Implicit Finite-Difference Schemes for the CIR Bond Pricing PDE
Summary
The document raises numerical questions about solving the Cox–Ingersoll–Ross short-rate model’s bond-pricing PDE with an implicit finite-difference scheme. It asks how to set a boundary at a large maximum rate, whether the grid point at zero requires a one-sided derivative, and how time and rate steps or model parameters affect positivity of computed prices.
It also considers upwind differences, choosing a right-sided or left-sided stencil according to whether the current rate is below or above the long-run mean. The text presents these as open questions rather than supplying a derivation, numerical results, or a validated scheme. Readers should therefore treat the proposed zero boundary and stencil choices as hypotheses to analyze, not established prescriptions. The document is useful as a problem statement about boundary conditions, discretization, and positivity in numerical fixed-income pricing.
Key ideas
- The document frames implicit finite differences as a way to approximate the CIR bond-pricing PDE.
- It asks whether bond value approaches zero at a sufficiently large short rate.
- It questions whether a one-sided derivative is needed at the zero-rate boundary.
- It raises positivity and step-size restrictions as stability concerns.
- It proposes rate-dependent upwinding but does not establish its positivity properties.
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Full text
# Implicit Scheme for Cox-Ingersoll-Ross Model PDE
# Implicit Scheme for Cox-Ingersoll-Ross Model PDE
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.
- 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).
- 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?
- 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.)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.