Zero-Coupon Bond Pricing in a CIR Jump-Diffusion Model
Summary
The document sets up a short-rate model with mean reversion, square-root diffusion volatility, and independent positive jumps whose sizes follow an exponential distribution. It presents the pricing PIDE for a security paying a function of the terminal rate and identifies the zero-coupon bond payoff as one. The bond price is represented in exponential-affine form, with deterministic coefficient functions that can be found by substituting that form into the PIDE and solving the resulting ordinary differential equations.
The discussion raises numerical solution as an alternative, but does not give a discretization method or boundary treatment. The only response points to a reference on affine models; it does not derive the coefficient equations or provide numerical results. Thus, the document outlines the model and analytic route, while leaving the derivation and implementation to further study.
Key ideas
- The short rate combines CIR-style mean reversion and diffusion with independent compound Poisson jumps.
- Jump sizes are assumed positive and exponentially distributed.
- The security price satisfies a partial integro-differential equation with a terminal payoff condition.
- A zero-coupon bond can be sought in exponential-affine form, reducing the pricing problem to equations for deterministic coefficients.
- The document does not specify numerical boundary conditions or a numerical scheme.
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# Pricing the discount zero-coupon bond under a jump-diffusion model
# 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
- 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.
- How can I approximate this PIDE by Numerical Methods? Indeed, I have no other boundary conditions.
## Answer by M. Jeunesse (score 1)
https://quant.stackexchange.com/a/27459
If I understood well, your model falls into the generic case of affine models.
This reference might help you : http://arxiv.org/pdf/1512.03677v1.pdfShown 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.