5 Cox-Ingersoll-Ross Model

This is a different short-rate model. You will not be able to solve it using the techniques we have learned so far, but you can characterize the mean and variance with some tricks. Let drt = a(brt) dt + ort dWt As with the Vasicek model, assume that a > 0. We will also assume that b>0. It is also usually assumed that 2ab 0. As a side note: in finance we call this Cox-Ingersoll-Ross model, because these authors used it to model the short-rate process. However, outside of finance this is called an Ornstein-Uhlenbeck process. (a) (5 points) Without do any math, briefly explain why r, will remain non-negative? There's no need for a formal justification. Hint: look at the volatility. What happens as rt gets close to 0? (b) (5 points) Try the same trick as with the Vasicek model. That is, define Xt = eat rt, and use Ito's Lemma to find the drift and diffusion of Xt. Hint: make sure there is no rt in your expression (you will need to substitute it out). You should get something of the form dXt = pt dt+otdWt, where ut depends only on time, and ot depends on time and Xt. (c) (10 points) What is the expected value E[Xt| Xo]? Remember that the It integral is typically a martingale. So, if you found dXt = t dt+otdWt, then and therefore Lunds + fo Xt = Xo + E[Xt| Xo] (d) (10 points) What is the variance var(Xt | Xo)? Hint: Remember the It isometry result. This gives you that Xo+ E var(Xt | Xo) E E[Hads Xo] | In this case, you should get something that looks like var (Xt| Xo) = E s dWs t [[ 0 ds | Xo] E [ J(s)X, ds | Xo] E f(s)X, ds | Xo Push the expectation inside of the integral, and integrate. That is, switch the order of the integral sign and the expectation: | Xo] = [B 0 * Elf(s)X, | Xo] ds (e) (5 points) Use your results to compute the mean and variance of rt conditional on ro. Note: you can do this even if you couldn't do all of (c) and (d). Just leave your answers in terms of E[X | Xo] and var(Xt| Xo) accordingly.

5 Cox-Ingersoll-Ross Model (35 points, medium) This is a different short-rate model. You will not be able to solve it uxing the techniques we have learned so far, but you can characterize the mean and variance with some tricks. Let drt=a(brt)dt+rtdWt As with the Vasicek model, assume that a>0. We will also assume that b>0. It is also usually assumed that 2ab2. As a side note; in finance we call this Cox-Ingersoll-Ross model, because these authors used it to model the short-rate process. However, outside of finance this is called an Ornstein-Uhlenbeck process. (a) (5 points) Without do any math, briefly explain why rt will remain non-negative? There's no need for a formal justification. Hint: look at the volatility. What happens as rt gets clooe to 0 ? Lemma to find the drift and diffusion of Xt. Hint: make sure there is no rt in your expression (you will need to substitute it out). You should get something of the form dXt=tdt+tdWt, where t depends only on time, and t depends on time and Xt. (c) (10 points) What is the expected value E[XX0] ? Remember that the Ito integral is typically a martingale. So, if you found dXt=tdt+tdWt, then Xt=X0+0tsds+0tsdWv and therefore E[XtX0]=X0+E[0tsdsX0] (d) (10 points) What is the variance var(XtX0) ? Hint: Remember the Ito iscmetry result. This gives you that var(XtX0)=E[0ts2dsX0] In this case, you should get something that looks like var(XtX0)=E[0tf(s)XdsX0] Push the expectation inside of the integral, and integrate. That is, switch the order of the integral sign and the expectation: E[0tf(s)XndsX0]=0tE[f(s)XsX0]ds (e) (5 points) Use your results to compute the mean and variance of rt conditional on r0. Note: you can do this even if you couldn't do all of (c) and (d). Just leave your answers in terms of E[XtX0] and var(XtX0) accordingly