image

The book begins with a brief note that highlights the difference between “expectation pricing” and “arbitrage pricing”. It gives an example of a bookmaker, someone who takes bets on horses. The bookmaker can always stay in the business by setting up odds based on the money at stake, rather than based on actual probabilities. If the book maker does a statistical analysis of horse performances,track conditions, historical data, etc.. and then sets the odds, there is always a possibility of a huge loss and getting wiped out. If the odds are quoted based on the amount bid on various horses, whatever be the outcome, he can always stay in business. This one little page in the preface is illustrative of the powerful technique to price financial instruments, i.e. “arbitrage pricing”.

Chapter 1

The first chapter is a shocker for someone coming from a pure statistics world where the expectation of a random variable is like a biblical term in the context of a “unbiased estimate of the random variable”. The expectation of a random variable is good but there is a problem with it if you apply directly to the financial instruments. The author takes a simple example of a forward contract that is priced at K via the expectation of the stock price with respect to real world measure, i.e. evaluate the expectation by taking log normal distribution of stock prices. This sounds reasonable as the expected value of the stock price should be equal to the forward price. But this kind of logic for pricing forwards is useless. There is another more powerful force that is the feature of markets, “arbitrage”. A forward contract payoff  can be replicated by going long the stock and borrowing cash from the money market. The forward price should be based on this replicating portfolio. Anything other price quote, there will be an arbitrage. Hence, SLLN(Strong Law of Large Numbers) or expectations based pricing is not wrong, but it is not enforceable. This is the case with any instrument in finance. You can’t use strong law and expectation to price them. The takeaway from this chapter is: If there is an arbitrage price, any other price is too dangerous to quote.


Chapter 2

The chapter starts off with a discrete price process where the stock behaves like a coin toss. Using a one period binomial model, the author shows that the price of the option ought to be time 0 value of a replicating portfolio of stocks and bonds. Any other price quote, there will be an arbitrage. The key ideas one needs to understand are:

  • The discounted price process is a martingale under risk neutral and actual probabilities
  • The claim process is a martingale on the tree
  • The fact that there are two equivalent martingales measures means that one can produce one random process from another.( This is mentioned as Binomial representation theorem). This representation guarantees a replicating strategy.

All the terms that are relevant to Binomial representation theorem are explained using examples and visuals. The terms defined are

  • Process
  • Measure -  For one not familiar to measure change on the same outcome space, this is quite a new thing. In fact the whole derivative pricing hinges on change of measure.
  • Filtration -  It is important to think about these terms than merely reading up the definitions and having a vague notion of filtration. Some questions that can aid a reader’s thinking process are :
    • What’s the need for introducing filtration?
    • What exactly does it mean when you come across the phrase,” Xi is Fi measurable”
    • How does one turn the phrase “ more information as time moves forward” in to more precise language using sigma algebra notation ?
    • Can you think of simple process and write down the filtration for it ?
  • Claim
  • Conditional Expectation operator – It has two parameters, the measure and the history. Ideally whenever you use the term expectation, you should always append the phrase ,”with respect to xyz measure”. May be outside quant fin, one usually never hears this phrase as typically the word expectation means that you are talking with respect to the real world measure.
  • previsible process – This is a process whose value does not depend on the future. Given a particular time, the value this process takes is exactly known. 
  • Martingale  – Basic definitions are given and the a connection between measure and martingale is explained. A measure needs to be attached to every martingale , much like a measure needs to be attached to expectation operator.

With the above terms explained, the chapter then goes on to explain “ Binomial representation theorem”, a theorem that is key to understanding derivative pricing. The continuous-time form of the theorem is stated and proved in the next chapter. Seeing the theorem in the discrete form gives a reader enough intuition to understand the details of the continuous time form.

The crux of the binomial representation theorem is that there are two martingales under the same measure, you can manufacture one from another. In the case of a discrete binomial tree model, the discounted price process and discounted claim process are both martingales under risk neutral measure. Hence you can hedge one with the other. The theorem states that such a hedge is possible. Basically scaling and shifting of one random process can create another process. Subsequently, the necessary condition for a portfolio to be a self-replicating portfolio is also derived. The logical conclusion of this chapter is in the form of two slogans.

