Stochastic calculus

5 lessons · 10 equations. Each lesson below gives its formulas and key rules; open the lesson for the full explanation.

Brownian motion: the defining properties and what follows

Building a Brownian bridge

Bt=Wt−tTWT,0≤t≤TB_t = W_t - \frac{t}{T}W_T,\quad 0 \le t \le T

Subtracting the straight line to the endpoint pins the path to zero at TT. The result has covariance s(T−t)/Ts(T - t)/T for s≤ts \le t, and it is how Monte Carlo engines fill in a path between points already simulated — the basis of the Brownian-bridge construction for quasi-random sampling.

Remember

  • Three properties: starts at zero, normal increments with variance equal to elapsed time, independent increments.

Hitting times, the reflection principle and the running maximum

Hitting one level before another

Pr⁡(hit b before −a)=aa+b\Pr(\text{hit } b \text{ before } -a) = \frac{a}{a + b}

For driftless Brownian motion started at zero between levels −a-a and bb. The probability is the distance to the *other* barrier over the total width.

The first-passage distribution

Pr⁡(τa≤t)=2(1−Φ ⁣(at))\Pr(\tau_a \le t) = 2\left(1 - \Phi\!\left(\frac{a}{\sqrt t}\right)\right)

For driftless standard Brownian motion and a level a>0a > 0 — the reflection principle restated as a statement about when the level is first reached. Differentiating gives the density a2πt3e−a2/2t\frac{a}{\sqrt{2\pi t^3}}e^{-a^2/2t}, whose heavy t−3/2t^{-3/2} tail makes the mean infinite.

The maximum and the endpoint together

Pr⁡(max⁡s≤tWs≥m, Wt≤w)=Pr⁡(Wt≥2m−w),w≤m\Pr\Big(\max_{s \le t} W_s \ge m,\ W_t \le w\Big) = \Pr(W_t \ge 2m - w),\quad w \le m

Reflecting after the first touch of mm maps paths that end below ww onto paths that end above 2m−w2m - w. This joint law is what prices barrier options with a strike: it counts paths that touched the barrier and still finished in the money.

Remember

  • Reflection: Pr⁡(max⁡s≤tWs≥a)=2Pr⁡(Wt≥a)\Pr(\max_{s\le t}W_s \ge a) = 2\Pr(W_t \ge a).

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

Itô’s lemma

df=(∂f∂t+μ∂f∂x+12σ2∂2f∂x2)dt+σ∂f∂x dWdf = \left(\frac{\partial f}{\partial t} + \mu\frac{\partial f}{\partial x} + \frac{1}{2}\sigma^2\frac{\partial^2 f}{\partial x^2}\right)dt + \sigma\frac{\partial f}{\partial x}\,dW

For dX=μ dt+σ dWdX = \mu\,dt + \sigma\,dW. It is a second-order Taylor expansion in which (dX)2(dX)^2 survives because it equals σ2dt\sigma^2 dt.

The Ornstein–Uhlenbeck process

dX=θ(xˉ−X) dt+σ dWdX = \theta(\bar{x} - X)\,dt + \sigma\,dW

The standard mean-reverting model, and the one every pairs-trading question ends up at.

Itô for two correlated processes

d(XY)=X dY+Y dX+dX dY,dW(1)dW(2)=ρ dtd(XY) = X\,dY + Y\,dX + dX\,dY, \qquad dW^{(1)}dW^{(2)} = \rho\,dt

The product rule gains a cross-variation term, which is non-zero only when both processes carry Brownian noise and their drivers are correlated. It is the source of the quanto adjustment and of every correlation term in multi-asset pricing.

Remember

  • (dW)2=dt(dW)^2 = dt is the single difference from ordinary calculus.

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

Bond prices in the Vasicek model

P(t,T)=A(τ) e−B(τ) rt,B(τ)=1−e−κτκ, τ=T−tP(t, T) = A(\tau)\,e^{-B(\tau)\,r_t},\quad B(\tau) = \frac{1 - e^{-\kappa\tau}}{\kappa},\ \tau = T - t

When the short rate follows an OU process, zero-coupon bond prices are exponential-affine in the current rate. B(τ)B(\tau) is the bond’s sensitivity to the short rate — a duration that saturates at 1/κ1/\kappa because mean reversion pulls distant rates back.

Remember

  • GBM solves to S0exp⁡[(μ−σ2/2)T+σWT]S_0\exp[(\mu - \sigma^2/2)T + \sigma W_T] and is lognormal.

Measure change: Girsanov, Feynman–Kac and the fundamental theorems

Feynman–Kac

∂V∂t+μ∂V∂x+12σ2∂2V∂x2−rV=0 ⟺ V=e−rTE[payoff]\frac{\partial V}{\partial t} + \mu\frac{\partial V}{\partial x} + \tfrac{1}{2}\sigma^2\frac{\partial^2 V}{\partial x^2} - rV = 0 \ \Longleftrightarrow \ V = e^{-rT}\mathbb{E}\big[\text{payoff}\big]

An expectation of a diffusion and a parabolic PDE are the same object. Either can be solved and the answer is identical.

Changing numeraire

dQSdQ=ST/S0BT/B0,V0=S0 EQS ⁣[VTST]\frac{d\mathbb{Q}^S}{d\mathbb{Q}} = \frac{S_T/S_0}{B_T/B_0}, \qquad V_0 = S_0\,\mathbb{E}^{\mathbb{Q}^S}\!\left[\frac{V_T}{S_T}\right]

Any positive traded asset can serve as the unit of account; prices divided by it are martingales under its own measure. Using the stock as numeraire turns the asset-or-nothing term of Black–Scholes into S0 QS(ST>K)=S0Φ(d1)S_0\,\mathbb{Q}^S(S_T > K) = S_0\Phi(d_1) in one line.

Remember

  • Girsanov changes the drift and leaves the volatility alone.

Detailed formula cards