Volatility of the Product of Correlated Lognormal Assets

Summary

The document derives the instantaneous volatility of the product of two correlated assets modeled with lognormal diffusion dynamics. Applying Itô’s product rule produces a return shock with contributions from both assets, including their correlation. Combining the shared and independent Brownian components gives product volatility equal to the square root of the sum of the two individual variance terms and twice their covariance contribution.

The derivation assumes continuous diffusion processes, constant volatilities and correlation, and zero drift in the setup. It addresses instantaneous proportional volatility of the product, rather than the volatility of the product’s level over a finite horizon. The questioner’s proposed expression is consistent with the derivation; the document does not discuss empirical estimation, jumps, or changing parameters.

Key ideas

• Itô’s product rule captures the interaction between the two asset processes.
• The product’s instantaneous variance includes both individual variances and a correlation term.
• The combined Brownian shock can be represented as a single shock scaled by the product volatility.
• The result relies on constant-parameter diffusion assumptions and does not cover jumps.

Tags

# Volatility of the product of two correlated asset following a log normal distribution


# Volatility of the product of two correlated asset following a log normal distribution












I am trying to solve the problem: Given two assets X and Y that follow a log normal distribution with volatility $\sigma_1$ and $\sigma_2$ respectively and with correlation $\rho$, what is the volatility of X*Y ?

I tried modelling X and Y as: $$dX_{t} = X_{t}(\mu dt + \sigma_{1}dW_{t}^{1})$$ $$dY_{t} = Y_{t}(\mu dt + \sigma_{2}(\rho dW_{t}^{1} + \sqrt{1-\rho^2}dW_{t}^{2}))$$ where $W_{t}^{1}$ and $W_{t}^{2}$ are independent. Assuming $\mu = 0$ and using Itô integration by parts formula for the dynamics of the process X*Y

$$d(XY)_{t} = X_{t}Y_{t}((\sigma_{1} + \sigma_{2}\rho)dW_{t}^{1} + \sigma_{2}\sqrt{1-\rho^2}dW_{t}^{2})$$ $$d(XY)_{t} = X_{t}Y_{t}d\hat{W}_{t}$$

Then I try to compute the variance of $\hat{W}_{t}$. I find that $\mathbb{V}(\hat{W}_{t}) = [(\sigma_{1} + \sigma_{2}\rho)^2 + \sigma_{2}^{2}(1- \rho^2)]t$

$d(XY)_{t} = X_tY_t\sqrt{(\sigma_{1} + \sigma_{2}\rho)^2 + \sigma_{2}^{2}(1-\rho^2)}dW_{t}$ where $dW = \frac{1}{\sqrt{(\sigma_{1} + \sigma_{2}\rho)^2 + \sigma_{2}^{2}(1-\rho^2)}}d\hat{W}_{t}$. I find that the volatility of XY is $\sqrt{(\sigma_{1} + \sigma_{2}\rho)^2 + \sigma_{2}^{2}(1-\rho^2)}$. Is that correct? I would like to get some advice on how to solve this type of problems.

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