Is measure theory really needed in finance?
Measure theory is an area of mathematics that was created to generalise the theory of integration to more exotic kinds of functions. It has applications in pure mathematics, particularly to the theory of differential equations – see, for example, Sobolev spaces. At some point, finance academics started using the language of measure theory when writing papers and textbooks about financial derivative pricing. However, the field of measure theory isn’t really necessary in finance because in almost all cases the functions that arise in derivative pricing in practice are nice, smooth functions that don’t require considerations from measure theory like sigma algebras. Couching derivative pricing in the language of measure theory simply makes it difficult for non-mathematicians to understand things which they would otherwise be able to understand.
The main concept from measure theory which is useful in derivative pricing is that of a “measure”. However, we only need a greatly simplified concept of it. In finance, a measure is basically just a smooth function which is used to weight or assign probabilities to different outcomes.
Futures convexity corrections
In this section, we discuss the convexity correction which accounts for the difference between forwards and futures prices. Later, we will identify another convexity correction that is required for late or early payment of interest (such as for swaps in arrears).
In this discussion, everything is with respect to the risk neutral measure, which takes into account investor’s risk preferences so that by definition expectations agree with market prices.
Consider modelling the short rate (instantaneous interest rate) \(r\) by a stochastic differential equation with time dependent drift \(\theta(t)\):
\[dr = \theta(t)dt + \sigma dz.\]
The forward rate \(F(t,T,T + \tau)\) between \(T\) and \(T+\tau\), as seen from time \(t\), is some function of the values of \(r(t)\) at each time \(t\). You’ll remember from calculus that when you want to find the rate of change of a function of a function, you use the chain rule, which generates an additional term. In Stochastic calculus, the chain rule is known as Ito’s Lemma, and it actually has three terms – it has an additional second order (or convexity) term that you don’t get in ordinary calculus. So, a function of a stochastic variable obeys its own stochastic differential equation which we can generate using Ito’s lemma. Because of the new third term, this new stochastic differential equation can have a drift term even if the original equation does not. In other words, even if \(\theta(t) = 0\), the forward rate equation can have a drift term, so that it’s expected value at time \(T\) is not the same as it’s value now at time \(t\). The precise equation for the expected future value is derived in this addendum to Hull.
We’ve seen that the forward rate is not a martingale under the risk neutral measure. In other words, it is not equal to its future expected value because the equation describing it has a drift term. By contrast, futures are martingales and are equal to their future expected values. The reason for this is simple. Because Futures are margined daily, their market value is by definition zero. So, since they cost nothing to buy, their expected future value, with respect to the risk neutral measure, must be zero as well. This means that the expected value tomorrow, or any future day, must be equal to today’s value.
This means that the equation derived in Hull for the expected future value of the forward, is also equal to the convexity correction between forwards and futures prices.
Change of numéraire
Changing the numéraire means changing the unit of measure used to value financial instruments. For example, changing to a different currency, or changing to the value of the same currency at some future time after it has appreciated due to the time value of money. We will refer to the wikipedia article on numeraires.
We consider the value of an asset \(S\) in terms of a numéraire \(M\). We assume the existence of a so-called risk neutral measure \(Q\) under which asset prices are martingales, that is the expected values at time \(T\) are the same as their current values:
\[\frac{S(t)}{M(t)} = E_Q \left[ \frac{S(T)}{M(T)} \right]\]
We define a new measure \(Q^N\) at time \(T\) by weighting all probabilities by the factor \(\frac{N(T)}{M(T)}\). Then we have the formula
\[E_{Q^N} \left[ \frac{S(T)}{N(T)} \right] = E_Q \left[\frac{N(T)}{M(T)} \frac{S(T)}{N(T)} \right] / E_Q \left[ \frac{N(T)}{M(T)} \right]\]
To understand this, note that when we switch back to calculating the expectation with respect to \(Q\), we have to add the weighting (or scaling) factor inside the expectation. In addition, because on the LHS everything is in terms of the numeraire \(N\), we have to have the division on the RHS so both sides are in the same numeraire (imagine a currency change). Note also that the expectation in the denominator is simply equal to the same expression evaluated at time \(t\), using the martingale property above.
The forward measure
Now we consider a particular example that we will need shortly. The time value of money means that a dollar today is worth more than a dollar in the future, leading to the use of discounting in pricing financial instruments. We let \(P(t,T)\) be the value at time \(t\) of a zero coupon bond which pays a cashflow of \(1\) at time \(T\) (effectively, this is the discount factor between \(t\) and \(T\)). Let’s choose a change of numeraire given by \(N(t) = P(t,T)\) and \(M(t) = P(t,T+\tau)\). Then we have
\[ \frac{N(t)}{M(t)} = \frac{1}{P(t,T,T+\tau)} = E_Q \left[ \frac{N(T)}{M(T)} \right], \]
where \(P(t,T,T+\tau)\) represents the forward rate as seen from time \(t\), and we have used the martingale property described above.
