Lawler — Stochastic Calculus: An Introduction with Applications

T1 — gregory-lawler (George Wells Beadle Distinguished Service Professor, university-of-chicago), © 2014, last revised 15 February 2023, 260pp, 8 chapters, hosted on the author's own university page. A maintained book given away rather than a set of notes: the file is literally named finbook.pdf, and the finance is where it points.

Routed here rather than to ../quant-trading-wiki by this spoke's boundary rule — the subject is the mathematics, and the option formula is one worked application of it. It is the corpus's first text on stochastic-processes and the source for stochastic-calculus, brownian-motion and martingale.

What it teaches, in order

Martingales in discrete time first (martingale: conditional expectation, optional sampling, the convergence theorem, integrals with respect to a random walk), then brownian-motion as a limit of those walks, built rather than assumed, and examined four ways — as a continuous martingale, as a Markov process, as a Gaussian process, as a self-similar process — with quadratic variation and the heat equation following.

Chapters 3 to 5 are the calculus: the stochastic integral, Itô's formula, diffusions, local martingales, Feynman-Kac, then change of measure and the Girsanov theorem, with the Black-Scholes formula, the martingale approach to pricing and the martingale representation theorem as sections 5.4 to 5.7. The last three chapters go past the finance: jump processes (Lévy, compound Poisson, the Lévy-Khinchin characterisation), fractional Brownian motion, and harmonic functions with complex Brownian motion.

Two books in one file, and a tool assumption

Lawler assumes "a post-calculus course in probability or statistics" and says plainly that a deep understanding needs measure theory. Rather than picking one audience he indents the measure-theoretic discussion in a different font, to be skipped or read. The spoke has seen split audiences before (lal-algebra-1 addressing undergraduates and graduates in one volume); this is the first source that solves it typographically.

He also assumes the reader "can write simple computer programs either using a language like C++ or by using software such as Matlab or Mathematica" — and the exercises use it, asking for a simulated Brownian path at step size Δt = 0.01 and an estimated hitting probability to compare against the reflection principle. On the spoke's computational tool assumed axis this sits opposite carroll-geometric-function-theory, which assumes none.

The stance on modelling, stated before any mathematics

The introduction sets out how applied mathematics is supposed to work — a model simplifies a real phenomenon, analysis derives consequences, predictions get checked, and a bad prediction means either the analysis or the model was wrong — and closes it with the line the rest of the book keeps earning:

The user of mathematics does not always need to know the details of the mathematical analysis, but it is critical to understand the assumptions in the model. No matter how precise or sophisticated the analysis is, if the assumptions are bad, one cannot expect a good answer.

This is a mathematician saying, in the preface of a finance-facing text, that the modelling assumption is the load-bearing part. It is the same warning ../quant-trading-wiki records from the other end, where black-scholes lists seven ideal conditions and pseudo-mathematics-financial-charlatanism shows what happens when nobody checks them.

What the Black-Scholes chapter actually derives

Worth stating precisely, because it is the mathematical content behind a formula that spoke holds as a finance result.

Stock price follows a geometric Brownian motion dSₜ = Sₜ[m dt + σ dBₜ] with a risk-free bond dRₜ = r Rₜ dt. Pricing the call at its discounted expected payoff is not the Black-Scholes answer, and Lawler says why: sell at that price, buy a bond with the money, and there is positive probability of losing. Instead the price is the value of a self-financing portfolio of stock and bond, rebalanced so that its value at expiry equals the payoff whatever the stock does. Apply Itô's formula to f(t, Sₜ), match the dBₜ terms, and the stock holding falls out as aₜ = f′(t, Sₜ) — the hedge ratio. Match the dt terms and the Black-Scholes equation appears.

Two observations he draws from it:

• The drift m is gone. Not simplified away: geometric Brownian motions with the same σ and different m are mutually absolutely continuous, so they agree about which events have probability one, and a hedge that works with probability one cannot depend on m. This is the mathematical content of the fact ../quant-trading-wiki records as the formula's most useful feature — the expected return on the stock does not appear in it.
• The equation is the naive one with m replaced by r. Which is what "risk-neutral pricing" means: the price is an expectation, but under a measure in which the stock drifts at the risk-free rate rather than its own.

Then the qualifier, in his own parenthesis: given r and σ "(which from a practical perspective is a big assumption), one can just plug into this formula."

Related

stochastic-processes · stochastic-calculus · brownian-motion · martingale · markov-chain · probability-theory · gregory-lawler · university-of-chicago · gamarnik-advanced-stochastic-processes · wu-introduction-stochastic-processes · mathematics · synthesis