MonteCarlo

Market Risk Simulation for Multi‑Currency Revenue Linked to Brent (EUR Book) This guide shows how to simulate monthly revenue risk when commodity prices are linked to Brent (in USD), the book currency is EUR, and you have revenue streams across EUR, USD, INR, and AUD markets. We model Brent with a mean‑reverting process and preserve correlation with FX using a Monte Carlo method. Assumptions: Prices are monthly and linked to Brent in USD. Brent follows a mean‑reverting (Ornstein–Uhlenbeck) process in log‑space. FX pairs are modeled in log‑space (e.g., GBM for simplicity) and jointly simulated with Brent via a correlation matrix. Book currency is EUR; portfolio revenue is aggregated in EUR. Why this setup works: ...

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: ...