Stochastic Rates Models

In a previous post, I introduced the Black-Scholes model, in which the price of an underlying stock is modeled with a stochastic variable that changes unpredictably with time. I’ve also discussed the basic model-independent rates products whose value can be determined at the present time exactly. However, to progress further with interest rate derivatives, we’re going to need to model interest rates more carefully. We’ve assumed rates are deterministic so far, but of course this isn’t true – just like stocks, they change with time in an uncertain manner, so we need to allow them to become stochastic as well.

One way of doing this is by analogy with the BS case, by allowing the short rate (which is the instantaneous risk-neutral interest rate r(t)) to become stochastic as well, and then integrating it over the required period of time to calculate forward rates.

A very basic example of this is the Vasicek Model. In this model the short rate is defined to be stochastic, with behaviour governed by the following SDE

    \[dr = k\ [\ \theta - r(t)\ ]\ dt\ +\ \sigma\ dW\]

where k\theta and \sigma are constants and dW is the standard Wiener increment as described before. This is marginally more complicated than the BS model, but still belongs to the small family of SDEs that are analytically tractable. Unlike stock prices, we expect rates to be mean-reverting – stock price variance grows with time, but we expect the distribution of rates to be confined to a fairly narrow range by comparison. The term in square brackets above achieves this, since if r(t) is greater than \theta then it will be negative and cause the rate to be pulled down, while if it is below \theta the term will be positive and push the rate up towards \theta.

Solving the equation requires a trick, which is instead of thinking about the rate alone to think about the quantity d\big(\ e^{kt}\cdot r(t)\ \big). This is equal to e^{kt}\cdot dr(t)\ + \ ke^{kt} dt \cdot r(t), and substituting in the incremental rate term from the original equation we have

    \[d\big(\ e^{kt}\cdot r(t)\ \big) = e^{kt} \Big(k\ [ \theta - r(t) ] dt + \sigma dW \Big) + \ e^{kt} dt \cdot r(t)\]

    \[d\big(\ e^{kt}\cdot r(t)\ \big) = e^{kt} \Big( k\cdot \theta\cdot dt + \sigma\cdot dW \Big)\]

note that the term in r(t) has been cancelled out, and the remaining terms can be integrated directly from a starting time s to a finishing time T

    \[\Big[\ e^{kt}\cdot r(t)\ \Big]_{t=s}^{t=T} = \int_{t=s}^{t=T} k\ \theta\ e^{kt}\cdot dt + \int_{t=s}^{t=T} e^{kt} \sigma\cdot dW\]

    \[e^{kT}\cdot r(T)\ - e^{ks}\cdot r(s)\ = \Big[\ \theta\ e^{kt}\ \Big]_{t=s}^{t=T} + \sigma \int_{t=s}^{t=T} e^{kt}\cdot dW\]

    \[r(T)\ = e^{-k(T-s)}\cdot r(s)\ + \theta\Big(\ 1 - e^{-k(T-s)}\ \Big) + \sigma \int_{s}^{T} e^{-k(T-t)}\cdot dW(t)\]

where r(s) is the initial rate. This is simply a gaussian distribution with mean and variance given by

    \[\mathbb{E} \big[\ r(T)\ |\ {\cal F}_{\rm s}\ \big] = e^{-k(T-s)}\cdot r(s)\ +\ \theta\Big(\ 1 - e^{-k(T-s)}\ \Big)\]

    \[\mathbb{V} \big[\ r(T)\ |\ {\cal F}_{\rm s}\ \big] = {\sigma^2 \over 2k} \Big(\ 1 - e^{-2k(T-s)}\ \Big)\]

Where the variance is calculated using the “Ito isometry

    \[\mathbb{V} \big[\ r(T)\ |\ {\cal F}_{\rm s}\ \big] = \mathbb{E} \big[\ \big(r(T)\big)^2\ |\ {\cal F}_{\rm s}\ \big] - \big(\ \mathbb{E} \big[\ r(T)\ |\ {\cal F}_{\rm s}\ \big]\ \big)^2\]

    \[= \Big( \sigma \int_s^T e^{-k(T-t)} dW(t) \Big)^2\]

    \[=\sigma^2 \int_s^T \Big( e^{-k(T-t)}\Big)^2 dt\]

    \[= {\sigma^2 \over 2k} \Big(\ 1 - e^{-2k(T-s)}\ \Big)\]

as stated above. Note that this allows the possibility of rates going negative, which is generally considered to be a weakness of the model, but the chance is usually rather small.




