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The purpose of the book is to illustrate various asset pricing concepts in a coin tossing world. Imagine a world where there is a money market instrument, an underlying security and various derivatives written on the underlying security. This world has a peculiar feature, i.e., the stock price can move up or down based on a coin toss. So, if one is looking at an option that has expiry 3 units from the current time, there are 8 outcomes of the stock price based on 8 coin toss realizations (HHH,HTH,HHT,HTT,THH,TTH,THT,TTT). Also, the probability of heads and tails are known upfront. One can’t think of a more simpler world for understanding option pricing. So, given this setting, the book explores the valuation of the following securities:

  • European Derivative Securities whose payoff is not path dependent

  • European Derivative Securities whose payoff is path dependent, like an Asian option or a lookback option

  • American Derivative Securities whose payoff is not path dependent

  • General American Derivative Securities

  • Perpetual put – an abstract instrument to illustrate various principles of American option pricing

  • Derivatives in the Fixed Income world.

The book starts off with a single period model where the payoff of a derivative security is replicated using a specific portfolio that is long some units of stock and money market. To replicate the portfolio, convenient variables are introduced which go by the name “risk neutral probabilities”. These are not the actual probabilities of the world but fictitious probabilities that help in solving a set of equations (that are used to obtain the hedge ratio and initial wealth needed to replicate the option payoff). Once the intuition is established using a one period model, the replicating portfolio is constructed dynamically for three periods. One sees that the risk neutral probabilities remain the same, and the framework that is applicable for a one period can be extended to multi-period model. The replication in the multi-period binomial model for a derivative security has one issue that is highlighted in the section titled “computational considerations”. If one needs to evaluate an option that has an expiry 100 units in to the future, one ends up with 2100 different coin toss realizations. This means that the final value of the option needs to be computed for 2100 coin toss realizations. Fortunately there is an easy way out. A flavor of Markov chain property is given in the first chapter itself, even though it is dealt at length in the subsequent chapters. The Markov property of the price process allows a computationally efficient way of pricing a derivative in a multi-period set up. There are two examples that illustrate the usefulness of Markov property. These two examples give an indication of the real power of Markov property. In the case of path dependent option, you can enlarge the state space to include variables that will make the new process possess the Markov property. I guess one can always come back to these two examples after one is comfortable with Markov property principles that are mentioned in subsequent chapters.

Chapter 2 gives enough math tools to work with the asset pricing model introduced in the first chapter. In the context of a coin toss model, basic concepts of probability are introduced such as the probability model, expectation, Jensen’s inequality, conditional expectation. Enough care is taken so that terms such as “Sigma Algebra”, “Lebesgue measure” are avoided. These terms usually demotivate a reader who is looking to gain intuition in to derivative pricing models. The key mathematical object discussed in this chapter is “Martingale”. Basic definitions and properties of martingales are illustrated using the price process in the binomial outcome world. The core idea is that the discounted wealth process is a Martingale. This coupled with the fact that the wealth process actually concurs with the payoff of the derivative security in every outcome, implies the discounted value of the derivative price process is also a Martingale.

The other aspect that is discussed in this chapter is the Markov property. Usually one comes across Markov chains definitions that have terms such as transition probabilities, initial distribution, etc. Nothing of that sort is mentioned here. May be the author already assumes some sort of background from the reader. Or maybe he does not want to unnecessarily introduce things that are not used anywhere in the book. The chapter defines Markov property using a conditional expectation definition. The Markov property becomes very important to reduce the computational complexity of the derivative pricing algorithm dealt in the first chapter. Using the multi-step-ahead Markov property, a recursive algorithm for a path dependent option is developed. If you start seeing things in the discrete-time world, I think it provides a good insight in to continuous-time world. For example, the recursive equations for the derivative in the discrete setting are equivalent to Partial differential equations in the continuous-time world. Feynman-Kac theorem that comes up in the Volume II of this book helps one move from a continuous-time analogue of risk-neutral pricing formula to PDEs. One might be comfortable with understanding Feynman-Kac theorem without ever coming across the discrete version. But I think the discrete time analogue of the same in the binomial model is going to be very useful for a lot of readers, to gain a better understanding of Feynman-Kac theorem.

Chapter 3 is all about Radon Nikodym derivative denoted by Z in the book. This variable is the bridge between evaluating expectation in the risk neutral world and the real world. Besides the price process and the derivative security price process, there is a process associated with Z. The way to manufacture this process is to from a ratio between the risk neutral probabilities and real world probabilities. Various properties of Z are explored such a Martingale property and these properties in turn help in evaluating the expectation under the real world measure. There is also a section on CAPM, which could have easily been pushed to appendix. Somehow I failed to appreciate the relevance of CAPM framework in this chapter.

Chapter 4 extends the concepts to American derivatives. This complex topic is dealt in a beautiful way. The multi-period binomial model introduced in the first chapter is extended to situations where the derivative can be exercised at any point in time before the expiry. The replication of a path dependent security is laid out in the chapter with enough numerical examples to get a sense of the algorithm. Stopping times are introduced as they are essential for the valuation of American derivative securities. Stopping time principle is useful in a lot of places and helps in reducing the computational steps in many problems. For example one can prove that a symmetric random walk is null recurrent using stopping time in just a few steps. The fact that Martingale stopped at a stopping time is a Martingale, submartingale (supermartingale) stopped at a stopping time retains its original property becomes critical in evaluating general American derivative securities. What’s the connection between stopping times and American options? One can think of the option being exercised at a time as a stopping time. The discounted derivative security price process of an American option is a supermartingale. It has a tendency to go down at exactly those moments when it should be exercised. Hence on way to get a grip on the price process is to think of all the stopping times till expiry and evaluate the risk neutral expectation of the discounted intrinsic payoff value at the stopping rule that makes this expectation attain its maximum. The chapter ends with the application of stopping rule principles to non-dividend paying American call option and shows that it not worth exercising the option at any point before expiry.

Chapter 5 talks about symmetric random walk and its properties. Well, the first time I came across symmetric random walk was in the context of Markov chains. While learning the concepts of classification of states, one usually comes across symmetric random walk as an example of null recurrent chain. There are many ways to prove that a symmetric random walk is a null recurrent chain. One can try to establish the probability that a chain hits a specific level in n steps. Then use limiting condition to show that the symmetric random walk will almost surely hit any level. Another way is to use stationary distribution of a countable Markov chain and prove that the expected time to hit a specific level is infinite and hence it is a null recurrent chain. The treatment in this book is different and infact elegant as it uses Martingales. Manufacturing a Martingale that has a symmetric random walk as a term is the key to efficient computation. The chapter provides a Martingale for a symmetric random walk and one can see that most of the calculations become pleasant. One can show that a state in symmetric random walk is null recurrent after wrangling with the relevant Martingale. The chapter subsequently introduces reflection principle, which is basically a time saving tool for someone who is working with random walks and Brownian motion. The chapter ends with an elaborate discussion of a hypothetical instrument, “Perpetual Put option”. The purpose of dealing with “Perpetual Put” option is to illustrate the concepts of Martingale and Stopping times. The last chapter deals with derivative securities in the fixed income world.

image Takeaway :

This is a book that deals with a world where the price of a security moves based on a coin toss. Limiting the number of periods to three, the author shows various techniques to value European and American Derivative securities. This book serves as a good foundation for someone entering in to the continuous-time world of derivative pricing.