think the key property that one needs to understand is that of “partial averaging”. In order to understand “partial averaging “' it is always better to write down the partial averaging condition , think about it for some time, and then try to verbalize the condition in simple words. By forcing oneself to write the condition in symbols and then translate the same in to words, one might get a proper understanding of conditional expectation. Martingales are mathematical objects defined in terms of conditional expectations. In option pricing it is important to manufacture a martingale from the terminal value of a derivative claim. The concept of filtration helps one manufacture a martingale from the claim. This process of creating a martingale by conditioning is described in the chapter. This chapter also gives a sample list of martingales to make the reader get an intuitive sense of these objects.

Chapter 3 : Ito Stochastic Integral

Ito Stochastic integrals have peculiar features and they cannot be integrated in the Riemann sense. The integrand is often a general random variable and the integrator is a stochastic process like a Brownian motion. For integrating with respect Brownian motion, one cannot use Riemann Stieltjes integration as the Brownian motion has unbounded variation. Hence an alternate route is taken to compute these integrals. In fact an Ito integral can only be computed in the mean square sense.

One needs to start with a sequence of non anticipating integrands and then extend the results of the stochastic integral for these non anticipating integrands to a general integrand. The whole magic happens in the Hilbert space where one can approximate the Ito integral of a general integrand with a limiting value of a sequence of Ito integrals of non anticipating integrands. Obviously this method is pretty cumbersome. For every integral if one needs to find approximating functions, evaluate the limiting value, it is like doing Riemann integration by partitioning for every single function. Like there are standard rules for Riemann calculus, there is one savior for Ito calculus, the “Ito’s lemma”. Thanks to Ito’s lemma, the evaluation of stochastic integral becomes relatively easy.

The exercises in the chapter are laid out in such a way that they reinforce the mean square convergence aspect of Ito integrals. The recipe for computing Ito integral is : a) formulate a sequence of non anticipating sequence of functions that converges in mean square to the general integrand b) write down the discrete stochastic integral for the sequence , and c) evaluate the limit of the discrete stochastic integrals. This converged value is the Ito integral value. Basically whenever you see Ito integral equals something, the equality sign should be interpreted in the mean square sense.

Chapter 4 : Ito Calculus

For computing Ito integrals, one of the most important tools is Ito’s lemma. This chapter covers Ito lemma in various shades and colors. Levy characterization of Brownian motion is stated so as to easily identify a Brownian motion. Basic recipe for simulating a Multivariate Brownian motion is given.

Chapter 5 : Stochastic Differential Equations

SDE for the following processes are described and solved

• Arithmetic Brownian Motion

• Geometric Brownian Motion

• Ornstein-Uhlenbeck SDE

• Mean Reversion SDE

• Mean Reversion with square root diffusion SDE

• Coupled SDE

Solving an SDE, as they say is part art and part science. A good guess is all that is required sometimes. However the chapter tries to give a generic framework for SDEs that are a combination of Arithmetic BM and Geometric BM. The chapter ends with Martingale representation theorem that basically says that, “If Brownian motion is the only source of randomness, then a continuous martingale can be expressed as a driftless SDE driven by Brownian motion”. This is the heart of option pricing framework. MRT guarantees that the claim process can be replicated. However the thing to keep in mind is that it only guarantees replication, it does not tell you the exact hedge.

Chapter 6 : Option Valuation

This chapter tries to do too many things, i.e. 1)PDE approach to option valuation, 2)Risk neutral approach to option valuation and 3) connecting PDE with martingale pricing using Feynman Kac. It is like squeezing in 120  pages of crystal clear treatment by Shreve( in his book on stochastic calculus) to 20 pages. I think this chapter needs to be rewritten so that it can at least give a good direction to the option pricing framework.

Chapter 7 : Change of Probability

This chapter is written very well. It starts off with a change of measure for discrete random variable. For a random variable taking countable values, one can adjust the probabilities in such a way that you will be able to shift the first moment of the random variable. Subsequently, a change of measure is done for a standard normal variable to shift its mean. These two examples are followed by changing the measure for a Brownian motion using Girsanov transformation. Unlike the simple cases, one needs to rigorously prove that the Brownian motion after a measure change results in another Brownian motion with change in drift. Enough examples are given so that the reader gets a good idea about, “how to move from one measure to another?” The application of Radon Nikodym derivative is shown via Importance sampling, a technique to produce robust simulation results. The concept of equivalent measures is also touched upon towards the end.

Chapter 8 : Numeraire Pricing

The last chapter like the chapter on option valuation covers too much ground and hence falls flat. An enormous number of topics are touched upon in 20 pages and ends up not doing justice to any of them. May be the author meant it that way so that a curious reader can explore things in other books.

Takeaway:

The title of the book appears daunting but the contents of the book are accessible to a pretty large audience. The math covered in this book does not require many prerequisites from the reader other than basic calculus and probability concepts. The background to all the discussion about Brownian motion is the option pricing application and hence can be read by most of the finance professionals who are looking to get a little deeper understanding of the math behind option pricing.