Deriving a Short-Rate PDE for Discounted Terminal Payoffs
Summary
The document considers a short rate following a Vasicek process and asks whether a zero-coupon bond pricing PDE also applies when the terminal payoff is a smooth function of the short rate. The proposed value is the risk-neutral expectation of that payoff discounted by the accumulated short rate, with the terminal condition set to the payoff function.
The included answer verifies the PDE by defining the discounted price process and using its martingale property. Applying Itô’s lemma yields a drift term combining time change, mean reversion, rate discounting, and diffusion curvature; setting that drift to zero gives the pricing equation. At maturity, the value equals the specified payoff. The argument assumes the stated short-rate dynamics and sufficient smoothness for the derivatives and Itô calculation. It establishes the generalization for a payoff depending on the terminal short rate, rather than detailing numerical solution methods or broader model calibration.
Key ideas
- Under the assumed short-rate model, the discounted expectation of a terminal payoff defines its time-t price.
- The terminal boundary condition is the payoff function evaluated at maturity.
- The discounted price process is a martingale under the risk-neutral measure.
- Itô’s lemma gives the pricing PDE by requiring the process drift to vanish.
- The derivation relies on smoothness and the specified short-rate dynamics.
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Full text
# PDE for Pricing Interest Rate Derivatives
# PDE for Pricing Interest Rate Derivatives
Suppose that interest rate $r(t)$ follows some short-rate models, say Vasicek, so that$dr = a(b-r) dt + \sigma dZ$, with constants $a,b,\sigma$.
It is well known that the price of zero-coupon bond $P(r,t)$ at current time $t$ maturing at $T$ with face value 1 follows (for example, see McDonald's Derivatives Markets, 3rd ed, p.758): $$\frac{\sigma^2}{2} \frac{\partial^2 P}{\partial r^2} + a(b-r) \frac{\partial P}{\partial r} + \frac{\partial P}{\partial t} - r P=0$$ with boundary condition $P(r,T) = 1$. Note that we could write $P(r,t) = \mathbb{E}^\mathbb{Q} \Big[ e^{- \int_t^T r(u) du} \big| F_t \Big]$ for all $t \leq T$.
Trying to generalize, for some smooth condition of $h(r,T)$ depending only on $r$ at $T$: If we define $Q(r,t) = \mathbb{E}^\mathbb{Q} \Big[ e^{- \int_t^T r(u) du} h(r,T) \big| F_t \Big]$ for all $t \leq T$, does the following PDE $$\frac{\sigma^2}{2} \frac{\partial^2 Q}{\partial r^2} + a(b-r) \frac{\partial Q}{\partial r} + \frac{\partial Q}{\partial t} - r Q=0$$ with boundary condition $Q(r,T) = h(r,T)$ hold?
## Answer by Gordon (score 4)
https://quant.stackexchange.com/a/30082
Note that \begin{align*} M(r_t, t) &\equiv Q(r_t, t) e^{-\int_0^t r_u du} \\ &=E\left(e^{-\int_0^T r_u du} h(r_T, T) \mid \mathscr{F}_t \right) \end{align*} is a martingale. Moreover, \begin{align*} dM &= \frac{\partial M}{\partial t}dt + \frac{\partial M}{\partial r} dr_t + \frac{1}{2}\frac{\partial^2 M}{\partial r^2}d\langle r, r\rangle_t\\ &=\left[\frac{\partial Q}{\partial t} e^{-\int_0^t r_u du} - r_t Q(r_t, t) e^{-\int_0^t r_u du}\right]dt+e^{-\int_0^t r_u du}\frac{\partial Q}{\partial r}dr+ \frac{\sigma^2}{2}\frac{\partial^2 Q}{\partial r^2}e^{-\int_0^t r_u du}dt\\ &=e^{-\int_0^t r_u du}\left[\left(\frac{\sigma^2}{2}\frac{\partial^2 Q}{\partial r^2} +a(b-r_t)\frac{\partial Q}{\partial r} -r_t Q(r_t, t) + \frac{\partial Q}{\partial t} \right)dt + \sigma \frac{\partial Q}{\partial r} dZ \right]. \end{align*} Therefore, \begin{align*} \frac{\sigma^2}{2}\frac{\partial^2 Q}{\partial r^2} +a(b-r_t)\frac{\partial Q}{\partial r} -r_t Q(r_t, t) + \frac{\partial Q}{\partial t}=0. \end{align*}Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.