# Pricing interest rate derivatives

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?

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.

• What would be an example of a semi-explicit solution? (Does it contain e.g. the standard Gaussian CDF?)
– J.G.
Sep 30, 2021 at 11:49
• The well-known solutions for the zero bond price you can google. If you want another example you should clearly define your derivative, and the model you want this to be sloved in. Sep 30, 2021 at 12:20
• You're right, thanks. Shreve discusses such results in Secs 5.6 and 6.5; for example, the latter contains the zero-coupon bond pricing formula $B=\widetilde{\Bbb E}\left[\left.e^{-\int_t^TRds}\right|\mathcal{F}\right]$, which is probably what you meant by semi-explicit.
– J.G.
Sep 30, 2021 at 14:13
• That formula is far from what I meant. Rather see Musiela & Rutkowski, Martingale Methods in Financial Modelling, ch. 12.3. Sep 30, 2021 at 14:38
• Thanks for the reference.
– J.G.
Sep 30, 2021 at 14:41