  1. There is a self-financing strategy in the binomial tree that duplicates any claim.
  2. The price of any derivative within the tree binomial tree model is the expectation of the discounted claim under the risk-neutral measure that makes the discounted stock a martingale.

 

Chapter 3

This chapter builds upon the principles covered in the previous chapter and moves to continuous-time domain. Given a discrete time model , one might think of extending the model to continuous time by applying limit conditions on tick times. However the limiting arguments are too dangerous to be used rigorously. What are continuous-time processes ? Continuous process can change at any point in time, can take values that can be expressed in arbitrarily fine fractions, and have no jumps. If you look at the literature on option pricing, the process that will stand out as the king of all processes is the “ Brownian motion” process.

It is sophisticated enough to produce interesting models and simple enough to be tractable. The first thing that strikes a reader is ,”What’s the connection between Brownian motion and stock prices ?” There is no relation of a stock price to Brownian motion at a global level, but when you get down to local behavior, the representation of a Brownian motion looks similar to a stock price movements. The point the authors make with the help of arguments and visuals is that “Brownian motion can’t be the whole story but locally Brownian motion looks realistic to a stock price movement”.

A intuitive way of understanding Poisson processes begins with applying scaling a discrete binomial process. Similarly , one can look to binomial process to help one get a good intuition of the Brownian motion. An easy way to understand Brownian motion is to take a symmetric random walk and then change the time scale. One can construct a random walk process in such a way that marginal and conditional distributions match as that of a Brownian motion. There are some peculiarities of the Brownian motion (BM)  that are highlighted such as

  • BM is a nowhere differentiable function
  • BM will hit any value with probability 1
  • Once BM hits a value, it immediately hits it again infinitely often
  • BM is a fractal

The chapter subsequently talks about the Geometric Brownian motion(GBM), the standard price process assumed for a stock in most of finance. The chapter then gives definitions for a stochastic process and introduces the symbolic notation via Stochastic differential equation. Ito’s calculus is then introduced. Ito’s lemma is mainly used to formulate a SDE from process or cull out the process from the SDE. The latter essentially means that it is a tool that is useful for solving SDE. Univariate and Bivariate Ito’s lemma are then introduced and some basic examples are given.

The highlight of this chapter is Cameron-Martin-Girsanov theorem. In order to understand this theorem, one needs to understand

  • Change of measure
  • Radon Nikodym derivative and Radon Nikodym process
  • Equivalent measures

Instead of jumping in to the continuous-time domain, the author takes time to go over Radon Nikodym process in a discrete setting using good visuals. These visuals are really a good way to illustrate the procedure of changing measure. The reason the process is important is that it gives a way to change a price process from one measure to another. It serves as the bridge between the expectation of a random claim under two different measures. In the context of derivative pricing, one typically deals with market measure and risk neutral measure. A few examples are given to illustrate the outcome of change of measure. It is made clear that the process outcomes are not changed, only the likelihood of those outcomes are changed. For example a Brownian motion with drift becomes a Brownian motion without drift etc.

After the explanation of all the relevant math tools, the chapter states Cameron-Martin-Girsanov theorem. The crux of the theorem is that given a drifting Brownian motion under a measure, it can be transformed in to a Brownian motion with out drift under a different measure. The link between the two is the Radon Nikodym process.

Finally Martingale representation theorem is stated , a continuous-time version of Binomial representation theorem. The essence of theorem is : Given two martingale processes under risk neutral measure , one can use one to manufacture other. The discounted price process is a martingale under risk neutral measure, the discounted claim process is a also a martingale under risk neutral measure and hence there is a replicating strategy that links between these two processes.

The existence of replication strategy is guaranteed by Martingale representation theorem. As in the case of discrete case, the theorem does not give you the exact strategy, it merely states that there is one. With all these concepts explained, the author discusses the famous Black Scholes pricing and Black Scholes PDE. In doing so, the author clearly states the three steps to replication. Instead of using symbols, I will write down the steps in words

  1. Find a risk neutral measure that makes the discounted stock price a martingale
  2. Form the discounted claim process under this risk neutral measure
  3. Find a previsible process such that a self replicating portfolio exists

To make the transition smooth, the chapter discusses the replication strategy in a world with no interest rates. It then moves in to the world with interest rates to show the basic replication strategy for a derivative security. The beauty of risk neutral valuation is that the formula remains more or less the same for a variety of derivative securities.

