Deriving Price Correlation from Correlated GBM Returns
Summary
The note derives the correlation between two stock prices at time t when the stocks follow correlated geometric Brownian motions. It expresses each price in terms of its initial value, drift, volatility, and a normally distributed Brownian component, then computes each price’s mean and variance along with the covariance between prices.
The resulting closed form uses the return correlation and both volatilities over the elapsed time: price correlation equals the exponential of their correlation-volatility-time product, minus one, divided by the product of the corresponding volatility-time standard deviation terms. The expression shows why price-level correlation depends on the horizon and volatilities, even when log-return correlation is specified. The result assumes GBM, constant parameters, and jointly normal Brownian drivers; it does not establish the relationship for real stocks or changing market conditions.
Key ideas
- Under correlated geometric Brownian motion, each stock price is lognormally distributed at a fixed time.
- Price-level correlation depends on return correlation, both volatilities, and the time horizon.
- The derivation obtains price means and variances before calculating covariance.
- The closed-form relationship relies on constant-parameter GBM and jointly normal drivers.
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Full text
# Correlation between stock prices given correlation between returns
# Correlation between stock prices given correlation between returns
assume I have two stocks with known volatilities and a known correlation coefficient of returns - does anyone know how to determine the correlation between the prices and NOT THE RETURNS
## Answer by RRL (score 9, accepted)
https://quant.stackexchange.com/a/45881
We can obtain a closed-form expression for price correlation given (log) return correlation when the two stocks follow geometric Brownian motion:
$$S_1(t) = S_1(0)e^{(\mu_1- \frac{1}{2} \sigma_1^2)t}e^{\sigma_1Z_1(t)},\\ S_2(t) = S_2(0)e^{(\mu_2- \frac{1}{2} \sigma_2^2)t}e^{\sigma_2Z_2(t)},$$
where $\text{corr}(Z_1(t),Z_2(t)) = E[Z_1(t)Z_2(t)]=\rho t$. The correlation of log returns over an interval of length $\delta t$ is
$$\text{corr}\left(\log \frac{S_1(t+\delta t)}{S_1(t)} , \log \frac{S_2(t + \delta t)}{S_2(t)} \right) = \rho \delta t$$
The price correlation is
$$\tag{*}\rho_{S_1S_2}=\frac{E[(S_1(t) - E(S_1(t))(S_2(t) - E(S_2(t))]}{\sqrt{\text{var}(S_1(t))}\sqrt{\text{var}(S_2(t))}}$$
Recalling that $E(e^{\sigma_1 Z_1(t)}) = e^{\frac{1}{2} \sigma_1^2 t}$, we obtain $$E(S_1(t)) = S_1(0)e^{\mu_1t}, \quad E(S_2(t)) = S_2(0)e^{\mu_2t} \\\text{var}(S_1(t)) = S_1(0)^2e^{2 \mu_1 t}( e^{\sigma_1^2t}-1), \quad \text{var}(S_2(t)) = S_2(0)^2e^{2 \mu_2 t}( e^{\sigma_2^2t}-1) $$
Note that
$$E[(S_1(t) - E(S_1(t))(S_2(t) - E(S_2(t))] = E[S_1(t)S_2(t)] - E(S_1(t)) E(S_2(t)) \\ = S_1(0)S_2(0)e^{\mu_1t}e^{\mu_2t}\left(e^{-\frac{1}{2}\sigma_1^2t}e^{-\frac{1}{2}\sigma_2^2t}E[e^{\sigma_1Z_1(t) + \sigma_2Z_2(t)}] - 1\right)$$
Substituting into (*) we obtain
$$\tag{**}\rho_{S_1S_2} = \frac{e^{-\frac{1}{2}\sigma_1^2t}e^{-\frac{1}{2}\sigma_2^2t}E[e^{\sigma_1Z_1(t) + \sigma_2Z_2(t)}] - 1}{\sqrt{ e^{\sigma_1^2t}-1}\sqrt{ e^{\sigma_2^2t}-1}}$$
Since $Z_1(t)$ and $Z_2(t)$ are both normally distributed with mean $0$ and variance $t$, it follows that $\sigma_1Z_1(t) + \sigma_2 Z_2(t)$ is normally distributed with mean $0$ and variance
$$\text{var}(\sigma_1Z_1(t)+\sigma_2Z_2(t)) = E[(\sigma_1Z_1(t)+\sigma_2Z_2(t))^2 \\ = (\sigma_1^2 + \sigma_2^2 + 2\rho \sigma_1\sigma_2)t$$
We then have
$$E[e^{\sigma_1Z_1(t) + \sigma_2Z_2(t)}] = e^{\frac{1}{2}\sigma_1^2t}e^{\frac{1}{2}\sigma_2^2t}e^{\rho\sigma_1\sigma_2t},$$
and after substituting into (**)
$$\rho_{S_1S_2} = \frac{e^{\rho\sigma_1\sigma_2t} - 1}{\sqrt{ e^{\sigma_1^2t}-1}\sqrt{ e^{\sigma_2^2t}-1}}$$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.