Affine Jump-Diffusion Models

Heston's stochastic volatility, Vasicek's interest rate, and Merton's jump-diffusion look like three unrelated models built by three different people for three different problems. They share a hidden structural feature that turns out to matter more than any of their surface differences: in every one of them, the drift, the variance, and the jump intensity are all straight-line — "affine" — functions of the current state. That single shared shape is exactly what makes each of them solvable in closed form, and it's why an entire family of new models keeps getting built the same way on purpose.

A recipe that scales in a straight line

Most recipes don't scale simply — doubling a cake recipe doesn't just double every quantity, because baking time and pan size interact in complicated ways. An affine recipe is the rare one that does scale simply: double the base quantity and every ingredient amount doubles too, with maybe one fixed extra pinch of salt added regardless of size. That linear-plus-constant structure is trivial to reason about no matter how large the batch gets. Affine models are built the same way: every rate governing the process is (a constant) plus (a constant times the current state) — nothing more exotic — and that keeps the whole machine solvable at any scale.

$$dX_t = \left(a_0 + a_1 X_t\right) dt + \sqrt{b_0 + b_1 X_t}\; dW_t + dJ_t, \qquad \lambda(X_t) = c_0 + c_1 X_t$$

In plain English: the drift is a constant $a_0$ plus a constant times the state, the variance under the square root is likewise constant-plus-linear, and even the rate at which jumps arrive, $\lambda$, is constant-plus-linear in the state. Nothing here is quadratic, exponential, or otherwise curved in $X_t$ — every governing rate is "affine," meaning linear plus a constant. That specific restriction is what makes the model's characteristic function solvable via a system of ordinary differential equations rather than requiring a full, generally-unsolvable partial differential equation.

Worked example 1: recognising the affine structure in Heston. Heston's variance process is $dv_t = \kappa(\theta - v_t)\,dt + \xi\sqrt{v_t}\,dW_t$. Rewrite the drift as $\kappa\theta - \kappa v_t$ — that's $a_0=\kappa\theta$ and $a_1=-\kappa$, both constants, times $1$ and $v_t$ respectively: affine. Take $\kappa=2$, $\theta=0.04$: the drift at $v_t=0.09$ (30% vol) is $2(0.04-0.09)=-0.10$, pulling variance down, while at $v_t=0.01$ (10% vol) it's $2(0.04-0.01)=0.06$, pulling variance up — in both cases a simple straight-line function of the current level, never a curve.

That mean-reverting pull toward $\theta$ is exactly the affine drift $\kappa(\theta-v_t)$ in motion — watch how it's always pulling proportionally back toward the same level, never bending the pull itself as the level changes.

Worked example 2: what breaks the affine property. Compare that to a hypothetical drift like $\kappa(\theta - v_t^2)$ (quadratic in $v_t$) or $\kappa\sinh(v_t)$ — plug in $v_t=0.09$ and $v_t=0.01$ and the ratio of drift responses no longer matches a straight line at all; $v_t^2$ at 0.09 is $0.0081$ versus $0.0001$ at $v_t=0.01$, an 81-times difference for a 9-times difference in $v_t$, whereas the affine drift above scales exactly proportionally. Once that straight-line relationship breaks, the elegant closed-form characteristic function breaks with it, and pricing falls back to slower numerical methods like Monte Carlo or finite differences.

What this means in practice

Once a model is recognised as affine, its characteristic function can be found by solving a small system of ordinary differential equations (the Riccati equations from Duffie-Pan-Singleton) instead of a full PDE, and priced efficiently through Characteristic Function Pricing and Carr-Madan FFT Pricing. This is why quant researchers building a new pricing model deliberately try to keep it affine whenever possible — it isn't a mathematical nicety, it's the difference between a model that prices in milliseconds and one that requires a full Monte Carlo run for every quote.