Pricing a Convertible Bond with Stock and Vasicek Rate Factors

Summary

The document presents a proposed closed-form value for a convertible bond under a two-factor setup involving the stock price and a Vasicek-like interest-rate factor. The cited response expresses the value in a Black-style form, using the stock price, a zero-coupon bond price, and a volatility term that combines stock and rate volatility. It also gives formulas for the bond price's time-dependent coefficients.

The material offers a pricing expression and its rate-model components, but the original question's decomposition into conversion option, straight bond, and premium is not directly resolved. The response points to a separate article for additional detail, so the excerpt alone does not establish assumptions such as default risk, conversion features beyond the simplified option payoff, or model calibration. The formula should be understood as a specialized model result rather than a complete treatment of convertible-bond valuation.

Key ideas

• The proposed valuation combines stock and interest-rate factors in a Black-style expression.
• The rate component is represented through a zero-coupon bond price under a Vasicek-like model.
• The formula includes a volatility term combining stock and rate volatility.
• The excerpt does not fully explain how its simplified option representation maps to a general convertible bond.

Tags

# Closed- solution for Convertible bond price two factor model


# 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?

## Answer by user16651 (score 2)

https://quant.stackexchange.com/a/31030

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:

- Towards non-equilibrium option pricing theory

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.