# 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?

• you mean $Q(t,T) = \mathbb{E}^{\mathbb{Q}}\left[e^{-\int_t^T r(u) du} h(r_T)\right]$ ? – MJ73550 Sep 8 '16 at 12:03
• Hi, MJ73550, yes I mean $h(r, T)$, to be terminal condition for $Q$. Thanks. – quant123 Sep 8 '16 at 13:15
• My question is whether $h$ depends only on the final value of $r$ or the whole path. – MJ73550 Sep 9 '16 at 5:32
• $h$ depends only on final value $r$ at $T$. – quant123 Sep 9 '16 at 7:09