We will now see how the Arbitrage Pricing Theorem yields a bond pricing formula in terms of the short rates. Indeed, the earliest interest rate models which date back to 1980s were short rate models. The most prominent among the two are the Vasicek model and the Cox–Ingersoll–Ross model. You know both are still very much in use. They both come with their strengths and their limitations. Limitations are that these models produce forward curves which are low parametric, and we have seen that this posses problems when fitting to a given term structure. We remedy this by adding a time-inhomogeneous term and that leads us to the hull-white extensions. We now focus on the risk neutral measure Q and assume, given an adapted short rate r. The Arbitrage Pricing Theorem implies that the discounted bond prices follow martingales and Q. And that has to hold for any maturity, capital T. Using the fact that the terminal value of the T bond is 1, and using the defining properties of a martingale being the conditional expectation of its terminal value, this leads us to this bond pricing formula. So that the T bond price is the conditional expectation under the risk neutral measure of the exponential of the time integrated short rate process. This is not very explicit and we ask what are the dynamics of the T bonds price processes that we get? While this will depend on the model for the short rate process, under the risk neutral measure Q and if on top of it we're interested in the dynamics of P, and the real world measure P, we also would need to specify the mark price of risk lambda, which brings us back from Q to P. The first model we consider is the Vasitek Model. The Q dynamics of the short rate is as shown here. This model has four parameters. Kappa is the mean reversion speed, theta is the mean reversion level and sigma is the volatility. The forth parameter, don't forget, is the initial value of the short rate which is not shown in the differention notation. As we will see on the following slides, this is a Gaussian process that mean reverse to the level theta. The figure on the right hand side shows a simulated trajectory for these parameters as shown here. The means reversion level is 8 %. The process starts at the value 6 % and we see that the mean reversion forces it to tend to the mean reversion level of 8%. This trend is super posed by the martingale part that is larger, the larger the sigma parameters is. We can solve the Vasicek differential equation shown on the previous slide for the short rate process r(t) using the variation of constants formula. It shows that the solution can be written explicitly in integral form as shown here . Again you may be familiar with this fact from real calculus for this part of it. Overall, you can verify the validity of this expression by just applying Ito formula. This expression also shows that r(t) is a Gaussian process with mean function equal to the deterministic part of it and the covariance function as shown here. When t tends to infinity, we see that the mean tends to theta and the variance when t1 is equal t2, is equal to t, tends to sigma square the value by 2 kappa. In deed, the limiting distribution of the short rates is normal with mean theta and variance sigma squared divided by two kappa. Now what about bond prices in the Vasicek Model? We proceed to calculating bond prices in two steps. The first step is to show that the time-integrated short rate process is a normal random variable and we determine its characteristics. We proceed by plugging in the solution for r(s) that we saw on the previous slide. And it gives us the expression here on the left hand side in black. We next manipulate these expressions in two step. First is, we change the order of integration as shown here. we apply that to both the drift term and the stochastic integral term. We then do for both parts as shown here in the first part multiply with e to the minus kappa t and multiply with e to the kappa t again. We then realize that this integral is equal to B cap T minus t where the function B is defined as integral of e to the minus kappa s. We further realize that e to the minus kappa t, r(0) is the first part of the expression for the short rate r(t) as shown on the previous slide. We get the other two parts from the drift integral and from the stochastic integral so that we can summarize and get this first to mend on the right hand side here. We also get these two terms as shown here. Using the dependence of the increments of the Brownian motion, we realize that the stochastic integral is independent of f(t) and it is normal with mean 0 and variance given by the integral of the square of the integrant with respect to d(u). The first 2 terms are measurable with respect to f(t) because the short rate process is adopted. These observations imply that the time-integrated short rate process, conditional on F(t) is normal distributed with mean given here and variance given here. The bond prize, remember, is the conditional expectation of the exponential of the time integrated short rate process with a minus sign We have just seen that this exponent is normal distributed. We know it's mean function and we know it's variance, so we can use the moment-generating function for the normal distribution plugging the ingredients and arrive at the expression as shown on the right hand side here where the function A is defined, here. Summarizing the Vasicek short rate model yields exponential afﬁne bond prices in the prevailing short rate r(t) as shown here. The functions A and B can be shown to solve the two coupled ordinary differential equations which are quadratic. This is why they are called Riccati equations along with initial conditions 0 and 0, for A and B. The next short rate model we consider is the Cox-Ingersell- Ross Model. In short, CIR Model. The Q dynamics of the short rate look similar to the dynamics that we have for the Vasicek model except that here, the volatility of the short rate process depends on the