SDEs: geometric Brownian motion, Ornstein–Uhlenbeck and the rest

SC · Chapter 412 min readAsked at Two Sigma, Citadel, Optiver, DE Shaw

Assumes Itô’s lemma and the computations you will be asked for.

After this lesson you should be able to

  • Recognise the standard processes from their SDEs.
  • Solve GBM and OU, and state their stationary behaviour.
  • Say which process suits which financial quantity.

A handful of stochastic differential equations cover nearly all of finance, and each is chosen for a structural reason: prices cannot go negative, spreads must revert, rates must stay positive. Recognising which constraint a process enforces is more useful than memorising its solution.

ProcessSDEEnforcesUsed for
Arithmetic BMdX=μ dt+σ dWdX = \mu\,dt + \sigma\,dWNothingSpreads, which can be negative
Geometric BMdS=μS dt+σS dWdS = \mu S\,dt + \sigma S\,dWPositivityEquity prices
Ornstein–UhlenbeckdX=θ(xˉ−X)dt+σ dWdX = \theta(\bar{x} - X)dt + \sigma\,dWMean reversionSpreads, log-volatility
Cox–Ingersoll–Rossdr=θ(rˉ−r)dt+σr dWdr = \theta(\bar{r} - r)dt + \sigma\sqrt{r}\,dWPositivity and reversionInterest rates
Hestondv=θ(vˉ−v)dt+ξv dWdv = \theta(\bar{v} - v)dt + \xi\sqrt{v}\,dWPositive varianceStochastic volatility
Merton jump-diffusionGBM plus a compound Poisson jumpFat tailsShort-dated skew
Table 4.1 · The named processes. CIR and Heston share a structure: the x\sqrt{x} diffusion vanishes as the process approaches zero, which keeps it positive without an artificial floor.

Derivation 4.2

Solving geometric Brownian motion

Apply Itô to the logarithm, which linearises the equation.

  1. d(ln⁡S)=(μ−σ22)dt+σ dWd(\ln S) = \left(\mu - \tfrac{\sigma^2}{2}\right)dt + \sigma\,dW

    The −σ2/2-\sigma^2/2 is the Itô correction.

  2. ln⁡ST=ln⁡S0+(μ−σ22)T+σWT\ln S_T = \ln S_0 + \left(\mu - \tfrac{\sigma^2}{2}\right)T + \sigma W_T

    Now an ordinary integral, since the right-hand side no longer involves SS.

ST=S0exp⁡ ⁣[(μ−σ22)T+σWT]S_T = S_0\exp\!\left[\left(\mu - \tfrac{\sigma^2}{2}\right)T + \sigma W_T\right]

Proposition 4.3

What the solution says

The log is normal, so the price is lognormal: strictly positive, right-skewed, with a mean above its median. The mean is S0eμTS_0e^{\mu T} exactly, and the median is S0e(μ−σ2/2)TS_0e^{(\mu - \sigma^2/2)T} — the gap between them grows with volatility and is the reason a typical outcome lags the expected one.

Holds when

  • Positivity comes free: an exponential cannot be negative, so no floor is needed.
  • The median lagging the mean is why compounded returns disappoint relative to arithmetic averages.
  • This is why an equity index can have a positive expected return and a most-likely outcome well below it.

Derivation 4.4

Solving Ornstein–Uhlenbeck

Use an integrating factor, exactly as for a linear ODE.

  1. d(eθtXt)=θxˉeθtdt+σeθtdWtd\left(e^{\theta t}X_t\right) = \theta\bar{x}e^{\theta t}dt + \sigma e^{\theta t}dW_t

    The drift term cancels, which is the point of the factor.

  2. Xt=xˉ+(X0−xˉ)e−θt+σ∫0te−θ(t−s)dWsX_t = \bar{x} + (X_0 - \bar{x})e^{-\theta t} + \sigma\int_0^t e^{-\theta(t-s)}dW_s
E[Xt]=xˉ+(X0−xˉ)e−θt,Var∞=σ22θ\mathbb{E}[X_t] = \bar{x} + (X_0-\bar{x})e^{-\theta t}, \qquad \mathrm{Var}_\infty = \frac{\sigma^2}{2\theta}

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SDEs: geometric Brownian motion, Ornstein–Uhlenbeck and the rest · QuantMax