Martingales (Discrete Time)

Motivation: why this matters in quant finance

A martingale is the mathematical formalisation of a "fair game": a sequence of random variables whose expected next value, given everything known so far, equals the current value. If you could predict the direction of a martingale's next step, it would not be a fair game — hence, no edge, no arbitrage, no free lunch. Every arbitrage-pricing theorem in quant finance ultimately reduces to "discounted asset prices are martingales under an appropriate measure."

Discrete-time martingales are where this intuition is cleanest to state and cleanest to prove. Before tackling continuous-time stochastic integration, every quant should master the discrete case: it sets the definitions (adapted, filtration, conditional expectation), it makes the computations explicit (a sum over a binomial tree, not a stochastic integral), and it is the exact framework underlying binomial-tree option pricing, the workhorse technique for American options and path-dependent exotics. The Cox-Ross-Rubinstein model is a discrete-time martingale under

\mathbb{Q}

; its limit is the risk-neutral continuous-time model that underlies Black-Scholes.

The discrete-time setting also delivers, for free, two of the deepest theorems in probability: the martingale convergence theorem (a bounded martingale converges a.s.) and the optional stopping theorem (you cannot cheat a fair game by timing your exit — more on this in the dedicated OST lesson). Neither has a cleaner entry point than here.

The informal idea

A sequence

(M_n)_{n \ge 0}

is a martingale with respect to a filtration

(\mathcal{F}_n)_n

if, at every step,

\mathbb{E}[M_{n+1} \mid \mathcal{F}_n] = M_n.

Three words tell the whole story: expected next equals current.

Read the left-hand side as "my best forecast of tomorrow's $M$ , given everything I know today." The martingale property says this forecast is exactly $M_n$ — not higher, not lower. Any edge you could exploit (a predictable drift) would violate it.

Two cousins appear constantly:

Submartingale: $\mathbb{E}[M_{n+1} \mid \mathcal{F}_n] \ge M_n$ — "tends to drift up." Asset prices under the real-world measure usually behave this way.
Supermartingale: $\mathbb{E}[M_{n+1} \mid \mathcal{F}_n] \le M_n$ — "tends to drift down." Short-position P&L on a positive-drift asset is super.

Sub and super are not as memorable as they sound: a submartingale increases on average ("supports from below") while a supermartingale decreases on average ("supports from above"). The terminology is a notational curiosity, not a mnemonic aid.

Formal definition

Fix a probability space

(\Omega, \mathcal{F}, \mathbb{P})

and a filtration

(\mathcal{F}_n)_{n \ge 0}

— an increasing sequence of

\sigma

-algebras,

\mathcal{F}_n \subseteq \mathcal{F}_{n+1}

. A sequence of random variables

(M_n)_{n \ge 0}

is a $(\mathcal{F}_n, \mathbb{P})$ -martingale if:

Adapted: Each $M_n$ is $\mathcal{F}_n$ -measurable. (Equivalently: $M_n$ is known given the information at time $n$ .)
Integrable: $\mathbb{E}[|M_n|] < \infty$ for every $n$ .
Martingale property: $\mathbb{E}[M_{n+1} \mid \mathcal{F}_n] = M_n$ almost surely, for every $n \ge 0$ .

Equivalent formulation: for every

m \le n

\mathbb{E}[M_n \mid \mathcal{F}_m] = M_m

a.s. (proved by iterating the one-step property using the tower property of conditional expectation).

Submartingales and supermartingales

Replace "

=

" in (3) by "

\ge

" to get a submartingale, by "

\le

" to get a supermartingale. A martingale is both; the reverse is not implied.

Elementary consequences

Constant expectation: For a martingale, $\mathbb{E}[M_n] = \mathbb{E}[M_0]$ for all $n$ . Proof: take unconditional expectations of the martingale property.
Jensen's inequality for convex functions: If $M_n$ is a martingale and $\phi$ is convex with $\mathbb{E}[|\phi(M_n)|] < \infty$ , then $(\phi(M_n))_n$ is a submartingale. (In particular, $M_n^2$ is a submartingale when it is integrable.)
Linear combinations: Sums of martingales (relative to the same filtration) are martingales.

Canonical examples

1. Simple symmetric random walk

Let $X_1, X_2, \ldots$ be i.i.d. with $\mathbb{P}(X_i = \pm 1) = 1/2$ , and let $S_n = X_1 + \cdots + X_n$ , $S_0 = 0$ , with filtration $\mathcal{F}_n = \sigma(X_1, \ldots, X_n)$ .

