Most of us have learnt probability theory at a certain stage of time in our lives — be it in school, undergraduate or postgraduate level or on the job. While understanding most concepts in probability may come naturally, there are a few ideas based on the probability theory that appear to be a bit fuzzy on the face of it. Martingales and measures is one such concept that appears to be esoteric, mainly owing to the fact that most literature around this delves into pure mathematics which may not be appealing for everyone.
In this article, I attempt to introduce this powerful and interesting concept of martingales and measures in simple words. I promise to keep the level of math to the minimum, so that this also palatable to a reader who is entirely new to this concept. I will try to explain things from an application maths perspective, because when we relate the underlying math to a certain end application, then things start to clear up considerably! Further, since most of my professional experience till date is in the financial industry, I will use examples from finance to explain this concept further.
So what are martingales really?
By definition, martingale may be defined as a stochastic process whose trajectories do not display any discernible trends. In other words, we can say that martingale process represents pure randomness.
Some basic notation: In probability theory, we come across a quantity called a tuple represented as:
(Ω, I, P)
Where, Ω is defined as a sample space;
I: is an information set (also called Filtration set in some texts)
P: Probability measure (we may view this loosely like a set)
Consider a continuous time stochastic process say G(t). Also consider an information set represented as I(t) where the (t) represents as an item that is indexed to time. Every stochastic process has this indexing to time, because in simple words, stochastic process may be interpreted to understanding the behavior ot a certain random variable across time steps.
The information set I(t) will keep on incrementing over time. For example, consider we are tracking the stock prices during a trading day, the information set will go on augmenting as we get access to more and more information on the stock price throughout the trading day.
If G(t) is included in the information set I(t) at each t > 0, then we can say that the process G(t) is I(t) adapted. i.e. the value of G(t) will be known given an information set I(t)
Applying the idea to a continuous time process
Let the process G(t) be a martingale. We know by the basic definition of martingale, it’s a process for which we cannot comment on what would be its trajectory/movement over the next time step. You will begin to appreciate the relevance of time i.e. (t) term introduced earlier. We relate the process evolution with time indexing. So if G(t) is a martingale, and assuming we are on time step “t” what is the best guess estimate of the value of G(t) over a small step “(t+u)”? We will borrow the idea of expectation operator to relate to understand this:
E[G(t+u)] = ?
By definition of martingale, we can say that the best estimate of the above equation is what we have now at time “t” i.e. G(t). Therefore, we say that
E[G(t+u)] = G(t)
Some may wonder, how exactly does the filtration set I(t) come into play in the above relation? The idea is that a process (in our case G(t)) is a martingale for a specific filtration set and a specific probability measure. We have not spoken about what is the meaning of a probability measure yet, but will do so shortly.
So technically correct representation of the above is
EQ [G(t+u)|I(t)] = G(t)
Where Q is the assumed probability measure, I(t) is the information set as of time “t”. As most of you must have guessed, this is a form of a conditional probability measure.
Why should we study martingales and measures for finance?
For quantitative finance in particular martingale theory provides a very powerful approach to solving certain activities that may be pretty complex to solve otherwise. In quant finance, we extensively use numerical techniques for solving problems, because it’s seldom the case that we have a nice closed form solution for a problem. Therefore, some problems may be attempted via lattice structures, whereas, majority of the other problems may be attempted via partial differential equation (PDE) approach OR martingales approach.
The PDE approach may appear to be more appealing owing to the fact that we can apply rules of calculus and attempt to solve a partial differential equation. For example, take Black Scholes Merton (BSM) PDE. We can leverage ideas of Ito’s and delta-hedging to arrive at the PDE. Subsequently, we can slice and dice the individual terms in the PDE to solve via a finite difference grid. This makes the whole approach very explicit and visible.
In contrast, the martingale theory is a bit more subtle. We need to combine our understanding of calculus with certain ideas from probability theory in order to allow us to solve problems via martingales. It’s this sublime nature of martingales that makes them so elegant. Further, the advantage is martingales lend themselves well for monte carlo simulations, thus increasing their attractiveness; as simulation happens to be a method of choice of a variety of purposes including pricing, interest rates modelling etc.
Equivalent measures and their applications
While the idea of equivalent measures requires some preparation on the probability theory, I will attempt to explain this through a simple example, so that we do not have to worry about the background math.
Most readers may be familiar with the idea of risk-neutral valuation. A fundamental concept that is used for pricing of financial instruments. For example, when we are pricing an option, and we say that USD X is the MTM of the option. This USD X price is technically called as the risk-neutral price of the option. Entire derivatives theory is based on the idea of risk neutrality. However, there are certain specific cases whereby we need to move away from risk neutrality into the real world.
In risk neutral world, all that we care about is the usage of risk free rate for our purposes. However, for applications like interest rate modelling, we need to move away from the assumption of risk free rates, because we then attempt to simulate interest rate paths in the real world. It is this transition between risk neutral ßà real world that make the idea of martingale very convenient. Through martingale theory, we attempt to move between measures that are equivalent to each other in probabilistic terms. Technically also called as the Equivalent Martingale Measure (EMM).
For instance let Q represent a risk neutral measure; and let P be defined as an “equivalent” real world measure. Martingale theory can allow us to switch between these two measures smoothly. Using our example of rates modelling, we can then attempt to figure out the risk premium that is to be adjusted above the risk free rate to aid out simulation exercise. We can roughly say that martingale theory allows us to move things from one world (where they may be difficult to solve) to another yet equivalent world (where they are easier to solve) and then finally return back to the original world they were in. This idea of equivalent measures help us make difficult problems easier to solve, which honestly is the main purpose of mathematics!
In this article, we discussed only the basic concepts of martingales via near zero math. This article was meant to be an introduction to this seemingly complex topic. We have also very briefly discussed two popular measures that one needs to work with in practice. But this article in general gives a convincing idea as to why it’s useful to have martingales as a part of your quant finance arsenal.
Readers who would like to learn this further, are suggested to brush up on some additional math before they delve deeper into this interesting and powerful concept of martingales and measures.