As we know the distribution of the short rate, we can calculate some other relevant quantities. Of primary importance are the Zero Coupon Bonds, which are required for calculation of forward interest rates. A ZCB is a derivative that pays $1 at a future time t, and we can price this using the Risk-Neutral Valuation technique.

According to the fundamental theorem of asset pricing, the current price of a derivative divided by our choice of numeraire must be equal to it’s future expected price at any time divided by the value of the numeraire at that time, with the expectation taken in the probability measure corresponding to the choice of numeraire.

In the risk-neutral measure, the numeraire is just a unit of currency, initially worth $1 but continually re-invested at the instantaneous short rate, so that its price at time t is $1 \cdot e^{\int_0^t r(t') dt}. Now, the price of the ZCB is given by

    \[{Z(0,t) \over \$1} = {\mathbb E}^{RN}\Big[\ {Z(s,t)\over \$1 \cdot e^{\int_{0}^s r(t') dt' }}\ |\ {\cal F}_0\ \Big]\]

    \[Z(0,t) = {\mathbb E}\Big[\ {Z(s,t)\cdot e^{-\int_{0}^s r(t') dt' }}\ |\ {\cal F}_0\ \Big]\]

Although the RHS is true at any time, we only know the value of the ZCB exactly at a single time at the moment – the expiry date t, at which it is worth exactly $1. Plugging in these values we have

    \[Z(0,t) = \$1 \cdot {\mathbb E}\Big[\ e^{-\int_{0}^t r(t') dt' }\ |\ {\cal F}_0\ \Big]\]

So the ZCB is given by the expectation of the integral of the rate over a period of time. Since the rate is itself gaussian, and an integral is the limit of a sum, it’s not surprising that this quantity is also gaussian (it’s effectively the sum of many correlated gaussians, which is also gaussian, as discussed before), but it’s rather tricky to calculate, I’ve included the derivation at the bottom of the post to same space here. The mean and variance are given by

    \[\mathbb{E} \big[\int_s^T r(t)\ dt \ |\ {\cal F}_{\rm s}\ \big] = \big(r(s) - \theta \big) \cdot B(s,T)\ +\ \theta\cdot\big( T- s \big)\]

    \[\mathbb{V} \big[\int^T_s r(t)\ dt \ |\ {\cal F}_{\rm s}\ \big] = {\sigma^2 \over k^2} \Big( (T-s) -B(s,T) - {k\over 2} B(s,T)^2 \Big)\]

where

    \[B(s,T) = {1\over k}\big( 1 - e^{-k(T-s)}\big)\]

The ZCB is given by the expectation of the exponential of a gaussian variable – and we’ve seen on several occasions that

    \[\mathbb{E} \big[\ \exp{(x)}\ |\ x\sim {\mathbb N}(\mu, \sigma^2)\ \big] = \exp{ \Big(\mu + {1\over 2}\sigma^2 \Big) }\]

So the ZCB prices are

    \[Z(s,T) = A(s,T)\ e^{-B(s,T)\ r(s)}\]

with B(s,T) as defined above and

    \[A(s,T) = \exp{ \Big[ \Big( \theta - {\sigma^2 \over 2 k^2}\Big)\Big( B(s,T) - T + s \Big) - {\sigma^2 \over 4 k^2} B(s,T)^2 \Big]}\]

and as these expressions only depend on the rates at the initial time, we can calculate ZCB bond prices and hence forward rates for any future expiry dates at any given time if we have the current instantaneous rate.

