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?

  • 1
    $\begingroup$ I doubt that there exists a closed form solution. $\endgroup$ – 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:

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  • $\begingroup$ Thanks Dear Maleki, I think this is c option price not convertible bond price $\endgroup$ – Fad F Nov 16 '16 at 13:51
  • $\begingroup$ convertible bond should equal to ( option price you shown + straight bond component). $\endgroup$ – Fad F Nov 16 '16 at 13:54
  • $\begingroup$ Upvoted. This looks correct, with the big caveats that actual convertible bonds are almost never European exercise and are often callable, that default risk is not considered here, and that you may want correlation between interest rate and equity stochastic processes. Using this simple formula for trading would be a good way to make your counterparties wealthy. $\endgroup$ – Brian B Jan 10 '18 at 15:50

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