The whole white model is a short rate model, and we're going to discuss partly what that is to begin with, but then also look at the details of this modeling framework.

So let's get started.

First of all, we want to describe what a short rate is and an introduction to short rate models in general.

We can think of short rates as the continuously compounded and annualized interest rate at which an entity can borrow money for an infinitesimally short period of time.

Now that's a bit abstract, so we have that illustrated here by means of an equation.

We have some price of a risk-free asset at the time t plus delta t, and that can be expressed as the price at the initial time point, Pt, times this factor here.

And this is an exponential of the short rate, Rt, times the time interval delta t.

So the short rate tells us how much this value gain is going to be over some short time interval if we invest in a risk-free asset.

And it is what's called continuously compounded, since these interest rate payments happen all the time.

It is not like in some other models where you have interest rate payments at some sort of frequency, but this compounds continuously.

And it is annualized in the sense that we need to express our interest rate in terms of some time interval.

We would say you get 10% per year, for example, or 10% per week, or something like that.

It's annualized, so we always talk about years when we talk about these interest rates.

So a short rate model then describes the future evolution of these short rates by generating a term structure.

So the short rates would vary over time.

You can see that Rt here has a time index to it, so the rates are not constant in the market.

And the short rate models describe how this Rt varies over time.

So with that, let's focus on the whole wide model now.

This model implies a mean reverting and normally distributed assumption on the interest rates.

And this gives us lognormal bond prices as a consequence.

We can see the stochastic differential equation that summarizes the whole wide model in this slide.

And we can see that Drt, so the difference in the rate, is given by this drift term here plus some Brownian motion.

And the drift term has two components.

First we have a theta of t here, and then we have a constant alpha times the rate itself.

Now this theta is, as you can see, a function of time.

And this is chosen explicitly so that the model fits with the term structure that is being observed in the market right now.

And that way you can calibrate this model to be accurate to the kind of term structures that are existing currently.

And then what the model gives you is a way to extrapolate, based on the current term structure, what they are going to be in the future.

The second part of this drift term is alpha Rt.

And alpha is a mean reversion parameter.

So it tells us how fast we are going to revert to the mean value of this process.

It is typically left as a user input and can, for example, be estimated from historical data.

The last part of this equation is sigma that you can see here, which is the volatility parameter.

And it is determined by a calibration to a set of caplets and swaptions that are tradable in the market.

So that is also something that would be derived from market data.

And, as we said in the beginning, theta is something that is being explicitly chosen so that this model matches the term structures that are existing in the market right now.

And that gives us essentially this expression, can be derived for what theta should be.

And so plugging this into theta would give you essentially a model that fits the current term structure.

And it uses the instantaneous forward rate at time 0 for the maturity t here as a component of this value.

So that is where the market values themselves actually make it into the model.

So that is fair enough.

We can see here an example of a realized process of the Holowite model.

So we can see here the short rate, which is what the model ultimately predicts over time.

And we can see a time in years on the x-axis here.

And as we extrapolate, something happens to this rate and it follows some sort of behavior that is stochastic in nature.

We can see the parameters that have been used for this evaluation here.

And this particular trajectory that we are analyzing now is going to be used for some of the comparisons coming up in the preceding slides.

Nevertheless, what you need in terms of parameterization are the parameters alpha and sigma that we discussed before.

You also need how long you want to predict.

So in this case, big T is 2 and the number of points that you want to simulate as well.

In addition to that, you need a yield curve or some sort of future rates or forward rates to actually plug into the model to calibrate the theta.

And this is based on a zero or a zero yield curve.

So that is an introduction to the Holowite model.

And if we look at some of the mathematical properties of the stochastic differential equation that we saw in the last slide, we can note the following.

So firstly, this can be integrated.

It can be integrated and solved for, which means that R of T can be explicitly derived to be the following value here.

So this is R of T expressed in terms of an earlier time s.

And this time s doesn't necessarily have to be, I mean, we can pick this to be zero if we want.

But also if we have another time, sort of halfway through, we want to extrapolate based on that, we can do so as well.

And we introduced a new term beta in this expression.

And that is explained here what that expression is.

And with this derivation, which we won't go over in detail, you essentially arrive at a normal distribution.

So that's something we commented on in the last slide, that the Holowite framework gives normal distributions for the short rates.

And we can see the mean and variance of this normal distribution outlined here.

Now what this means is if we're standing at the starting point, let's say, and we want to estimate what the short rate is going to be in one year, that's going to be normally distributed across or around some mean.

We don't have the mean plotted here, but let's say the mean is up here, and then you would have some variance outside of that.

And so you could estimate a given point in this yield curve or in this short rate curve, if you wanted to.

In order to make this a bit more tangible, we're going to assume that theta is a constant.

So I said before that you would calibrate theta based on market parameters, and that would ultimately be a function of time.

But to make some of these properties a bit more analytical, we're going to set it to be constant for a second.

We can see that if it's constant, the expression for r of t becomes a bit more simple.

It becomes the following.

And just like before, this would be normally distributed.

And if we compare or if we now generate the expected value of the rate at a given time, and also the variance of this, we would get these two expressions.

Now these are a bit more approachable than the ones we saw on the last slide here.

Also, I made the change that rather than extrapolating from another time s, I simply set them to be zero in this example.

So we can see the variance predicated on, well, the filtration at zero.

If you don't know what that is, never mind, but nevertheless, we don't extrapolate based on another time point.

We just take the starting point as our baseline.

And we can see these two analytical expressions here, and I've plotted the corresponding curves in these two figures.

