Boundary Condition for Convertible Bond under Two-factor Model Interest Rate

Question

I want to find Boundary conditions for Convertible Bond under Two-factor Model Interest Rate.The portfolio contains stock where stochastic differential equation for the stock price is \begin{align} ds_t=rS_t+\sigma S_tdW_1(t) \end{align} where $\sigma$ is constant and dynamics of $r$ as follow \begin{align} dr_t=\kappa(\theta-r_t)dt+\Sigma dW_2(t) \end{align}

Answer 1

For a two-factor option pricing model with underlying variables $S$ and $r$ defined as above, if we assume there is no correlation between the two Wiener processes $W_1$ and $W_2$, one finds the generalized Black-Scholes PDE \begin{align} V_t+\frac{1}{2}\sigma^2V_{SS}+r\,S\,V_S-r\,V+\frac{1}{2}\Sigma\,^2\,V_{rr}+\kappa(\theta-r)V_r=0 \end{align} This equation is subject to initial and boundary conditions. Generally speaking, derivative pricing models for different financial scenarios may share a similar pricing partial differential equation (PDE) with adjusted parameters and boundary conditions.Boundary conditions defining two portfolios will be considered. The first set of conditions will describe a European call stock option. The second set of conditions models a convertible bond. The stock price $S$ and interest rate, $r$.

1. At the maturity time T, the call option price will be the payoff function \begin{align} V(S,r,T)=\max\{S-K,0\} \end{align}
2. At $S = 0$, the option is worthless: \begin{align} V(0,r,T)=0 \end{align}
3. For large stock price $S_{\max}$, it is almost certain that the bond will be converted to one share of the stock. Hence \begin{align} V(S_{\max},r,t)=S_{\max} \end{align}
4. When $r_t$ is infinitely large, the bond component tends to zero. Since we do not enforce any time-dependent constraints of puttable and callable features, the upper bound and the lower bound to the price of the convertible bond are $\max\{S,\infty\}$ and $\max\{S,0\}$ respectively. Therefore we define the boundary condition as \begin{align} V(S,r_{\max},t)=\min\left\{\max\{S,\infty\},\max\{S,0\}\right\}=S \end{align}
5. For a very small interest rate, we use homogeneous Neumann condition suggested by Bermudez and Nogueiras.