Discounted prices are martingales under Q

Which probability turns the price of an option into an average of its payoff?

The one under which the discounted stock price $\tilde S_t = S_t/(1+r)^t$ is a martingale, a fair game against the bond. Call it $\mathbb Q$. A portfolio that copies an option then starts at the $\mathbb Q$-average of the option's discounted payoff, whatever the real probabilities are.

Before you start · 3 quick checks on discounting, fair games and conditional expectation on a tree, about 3 min

Setup: a binomial market, discounted prices and probabilities on paths

Market. As in step 6: time runs over $n$ periods, $t = 0, 1, \dots, n$. The stock starts at $S_0 \gt 0$, and each period its price is multiplied by $u$ (an up move) or $d$ (a down move), with $0 \lt d \lt u$. The bond grows by the factor $1+r$ each period, with $r \gt -1$ and $d \lt 1+r \lt u$. The market is frictionless, the stock pays no dividends, any real number of shares can be bought or sold short, and anyone can lend or borrow any amount at the rate $r$.
Information. A path is the list of the $n$ moves. $\mathcal F_t$ is the information in the first $t$ moves, and a node at time $t$ is one possible list of the first $t$ moves.
Discounted prices. $\tilde S_t = S_t/(1+r)^t$. A dollar put in the bond at time 0 is worth $(1+r)^t$ dollars at time $t$, so $\tilde S_t$ is the stock price counted in units of that bond.
Probabilities on paths. A probability on the $2^n$ paths has, at each node before time $n$, a conditional probability of an up move. For $0 \lt \pi \lt 1$, $\mathbb P_\pi$ is the probability under which each move is up with probability $\pi$, independently of the others, and $\mathbb E_\pi$ is its expectation. The real probability $\mathbb P$ gives every path a positive probability; nothing else about it is assumed. In stage 1 it is $\mathbb P_p$ for a real up-probability $p$ that you choose.
Martingale measure. A probability $\mathbb Q$ on the paths that gives every path a positive probability and makes $\tilde S$ a martingale: $\mathbb E^{\mathbb Q}[\tilde S_{t+1} \mid \mathcal F_t] = \tilde S_t$ at every node before time $n$. Giving every path positive probability makes $\mathbb Q$ equivalent to the real probability: the two agree on which paths can happen. On a tree one step at every node is enough (step 11). It is also called a risk-neutral measure.
Options. A European option pays only at time $n$ and has no cash flows before then. It can be bought or sold (written) in any amount at one price, with no bid-ask spread, and the market of the stock, the bond and the option has no arbitrage. Then, as in step 6, its value at every node is the cost there of the portfolio that copies it. The call of stage 1 and the call and put of stage 2 are valued this way, at the root and at time 1.
Self-financing strategies (step 15). A strategy holds $\Delta_t$ shares from time $t$ to $t+1$, chosen from the first $t$ moves, and keeps the rest of its value, $V_t - \Delta_t S_t$, in the bond. No cash is added or removed after time 0, so $V_{t+1} = \Delta_t S_{t+1} + (1+r)(V_t - \Delta_t S_t)$. Dividing by $(1+r)^{t+1}$ gives the discounted gains identity, with $\tilde V_t = V_t/(1+r)^t$:

\[ \tilde V_{t+1} - \tilde V_t = \Delta_t\big(\tilde S_{t+1} - \tilde S_t\big). \]

Fine print: conditional expectations on a tree, the rate convention and the unit of account

At a node at time $t$, $\mathbb E^{\mathbb Q}[Y \mid \mathcal F_t]$ is the average of $Y$ over the paths through that node, weighted by their $\mathbb Q$-probabilities. For a value known one period later it is the average of its two next values with weights $\pi$ and $1-\pi$, where $\pi$ is the conditional up-probability at the node. Under a probability that gives every path positive probability, each equation between conditional expectations holds at every node, not only with probability 1.

The page uses simple growth $1+r$ per period. With a rate $\rho$ quoted with continuous compounding, for periods of length $\Delta t$, replace every $1+r$ by $e^{\rho\Delta t}$. All prices are hypothetical.

The bond is the unit of account here: prices are divided by the bond's value. Another traded asset whose price stays positive could serve as the unit instead, and the probability that makes prices fair may then change; that is the subject of a later page on changing the numeraire.

1 See

Search for the fair probability

In Market 1 the stock starts at $100 and moves by $u = 1.2$ or $d = 0.9$ in each of 3 periods, and the bond pays $r = 2\%$ per period. At each node before time 3, the gap is the expected change in the discounted price over the next period, $\mathbb E_\pi[\tilde S_{t+1} \mid \mathcal F_t] - \tilde S_t$. Make your prediction first; the tree opens after it.

