By GormGeier on May 4th, 2015
Martingale methods are a staple in financial mathematics and it allows us to deal with a much broader set of cash flows than the PDE-approach, which I covered in an earlier post. Here I try as best as possible to give a simple walkthrough of the concepts and ideas behind it.
First of all a martingale is defined as some stochastic process in where the expected future value of the process is equal to its current value. Assume we define some price process as following:
Lets take a conditional expectation for :
which will be a martingale if and only if , i.e. the process has no deterministic drift.
The first fundamental theorem of asset pricing now states that if we can find an equivalent martingale measure, there exists no arbitrage.
This has a simple explanation, but first we require some definitions.
First, arbitrage can be defined in two ways:
- we can construct a portfolio of zero value today, which has a zero probability of a negative future value and a positive probability of positive future value
- we can construct a portfolio with positive value today and zero probability of a future negative value.
In our case the first situation is what we really care about, but either way we would make a riskless profit.
Second, two probability measures are equivalent if they share the same sets of zero-probability events. Intuitively, this is related to Girsanov’s theorem, which states that we can move from one measure to another equivalent measure by changing the drift of the process only, i.e.
where and are Brownian motions under probability measures and respectively. Changing measure basically implies changing the mean of the Brownian motion.
The transition is also visualized by the Radon-Nikodym derivative, which for the case outlined above is a random variable with dynamics defined by:
This formally defines the transition between the and measure. Furthermore, we can now find the expectation of a random variable, , by using this variable, such that:
i.e. the expectation of the variable under the -measure is equal to the expectation of the product between and under the normal, -measure. Equally, therefore the probability of any set occurring is given by:
where is the indicator function for the set
In the case two measures do not have the same sets of zero-probability events this relationship cannot hold. Per definition, the sum of all probabilities for all possible outcomes must sum to one. If the set of outcomes with positive probabilities differ between two measures, the above function will not hold, simply because there are outcomes under one of the measures that are not accounted for. So, by extension of the equation above, denote the whole outcome space under as and under . Furthermore, define the indicator function for all potential outcomes for as and assume that the smaller of the two sets is (this assumption can be reversed and by reversing the arguments below we would arrive at the same conclusion). For the two measures to be equivalent, we must have that:
Or simply the two sets and are the same. In the case that the sets of zero-probability events are different under the two measures this will obviously not hold since the probabilities will not sum to one.
And here comes the point:
The reasoning above implies that the set of arbitrage portfolios must also be invariant when we change between two equivalent measures, since if all assets under some equivalent measure are martingales no portfolios allow arbitrage. Equally, arbitrage cannot exist under any other, equivalent measure either, which is also why the price of any derivative is independent of which equivalent measure we use to calculate it – it is guaranteed arbitrage free. Thus, if we can find an EMM, arbitrage is not possible since all arbitrage portfolios must be in the zero-probability set.
We can now price any derivative by finding an equivalent martingale measure under which all asset prices are martingales, since any derivatives written on these will also be martingales. And we also know that the price we find using the method is correct, since the price cannot be arbitraged under either the real-world measure or any other equivalent measure.