Solution: Building the Probability Space for a Two-Step Binomial Tree

Exercise: Building the Probability Space for a Two-Step Binomial Tree

Part 1: Sample space

\Omega = \{uu,\; ud,\; du,\; dd\}

Each element

\omega \in \Omega

is a complete path — a sequence of moves over both time steps.

\omega = ud

means the stock moved up at step 1 and down at step 2, ending at

S_2 = 100 \times 1.1 \times 0.9 = 99

. These four paths are the four ways the world can evolve under this model.

The terminal stock prices are:

$\omega$	$S_1$	$S_2$
$uu$	110	121
$ud$	110	99
$du$	90	99
$dd$	90	81

Note:

ud

and

du

give the same terminal price (

S_2 = 99

) but are distinct paths — the model tracks the entire trajectory, not just the endpoint.

Part 2: Sigma-algebra

With $|\Omega| = 4$ , the power set $\mathcal{F} = 2^\Omega$ has $2^4 = 16$ elements:

\mathcal{F} = \{\emptyset,\; \{uu\},\; \{ud\},\; \{du\},\; \{dd\},\; \{uu,ud\},\; \{uu,du\},\; \{uu,dd\},\; \{ud,du\},\; \{ud,dd\},\; \{du,dd\},\; \{uu,ud,du\},\; \{uu,ud,dd\},\; \{uu,du,dd\},\; \{ud,du,dd\},\; \Omega\}

In practice, the full power set is manageable for finite $\Omega$ . The richness of $\mathcal{F}$ matters when we introduce filtrations: $\mathcal{F}_0 \subset \mathcal{F}_1 \subset \mathcal{F}_2 = \mathcal{F}$ captures what is known at each time step.

Part 3: Real-world measure $\mathbb{P}$

With independent up/down moves at probability $p = 0.6$ :

\mathbb{P}(\{uu\}) = 0.6^2 = 0.36

\mathbb{P}(\{ud\}) = 0.6 \times 0.4 = 0.24

\mathbb{P}(\{du\}) = 0.4 \times 0.6 = 0.24

\mathbb{P}(\{dd\}) = 0.4^2 = 0.16

Check: $0.36 + 0.24 + 0.24 + 0.16 = 1.00$ . ✓

Part 4: Risk-neutral measure $\mathbb{Q}$

The risk-neutral condition at any node: $e^{-r}[q \cdot S_t u + (1-q) \cdot S_t d] = S_t$ , which simplifies to:

q = \frac{e^r - d}{u - d} = \frac{e^{0.05} - 0.9}{1.1 - 0.9} = \frac{1.0513 - 0.9}{0.2} = \frac{0.1513}{0.2} \approx 0.7565

With $q \approx 0.7565$ and $1 - q \approx 0.2435$ , applying independence across steps:

\mathbb{Q}(\{uu\}) = q^2 \approx 0.5723

\mathbb{Q}(\{ud\}) = q(1-q) \approx 0.1842

\mathbb{Q}(\{du\}) = (1-q)q \approx 0.1842

\mathbb{Q}(\{dd\}) = (1-q)^2 \approx 0.0593

Check: $0.5723 + 0.1842 + 0.1842 + 0.0593 = 1.00$ . ✓

Part 5: Comparison of $\mathbb{P}$ and $\mathbb{Q}$

For

A = \{uu\}

\mathbb{P}(A) = 0.36

and

\mathbb{Q}(A) = 0.5723

. They are not equal.

Are they equivalent? Two measures are equivalent if they agree on null sets:

\mathbb{P}(A) = 0 \Leftrightarrow \mathbb{Q}(A) = 0

for all

A \in \mathcal{F}

. In this finite model,

\mathbb{P}

and

\mathbb{Q}

both assign positive probability to every path (since

0 < p < 1

and

0 < q < 1

). So neither measure has any null events except

\emptyset

. Therefore $\mathbb{P}$ and $\mathbb{Q}$ are equivalent.

Equivalent measures agree on what is possible and what is impossible — they just disagree on how likely things are. In continuous-time finance,

\mathbb{P} \sim \mathbb{Q}

(equivalent measures) is precisely the condition that guarantees Girsanov's theorem applies and that the change of measure from

\mathbb{P}

\mathbb{Q}

is well-defined. If the measures were not equivalent — if some paths had positive

\mathbb{P}

-probability but zero

\mathbb{Q}

-probability — no consistent no-arbitrage price would exist for instruments whose payoff depended on those paths.

Takeaways

A multi-step binomial tree is a fully explicit probability space: $\Omega$ is the set of paths, $\mathcal{F}$ is the power set, and both $\mathbb{P}$ and $\mathbb{Q}$ are valid probability measures on this triplet.
The risk-neutral measure $\mathbb{Q}$ makes discounted prices martingales. It is uniquely determined by the no-arbitrage condition, not by the real-world dynamics.
$\mathbb{P}$ and $\mathbb{Q}$ are equivalent in complete markets — they share the same null sets. This equivalence is the content of "no-arbitrage implies equivalent martingale measure" and is the rigorous foundation for risk-neutral pricing.
Individual paths $ud$ and $du$ are distinct elements of $\Omega$ even when they produce the same terminal stock price. Filtrations distinguish them by tracking which information has been revealed at each step.