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 for 2020 is here as a PDF file, and here in HTML. The information it contains is also given below.
We will meet virtually, via Zoom, on Tuesdays and Thursday, from 10.00 until 13.00 Montreal time. I know that some people, including me, are not in that time zone, and so, if you are elsewhere, be sure to know what the time difference is. Judging by the first class, though, everyone is aware of this - no flat earthers!
I received word today (June 9) that you may now complete the Mercury course evaluation, at this site. I need a minimum of five (5) responses for the results to be made available.
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
On May 5, we went through Chapter 1 of the first volume of Shreve's books, on the binomial asset pricing model. This let us begin to talk about the principle of no-arbitrage pricing, and see how a derivative security could be priced by a recursive argument.
Then we moved on to the second volume, and started with Chapter 1. We saw formal definitions of sigma-algebras, and probability measures, in particular Lebesgue measure on he [0,1] interval, acting on the Borel sigma-algebra. Random variables were defined as measurable mappings from the outcome space to the real line, and this let us define the induced measure for a random variable. We concluded by defining the density of such an induced distribution. This took us through section 1.1 of the book and a good way through section 1.2.
On May 7, we started with section 1.3, on expectations and integrals, distinguishing between Riemann and Lebesgue integrals. Section 1.4 deals with convergence of integrals and expectations, and states the two main sufficient conditions for convergence of integrals or expectations of an almost surely or almost everywhere convergent sequence, namely the monotone convergence theorem and the dominated convergence theorem. We then started on section 1.5, and looked quickly at the "standard machine" as a technique of proof.
We began on May 12 by going over a proof using the "standard machine". The results prove are not surprising. They just show formally that what seems obvious is in fact true. The next section of the textbook deals with the very important topic of change of measure. The key concept is the Radon-Nikodým derivative. We saw how to change the measure in order to convert an uncentred normal random variable into a centred one.
In Chapter 2, we looked at stochastic processes, filtrations, and how a stochastic process can be adapted to a filtration. Any random variable can be measurable with respect to a sigma-algebra. This led on to a formal definition of independence, of events, of sigma-algebras, and of random variables. We finished with Theorem 2.2.7, which provides a number of equivalent conditions for independence of two random variables.
After a look at moment-generating functions and cumulant-generating functions, we resumed our study of Chapter 2 on May 14. We saw what was meant by an expectation conditional on a sigma-algebra, and how such a thing is defined by two properties: measurability and partial averaging. We proved lots of properties of conditional expectations: taking out what is known, iterated conditioning, and also a result that Shreve calls the Independence Lemma. That chapter concludes by the definitions of martingales and Markov processes.
In Chapter 3, we looked at the symmetric random walk, characterised by independent increments, which leads to the martingale property. By scaling the symmetric random walk, a sequence of stochastic processes can be defined that tends to a continuous-time stochastic process called Brownian motion. The quadratic variation of these processes was defined and calculated.
On May 19, we started by deriving the limiting distribution of the scaled symmetric random walk. This was to provide us with the distribution of Brownian motion. We used a method based on the cumulant-generating function, which gave us the normality that would be guaranteed if we used an argument based on the central-limit theorem. As an aside, the limit was also found for the binomial model. It turned to be a log-normal distribution. Then properties of Brownian motion were developed formally - independent normal increments, expectation zero and variance equal to the length of the interval. This led to a discussion of quadratic variation, after defining first-order variation. Brownian motion is unlike continuously differentiable functions in that it has non-zero quadratic variation. We laid the ground for stochastic calculus by developing notation to express the fact that Brownian motion accumulates quadratic variation at a rate of one per unit time.
Then we saw that Brownian motion is a Markov process. The first-passage time was introduced and its distribution characterised in two ways, one that uses the exponential martigale, the other the reflection principle. All this material is in Chapter 3.
We continued work on the first passage time and also the maximum-to-date variable on May 21. This was more or less the end of Chapter 3.
Chapter 4 starts with the Itô integral. It is first defined for simple integrands, that is piecewise-constant integrands. Various properties of these integrals were developed. Then a sequence of approximating simple integrands was constructed that converges to a general adapted integrand. The properties established for a simple integrand go over easily to a general integrand.
A special case is the integral of Brownian motion with respect to Brownian motion. This example shows how the non-zero quadratic variation of the integrator Brownian motion leads to a result different from what one gets with a differentiable integrator. Study of this and other examples led to the celebrated Itô-Doeblin formula, especially the formula in differential form. Some applications of the formula showed that it correctly reproduces a great many results that were earlier derived more laboriously.
We began on May 26 by looking at two interest-rate models, the Vasicek model and the Cox-Ingersoll-Ross model. This gave us the opportunity to see how to derive properties of processes defined by a stochastic differential equation. Then we embarked on the first attempt to do option pricing in continuous time - the Black-Scholes-Merton model. With the assumption that the price of the underlying asset evolves according to a geometric Brownian motion, we ended up with a second-order partial differential equation for the function that gives the value of a derived security as a function of time and the current price of the underlying asset. This function was derived explicitly for a European call option.