Although we can calculate a discount curve for a given set of parameters, the Vasicek model can’t calibrate to an initial ZCB curve taken from the market, which is a serious disadvantage. There are more advanced generalisations which can, and I’ll discuss some soon, but they will use all of the same tricks and algebra that I’ve covered here.

I’ve written enough for one day here – in later posts I’ll discuss changing to the t-forward measure, in which the ZCB Z(0,t) forms the numeraire instead of a unit of currency, which simplifies many calculations, and I’ll use it to price caplets under stochastic rates, and show that these are equivalent to european options on ZCBs.

An alternate approach to the short-rate model approach discussed today which is very popular these days is the Libor Market Model (LMM) approach, in which instead of simulating the short rate and calculating the required forwards, the different forwards required are instead computed directly and in tandem – I’ll look further at this approach in another post.

 

Here is the calculation of the distribution of the integral of the instantaneous rate over the period s to T:

    \[r(t)\ = e^{-k(t-s)}\cdot r(s)\ + \theta\Big(\ 1 - e^{-k(t-s)}\ \Big) + \sigma \int_{s}^{t} e^{-k(t-u)}\cdot dW(u)\]

    \[\int_s^T r(t)\ dt = \int_s^T \Big[\ e^{-k(t-s)} r(s)\ + \theta\Big(\ 1 - e^{-k(t-s)}\ \Big) + \sigma \int_{s}^{t} e^{-k(t-u)} dW(u)\Big] dt\]

and splitting this into the terms that contribute to the expectation and the variance we have

    \[\mathbb{E} \big[\int_s^T r(t)\ dt \ |\ {\cal F}_{\rm s}\ \big] = \int_s^T \Big[\ e^{-k(t-s)} r(s)\ + \theta\Big(\ 1 - e^{-k(t-s)}\ \Big)\Big] dt\]

    \[= r(s) \int_s^T e^{-k(t-s)} \ dt + \theta \int_s^T \Big(\ 1 - e^{-k(t-s)}\ \Big) dt\]

    \[= r(s) \Big[ {1 \over -k}\ e^{-k(t-s)} \Big]^T_s + \theta \Big[ t - {1 \over -k}\ e^{-k(t-s)}\ \Big]^T_s\]

    \[= \big(r(s) - \theta \big)B(s,T)\ +\ \theta\cdot\big( T- s \big)\]

to calculate the variance, we first need to deal with the following term

    \[\int_s^T \Big[\ \sigma \int_{s}^{t} e^{-k(t-u)} dW(u)\Big] dt\]

we use stochastic integration by parts

    \[\sigma \int_s^T \int_{s}^{t} e^{-k(t-u)} dW(u)\ dt = \int_s^T e^{-kt}\big(\int_{s}^{t} e^{ku} dW(u)\big)\ dt\]

    \[= \sigma\int_s^T \big(\int_{s}^{t} e^{ku} dW(u)\big)\ d_t \Big( \int^t_s e^{-kv}dv \Big)\]

    \[= \sigma\Big[ \int_{s}^{t} e^{ku} dW(u)\cdot \int^t_s e^{-kv}dv\ \Big]^T_s - \sigma\int_s^T dW(t)\Big( e^{kt} \int^t_s e^{-kv}dv\ \Big)\]

    \[= \sigma\int_{s}^{T} e^{ku} dW(u)\cdot \int^T_s e^{-kv}dv - \sigma\int_s^T dW(t) e^{kt} \Big( \int^T_s e^{-kv}dv - \int^T_t e^{-kv}dv \Big)\]

    \[= \sigma\int_s^T e^{kt} \Big(\int^T_t e^{-kv}dv\Big) dW(t)\]

    \[= {\sigma \over k}\int_s^T \big(1 - e^{-k(T-t)}\big)\ dW(t)\]

