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This page is for the course entitled Models for Financial Economics. The course is offered in the spring, in the months of May and June.
Class Notices:
The class outline is here as a PDF file, and here as HTML. The information it contains is also given below.
The course is directed to students who wish to learn the mathematical techniques used in modern finance theory. The course will also include the basic theory of asset pricing, in particular, the pricing of derivative assets, such as options. If time permits, more elaborate models will also be discussed. The introductory material starts with measure theory, a topic not always treated in courses of mathematics for economists. Measure theory is however a necessary prerequisite for the sort of probability theory needed for financial applications. In particular, we will treat stochastic processes in continuous time, of which the simplest example is Brownian motion.
A brief list of the foundational topics we will treat is as follows.
On the more applied side, we will consider
We will follow the two-volume set entitled Stochastic Calculus for Finance, by Steven Shreve, in the Springer Finance series. The first volume contains no sophisticated mathematics, but allows readers to develop valuable intuition by a detailed treatment of the so-called binomial model, the simplest of all models of derivative pricing. We will make use of many of the examples in that volume. The second volume is where most of the material for the course is to be found. It combines mathematical developments with some quite sophisticated financial models.
I have from time to time drawn attention to misprints and errors in Volume 2 of Shreve's book. I have located Shreve's own list of errata, which is in fact a lot more comprehensive than my own observations would have led me to think. Here is the PDF file containing these errata .
Log of material covered:
Well, we covered nothing on May 2nd, because no one showed up for class. Let's hope we can do better next week.
On May 7th, we had the first actual class. We began with Volume 1 of Shreve's textbook, looking at the One-Period Binomial Asset Pricing Model. This model is simple, but it allows us to understand the essentials of more complicated asset-pricing models.
A fundamental principle is that of no arbitrage. If one wants to price a derivative security, that is, an asset the payoff of which is a deterministic function of the price of an underlying asset, it may be possible to imagine a hedging portfolio, which gives exactly the same payoff as the derivative security in every possible state of the world. If so, then the value of the hedging portfolio must be equal to that of the derivative security in order to avoid arbitrage opportunities.
Options can be written on the underlying asset, either call options or put options. The former are options to buy a unit of the underlying at the strike price, the latter to sell one unit. The idea of no-arbitrage pricing is that the return on the option can be replicated by a hedging portfolio, which contains only the two assets, the risky underlying asset and the risk-free asset.
In each discrete period in the binary model, the market price of the underlying asset, which we may call the stock, can move to only two possible new values, one greater and one less than the price at the beginning of the period. If there are several periods, the value of the option is found by backward recursion from the possible returns at maturity. In this case, we speak of the infinite coin-toss model, where what happens in each period can be modelled as resulting from a coin toss giving heads (H) or tails (T).
In Chapter 2 of Volume 1, finite probability spaces are considered. Of these, the one of most interest is the finite coin-toss model, which is just the multi-period binary model. In this simple context, we can define a real-valued random variable as a mapping from the outcome space, denoted by Ω, to the real line or a subset of the real line. Subsequently, if a probability structure is defined on the outcome space, this induces a probability distribution for the random variable.
Then, still with a finite probability space, we may define the expectation of a random variable, and, with a random sequence, we may define conditional expectations. As an aside, we looked at Jensen's inequality. This led on to the concept of a martingale, a formalisation of a fair game. A result that will have a parallel in continuous time is that the sequence of discounted values of an asset is a martingale under a risk-neutral measure.
We finished our study of Volume 1 on May 9th. The only topic from there that we considered was the definition of a Markov process. Prediction of the future of a Markov process depends only on the current state of the process - the history of the process contributes no further information.
In Volume 2, we began at the beginning, and covered most of the first three sections of the chapter, which deals with probability in the abstract. A probability space is a triple (Ω,𝓕,P), where Ω is the outcome space, 𝓕 is a σ-algebra the elements of which are subsets of Ω, and P is a probability measure. Random variables are mappings from Ω to somewhere else, in the simplest case the real line. Ω comes equipped with 𝓕, and the real line with its σ-algebra, the Borel σ-algebra. An essential property of a well-defined random variable is that inverse images of Borel sets must belong to 𝓕.