\mathbb{E}[S_{n+1} \mid \mathcal{F}_n] = S_n + \mathbb{E}[X_{n+1} \mid \mathcal{F}_n] = S_n + \mathbb{E}[X_{n+1}] = S_n.

A biased walk with $\mathbb{P}(X_i = +1) = p > 1/2$ is a submartingale (drifts up); a biased walk with $p < 1/2$ is a supermartingale.

2. Product-of-mean-one

Let $Y_1, Y_2, \ldots$ be independent with $\mathbb{E}[Y_i] = 1$ , and set $M_n = Y_1 Y_2 \cdots Y_n$ , $M_0 = 1$ .

\mathbb{E}[M_{n+1} \mid \mathcal{F}_n] = M_n\cdot \mathbb{E}[Y_{n+1}] = M_n.

This is a martingale. The Doléans-Dade exponential (GBM's $\exp(\sigma W_t - \tfrac{1}{2}\sigma^2 t)$ ) is the continuous-time relative.

3. Gambler's wealth under a fair game

A gambler starts with wealth $X_0$ . At each round they bet some amount $B_n$ (a predictable wager, i.e. $B_n$ is $\mathcal{F}_{n-1}$ -measurable) on a fair coin, winning or losing $B_n$ with equal probability. Then

X_{n+1} = X_n + B_n\cdot \xi_{n+1}, \qquad \xi_{n+1} \in \{\pm 1\}, \mathbb{E}[\xi_{n+1}] = 0,

and the predictability of $B_n$ gives:

\mathbb{E}[X_{n+1} \mid \mathcal{F}_n] = X_n + B_n\cdot \mathbb{E}[\xi_{n+1}] = X_n.

(X_n)

is a martingale. The profound lesson: no betting strategy turns a fair game into a profitable one. You can choose

B_n

however cleverly you like (bet more after wins, bet less near ruin, double-down Martingale strategy) — all of them preserve the martingale property, so

\mathbb{E}[X_n] = X_0

forever.

4. Discounted binomial stock price under $\mathbb{Q}$

In the Cox-Ross-Rubinstein tree, $S_{n+1} = S_n\cdot u$ or $S_n\cdot d$ ( $u > 1 > d > 0$ ) with risk-neutral probabilities $p^* = (1 + r - d)/(u - d)$ and $1 - p^*$ . Then the discounted price $\tilde S_n = e^{-rn\Delta t}S_n$ satisfies:

\mathbb{E}^{\mathbb{Q}}[\tilde S_{n+1} \mid \mathcal{F}_n] = \tilde S_n.

This is the fundamental theorem of asset pricing in discrete time: a market is arbitrage-free iff there exists a measure under which discounted prices are martingales.

Martingale transforms and the fair-game principle

Martingale transform. Given a martingale

(M_n)_n

and a predictable sequence

(H_n)_n

(i.e.

H_n

\mathcal{F}_{n-1}

-measurable), define

(H \cdot M)_n = \sum_{k=1}^n H_k(M_k - M_{k-1}), \qquad (H \cdot M)_0 = 0.

This is a discrete analogue of the stochastic integral. If

(H_n)

is bounded (or more generally, makes everything integrable), then

(H \cdot M)_n

is a martingale — a result often called the discrete Itô integral theorem.

Financially, $H_n$ is "how much of asset $M$ you hold between times $n-1$ and $n$ ." The martingale transform is the cumulative P&L of that position. The theorem says: no predictable trading strategy on a martingale can generate expected profit. Every risk-neutral pricing argument uses this fact.

Key theorems in discrete time

Martingale convergence theorem (Doob)

If $(M_n)$ is an $L^1$ -bounded martingale (i.e. $\sup_n \mathbb{E}[|M_n|] < \infty$ ), then $M_n$ converges almost surely to an integrable random variable $M_\infty$ . Same holds for $L^1$ -bounded super- and submartingales.

Proof idea (upcrossing inequality). Count the number of times the trajectory crosses a strip

[a, b]

; the expected number is bounded by

\mathbb{E}[(M_n - a)^+]/(b-a)

. Finite crossings between every pair of rationals

\Rightarrow

no oscillation

\Rightarrow

convergence.

Practical content. A bounded-in-

L^1

martingale must settle down. This is why Galton-Watson branching processes with fair reproduction either die out or explode (they cannot fluctuate forever), and why gambler's-ruin problems have clean terminal distributions.