So if we begin by analyzing the expected value, you can see that down here, essentially, is the starting value.

And unsurprisingly, as we let t approach zero, r of t converges to r of zero.

I mean, that's the starting point value we use for this model, so we would expect that to converge.

In this case, r of zero is a small value here, so that's why we're close to zero.

And as we then let t tend to infinity, if we allow theta to be constant or set it to be constant, what's going to happen with the convergence is that the expected value here is going to approach sigma over alpha.

That's going to be the long-term mean of this process.

And now depending on what these two parameters are, so depending on how volatile your model is calibrated to be, and then also depending on the mean inversion parameter here, you're going to get some sort of stable equilibrium for the mean of this process.

Now, if we look at the variance, there are two points that I want to raise.

First of all, we have a limit similar to what we have for the expectation value, and that is in this case sigma squared over two alpha.

What this means is that as t goes to infinity, so as we look infinitely far into the future, we don't get more and more uncertainty beyond a certain point.

So let's say we're comparing, I mean, 25 years here to let's say 50 years in the future.

There would not be a significant difference in how much uncertainty we have in the rate over those time periods.

So after a while, due to the convergence behavior of this model, the rates don't just explode in variability.

So that's a good thing.

We can see also that in the beginning here, we have some form of slope of this curve in variance.

So as we're very close to zero, the variance is zero of the rate.

We can sort of see that intuitively as well, because if we are at time zero, we know exactly what the rate is going to be because we can measure it, and it's directly implied from the market, so we don't have any uncertainty, but uncertainty grows with time.

And how fast it grows with time is given by the tangent here, and we can see that if we go through time at the starting point, then that will come out to be sigma squared.

So that's how fast the uncertainty grows in the beginning.

Okay, so now we looked at some analytical properties of this.

Let's actually see what term structures we get as a consequence.

So we have a graph here of different term structures, and by taking the weighted average of interest rates that prevailed over any one period, we can obtain the effective interest rate for that specific period.

And if we plot these over time, that's how we obtain a yield curve.

So we can see a number of different yield curves here, and they look a bit different depending on when we start to analyze these, because they are ultimately dependent on the short rate, and because the short rate varies over time, so will these yield curves.

And with the trajectory of the whole white model that we saw in the beginning, these are essentially the different yield curves you would get for different points in time.

So we have the initial yield curve here, then we have the yield curve after 6 months, and we have the yield curve after 12 months, and so on.

And depending on which instantiation, which time point you start looking at the yield curve, it's going to look slightly different.

So that's the first point, I would say.

And we can see that there is some variability between the curves as well, which seems to be at around 10% of the value or so.

And one more thing to notice, as the time to maturity here grows very large, these yield curves tend to converge.

And this is expected if we think about it, because we have mean reverting behavior exhibited by the whole white model, and so we would expect these rates to converge over long time periods to essentially the same value.

And as such, the yield of a zero-coupon bond in that case would come out to be the same value as well.

And it is the yield that we are plotting on the y-axis here.

Okay, so I spoke about the variance between these curves, and let's look at the actual expression for this variance as well.

So we can see here the volatility term structure, and that is given by the expression here.

Now there is one thing that is very noticeable about this, and it is that the graphs that we are plotting here, these plots are essentially of this volatility term structure for the instantiations of the model that we discussed before.

We can see that these are the same, regardless of what time period we are considering, or rather regardless of what starting point we are considering.

And the reason for that is that this expression here of the volatility term structure only depends on big T minus small t.

It doesn't depend on small t explicitly, and so there is no dependence on when we actually start to analyze this.

And so we could see that the volatilities of this term structure sort of narrow in on itself, as explained by the graph that we can see here, where the spot rate volatility goes to, well, becomes smaller over time.

Okay, we have some limits here as well for the volatility term structure, and we can see that as we get to the starting point, so as big T minus small t goes to zero, this volatility term structure goes to sigma.

And the sigma value that we have in these parameters is 0.01, and you can see also that the spot rate volatility here goes to 0.01 in this limit.

And as we let t go to infinity, the expression for this goes to zero, and that's also observed by this sort of tapering off or exponential decay of the volatility term structure.

Okay, cool.

So then I also wanted to speak quickly about the bond pricing under the whole-white model, specifically for a zero-coupon bond.

So after a lot of analytical calculations, you can arrive at the following price for a zero-coupon bond under the whole-white model.

And we're not going to go over the calculations, but what I want to highlight is that the distribution of this is going to be log-normal.

So the rates in the whole-white model are normal, and as a consequence, the bond prices are going to be log-normal.

And we can see in the graph down here, the zero price for a zero-coupon bond with nominal $1 over, well, as a function of the time to maturity.

So naturally, as the time to maturity increases, the prices of these are going to decrease, because we're getting paid $1 at some point in the future, and the longer away that is, the more we're going to discount it.

And we can see the different instantiations here that we saw before as well, but now reflected in terms of their effect on the zero price for a nominal of $1.

And we have some variability between these, not too much though.

So one thing that's important to note is if you calibrate a model like this to, well, market parameters, the level of sophistication that you need is quite dependent on the application.

For a lot of applications, a quite simplistic model might do, because ultimately, the variance in terms of bond prices is not going to be very big under normal market conditions.

Nevertheless, if you need that extra granularity, a sophisticated model like the whole-white model might be, well, appropriate.

Okay, so we are able to price bonds as well with this framework, and that's sort of a key takeaway.

And the last thing I want to note is the tendency for this model to give negative interest rates, because we said before that the rates are going to be normally distributed, and if you have something that's normally distributed, you would know that you can also get negative values from that distribution.