square root of the short rate itself. Again we have the parameters, theta, the mean diversion level, kappa, the mean diversion speed and sigma, the volatility. For technical reasons, we have to assume that kappa times theta is non-negative. In contrast to the Vasicek model, we can not write down the solution to this stochastic differentiation equation. but it is a mathematical fact that there exists a unique no negative solution for any no negative initial values. We illustrate a similarity trajectory of the CIR short rate process for the same parameter theta, kappa and sigma as for the Vasicek model. Again we start at the level of 6% Again we have mean reversion but this time, the volatility is smaller because we use the same sigma which has to be multiplied with the square root of the short rate to give the total volatility contribution. These parameters of course, have only illustrative character. They would have to be calibrated to specific given market data conditions. But we cannot show it from first principles as we did in the Vasicek model, we claim that the CIR short rate model also yields exponential afﬁne bond prices in the prevailing short rate as shown here. The function A and B solve again, a (inaudible) system of differential equations which are quadratic. So Riccati equations, along with initial conditions 0 and 0 as was the case for the Vasicek short rate model, the solutions A and B for this Riccati equations are given in close form. They are shown on the slide here. We now prove the claim that the CIR model yields exponential afﬁne bond price and we use stochastic calculus. The claim is written here in black. We multiply both sides of this claimed equation by the red factor. This way, we get rid of the small t dependence of the argument in the conditional expectation on the left hand side. We call this argument M(T). Just looking at the left hand side, we realize we have a martingale. We label the expression on the right hand side by M(t). By Ito's formula, this can be seen to be an Ito process. We also see because of the initial conditions A and B being 0, that M(T), in deed coincides with the argument on the left hand side. Hence, in order to prove equality between these two sides, it is now enough to prove that M(t) is a martingale. We do this by applying integration by parts, and Ito formula, to M. A few lines of stochastic calculus shows us that the drift of the M which we already divided by M because it factors out, equals the expression that is given here. Now both components of this sum are zero, and this is due to the fact that A and B satisfy the Riccati equation shown in the claim. This proves indeed that M has zero drift, in other words, M is a martingale. Both models, the Vasicek and the CIR models yield closed-form expressions for the bond prices. It can also be shown that they yield analytic expressions for auction prices and bonds. This explains why they are very popular. This models also have a weakness. In both models the forward curve is given in closed form in terms of the functions A and B. Now the functions A and B are determined by the parameters kappa, theta, and sigma. If we count the initial values of the short rate as a fourth parameter, it shows us that the initial forward curve produced by either model, the Vasicek and the CIR is for parametric. The problem that comes with it is that these models do not fit the data in general. It is illustrated here for the CIR model with these parameters, the red is the implied initial forward curve which cannot match the observed data points in general because of its low degree of freedom Fortunately, there's an easy solution to it. The idea is to add a time-inhomogeneous and its component to it with infinity degree of freedom in such a way that you preserve the analytic tractability of the model on the other hand. Here is how the idea works. It is illustrated for the Vasicek model but it will equally-well work for the CIR model, and in fact, for any diffusion short rate models. We need 2 ingredients. The first is an auxiliary simple Vasicek model as show here. We assume that initial short rate matches the data. In blue we indicate a given initial forward curve. As we well know, the Vasicek model yields closed-form exponential afﬁne bond prices. For functions A tilde and B tilde, given in terms of solutions of Riccatic equations. The second ingredient is a deterministic function Phi T with 0 initial value. We then get a time-inhomogenous short rate model r(t), as the sum of r(t) plus the auxiliary simple Vaiscek model r tilde. Notice already that these short rate models matches the initial short rates by construction because Phi 0 is 0, and r tilde 0, is the desired initial value. The bond prices are still given in closed form. This is a simple consequence of this observation done here. From this, we easily infer that the model implied initial forward curve is given a shown here. And now, it is evident that we will obtain a perfect fit of the given initial forward curve whenever we choose the right function Phi as the difference of the given initial forward curve minus model implied forward curve from the auxiliary Vasicek model. What can we say about the dynamics of the time-inhomogeneous short rate process r(t). Let's illustrate these for the Vasicek short rate model. So assume r tilde is the auxiliary Vasicek short rate model as seen on the previous slides. We then simply differentiate the sum of Phi(t) plus r tilde(t), to get the differential of r in this form. We then r tilde by r minus Phi, and it follows that r(t) has a decomposition as shown here where now the theta is a t-dependent function given in terms of Phi, this way. It becomes evident that we just obtained an extension of the Vasicek model by making this constant theta a time-dependence theta. This extension is also called the Hull-White Model. In sum, the Hull-White Model, yields closed-form exponential afﬁne bond prices and at the same time matches any given initial forward curve.