Optional stopping theorem (light version)

(M_n)

is a martingale and

\tau

is a bounded stopping time (i.e.

\tau \le K

for some constant), then

\mathbb{E}[M_\tau] = \mathbb{E}[M_0].

The full theorem, under various integrability / boundedness hypotheses, is the subject of the dedicated Optional Stopping Theorem lesson. The one-line summary: you cannot beat a fair game by choosing a clever exit time.

Doob's maximal inequality

For a non-negative submartingale $(M_n)$ and $\lambda > 0$ :

\mathbb{P}\!\left(\max_{0 \le k \le n} M_k \ge \lambda\right) \le \frac{\mathbb{E}[M_n]}{\lambda}.

This gives pathwise tail bounds for martingale maxima — far stronger than marginal Chebyshev, because it controls the whole trajectory with just the terminal expectation.

Worked example — binomial-tree option price as a discounted-payoff martingale

Under risk-neutral $\mathbb{Q}$ in a CRR tree, the fair price of any $\mathcal{F}_N$ -measurable contingent claim $C_N$ is:

C_0 = \mathbb{E}^{\mathbb{Q}}[e^{-rT}C_N].

Equivalently, the discounted value process $\tilde V_n = e^{-rn\Delta t}\mathbb{E}^{\mathbb{Q}}[e^{-r(N-n)\Delta t}C_N \mid \mathcal{F}_n]$ is a $\mathbb{Q}$ -martingale, and its terminal value is $\tilde C_N$ . The backward-induction pricing algorithm:

V_n = e^{-r\Delta t}\left(p^* V_{n+1}^u + (1 - p^*) V_{n+1}^d\right), \qquad V_N = C_N,

is exactly the recursion implied by the martingale property. Pricing an option = computing a conditional expectation of a discounted payoff. This is the cleanest vanishing point to see why the abstract definition matters.

Common confusions and pitfalls

"A martingale has no trend." It has no trend given past information. The martingale

\mathbb{E}[Y \mid \mathcal{F}_n]

for some eventual

Y

can look wildly trendy before information arrives — the forecast updates, the updated forecast is the martingale. No trend conditional, possibly strong trend unconditional.

"Stock prices are martingales in the real world." They are martingales under an artificial probability measure

\mathbb{Q}

— the risk-neutral measure — chosen to make discounted prices fair. Under the real-world measure

\mathbb{P}

, prices have a positive drift (that is the whole point of investing). Confusing the two collapses the distinction between the real and risk-neutral economies.

"Martingales converge." Only if they are bounded in

L^1

. An unbounded martingale can oscillate forever — e.g. the symmetric random walk, which is a martingale and has a.s. unbounded oscillation.

"You can beat a martingale with a smart betting strategy." Famously: the "martingale strategy" (double your bet after every loss). Under a strict bankroll and finite number of rounds it is not a martingale but a supermartingale (costs ruin probability

\approx 1/2^N

); the expected P&L is indeed zero. Doubling up is not a free-money generator — it converts variance into a small chance of catastrophic loss.

"Submartingales are special martingales." They are a weaker class. Every martingale is a submartingale, but most submartingales are not martingales. Know the direction.

"Adapted = measurable." Adapted means

\mathcal{F}_n

-measurable for the specific filtration $(\mathcal{F}_n)$ of interest. A process can be adapted to a coarse filtration but not a fine one, or vice versa. Always say "adapted to which filtration."

Where this goes next

Stopping Times: The formal notion of "a strategy that decides when to exit." Required to state the optional stopping theorem.
Optional Stopping Theorem: The fair-game principle, in sharpened form. Provides the machinery to solve gambler's-ruin problems and boundary-crossing probabilities in closed form.
Brownian Motion: The continuous-time martingale par excellence. Every theorem in this lesson has a Brownian analogue.
Itô's Lemma: The continuous-time analogue of "functions of martingales are (usually) not martingales" — and the formula that tells you exactly which drift correction is needed to make them martingales again.
Martingale I and Martingale II: Applications to arbitrage pricing and the fundamental theorem of asset pricing.
Change of Measure: The technique that turns the real-world drift of a stock price into the risk-free rate under $\mathbb{Q}$ , making discounted prices martingales.

References

Lawler, G. F. (2023). Stochastic Calculus: An Introduction with Applications. Ch. 1 §1.2 (Martingales), §1.3 (Optional sampling theorem), §1.4 (Martingale convergence theorem), §1.5 (Square integrable martingales).