The probability measure on (Ω, 𝓕) can induce another measure on (R,𝓑), for a random variable X. This is the distribution measure μ(X). It characterises all the probabilistic properties of X. Another equivalent characterisation is given by the cumulative distribution function (CDF). Sometimes a density exists as well.
Combined with the distribution measure, a random variable may have an expectation. It can be defined in a quite abstract setting by a Lebesgue integral. In some cases the expectation may be infinite.
Section 1.4 deals with convergence of a sequence of integrals and a sequence of expectations of random variables. For the latter, the kind of convergence that suits us best is almost-sure convergence. An example was given of a sequence of densities of normal random variables, with expectation zero and variance 1/n, and n → ∞. The integrals of these densities are all equal to 1, but the densities themselves converge to zero almost everywhere, and so the limit of the integrals differs from the integral of the limiting function.
On May 14, we began with something we had skipped over, namely the relation between the Riemann and Lebesgue integrals. The condition needed for them to be equal is that the set of points at which the integrand is discontinuous has Lebesgue measure zero.
The rest of our time that day we finished Chapter 1 of Shreve. We had seen that the limit of integrals of a sequence of functions is not necessarily equal to the integral of the limiting function. Two theorems let us arrive at the opposite conclusion; monotone convergence and dominated convergence.
Section 1.5 gives various results that are well known, but need proof in the abstract context. The very valuable technique of proof laid out here is called the standard machine. It starts by trying to prove the desired result for an indicator function, and then, in three further steps, extends the result to non-negative simple functions, then to general non-negative Borel-measurable functions, and finally to Borel-measurable functions that can have positive and negative values.
The last section, 1.6, of the chapter introduces some valuable concepts. Starting from some probability measure on a probability space, the measure can be changed by use of a random variable that is non-negative almost surely, and has expectation 1 under the first measure. This random variable is called the Radon-Nikodým derivative. If it is almost surely strictly positive, it defines a measure that is equivalent to the original one, by which is meant that two measure agree on which sets have measure zero.
An important example is given whereby a normal variable Y that is equal to a standard normal variable X plus a constant θ ends up with the standard normal distribution, with expection 0, under the new measure. The Radon-Nikodým derivative random variable is a deterministic function of the variable X. Section 1.6 concludes with the statement of the Radon-Nikodým theorem, which says that any two equivalent probability measures have a Radon-Nikodým derivative that is almost surely positive and has unit expectation.
Assignments:
Here is the link to the first assignment, dated May 9. The assignment is due on Thursday May 16. The easiest way to submit your assignment is by sending it to me by email.
Ancillary readings:
Here is the note I prepared as an alternative to Shreve's discussion of forward and future contracts, in the hope that it would be more understandable.
I mentioned in class that there is a construction of Brownian motion by a sequence of stochastic processes that converge almost surely to the Brownian motion. The gory details of this, and a few other things, can be found in this link
I found this review of Shreve's texts, written by Darrell Duffie of Stanford. In it, you will read how good a set of two texts these books are!
By chance I came across an article (in French) written by a Parisian probabilist on the "History of Martingales". It gives a fairly complete account of the numerous senses of the word "martingale", and explains the best modern theories as to why the word means what it does in Probability theory. The article is well written and amusing, as well as being scholarly. It can be found here as a PDF file.
The article found by following this link, by Jarrow and Protter, gives a history of the development of stochastic calculus and its application to mathematical finance. It includes the sad tale of Doeblin, and explains why a Frenchman had a German name.
In order to encourage the use of the Linux operating system, here is a link to an article by James MacKinnon, in which he gives valuable information about what software is appropriate for the various tasks econometricians wish to undertake.
To send me email, click here or write directly to
Russell.Davidson@mcgill.ca.
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URL: https://russell-davidson.research.mcgill.ca/e765/