The master SDE and its catalog

Every model below is a specific $(\mu, \sigma)$ choice — possibly with jumps, multiple dimensions, or path memory — plugged into one template, then labeled for a particular asset class.

The master template (one-dimensional, Markov, Itô version) is one equation:

$$dX_t \;=\; \mu(X_t, t)\, dt \;+\; \sigma(X_t, t)\, dW_t$$

A "model" in this framework is determined entirely by your choice of the pair $(\mu, \sigma)$, plus, if needed, a jump term or a path-memory kernel. The state variable $X_t$ is just a real number; what it represents in the world — a short rate, a stock price, an instantaneous variance, a default intensity — and which market it gets used in is interpretation attached after the math is done.

There are progressively more general master templates, and each model below sits at one of four levels:

Level 1 — 1D, Markov, Itô. Level 2 — $n$-dim, Markov. Level 3 — with jumps. Level 4 — path-dependent / non-Markov.

Level 1 — one-dimensional Markov Itô SDEs

VasicekLevel 1

$\mu(x)$$\kappa(\theta - x)$

$\sigma(x)$$\sigma_0$

$X_t$short rate $r_t$

Domainrates

Simplest mean-reverting Gaussian model. Rate can go negative — historically the model's "flaw," practically a feature post-2014 in EUR/JPY.

Ho-LeeLevel 1

$\mu(x,t)$$\theta(t)$

$\sigma(x)$$\sigma_0$

$X_t$short rate $r_t$

Domainrates

No mean reversion; $\theta(t)$ is calibrated so the model fits today's yield curve exactly.

Hull-White (extended Vasicek)Level 1

$\mu(x,t)$$\theta(t) - \kappa x$

$\sigma(x)$$\sigma_0$ (or $\sigma(t)$)

$X_t$short rate $r_t$

Domainrates

Combines Vasicek's mean reversion with Ho-Lee's curve-fitting trick. The desk-standard one-factor rates model.

CIR (Cox-Ingersoll-Ross)Level 1

$\mu(x)$$\kappa(\theta - x)$

$\sigma(x)$$\sigma_0\sqrt{x}$

$X_t$short rate, instantaneous variance, or default intensity

Domainratesequity volcreditcommodity vol

The clearest example of one SDE serving many domains. Non-central chi-squared distribution; $X \geq 0$ under Feller condition $2\kappa\theta \geq \sigma_0^2$.

CIR++Level 1 + shift

SDE$dY_t = \kappa(\theta - Y_t)\,dt + \sigma_0\sqrt{Y_t}\,dW_t$, $r_t = Y_t + \varphi(t)$

$X_t$short rate $r_t$

Domainrates

CIR with a deterministic shift $\varphi(t)$ that calibrates the model to today's curve. Shifted variants were the practical workaround when EUR rates went negative.

Black-Scholes / GBMLevel 1

$\mu(x)$$\mu_0\, x$

$\sigma(x)$$\sigma_0\, x$

$X_t$stock price $S_t$ (also FX spot, commodity spot)

DomainequityFXcommodity

Under the risk-neutral measure $\mu_0$ is replaced by the cost-of-carry (risk-free rate for stocks, $r-r_f$ for FX, $r-y$ for commodities). Lognormal distribution; $X > 0$ always.

Black-KarasinskiLevel 1 (OU on $\ln X$)

SDE$d(\ln X_t) = \kappa(\theta(t) - \ln X_t)\,dt + \sigma_0\, dW_t$

$X_t$short rate $r_t$

Domainrates

OU dynamics on the log of the rate. Guarantees $r > 0$; not affine (no closed-form bond prices). Common pre-2008, less so once negative rates appeared.

DothanLevel 1

$\mu(x)$$a\, x$

$\sigma(x)$$\sigma_0\, x$

$X_t$short rate $r_t$

Domainrates

GBM applied to the short rate. Rate stays positive; not affine. Mostly of historical interest.

