# Closed- solution for Convertible bond price two factor model

I am trying to find the closed- solution of convertible bond $V(s,r,t)$ under Vasicek model of two factor model of PDE shown in below link Ito lemma of Convertible Bond under Two-factor Model Interest Rate.

I think that:

$$V(s,r,t) = \text{conversion option} + \text{straight bond} + \text{premium}$$

Is this right?

• I doubt that there exists a closed form solution. – Gordon Nov 15 '16 at 21:41

Let $r_t=r_0+x_t$ where $r_0$ is a constant. We have $$V(S,x,t)=SN(d_1)-KP(x,t)N(d_2)$$ where $$d_1=\frac{\ln(S/K)-\ln P(x,t)+\frac12\widehat{\sigma}\tau}{\widehat{\sigma}\sqrt{\tau}}$$ and $$d_2=d_1-\widehat{\sigma}\sqrt{\tau}$$ and $$\widehat{\sigma}=\sigma^2+\Sigma^2$$ The zero coupon bond pricing in terms of Vasicek-like rates is $$P(x,t)=A(t,T)e^{-xB(t,T)}$$ where $$B(t,T)=\frac{1-e^{-\kappa\tau}}{\kappa}$$ and $$A(t,T)=\exp\left(-r_0\tau-\frac12\frac{\Sigma^2}{\kappa^2}\left(-\tau-\frac{2}{\kappa}\left(e^{-\kappa\tau}-1\right)+\frac{1}{2\kappa}(e^{-2\kappa\tau}-1)\right)\right)$$ For more details, read this article: