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How Wiener Process Correlation Maps to Model Variable Correlation

Article Quant Q&A · Author: Markov
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Summary

The document explains the role of correlation between Wiener process increments in stochastic asset models. In a two-stock Black Scholes setup, correlated Brownian shocks produce the same instantaneous correlation between the stocks’ proportional returns, while the covariance also depends on their volatilities. This relationship is shown through the covariance and variance terms, rather than assuming that correlation between process levels is the relevant quantity.

A second example uses the Heston stochastic volatility model. It shows how correlation between the price and variance shocks governs the covariance of their instantaneous innovations. These examples clarify how shock correlation carries into modeled economic variables. The claims concern instantaneous changes under the displayed diffusion equations; they do not imply that finite horizon price levels or returns will always have correlation equal to the shock parameter, especially under changing volatility, drift, or more complex model dynamics.

Key ideas

  • Brownian correlation describes dependence between instantaneous shocks in a stochastic model.
  • In a two asset diffusion, it equals the instantaneous correlation of proportional returns.
  • The covariance of asset returns also depends on each asset’s volatility.
  • In the Heston example, shock correlation links innovations in price and variance.
  • Instantaneous relationships should not be assumed to hold unchanged for finite horizon price levels.

Tags

StatisticsEquitiesVolatility
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# Why we introduce correlations between Wiener processes?


# Why we introduce correlations between Wiener processes?












Wiener processes are used to model various assets, and I wonder why we are introducing correlations between the Wiener processes and what is the interpretation? Because when the correlations between two Wiener processes is $\rho$, the the correlations between for example stock prices in Black Scholes model will not be equal $\rho$.

## Answer by Kevin (score 7, accepted)

https://quant.stackexchange.com/a/70223

Suppose we model two stocks by \begin{align*} \text{d}S_1 &= \mu_1S_1\text{d}t+\sigma_1S_1\text{d}W_1 \\ \text{d}S_2 &=\mu_2S_2\text{d}t+\sigma_2 S_2\text{d}W_2 \end{align*} where $\text{d}W_1\text{d}W_2=\rho \text{d}t$. Then, $\rho$ indeed measures the correlation of instantaneous returns: \begin{align} \mathbb{C}\text{ov}\left(\frac{\text{d}S_1}{S_1},\frac{\text{d}S_2}{S_2}\right) = \sigma_1\sigma_2\mathbb{C}\text{ov}\left(\text{d}W_1,\text{d}W_2\right)=\sigma_1\sigma_2\rho\text{d}t. \end{align} Thus, \begin{align} \mathbb{C}\text{orr}\left(\frac{\text{d}S_1}{S_1},\frac{\text{d}S_2}{S_2}\right)=\frac{\mathbb{C}\text{ov}\left(\frac{\text{d}S_1}{S_1},\frac{\text{d}S_2}{S_2}\right)}{\sqrt{\mathbb{V}\text{ar}\left[\frac{\text{d}S_1}{S_1}\right]\mathbb{V}\text{ar}\left[\frac{\text{d}S_2}{S_2}\right]}} = \frac{\sigma_1\sigma_2\rho\text{d}t}{\sigma_1\sigma_2\text{d}t}=\rho. \end{align} You can equally consider the correlation between log-returns, $\text{d}\ln(S_1)$ and and $\ln(S_2)$. Note that I did not need to assume that drift or volatility are constant.

This also works for other model set-ups. Consider the Heston (1993) stochastic volatility model \begin{align*} \text{d}S &= \mu S\text{d}t+\sqrt{v}S\text{d}W_1 \\ \text{d}v &=\kappa(\bar{v}-v)\text{d}t+\xi \sqrt{v}\text{d}W_2 \end{align*} where $\text{d}W_1\text{d}W_2=\rho \text{d}t$. Then, the covariance between innovations (changes) in stock prices and variances is \begin{align} \mathbb{C}\text{ov}\left(\text{d}S,\text{d}v\right) = S\xi v\mathbb{C}\text{ov}\left(\text{d}W_1,\text{d}W_2\right)=S\xi v\rho\text{d}t. \end{align} Thus, \begin{align} \mathbb{C}\text{orr}\left(\text{d}S,\text{d}v\right)=\frac{\mathbb{C}\text{ov}\left(\text{d}S,\text{d}v\right)}{\sqrt{\mathbb{V}\text{ar}\left[\text{d}S\right]\mathbb{V}\text{ar}\left[\text{d}v\right]}}= \frac{S\xi v\rho\text{d}t}{S\xi v\text{d}t}=\rho. \end{align}

Thus, $\rho$ measures not only the correlations between changes in Brownian motions, but these correlations typically penetrate through to economic variables which are the actual variables of interest in the model.

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.