Pricing interest rate derivatives

Question

In Sec. 3.2 here, Mandel deduces the price $P$ of a derivative on an interest rate $r$ obeys a PDE of the form$$\frac{\partial P}{\partial t}+\frac{1}{2}\beta^{2}\frac{\partial^{2}P}{\partial r^{2}}+\left(\alpha-\beta\lambda\right)\frac{\partial P}{\partial r}-rP=0.$$(Does this model have a name?) I attempt a solution here (for the case where $\alpha,\,\beta$ are constant), as I didn't find one in the cited text, Shreve 2010.

I start with a Fourier transform viz. $P(t,\,r)=\int_{\Bbb R}\widetilde{P}(t,\,k)e^{ikr}dk$ so $\int_{\Bbb R}f(t,\,k)e^{ikr}dk=0$ with$$f:=\frac{\partial\widetilde{P}}{\partial t}-\frac12\beta^2k^2\widetilde{P}+(\alpha-\beta\lambda)ik\widetilde{P}-i\frac{\partial\widetilde{P}}{\partial k}.$$A separable solution $\widetilde{P}=K(k)T(t)$ of $f=0$ gives constant $\rho:=\frac1T\frac{\partial T}{\partial t}$, whence$$\frac1K\frac{\partial K}{\partial k}=(\alpha-\beta\lambda)k+i\left(\frac12\beta^2k^2-\rho\right).$$Since $\ln K$ is cubic in $k$, I expect $P$ isn't analytic in $r$. Is this model solved with numerical methods (especially when $\alpha\,\beta$ are functions of $t,\,r$)? If so, which ones are recommended?

Answer 1

Mandel assumes that $\alpha,\beta$ are functions of $t$ and $r$ and the market price of risk $\lambda$ is a function of $t\,.$ This model is a Markovian short rate model. Other than that, it is too general to have a name. The following named models are special cases: \begin{align} &\alpha(t,r(t)) & \beta(t,r(t)) & & \text{ Name }\\[3mm] \hline &a(t)(\theta(t)-r(t))&\sigma(t)& & \text{ Hull-White }\\[3mm] &a(t)(\theta(t)-r(t))&\sigma(t)\sqrt{r(t)}& &\text{ Cox-Ingersoll-Ross }\\[3mm] &a(t)(\theta(t)-\log r(t))\,r(t)& \sigma(t)\,r(t)& &\text{ Black-Karasinski} \end{align} The CIR and HW models are known to have a semi explicit solutions for the zero coupon bond $P(t,r(t))\,.$ If you want $P$ to be an arbitrary derivative on the interest rate there is no hope to find a semi explicit solution always.