Related to the issue that I have raised here, I am facing another question. As the rule here is 1 question / 1 post, I take the opportunity to ask it below:

By exploring StackExchange, I noticed the existence of these three threads:

Girsanov Theorem for Quanto/Compo adjustment (i)

Quanto/Compo adjustments - Product of two geometric brownian motion (ii)

Multivariate Ito problem $M_{t}=\frac{X_{t}}{Y_{t}}$ (iii)

However, I have the feeling that all deals the same question with a different solution. Indeed, it appears in (i) that the expectation of $Y_tX_t = Y_t/\tilde{X}_t$ under $\mathbb{Q}^d$ is found by "a proxy": $Y_t/\tilde{X}_t$ being lognormally distributed (ratio of two lognormals) with mean $$ \mu = \ln(Y_0/\tilde{X}_0) + \left(r_d - \frac{\sigma^2_X - 2\rho\sigma_\tilde{X}\sigma_Y + \sigma_Y^2}{2}\right)t $$ and variance $$ \sigma^2 = \left(\sigma_\tilde{X}^2 - 2 \rho \sigma_\tilde{X} \sigma_Y + \sigma_Y^2 \right)t $$

On the other hand, in (ii), the correlation factor disappear from the numerator of the lognormal mean. By replacing $a$ and $b$ with the notation from (i), this is equivalent to: $$ \mu = \ln(Y_0/\tilde{X}_0) + \left(r_d - \frac{\sigma^2_X + \sigma_Y^2}{2}\right)t $$

Finally in (iii), the volatility of the ratio of stochastic processes becomes $$ \sigma = \left(\sigma_\tilde{X}^2 - \rho \sigma_\tilde{X} \sigma_Y + \sigma_Y^2 \right)^{0.5} $$

Are all those notations really equivalent ? Would you mind to explain me quickly from where the "fork" of solutions comes from ?


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