Chapter 4

The most important lesson that this chapter tries to impart is that “Martingales are tradables” and "Non Martingales are non tradables”. The chapter considers various cases where the derivative is written on something that is not directly tradable. In the case of the foreign exchange process one needs to convert from a non-tradable cash process to a tradable discount bound process. For dividend paying equities, the model process needs to be changed so that dividends are reinvested. For bonds, the coupons need to be reinvested in the numeraire process. Underlying all this is a tradable/non tradable distinction. Unless you create a tradable that is martingale, there is no way one can use CMG theorem.

This distinction between tradable and non tradables is made concrete by connecting it with martingales. Through nice and easy no arbitrage arguments, the chapter proves that

  • Martingales are tradables
  • Non tradables are Non Martingales

There is an interesting connection between CGM theorem and market price of risk. If there are x number of tradable assets, it means that the discounted price process for the assets should be martingales under same measure Q. This means all tradables in a market should have the same market price of risk. The market price of risk is actually the drift change of the underlying Brownian motion given by CMG theorem. If we write the SDEs in terms of Q(risk neutral measure) Brownian motion, then the asset is tradable if and only if its market price of risk is zero.

The above statements become important in a one factor model when one is trying to create a replicating portfolio of a claim. There are times when the claim is a function of non tradable , for example in the case of a stock paying continuous dividends. If a call option is written on this stock, you cannot hedge the position with the original price process. So, you try to find a function of that non tradable that is tradable and then use that function to work with the CGM framework. In the one factor model world where there are two independent tradables, the cash bond and the stock, all the other tradable are nothing but a linear combination of the two assets. Again the assumption here is that there is a single source of randomness.

The key idea that one needs to get from this chapter is that there are claims that are written on non tradables( the classic example is that of zero coupon bond that is a claim on the interest rate, which is a non tradable), and one needs to formulate an appropriate function of the tradable and use CGM framework to form a replicating portfolio.The chapter ends with a discussion of quantos.Quantos are interesting derivative contracts whose derivative payoffs are paid off in a different currency. As with any exercise of valuing derivatives, one must observe the tradables and the number of Brownian motions that are driving the processes. Once the tradables are identified, the procedure is similar to the one that is used through out the book,i.e cut the drift of the Brownian motions using Girsanov theorem, form the discounted claim process and use Martingale representation theorem. The takeaway from the section on Quantos is that since there is a measure under which dollar tradables are martingales, one can price quanto options.

Chapter 6: Bigger Models

The chapter starts off by considering a generalized GBM where the drift and volatility parameters are dependent on the previsible processes. Despite these relaxations, there is no change to the procedure to value a derivative on the stock. The same three steps to replication can be used to value an option. The flip side of making fewer assumptions is that you don’t end up with closed form solutions for derivative prices. 

Further generalization of the GBM model is done by allowing an n-dimensional Brownian motion. To deal with n-dimensional volatility processes, there is a crash course given on n-factor Ito’s lemma and n factor Martingale representation theorem. The price one needs to pay for allowing n dimensional Brownian motions is that there are restrictions on the existence of Martingale measure. Once these restrictions, termed as “market price of risk equations” are satisfied, the three step replication framework works like a charm. The last section of the book is actually the section that fits all the pieces of jigsaw puzzle together. It states the arbitrage-free and completeness theorem that is the basis on which “three step procedure” works. 

imageTakeaway:

Risk neutral pricing technique comprises three main steps, i.e. 1) Finding a measure under which tradables are Martingales, 2) Constructing a claim process under the measure found in the previous step, 3) Using Martingale representation theorem to form a self replicating portfolio. This framework is used to value many types of derivative contracts, in each case tweaking some aspect but retaining the overall philosophy. The highlight of the book is that the authors emphasize the three step framework over and over again at so many places that it becomes your natural way of thinking about any derivative instrument pricing.

Advertisements