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Numerical SDE Methods for Interest Rate Dynamics This post shows how to simulate short-rate models for interest rates and use them in Monte Carlo pricing/valuation. We cover discretization choices (Euler–Maruyama, Milstein, exact), correlation handling, and practical tips for stability and accuracy. Scenarios: Path‑dependent discounting for cashflows: $DF = \exp\big(-\int r_t dt\big)$. Pricing/valuation under short‑rate models: Vasicek/OU, CIR, and a note on Hull–White. Monthly or finer time steps with correlated factors. Models Vasicek (Ornstein–Uhlenbeck on $r$): $\mathrm{d}r = \kappa(\theta - r)\mathrm{d}t + \sigma \mathrm{d}W_t$. Mean‑reverting, Gaussian; can be negative; has exact discretization in discrete time. CIR (Cox–Ingersoll–Ross): $\mathrm{d}r = \kappa(\theta - r) \mathrm{d}t + \sigma\sqrt{r} \mathrm{d}W_t$. Mean‑reverting, strictly positive if Feller: $2\kappa\theta \ge \sigma^2$; has exact transition via noncentral $\chi^2$. Hull–White (time‑dependent Vasicek): $\mathrm{d}r = a(t)\big(b(t) - r\big) \mathrm{d}t + \sigma(t) \mathrm{d}W_t$. Matches an initial yield curve; can simulate with Euler or semi‑exact integrator. Discretization methods Euler–Maruyama (EM): strong order 0.5; simple and widely used. Milstein: strong order 1.0 for scalar SDEs; adds a diffusion derivative term (useful for √r in CIR). Exact schemes (when available): Vasicek/OU in discrete time has a closed form. CIR has an exact transition: $r_{t+\Delta} = c \cdot \chi^2_{\nu}(\lambda)$ for appropriate $(\nu, \lambda, c)$. Guidance: ...