2 Try

The edge of the range, and returns under Q

Expected return over the first period, from the root to time 1, in Market 1. The values of the call and the put at time 1 are their values at those nodes, as in the setup. Percentages are shown to 2 decimals; ≈ marks a rounded value.
Asset	Price at the root	Under $\mathbb Q$	Under the real p

3 Prove

One measure, and every portfolio is fair under it

Theorem (the martingale measure of a binomial tree)

Assumptions. The setup above: $n \ge 1$, $S_0 \gt 0$, $0 \lt d \lt u$, $r \gt -1$ with $d \lt 1+r \lt u$, information $\mathcal F_t$ from the first $t$ moves, and self-financing strategies with the discounted gains identity.

Conclusions.

A probability $\mathbb Q$ on the paths that gives every path positive probability makes $\tilde S$ a martingale exactly when its conditional up-probability at every node is $q = \frac{(1+r)-d}{u-d}$. So there is exactly one martingale measure, and it gives a path with $k$ up moves the probability $q^k(1-q)^{n-k}$.
Under it, the discounted value $\tilde V_t = V_t/(1+r)^t$ of every self-financing strategy is a martingale.
If a self-financing strategy is worth $X$ at time $n$ on every path, then $V_0 = \mathbb E^{\mathbb Q}[X]/(1+r)^n$. No real probability enters.

Lines 1 and 2 are claims at a node, for every probability $\mathbb Q$ that gives each path positive probability. Lines 3 and 4 are each one claim about every self-financing strategy at once, so a candidate line that fails for some strategy is false, even if it holds for one in particular: a strategy that keeps all its money in the bond, for example, has $\Delta_t = 0$ at every node.

Scratch work, not part of the proof.

Going further: the same idea in continuous time

In the Black–Scholes model (step 16) the stock follows $dS_t = \mu S_t\,dt + \sigma S_t\,dW_t$ under the real probability, with constants $\mu$, $\sigma \gt 0$ and $r$. Set $\theta = (\mu - r)/\sigma$, the market price of risk, and $W^{\mathbb Q}_t = W_t + \theta t$. Girsanov's theorem, the subject of a later page, gives a probability $\mathbb Q$ with the same events of probability 0 as the real one, under which $W^{\mathbb Q}$ is a standard Brownian motion on $[0, T]$. Then

\[ dS_t = rS_t\,dt + \sigma S_t\,dW^{\mathbb Q}_t, \] \[ \tilde S_t = e^{-rt}S_t, \qquad d\tilde S_t = \sigma\tilde S_t\,dW^{\mathbb Q}_t, \]

by Itô's lemma (step 13): the discounted price has no $dt$ term under $\mathbb Q$. Only the drift changes; $\sigma$ is the same under both probabilities.

In continuous time, no $dt$ term makes a process a local martingale, which is weaker than a martingale. A local martingale can lose mean over time: the reciprocal of the distance from the origin of a three-dimensional Brownian motion started away from the origin has no $dt$ term, yet its mean falls. An Itô integral $\int_0^t H_s\,dW^{\mathbb Q}_s$ is a true martingale when $\mathbb E^{\mathbb Q}\int_0^T H_s^2\,ds \lt \infty$ (step 12). For the stock, $\tilde S_t = S_0\exp\big(\sigma W^{\mathbb Q}_t - \tfrac12\sigma^2 t\big)$ and $\mathbb E^{\mathbb Q}[\tilde S_t^2] = S_0^2e^{\sigma^2 t}$, so the condition holds with $H = \sigma\tilde S$, and $\tilde S$ is a true martingale. For a general strategy, lines 3 and 4 of the proof need a condition of the same kind on the strategy. Continuous-time pricing imposes one, for example $\mathbb E^{\mathbb Q}\int_0^T (\Delta_t\sigma\tilde S_t)^2\,dt \lt \infty$, and it rules out doubling strategies, which bet ever larger amounts to win with probability 1.

Key point Fairness of the discounted stock pins down one conditional up-probability, $q$, at every node. The discounted gains identity passes that fairness to every self-financing portfolio, so the copy of a claim $X$ costs $\mathbb E^{\mathbb Q}[X]/(1+r)^n$. The real probabilities decide how likely each path is and what the expected returns are, not what copies cost. The next step of the path (step 18), the Black–Scholes formula, computes this average for the continuous-time model.

4 Apply

Problems

All prices are hypothetical. Growth factors and rates are per period unless a problem says otherwise, and the markets are frictionless.

Discounted prices are martingales under Q

Search for the fair probability

The real probability and the price

The edge of the range, and returns under Q

One measure, and every portfolio is fair under it

Problems

Further reading