M.4.4 Stochastic Differential Equations & Simulation

Basic Prerequisites: 03 · Itô Integral & Doeblin.

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1. Intuition & Practical Objective

An SDE is a definition of a stochastic process by its differential:

$$d{X}_t=\mu (t,X_t)dt+\sigma (t,X_t)d{W}_t,$$

read rigorously as an integral equality $X_t=X_0+{\int }_0^t\mu ds+{\int }_0^t\sigma dW$. The Drift $\mu$ sets the expected growth; Diffusion $\sigma$ sets the random spread and (because then $(dX{)}^2={\sigma }^{2}dt$) feeds back into Itô–Doeblin computations.

Practical objective: three workhorse SDEs drive all of quantitative finance -

• GBM $dS=\mu Sdt+\sigma SdW$ (equities, indices; closed-form lognormal solution),
• Vasicek $dR=\kappa (\theta -R)dt+\sigma dW$ (Gaussian mean-reversion; interest rates, closed form),
• CIR $dR=\kappa (\theta -R)dt+\sigma \sqrt{R}dW$ (mean-reversion with vol proportional to $\sqrt{R}$; no closed form, stays nonnegative under the Feller condition).

The whole point of this page: know which SDEs have exact transitions (simulate without error) vs which need discrete schemes (Euler–Maruyama, Milstein) - and what each scheme gets wrong.

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2. Mathematical Ground Truth & Derivations

2.1 GBM - closed-form solution (Shreve I §15.3; Björk Prop 5.2; Glasserman §3.2)

$$d{S}_t=\mu S_{t}dt+\sigma S_{t}d{W}_t⟹S_t=S_0{e}^{\sigma W_t+\left(\mu -\frac{1}{2}{\sigma }^2\right)t},\mathbb{E}[S_t]=S_0{e}^{\mu t}.$$

Exact transition (Glasserman eq 3.20–3.22), correct on every grid point with no discretization error:

$$S(t_{i+1})=S(t_i)e^{\left(\mu -\frac{1}{2}{\sigma }^2\right)\Delta t+\sigma \sqrt{\Delta t}{Z}_{i+1}},Z_{i+1}\sim N(0,1).$$

The Euler–Maruyama scheme $S_{i+1}=S_i(1+\mu \Delta t+\sigma \sqrt{\Delta t}Z)$ has an easier-to-see but biased form, especially for coarse $\Delta t$ - the “simulate exact, not Euler, whenever possible” rule.

2.2 Vasicek - Gaussian, closed form (Shreve II Ex 4.4.10; Glasserman §3.3)

$$d{R}_t=\kappa (\theta -R_t)dt+\sigma d{W}_t⟹R_t=e^{-\kappa t}{R}_0+\theta (1-e^{-\kappa t})+\sigma e^{-\kappa t}{\int }_0^t{e}^{\kappa s}d{W}_s.$$

As a Gaussian process: mean $e^{-\kappa t}{R}_0+\theta \left(1-e^{-\kappa t}\right)$, variance $\frac{{\sigma }^2}{2\kappa }\left(1-e^{-2\kappa t}\right)\to \frac{{\sigma }^2}{2\kappa }$. Exact transition (Glasserman eq 3.43–3.45):

$$R_{t_{i+1}}\sim N(e^{-\kappa \Delta t}{R}_{t_i}+\theta (1-e^{-\kappa \Delta t}),\frac{{\sigma }^2}{2\kappa }(1-e^{-2\kappa \Delta t}\left)\right).$$

Long-run mean $\theta$; can go negative (unlike CIR).

2.3 CIR - no closed form, nonnegative (Shreve II Ex 4.4.11; Glasserman §3.4)

$$d{R}_t=\kappa (\theta -R_t)dt+\sigma \sqrt{R_t}d{W}_t.$$

Expectation identical to Vasicek $\mathbb{E}[R_t]=e^{-\kappa t}{R}_0+\theta \left(1-e^{-\kappa t}\right)$; the variance grows to $\theta {\sigma }^2/(2\kappa )$. The diffusion $\sigma \sqrt{R}\to 0$ at the origin, so with $\theta >0$ the drift pushes back up - stays nonnegative. Feller condition $2\kappa \theta \ge {\sigma }^2$ ⇒ strictly positive (Glasserman §3.4).

Because there is no closed-form transition, CIR simulation uses exact noncentral-${\chi }^2$ sampling (Glasserman eq 3.105: $d\equiv \frac{4\kappa \theta }{{\sigma }^2}$ “degrees of freedom”, $c\equiv \frac{{\sigma }^2(1-e^{-\kappa \Delta t})}{4\kappa }$, noncentrality $\lambda \equiv \frac{R_{t_i}{e}^{-\kappa \Delta t}}{c}$) or a well-behaved Milstein scheme; plain Euler undershoots below zero.

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3. Computational Implementation - exact vs Euler, and the Feller cliff

Stdlib block that (a) confirms GBM exact-MC mean converges to $S_0{e}^{\mu T}$; (b) shows Euler–Maruyama’s coarse-$\Delta t$ drift vs the exact-transition value; (c) draws the Vasicek exact transition and checks the stationary variance; (d) contrasts the Euler and Milstein CIR steps (with the Feller condition here satisfied, both stay positive - the negative-crossing case needs $2\kappa \theta <{\sigma }^2$).

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4. Failure Modes & First-Principles Breakdowns

1. Euler for everything. Euler has strong- and weak-discretization error $O(\Delta t)$; for GBM/Vasicek it is strictly inferior to the exact transition which costs the same. Coarse Euler systematically misprices moments (see §3: $98.34$ vs $148.33$ on the same-scale draw).
2. CIR Euler going negative. With $\sigma \sqrt{r}$ the diffusion vanishes at $0$ only in the SDE - a discrete Euler passes below zero and $\sqrt{\max (r,0)}$ silently “helps.” Milstein’s extra $\frac{1}{4}{\sigma }^2\Delta t((Z{)}^2-\Delta t)$ term (or exact ${\chi }^2$ sampling) respects the boundary. Violating the Feller condition without an exact scheme is a guaranteed bias.
3. Vasicek for rates you must keep positive. Vasicek is Gaussian and reaches negative values with positive probability. Using it for a process that must stay $\ge 0$ (e.g. spot rates near zero, or modelling a log-domain variable) is a model error - switch to CIR or a shifted/lognormal model.
4. Forgetting the $\frac{1}{2}{\sigma }^2$ when using the exact GBM transition. The transition is not $e^{\mu \Delta t+\sigma \sqrt{\Delta t}Z}$ - leaving out $-\frac{1}{2}{\sigma }^2$ inflates $\mathbb{E}[S]$ to $S_0{e}^{(\mu +\frac{1}{2}{\sigma }^2)T}$. Root cause: $(dW{)}^2=dt$ (03 · Itô).

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5. References

• Glasserman, Monte Carlo Methods in Financial Engineering
• Shreve, Stochastic Calculus for Finance II, Ex 4.4.8 (GBM), Ex 4.4.10 (Vasicek), Ex 4.4.11 (CIR, closed vs open form).
• Björk, Arbitrage Theory in Continuous Time, Prop 5.2 (GBM), Prop 5.3 (linear SDE)

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6. Connected Graph Bridges

• Back: 03 · Itô Integral & Doeblin
• Forward: 05 · Girsanov & Risk-Neutral · Index Hub
• Application: BSM Pricing Formula · Interest-Rate & Term-Structure Models · Econometrics & Time Series