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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.

kakarito
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  • Is it not just $\sqrt{\sigma_1^2 + \sigma_2^2+2\rho\sigma_1\sigma_2}$? Based on a simple and hand-wavy argument about projecting both processes out to time $t=1$ and treating each as simple lognormal distribution. The harder part might be proving that $X_tY_t$ is in fact a geometric Brownian motion with drift. Although maybe your integration by parts does that. – Jamie Ballingall Feb 23 '23 at 04:27
  • Also, could you clarify what $\sigma$, as opposed to $\sigma_1$ and $\sigma_2$, represents in your question? – Jamie Ballingall Feb 23 '23 at 04:30
  • @JamieBallingall it was a typo, sorry. It was supposed to be $\sigma_{2}$. I edited the question. The quantity above expands to what you wrote in your first comment. And I'm also not sure if that argument sufficient – kakarito Feb 23 '23 at 12:47

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