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Rearranging Expected Stock Price in a One-Step Binomial Model

Article Quant Q&A · Author: James
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Summary

The document asks how to transform the expected stock price equation in a one-step binomial model. Under the risk-neutral measure, the stock moves up by factor u with probability p and down by factor d with probability one minus p. The expected terminal stock price is the probability-weighted sum of those two outcomes, each multiplied by the initial stock price.

The answer expands the expression and groups the terms containing p, yielding the initial stock price times the down factor plus the probability times the initial stock price and the difference between the up and down factors. This is a straightforward algebraic rearrangement; it does not derive the risk-neutral probability, discount the expectation, or explain option valuation. The notation in the question and answer uses a letter that may resemble a lowercase letter o for the initial price, but the intended quantity is the starting stock price.

Key ideas

  • A one-step binomial model has an up outcome and a down outcome.
  • The expected terminal stock price is the probability-weighted average of the two outcomes.
  • Collecting the terms involving the risk-neutral probability gives a compact rearrangement.
  • The algebra alone does not derive the risk-neutral probability or price a derivative.

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OptionsDerivatives pricingStatistics
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# Risk-Neutral probability deduction


# Risk-Neutral probability deduction












Could anyone show me how to get the second row equation from the first row equation please? For each letter, $p$ is the risk-neutral probability in the risk-neutral world, $u$ is the up factor for the stock, and $d$ is the down factor for the stock, S0 is the beginning stock price. The equation is based on a one-step binomial tree model.

The textbook referred to is Options, futures, and other derivatives by John Hull 10th. Thank you guys.

## Answer by Bob Jansen (score 2)

https://quant.stackexchange.com/a/61208

Just expand the term:

\begin{align} \mathrm{E}(S_T) &= pS_o u + (1 - p)S_0 d \\ &= pS_o u - pS_0 d + S_0 d \\ &= pS_o (u - d) + S_0 d \end{align}

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