I am considering the PDE for the price of a bond $V(r,t)$ with maturity $T$ under the Cox-Ingersoll-Ross model,

$$V_t+\frac12\sigma^2rV_{rr}+\nu(\theta-r)V_r-rV=0\quad r>0, t\in(0,1)$$

with terminal condition $V(r,T)=1$ and $\sigma$, $\nu$ and $\theta$ are strictly positive constants. I have a few questions about the implicit scheme, which for brevity I will just express as $V^{(n+1)}_j=a_jV^{(n)}_{j-1}+b_jV^{(n)}_j+c_jV^{(n)}_{j+1}$ first.

  1. I am trying to think of a suitable Dirichlet boundary condition for $r\to\infty$, or rather in this case $r\to r_{\text{max}}$. Intuitively, just as how $V\to0$ for $S\to\infty$ for the Black-Scholes equation, I am also thinking that $V\to0$ for $r\to\infty$ here. However, I am not sure how to justify it (the Black-Scholes has an analytic argument that I understand, but I have not found any literature about this CIR PDE). 

  2. Is it necessary for a right-sided first order finite difference at $r=0$ to replace the central difference? Intuition again tells me that this is a requirement for positivity of $V$ $\forall$ $n$, is this true? Are there further conditions should I be enforcing on $\Delta t$, $\Delta r$ and the constants?

  3. I was told we can perform an upwind discretisation with right-sided (respectively left-sided) differences for $r_n\leq\theta$ (respectively $r_n\geq\theta$) to aid enforcement of positivity of $V$. I don't see that at all. What does the scheme look like and how does it ensure positivity?

(NB: I have shifted this question from MSE.)


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