Skip to content
All library documents

Verifying the Discounted Stock Martingale in a Two-State Model

Article Quant Q&A · Author: Wolfy
Read the original

Summary

This note shows how to verify the martingale property of the discounted stock in a one-period, two-state pricing model. The pricing measure assigns probabilities to the up and down states using the initial stock price, the discount factor, and the two possible terminal stock values. Substituting those probabilities into the expected discounted terminal stock and combining the terms yields the initial discounted stock price.

The calculation connects risk-neutral pricing probabilities with the martingale condition for discounted assets. The construction requires the state weights to be nonnegative for the formal measure to be a probability measure, as the question itself observes. The answer demonstrates the algebra for this simple setting; it does not establish the result for general multi-period or continuous-time models, or discuss broader conditions for absence of arbitrage. It is a focused instructional derivation rather than an empirical analysis or trading strategy.

Key ideas

  • The pricing measure weights the up and down states using the initial price and terminal stock values.
  • Taking the expected discounted terminal stock under those weights gives the initial discounted stock price.
  • The weights must be nonnegative for the pricing measure to be a probability measure.
  • The derivation addresses a one-period, two-state model and does not cover general settings.

Tags

Derivatives pricingStatistics
Full text
# Showing the discounted stock is a martingale


# Showing the discounted stock is a martingale












Background Information:

This question follows from here

It is tempting to write $$V_0(X) = \beta\left[\left(\frac{\beta^{-1}S_0 - S_1(d)}{S_1(u) - S_1(d)}\right)X(u) + \left(\frac{S_1(u) - \beta^{-1}S_0}{S_1(u) - S_1(d)}\right)X(d)\right]$$ as

$$V_0(X) = E_Q[\beta X]$$ where the expectation is taken with respect to the new purely formal probability measure $Q$ defined by $$Q(u) = \frac{\beta^{-1}S_0 - S_1(d)}{S_1(u) - S_1(d)}$$ and $$Q(d) = \frac{S_1(u) - \beta^{-1}S_0}{S_1(u) - S_1(d)}$$

Note that $Q(u) + Q(d) = 1$; $Q$ will be a probability measure provided these values are non-negative.

Question:

> The quantity $\frac{S}{B}$ is called the discounted stock price. It will be useful later to notice that the pricing measure $Q$ has a special property with respect to the discounted stock price: it makes the discounted stock a martingale, i.e.,

$$\frac{S_0}{B_0} = E_{Q}[S_1/B_1]$$

I am curious on how to verify this. I assume we need to show both sides but I am not sure how to proceed.

## Answer by Gordon (score 1, accepted)

https://quant.stackexchange.com/a/30894

In your setting, $\beta = B_0 B_1^{-1}$. Then \begin{align*} E_Q(S_1/B_1) &= Q(u)S_1(u)/B_1 + Q(d)S_1(d)/B_1\\ &=\frac{\beta^{-1}S_0 - S_1(d)}{S_1(u) - S_1(d)}S_1(u)/B_1 + \frac{S_1(u) - \beta^{-1}S_0}{S_1(u) - S_1(d)}S_1(d)/B_1\\ &=\frac{\beta^{-1}S_0S_1(u)/B_1 - S_1(d)S_1(u)/B_1 + S_1(u)S_1(d)/B_1-\beta^{-1}S_0S_1(d)/B_1}{S_1(u) - S_1(d)}\\ &=\beta^{-1}S_0/B_1\\ &=\frac{S_0}{B_0}. \end{align*}

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.