and we’re now in a position to try and find the variance of the integral

    \[\mathbb{V} \big[\int^T_s r(t)\ dt \ |\ {\cal F}_{\rm s}\ \big] = {\mathbb E}\Big[\ \Big({\sigma \over k}\int_s^T \big(1 - e^{-k(T-t)}\big)\ dW(t)\Big)^2\ \Big]\]

    \[= {\mathbb E}\Big[\ {\sigma^2 \over k^2}\int_s^T \big(1 - e^{-k(T-t)}\big)^2\ dt\ \Big]\]

where Ito’s isometry has been used again, and several more lines of routine algebra leads to the result

    \[= {\sigma^2 \over k^2} \Big( (T-s) -B(s,T) - {k\over 2} B(s,T)^2 \Big)\]

Bootstrapping the Discount Curve from Swap Rates

Today’s post will be a short one about calculation of discount curves from swap rates. I’ve discussed both swaps and discount curves in previous posts, you should read those before this one or it might not make much sense!

Although bonds can be used to calculate discount bond prices, typically swaps are the most liquid products on the market and will go to the longest expiry times (often 80+ years for major currencies), so these are used to calculate many of the points on the discount curve [and often both of these can be done simultaneously to give better reliability].

In the previous post on swaps, I calculated the swap rate X that makes swaps zero-valued at the current time

    \[X = {Z(0,t_0) - Z(0,t_N) \over \sum_{n=0}^{N-1} \tau \cdot Z(0,t_{n+1})}\]

where the t‘s here represent the fixing dates of the swap (although payment is made at the beginning of the following period, so the n‘th period is received at t_{n+1}.

Consider the sequence of times \{t_0 = 0, t_1, t_2, \dots , t_n\} for which a sequence of swaps are quoted on the markets, with swap rates X_i for the swap running from t_0 up to t_i. We can back out the discount factor at each time as follows:

    \[X_0 = {Z(0,0) - Z(0,t_1) \over \tau \cdot Z(0,t_1)}\]

    \[\Rightarrow Z(0,t_1) = {1 \over 1 + \tau X_0 }\]

    \[X_1 = {Z(0,0) - Z(0,t_2) \over \tau \big(Z(0,t_1) + Z(0,t_2)\big) }\]

    \[\Rightarrow Z(0,t_2) = {1 - \tau \cdot X_1 \cdot Z(0,t_1) \over 1 + \tau X_1}\]

and we can see from this the general procedure, calculating another ZCB from each successive swap rate using the expression

    \[Z(0,t_i) = {1 - \sum_{i=1}^{i-1} \big(\tau \cdot X_i \cdot Z(0,t_i) \big) \over 1 + \tau X_{i-1}}\]

These swaps and ZCBs are called co-initial because they both started at the same time t_0.

Now imagine that instead the swaps X_i have the first fixing at time t_i and their final fixing at time t_n for 0 \leq i < n – such swaps are called co-terminal swaps as they start at different times but finish at the same one. Once again we can calculate the discount factors up to a constant factor, this time by working backwards:

    \[X_{n-1} = {Z(0,t_{n-1}) - Z(0,t_n) \over \tau \cdot Z(0,t_n)}\]

    \[\Rightarrow Z(0,t_{n-1}) = Z(0,t_n)\cdot\big( { 1 + \tau X_{n-1} }\big)\]

    \[X_{n-2} = {Z(0,t_{n-2}) - Z(0,t_n) \over \tau \big(Z(0,t_{n-1}) + Z(0,t_n)\big) }\]

    \[\Rightarrow Z(0,t_{n-2}) = Z(0,t_{n})\big( 1 + \tau X_{n-2} ( 1 + \tau X_{n-1}) \big)\]

and so on, the dcfs can be backed out.

To specify the exact values of co-terminal swaps, we need to know at least one dcf exactly. In general the co-initial case will also require this – I implicitly assumed that they started fixing at t=0 where we know Z(0,0) = 1, but for general co-initial swaps we would also have this issue.