Short-Rate Models and When Interest Rate Derivatives Have Closed-Form Prices
Summary
The document concerns a pricing partial differential equation for a derivative whose underlying variable is an interest rate. Its general specification uses drift and volatility functions of time and the short rate, together with a market price of risk. The accepted explanation classifies this as a Markovian short-rate model rather than a single named model, because the functions are left broadly unspecified.
It identifies Hull–White, Cox–Ingersoll–Ross, and Black–Karasinski as special cases with particular drift and volatility forms. For zero-coupon bonds, Hull–White and CIR have semi-explicit solutions; an arbitrary interest-rate derivative does not necessarily admit one. The original question attempts a Fourier-transform separation for constant coefficients and asks about numerical methods for more general coefficients, but the response does not recommend a numerical scheme or validate that derivation. The main practical lesson is that solvability depends on the specific model and payoff.
Key ideas
- A model with a Markovian short rate and general time- and rate-dependent coefficients is too broad to have one specific name.
- Hull–White, CIR, and Black–Karasinski are special cases defined by different drift and volatility functions.
- Some short-rate models provide semi-explicit prices for zero-coupon bonds.
- Arbitrary interest-rate derivatives do not generally have a semi-explicit pricing solution.
- The document raises numerical solution questions but does not prescribe a method.
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Full text
# Pricing interest rate derivatives
# 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?
## Answer by Kurt G. (score 4, accepted)
https://quant.stackexchange.com/a/68144
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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