Martingales and discrete conditional pricing

Definition Fix a finite sample space $\Omega$, a physical probability $\mathbb{P}$ with $\mathbb{P}(\omega) > 0$ for every state $\omega$, and trading dates $t = 0, 1, \dots, T$. A filtration is a family of $\sigma$-algebras on $\Omega$ with

where $\mathcal{F}_t$ is the information available at date $t$. On a finite $\Omega$ every $\sigma$-algebra is generated by a partition of $\Omega$ into nonempty blocks, its atoms, and the inclusions say that each block of $\mathcal{F}_{t+1}$ lies inside a single block of $\mathcal{F}_t$. In a tree the blocks of $\mathcal{F}_t$ are the nodes at date $t$. A process $X = (X_t)$ is adapted if each $X_t$ is $\mathcal{F}_t$-measurable, which on a finite $\Omega$ means that $X_t$ is constant on every block of $\mathcal{F}_t$. Throughout, $\mathcal{F}_0 = \{\emptyset, \Omega\}$, so every adapted quantity at date $0$ is a number.

Definition Let the traded assets have adapted price processes $S^0, S^1, \dots, S^d$, where $S^0_t = B_t$ is the money-market account. A trading strategy is a process $\theta_t = (\theta^0_t, \dots, \theta^d_t)$ for $t = 1, \dots, T$, where $\theta^i_t$ is the number of units of asset $i$ held over the interval from date $t - 1$ to date $t$. The strategy is predictable if each $\theta_t$ is $\mathcal{F}_{t-1}$-measurable, so the holding for an interval is fixed with the information available when the interval starts. Holdings are real numbers of either sign, so short positions and borrowing are allowed. For assets without dividends the value of the strategy is $V_0 = \theta_1^{\top} S_0$ and $V_t = \theta_t^{\top} S_t$ for $t \ge 1$, and the strategy is self-financing if

for $t = 1, \dots, T - 1$, so that rebalancing at date $t$ neither adds nor withdraws cash. Predictability is part of what makes a strategy admissible in the arbitrage model. A rule whose holdings depend on the move that is about to happen is not a strategy of the model, whatever profit it shows on paper.

Definition Let $\mathbb{Q}$ be a probability on $\Omega$ with $\mathbb{Q}(\omega) > 0$ for every state, as for a pricing measure equivalent to a full-support $\mathbb{P}$, and let $(\mathcal{F}_t)$ be a filtration. An adapted process $M = (M_t)_{t=0}^{T}$ is a $\mathbb{Q}$-martingale with respect to $(\mathcal{F}_t)$ if $\mathbb{E}^{\mathbb{Q}}[\lvert M_t \rvert] < \infty$ for every $t$ and

for $t = 0, \dots, T - 1$. On a finite $\Omega$ integrability is automatic, and the condition says that on every block $A$ of $\mathcal{F}_t$ the $\mathbb{Q}$-weighted average of $M_{t+1}$ over $A$ equals the value of $M_t$ there,

Repeated use of the tower property gives $\mathbb{E}^{\mathbb{Q}}[M_s \mid \mathcal{F}_t] = M_t$ whenever $t \le s$, and since $\mathcal{F}_0$ is trivial, $\mathbb{E}^{\mathbb{Q}}[M_t] = M_0$ for every $t$. The property depends on the measure and on the filtration. It does not require independent increments, bounded variance or limited losses, and a constant unconditional mean is not enough, because the identity must hold on every block.

Definition The money-market account $B$ is a strictly positive adapted process with $B_0 = 1$, the value at date $t$ of one unit of currency deposited at date $0$ and rolled over at the one-period rates. When the rate of each period is known at its start, $B_t$ is $\mathcal{F}_{t-1}$-measurable, and it is still random when rates are random. The discounted price process of an asset with price $S_t$ is

the value of the asset measured in units of the account. A pricing measure for the numeraire $B$ is a probability $\mathbb{Q}$ equivalent to $\mathbb{P}$ under which the discounted price process of every traded asset without dividends is a $\mathbb{Q}$-martingale,

When the account grows by a constant gross factor $R$ per period, so that $B_t = R^t$, this says $\mathbb{E}^{\mathbb{Q}}[S_{t+1} \mid \mathcal{F}_t] = R\,S_t$, and a positive price is not itself a $\mathbb{Q}$-martingale unless $R = 1$. Assets that pay dividends need the gain process of the last card.

Definition Assume a finite frictionless market with a strictly positive money-market account $B$ and a pricing measure $\mathbb{Q}$ for $B$. Let $X$ be a payoff in currency at date $T$ that is $\mathcal{F}_T$-measurable and attainable, which means that some self-financing predictable strategy has value $X$ at $T$. In a complete market such as the binomial tree every payoff is attainable. The arbitrage-free value of $X$ at date $t$, for $t = 0, \dots, T$, is then

so that $V_T = X$. Inside the expectation the payoff is measured in account units at $T$, and multiplying by $B_t$ converts the result to currency at $t$. When rates are random, $B_T$ is random and must stay inside the expectation. It is the value of every replicating strategy, and in a tree it agrees with backward induction one step at a time. For a claim that is not attainable, the same formula under one chosen $\mathbb{Q}$ gives one consistent value among many, as Market completeness and nonunique prices explains.

The discounted value is a $\mathbb{Q}$-martingale. Write $Y = X/B_T$, so that $V_t/B_t = \mathbb{E}^{\mathbb{Q}}[Y \mid \mathcal{F}_t]$. For an integrable $Y$ and $\sigma$-algebras $\mathcal{G} \subseteq \mathcal{H}$ the tower property says

and with $\mathcal{H} = \mathcal{F}_{t+1}$ and $\mathcal{G} = \mathcal{F}_t$ it gives

The proof uses only the integrability of $Y$, automatic on a finite $\Omega$, and the nesting of the information. It does not make the undiscounted value $V_t$ a martingale, and it says nothing about $\mathbb{P}$.

Definition Let an asset have ex-dividend price $S_t$ and pay the dividend $D_t$ at dates $t = 1, \dots, T$, where $D_t$ is an $\mathcal{F}_t$-measurable amount in currency, possibly random or negative, and $S_t$ is the price just after $D_t$ is paid. Its discounted cumulative gain process is

for $t = 0, \dots, T$, so $G_0 = S_0$. Each dividend is divided by the account value on its own payment date. A pricing measure for $B$ makes the gain process of every traded asset a $\mathbb{Q}$-martingale, which over one step reads

The reason is a self-financing strategy. Hold one unit of the asset and deposit each dividend in the account when it is paid. The dividend $D_k$ buys $D_k/B_k$ units of the account, a number that never changes afterward, so at date $t$ the position is worth

in currency. Its discounted value is $G_t$, which without dividends reduces to $S_t/B_t$. The discounted ex-dividend price alone falls in conditional $\mathbb{Q}$-mean by $\mathbb{E}^{\mathbb{Q}}[D_{t+1}/B_{t+1} \mid \mathcal{F}_t]$ over each step, so with positive dividends it is not a $\mathbb{Q}$-martingale.