CEV (Cox 1975)Level 1

$\mu(x)$$\mu_0\, x$

$\sigma(x)$$\sigma_0\, x^{\beta}$

$X_t$stock price $S_t$ (or forward $F_t$ in rates/FX)

DomainequityratesFX

The $\beta$ exponent interpolates Gaussian ($\beta=0$), square-root ($\beta=\tfrac{1}{2}$), and lognormal ($\beta=1$) in one parametric family. Heavily used as the forward-rate component of SABR.

3/2 model (variance)Level 1

$\mu(x)$$x\,\kappa(\theta - x)$

$\sigma(x)$$\sigma_0\, x^{3/2}$

$X_t$instantaneous variance $v_t$

Domainequity vol

Alternative to CIR for variance; quadratic drift and a higher diffusion power produce steeper at-the-money vol skew. Used for VIX-style products.

Level 2 — multi-dimensional Markov Itô SDEs

G2++ (two-factor Hull-White)Level 2

$dX^1_t = -a\, X^1_t\, dt + \sigma_1\, dW^1_t$
$dX^2_t = -b\, X^2_t\, dt + \sigma_2\, dW^2_t,\quad \text{corr}(dW^1, dW^2) = \rho$

Labelshort rate $r_t = X^1_t + X^2_t + \varphi(t)$

Domainrates

Two latent Gaussian factors plus a deterministic shift that fits the curve. Workhorse for Bermudan swaption pricing.

Longstaff-SchwartzLevel 2

$dX^1_t = \kappa_1(\theta_1 - X^1_t)\,dt + \sigma_1\sqrt{X^1_t}\, dW^1_t$
$dX^2_t = \kappa_2(\theta_2 - X^2_t)\,dt + \sigma_2\sqrt{X^2_t}\, dW^2_t$

Labelshort rate $r_t = X^1_t + X^2_t$; one factor read as rate, other as volatility

Domainrates

Two independent CIR factors. The 2D square-root analog of G2++.

HestonLevel 2

$dS_t = \mu_0\, S_t\, dt + \sqrt{v_t}\, S_t\, dW^S_t$
$dv_t = \kappa(\theta - v_t)\,dt + \xi\sqrt{v_t}\, dW^v_t,\quad \text{corr}(dW^S, dW^v) = \rho$

Label$S_t$ stock, $v_t$ instantaneous variance

Domainequity volFX volindex options

Stock equation is GBM with stochastic vol. Variance equation is exactly CIR — same $(\mu,\sigma)$ pair as the rates model, with $X$ relabeled as variance.

SABRLevel 2

$dF_t = \alpha_t\, F_t^{\beta}\, dW^F_t$
$d\alpha_t = \nu\, \alpha_t\, dW^\alpha_t,\quad \text{corr}(dW^F, dW^\alpha) = \rho$

Label$F_t$ forward (rate or price), $\alpha_t$ stochastic volatility

DomainratesFXequity

CEV-type forward with lognormal stochastic vol. The market standard for swaption and caplet smiles; also widely used in FX.

Chen (three-factor)Level 2 ($n=3$)

$dr_t = \kappa_r(\theta_t - r_t)\,dt + \sqrt{v_t}\, dW^r_t$
$d\theta_t = \kappa_\theta(\bar{\theta} - \theta_t)\,dt + \sigma_\theta\sqrt{\theta_t}\, dW^\theta_t$
$dv_t = \kappa_v(\bar{v} - v_t)\,dt + \sigma_v\sqrt{v_t}\, dW^v_t$

Label$r_t$ short rate, $\theta_t$ stochastic long-run mean, $v_t$ stochastic volatility

Domainrates

Three-factor affine model with level, central tendency, and volatility as separate factors — matches the empirical three-factor PCA structure of yield curves.