The notion of a forward contract was introduced, and this led to the property called put-call parity, which gave us the function that gives the value of a put option without needing to solve the BSM PDE again.
Chapter 4 was finished off by extending Itô-Doeblin to the case of a vector of Brownian motions, and by looking at Levy's theorem, which shows that a set of necessary conditions for Brownian motion are also sufficient.
In Chapter 5, the first thing was Girsanov's theorem, which allows a change of measure that converts a displaced Brownian motion into ordinary Brownian motion.
Chapter 5 is long, and, in its first part, leads up to the two Fundamental Theorems of Asset Pricing. We started on May 28 with Girsanov's theorem, after which comes the martingale representation theorem, which is a little unsatisfactory, because it states the existence of something without telling you how to construct it. Its unenlightening proof is omitted from Shreve's text. Both of these theorems are given again in a multivariable context, in which we can speak of d-dimensional Brownian motion.
This makes it possible to construct market models with many assets. In this context, a risk-neutral measure is defined in such a way that the discounted prices of all of the assets are martingales, and so therefore are any portfolios constructed with them and the money market. It is shown that models exist with no risk-neutral measure, but in that case the model necesarily contains an arbitrage. The first fundamental theorem states that, contrariwise, the existence of a risk-neutral measure excludes arbitrage, and leads to the risk-neutral pricing formula for derivative securities. We were able to use this in order to get a complete solution of the BSM model, without recourse to the second-order PDE.
The second fundamental theorem deals with uniqueness or non-uniqueness of the risk-neutral measure. It states that any arbitrary derivative security can be hedged if and only if the risk-neutral measure is unique.
On the 2nd of June, we took up the material in Chapter 5 of Shreve after the two fundamental theorems of asset pricing. First, we went back to a model with only one underlying asset, this time assuming that it pays dividends, either continuously or at fixed discrete times. Almost nothing in the theory is changed if we assume that dividend payments are reinvested, but the stock price is no longer a martingale. This leads to some changes to the solution of the BSM model, but these amount simply to replacing the initial value of the asset price by that price suitably discounted.
Next, the consequences of a supposed random interest process for the bond market were studied. This led to a formulation of the forward price of a risky asset at time t as a function of the asset price and the price of a bond maturing at the same time as the forward contract, both at time t.
The nature of a futures contract and the stochastic process called the futures price was studied next. The futures price turns out to be a martingale. This conclusion comes from specifying that, at maturity, the futures price is equal to the asset price, and that, at earlier times, the value of the futures contract is zero. This can prevent the danger of default on a forward contract. The forward-futures spread vanishes with a constant interest rate, but is otherwise proportional to the covariance of the discount process with the asset price.
In Chapter 6, we defined the concept of a stochastic differential equation, and quoted the existence and uniqueness theorem for solutions of such an equation along with an initial condition. This led to the important conclusion that solutions to stochastic differential equations are Markov processes.
On June 4, we embarked on the Feynman-Kac theorem, which effects the link between stochastic differential equations (SDE) and partial differential equations (PDE). The theorem comes in two forms if there is just one SDE, plain and discounted. For both, the principle for deriving a PDE from an SDE is (i) find the martingale, (ii) compute its differential by Itô-Doeblin, (iii) set the dt term equal to zero. We followed this principle for a range of interest-rate models, and got explicit solutions for the Hull-White model and the Cox-Ingersoll-Ross model. We were also able to price call options on a bond.
The multidimensional Feynman-Kac theorem is a straightforward extension, and it lets us consider sets of coupled SDEs. It is important to note that the solutions to these sets are vector Markov processes; the individual components in general are not. The example of this was an Asian option, the payoff of which depends on the whole history of the underlying asset price, not just its value at the maturity of the option. This led to a pair of SDEs, and one PDE for the option value, and another for its discounted value. We were able to derive the delta-hedging rule for this option.
We skipped Chapter 7 on exotic options, and moved on to Chapter 8 on American options. These have the particular property that they may be exercised at any time up to and including maturity. It is not necessary to wait for the expiration of the option before exercise. Study of American options entails study of the concept of a stopping time. We embarked on that study.
Note that after Chapter 8, we will jump ahead to the last chapter in the book, on jump processes.
We managed to cover almost all of Chapter 8 on June 9. We began by showing that the first-passage time of a continuous stochastic process is a stopping time in the sense of the definition of a stopping time. Next came the study of the perpetual American put, a purely imaginary option, but one that allows for many exact solutions. We found that the optimal time to exercise the option was the first-passage time of the underlying asset price to some level below the strike price. It was possible to get an analytic formula for the optimal level. There followed two discussions of the put price so defined, one analytic, the other probabilistic. We ended up with a characterisation in terms of the linear complementarity conditions, and another in terms of a supermartingale and a stopped martingale.
