Stochastic calculus

Calculus along a random path. lawler-stochastic-calculus frames it as one change to ordinary calculus: where a differential equation says df = C(t, f(t)) dt — the function moves along a straight line of known slope for an infinitesimal time — a stochastic differential equation says

dXₜ = m(t, Xₜ) dt + σ(t, Xₜ) dBₜ

which reads as: at time t, the process is evolving like a brownian-motion with drift m and variance σ², both allowed to depend on where it currently is.

The integral comes first

In ordinary calculus one defines the derivative, then the integral, then relates them. Here the derivative is not easily defined at all — a Brownian path has none — so the subject is built on the integral instead: ∫₀ᵗ Aₛ dBₛ, defined for simple processes and extended by limits, and Xₜ is called a solution of the equation above when it equals the corresponding sum of integrals. Working backwards from the integral is a structural choice, not a presentational one.

The integral inherits the properties one would want — linearity, additivity over intervals, continuous paths — plus the variance rule Var[Zₜ] = ∫₀ᵗ E[Aₛ²] ds, and when that variance is finite the integral is a square-integrable martingale. When it is not, the object still exists but the martingale property can fail, which is the local-martingale distinction Lawler takes up in chapter 4.

Itô's formula

The chain rule of the subject, and the place the extra term appears. For f twice continuously differentiable and Bₜ a standard Brownian motion:

f(Bₜ) = f(B₀) + ∫₀ᵗ f′(Bₛ) dBₛ + ½ ∫₀ᵗ f″(Bₛ) ds

or in differential form df(Bₜ) = f′(Bₜ) dBₜ + ½ f″(Bₜ) dt. The second-order term survives for the reason recorded on brownian-motion: quadratic variation over [0,t] is t, not zero, so the y² term of the Taylor expansion contributes at first order instead of vanishing. Ordinary calculus is the special case where the path is smooth enough for that term to disappear.

Read it as a statement about processes rather than functions: Yₜ = f(Bₜ) is itself a Brownian motion with drift f″(Bₜ)/2 and variance f′(Bₜ)². Applying a function to a fair game generally does not leave a fair game, and the formula says exactly how much it bends.

What is built on it

• Diffusions and Feynman-Kac — the bridge to partial differential equations: an expectation over paths solves a PDE, and a PDE solution can be computed by simulating paths.
• Change of measure and Girsanov — how to give a Brownian motion a different drift by changing the probability measure rather than the process, which is the engine of risk-neutral pricing (martingale).
• Jump processes — the same integration theory rebuilt for Lévy and compound Poisson processes, where the path is not continuous.
• Numerics — the stochastic Euler method, X((k+1)Δt) = X(kΔt) + Δt·m + √Δt·σ·N, with N standard normal. The √Δt is Brownian scaling. Most SDEs cannot be solved in closed form, and Lawler's exercises expect the reader to simulate.

The Black-Scholes derivation as a worked example

Recorded in full on lawler-stochastic-calculus. In outline: a self-financing portfolio of stock and bond, rebalanced continuously to match the option payoff at expiry; Itô's formula applied to its value; the dBₜ terms matched to give the hedge ratio aₜ = f′(t, Sₜ); the dt terms matched to give the pricing PDE. The drift of the stock disappears, and the resulting equation is the naive discounted-expectation one with the drift replaced by the risk-free rate.

That is the mathematics behind what ../quant-trading-wiki records as finance: black-scholes's hedge argument, the delta of options-greeks, and the observation on option-pricing that the expected return on the stock is not among the formula's inputs.

Related

stochastic-processes · brownian-motion · martingale · probability-theory · real-analysis · lawler-stochastic-calculus · gamarnik-advanced-stochastic-processes · mathematics · synthesis