Level 3 — Itô SDEs with jumps

Merton (1976) jump-diffusionLevel 3

$dS_t = \mu_0\, S_{t^-}\, dt + \sigma_0\, S_{t^-}\, dW_t + S_{t^-}(e^{Y} - 1)\, dN_t$

Jumps$N_t$ Poisson with intensity $\lambda$; $Y \sim \mathcal{N}(\mu_J, \sigma_J^2)$

Labelstock price $S_t$

DomainequityFX

GBM plus i.i.d. lognormal jumps. Closed-form option prices via a Poisson-weighted sum of Black-Scholes terms.

KouLevel 3

FormMerton structure with double-exponential log-jump distribution

Labelstock price $S_t$

Domainequitycredit (KMV-style)

Asymmetric, fat-tailed jumps. Better fit to observed return distributions than Merton's symmetric Gaussian jumps.

BatesLevel 3 (2D, jumps in $S$)

$dS_t = \mu_0\, S_{t^-}\, dt + \sqrt{v_t}\, S_{t^-}\, dW^S_t + S_{t^-}(e^Y - 1)\, dN_t$
$dv_t = \kappa(\theta - v_t)\, dt + \xi\sqrt{v_t}\, dW^v_t$

Label$S_t$ stock, $v_t$ variance

Domainequity volindex options

Heston plus Merton-style jumps in the stock. The standard "smile + crash" extension for equity index options like SPX.

Affine jump-diffusion (Duffie-Pan-Singleton)Level 3, general $n$

Formdrift and variance affine in $X$; jump intensity affine in $X$

Labeldeliberately abstract

Domainratesequity volcreditcommodity

The general framework that contains CIR, Vasicek, Heston, Bates, and their multi-factor jump cousins. Pricing reduces to Fourier inversion of an exponential-affine characteristic function.

Level 4 — path-dependent / non-Markovian

HJM (Heath-Jarrow-Morton)Level 4 (or Level 2 in curve form)

$df(t,T) = \alpha(t,T)\, dt + \sigma(t,T)\, dW_t$
$\alpha(t,T) = \sigma(t,T)\int_t^T \sigma(t,u)\, du$ (no-arbitrage drift)

Label$f(t,T)$ instantaneous forward rate

Domainratescredit (HJM-style spread curves)

The short rate $r_t = f(t,t)$ is generally non-Markovian — a functional of the path of $W$. Natural state is the entire forward curve.

LIBOR Market Model (BGM)Level 2 / 4

$dL_k(t) = \mu_k(L, t)\, L_k\, dt + \sigma_k(t)\, L_k\, dW^k_t$

Label$L_k(t)$ forward LIBOR for accrual period $[T_k, T_{k+1}]$

Domainrates

Each forward rate is lognormal under its natural measure; drifts under a common measure depend on the other LIBORs. The market standard for cap, floor, and Bermudan-swaption pricing.

Rough HestonLevel 4

$v_t = v_0 + \dfrac{1}{\Gamma(H+\tfrac{1}{2})}\!\int_0^t (t-s)^{H-\tfrac{1}{2}}\!\bigl[\kappa(\theta - v_s)\, ds + \nu\sqrt{v_s}\, dW^v_s\bigr]$
$dS_t = \sqrt{v_t}\, S_t\, dW^S_t,\quad H < \tfrac{1}{2}$

Label$S_t$ stock, $v_t$ variance

Domainequity vol

Variance is a stochastic Volterra equation — depends on the whole path of $W^v$. No finite-dimensional Markov state. Captures the empirical "roughness" of realized vol better than Heston.

Rough BergomiLevel 4

$v_t = \xi_0(t)\, \exp\!\bigl(\eta\, W^H_t - \tfrac{1}{2}\eta^2\, t^{2H}\bigr)$
$dS_t = \sqrt{v_t}\, S_t\, dW^S_t$

Label$S_t$ stock, $v_t$ variance

Domainequity vol

$W^H$ is fractional Brownian with Hurst $H<\tfrac{1}{2}$. Forward variance curve specified explicitly; popular for SPX vol surface fitting.