The new measure associated with this numeraire change is called the \(T + \tau\)-forward measure, and expectations with respect to this measure are denoted by \(E^{T + \tau}\).
Vanilla swap pricing and the forward measure
For a vanilla swap, the rate fixes at the start of each period, but the payment is settled at the end of each period.
Pricing a vanilla swap requires no complex modelling, as the valuation just involves computing the interest payments as if the future interest rates were equal to their current forward values. This is possible because the expected future value of the forward rate (with respect to the forward measure) is equal to its present value. In technical language, the forward rate is a “martingale”.
Consider a swap period starting at time \(T\) and ending at time \(T + \tau\), as seen from the current time \(t\). Let \(F(t,T,T + \tau)\) denote the forward interest rate over this period. The Martingale property means that
\[E^{T + \tau}\{F(T,T,T+\tau|\mathcal{F}_t\}=F(t,T,T+\tau).\]
The “filtration” \(\mathcal{F}_t\) simply indicates that the expectation is as seen from time \(t\), and we neglect it in what follows. The notation \(E^{T + \tau}\) means that the expectation is taken with respect to the (\(T + \tau\))-forward measure.
A clear explanation of this fact can be found in Brigo & Mecurio [1]. Rearranging the definition of a (simply compounded) forward rate slightly, we have
\[F(t,T,T + \tau)P(t,T + \tau) = \frac{1}{Y(T, T+\tau)}(P(t,T) – P(t,T+\tau)),\]
where \(Y\) represents the year fraction.
We now consider taking the expectation of both sides. Because of the discount factor on the LHS which scales all possible forward values by a constant factor, we are taking the expectation of the forward rate with respect to the (\(T + \tau\))-forward measure. On the other hand, the RHS is just the difference between the today-prices of two zero coupon bonds of different maturities. Since the latter are real assets trading in the market, their expected value must be equal to their current value. It follows that the same is true for the left hand side, i.e. the forward rate scaled by the discount factor.
Swap in arrears pricing
Swaps in arrears differ from vanilla swaps in that the both fixing and settlement occur at the start of each period. In this case, the expected future interest rate is no longer equal to the current forward value, necessitating a so-called convexity correction. The value of a swap in arrears is given by
\[V(t) = P(t,T) E^T \left[ F(t,T,T + \tau) \right].\]
For swaps in arrears, the expectation is taken at \(T\) instead of \(T + \tau\), so we can’t use the martingale property from the previous section. Using the change of numéraire formulae in a previous section, we can change to the (\(T + \tau\))-forward measure which introduces a discount factor inside the expectation as follows:
\[V(t) = P(t,T) E^{T+\tau} \left[ \frac{F(t,T,T + \tau)}{P(T,T+\tau)} \right] P(t,T,T+\tau)\]
\[= P(t,T+\tau) E^{T+\tau} \left[ \frac{F(t,T,T + \tau)}{P(T,T+\tau)} \right].\]
Because of the factor of \(1/P(T,T+\tau)\) inside the expectation, we can’t use the martingale property of forward rates here.
Attempting the quantify the magnitude of this correction is known as the “convexity correction”. It’s an amount that gets added to the forward rates before using them to price swaps, and calculating it requires that you first make a choice of stochastic interest rate model.
Why Forward Rate Agreements (FRAs) don’t require a convexity adjustment.
A FRA is similar to a single period swap, but not quite. A key difference is that the rate is fixed in advance, at the start of the accural period, just like a swap in arrears. However, unlike a swap in arrears, FRAs do not require a convexity adjustment, and one naturally wonders why. The answer lies in the second key difference between a FRA and a swap – for a FRA, the floating leg is discounted by the forward floating rate between the start and end dates of the accrual period. This discount factor, as seen from time \(t\), is
\[ P(t,T,T+\tau) = P(t,T+\tau)/P(t,T).\]
The value of a FRA is given by
\[V(t) = P(t,T) E^T \left[ F(t,T,T + \tau) P(T,T+\tau) \right],\]
which differs from a swap in arrears by the discount factor inside the expectation.
Note that if we now change to the (\(T + \tau\))-forward measure, as we did above for swaps in arrears, the new discount factor \(P(T,T+\tau)\) would exactly cancel with the change of measure \(1/P(T,T+\tau)\). Thus, FRAs don’t require a convexity correction.
Girsanov’s theorem
We’ve seen above how forwards are martingales under the forward measure, but have a drift term under the risk neutral measure. So it seems like one could get rid of a drift term by an appropriate measure, or conversely, change measure at the expense of adding a drift term. Girsanov’s theorem is the formalization of this idea.
[1] Brigo & Mecurio, Interest Rate Models – Theory and Practice, Springer Finance 2007