# Pricing the discount zero-coupon bond under a jump-diffusion model

I am going to get the price of a zero coupon bond in a jump-diffusion model. The dynamic of interest rate as follow $$dr_t=\kappa(\theta-r_t)dt+\sigma\sqrt{r_t}\,dW_t+d\left(\sum\limits_{i=1}^{N_t}\,J_i\right)$$ where $N_t$ represents a Poisson process with constant intensity rate $\lambda>0$ and $\{J_i\}_{i=1}^{\infty}$ denotes the magnitudes of jump, which are assumed to be i.i.d. random variables with distribution $f_J$ independent of $W_t$ and $N_t$. Moreover,$W_t$ is assumed to be independent of $N_t$. In addition the jump sizes $\,J_i$ has an exponential distribution with density: $${{f}_{J}}(\chi )=\left\{ \begin{matrix} \eta {{e}^{-\eta\,\chi}}\,,\,\,\chi >0\, \\ 0\,\,\,\,\,\,\,\,,\,\,\,\,o.w. \\ \end{matrix} \right.$$ where $\eta > 0$ is an constant. I can prove the arbitrage-free price at time $t$ of a traded interest rate security with paying $H(r)\in\,\mathcal{L^1}(\Omega\,,\,\mathcal{F}_T\,,\,Q)$ and maturity $T$ satisfies the following parabolic partial integro differential equation $$\frac{\partial F}{\partial t}+\frac{1}{2}{{\sigma }^{2}}r\frac{{{\partial }^{2}}F}{\partial {{r}^{2}}}+\kappa (\theta -r)\frac{\partial F}{\partial r}-rF+\lambda \int_{-\infty }^{\infty }{(F(t,r+\chi ,T)-F(t,r,T)d\chi =0}$$ with boundary condition $F(T,r,T)=H(r)$. Obviously in the case of zero-coupon bond we have $$H(r)=P(T,r,T)=1$$

### My Challenge

1. I want to solve this PIDE leading to the bond pricing formula but I have no good idea. I know $$F(t,r,T)={{E}}^{\mathbb{Q}}\left[ {{e}^{-\int_{t}^{T}{{{r}_{s}}ds}}}|{\mathcal{F}_{t}} \right]=\exp \left[ A(T,t)-B(t,T){{r}_{t}} \right] \,\,\,\,\,\,(1)$$ but I can't extract these deterministic functions. Indeed, I substitute $(1)$ into PIDE. I then have a system of two ordinary differential equations that determine the coefficient functions.

2. How can I approximate this PIDE by Numerical Methods? Indeed, I have no other boundary conditions.

• Hi @Behrouz Maleki, although this won't answer your question (I'm sorry, not very familiar with interet rates problems, but I'll look at it later), have you tried a Fourier transform approach since the model is affine? See here pp 60 and following for instance? uniroma2.it/ppg/im/tesi/dominedo-tesi.pdf – Quantuple Jun 6 '16 at 10:29
• did it help? For your first problem, what is wrong exactly? You can't solve the ODEs? For the second, if you know $F(t,r;T) = \exp[A(T,t)-B(t,T)r_t]$, can't you derive Dirichlet conditions e.g. $\lim_{r\rightarrow 0} F(t,r;T) = \exp[A(T,t)]$, and something similar for the upper bound of your space domain? Well you'll probably tell me that you don't know $A(t,T)$ nor $B(t,T)$ at that point... – Quantuple Jun 6 '16 at 14:34
• I think the ODE is Ricatti equation but it's so complicated dude. – user16651 Jun 6 '16 at 14:39
• Yes that's what I thought too. That's why Fourier pricing seems better. If you have the characteristic function of the CIR part alone, you just multiply it by the char fun of the compound Poisson process using independence. – Quantuple Jun 6 '16 at 14:57

• In the case of $X$ affine process, you can compute $E(\exp(iuX_t+iv\int_0^tX_s ds))$ – MJ73550 Jun 7 '16 at 9:38