The finite-expiration put is of course more realistic, but it is not possible (as far as we know!) to get analytic expressions for the put value function and the curve that separates the continuation set and the stopping set, although they can be solved for numerically. With them, we can again derive an analytic and a probabilistic characterisation of the put price, with linear complementarity conditions and a supermartingale and stopped martingale.
The American call is much simpler to analyse, since it turns out that the right to exercise early is worthless unless the underlying asset pays dividends. Except for that case, therefore, the American call is equivalent to a European call.
We started on June 11 by finishing Chapter 8, with the section on American call options on an underlying asset that pays dividends. The possibility of early exercise is useless except just before the times when dividends are paid. At such times, it may be better to exercise if the underlying price is high enough.
Chapter 11 is long, and we will not be able to cover all of it. We began with the Poisson process, which is the basic process in terms of which a jump process is defined. This process can be developed by using the exponential distribution in order to define a sequence of consecutive waiting times. The Poisson process counts the number of completed waiting times.
A compound Poisson process jumps whenever an underlying Poisson process jumps, but, instead of jumping up by 1, it jumps by a random amount. These amounts are IID, and are independent of the Poisson process itself. We examined many properties of both the exponential and Poisson distributions, making extensive use of moment-generating functions.
An interesting theorem describes a decomposition of a compound Poisson process into a set of mutually independent Poisson processes. The theorem states among other things that the sum of a set of independent Poisson processes is itself a Poisson process, with intensity equal to the sum of the intensities of the summand processes. A compound process can be decomposed in a similar way.
We began the study of the stochastic calculus where a stochastic integral has an integrator that is a jump process, that is, the sum of a continuous process, in fact an Itô process, and a pure jump process.
The last class took place on June 16. We covered as much of Chapter 11 on jump processes as time permitted. This started with stochastic integrals with an integrator process with jumps. If the integrator is a martingale, then, for the integral to be a martingale as well, it is necessary for the integrand to be left-continuous although both the integrator and the integral are right-continuous.
In order to develop stochastic calculus with jumps, quadratic variation is the starting point. Non-zero quadratic variation can arise both from Brownian motion and also from a pure jump process. But the cross variation of these two sources is zero. The Itô-Doeblin formula that results usually must be expressed in integrated form, as it is not always possible to find the differential of a process with jumps. Sometimes, however, it is possible, and this simplifies things. An example of this is the geometric Poisson process, which turns out to be a martingale. It can be used as the basis of a model of an underlying asset price.
The intensity of a Poisson process can be changed by changing the measure. For a compound Poisson process, the intensities of the individual independent Poisson processes that constitute the compound process can also be changed, and this implies a change in the distribution of the jump sizes.
Working out the value of a European call on an asset with a geometric Poisson price process follows steps very similar to those in the probabilistic derivation of the Black-Scholes-Merton solution. An exact solution can be found, and it can be shown that is satisfies a differential-difference equation. It can be hedged, and the risk-neutral measure is unique. But if the asset price process is driven by a Brownian motion and also a compound Poisson process, there are so many sources of randomness that the risk-neutral measure is no longer unique. To obtain a complete market, in which all derivative securities can be hedged, it is necessary to have as many non-redundant assets as sources of randomness. For instance, price processes with one Brownian motion and a compound Poisson process with two components needs three assets before a unique risk-neutral measure can be found.
Recordings
Click here for the video/audio recording on the class on May 5. I'm sorry that, for some unknown reason, the recording stopped after the break. If all you need is the audio recording (smaller file, shorter download time), click here.
The recording from the May 7 class can be accessed from this link. Audio only is here.
For May 12, here is the link, and, for audio only here.
For May 14, here is the link, and, for audio only here.
For May 19, here is the link, and, for audio only here.
For May 21, here is the link, and, for audio only here.
For May 26, here is the link, and, for audio only here. I must apologise for not restarting the recording after the break. I was distracted by the trouble with the sound.
For May 28, here is the link, and, for audio only here.
For June 2, here is the link, and, for audio only here.
For June 4, here is the link, and, for audio only here.
For June 9, here is the link, and, for audio only here.
For June 11, here is the link, and, for audio only here.
For June 16, here is the link, and, for audio only here.
Notes
Here is the link to the notes on various topics prepared to complement or explain the material in Shreve's book.
Assignments:
This link is to the first assignment. It is due on Thursday May 21.
This link is to the second assignment. It is due on Thursday May 28.
This link is to the third assignment. It is due on Thursday June 4.
This link is to the fourth assignment. It is due on Thursday June 11.
This link is to the fifth assignment. It is due on Thursday June 18.
Final Exam:
The Final Exam is now available. It should be completed and returned to me by Thursday June 25 at the latest.
Other links:
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.
The picture I found of Catherine Doléans-Dade is contained in a set of slides prepared by B. Hajek from Urbana-Champaign. It contains interesting biographical material on this excellent female mathematician, and some background on her work.
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.arts.mcgill.ca/e765/