Discounted Stock Prices, Risk-Neutral Growth, and Interest-Rate Assumptions
Summary
The document defines a discounted stock price as the stock’s value divided by the money market account, or, with a constant rate, as the stock price multiplied by the discount factor. Under the risk-neutral measure, this discounted price is a martingale. The martingale property implies that the expected future stock-to-current-stock ratio matches the expected growth of the money market account when rates are deterministic.
The responses show this by conditioning on the present information, applying the martingale equality, and then taking the initial time. A second explanation decomposes the stock return into its discounted value and the bank-account growth factor, pointing out that a covariance term can affect the result when interest rates are stochastic. Thus the equal-growth conclusion depends on the rate assumptions and measure being used; the document is an introductory explanation and does not develop a pricing model or examine other asset distributions.
Key ideas
- A discounted stock price is the stock price measured relative to the money market account.
- Under the risk-neutral measure, the discounted stock price is a martingale.
- With deterministic interest rates, expected stock growth matches the bank account’s growth.
- Stochastic interest rates can introduce covariance that prevents the same equality from following directly.
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# Discounted Stock Price
# Discounted Stock Price
I have the following Question :
Prove that under the risk-neutral probability p the stock and the banjaccount have the same average rate of growth. In other words, if $ S_0 , S_N $ are the initial and final stock prices and $B_0 , B_N $ the initial and final bank prices , show that :
$$ E[S_N / S_0 ] = E[B_N / B_0 ] = c $$
Hint : The discounted stock price is a martingale under P.
Could you explain to me what is the discounted stock price ?
## Answer by Olórin (score 1)
https://quant.stackexchange.com/a/17463
Hope you will not mind if I place myself in continuous time. The discounted stock price at $T$ is $e^{-rT}S_T$. As you know that it is a martingale, you have that $\mathbf{E}^{\mathbf{P}}[e^{-rT}S_T | \mathscr{F}_t] = e^{-rt} S_t$ when $t\leq T$ which you can rewrite as $\mathbf{E}^{\mathbf{P}}\left[\frac{e^{-rT}S_T}{e^{-rt} S_t} | \mathscr{F}_t\right] = 1$ or $\mathbf{E}^{\mathbf{P}}\left[\frac{S_T}{S_t} | \mathscr{F}_t\right] = e^{r(T-t)}$ and $e^{r(T-t)}$ is but $\mathbf{E}^{\mathbf{P}}\left[\frac{B_T}{B_t} | \mathscr{F}_t\right]$. Finally, taking $t=0$ gives you the equality you are looking for.
## Answer by Gordon (score 0)
https://quant.stackexchange.com/a/18476
Let $S_t$ and $B_t$ be respectively the stock price and the money market account value at time $t$. Then $S_t/B_t$ is called the discounted stock price. Note that \begin{align*} E\left(\frac{S_N}{S_0}\right) &= E\left(\frac{S_N}{B_N} \frac{B_N}{B_0}\right)\frac{B_0}{S_0}\\ &= E\left(\frac{S_N}{B_N}\right) E\left(\frac{B_N}{B_0}\right)\frac{B_0}{S_0} + Cov\left(\frac{S_N}{B_N}, \frac{B_N}{B_0}\right)\frac{B_0}{S_0}, \end{align*} where $Cov(\,)$ is the covariance operator. If the interest rate is deterministic, then \begin{align*} E\left(\frac{S_N}{S_0}\right) &= E\left(\frac{S_N}{B_N}\right) E\left(\frac{B_N}{B_0}\right)\frac{B_0}{S_0}\\ &= E\left(\frac{B_N}{B_0}\right). \end{align*} However, if the interest rate is stochastic, this conclusion may not be true.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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