X and Y are geometric Brownian motions with drifts mu_1, mu_2, volatilities sigma_1, sigma_2 and instantaneous correlation rho. Give the SDEs and closed forms for XY and X/Y, including the effective volatility of each.

X and Y are geometric Brownian motions with drifts mu_1, mu_2, volatilities sigma_1, sigma_2 and instantaneous correlation rho. Give the SDEs and closed forms for XY and X/Y, including the effective volatility of each.

Approach: Apply the two dimensional Ito product and quotient rules, keeping the cross variation term d[X,Y] = rho sigma_1 sigma_2 X Y dt, then read off the drift and the variance of the combined diffusion.

d(XY)/(XY) = (mu_1 + mu_2 + rho sigma_1 sigma_2) dt + sigma_1 dW_1 + sigma_2 dW_2 with effective volatility sqrt(sigma_1^2 + sigma_2^2 + 2 rho sigma_1 sigma_2), and d(X/Y)/(X/Y) = (mu_1 - mu_2 - rho sigma_1 sigma_2 + sigma_2^2) dt + sigma_1 dW_1 - sigma_2 dW_2 with effective volatility sqrt(sigma_1^2 + sigma_2^2 - 2 rho sigma_1 sigma_2). The Ito product rule d(XY) = X dY + Y dX + d[X,Y] supplies the cross term rho sigma_1 sigma_2 XY dt for the product, which is the quanto adjustment that appears when a payoff is converted at a correlated rate. For the quotient, apply Ito to f(x,y) = x/y: the term f_yy (dY)^2/2 = (x/y^3)(sigma_2^2 y^2)/2 gives + sigma_2^2 and the cross partial gives -rho sigma_1 sigma_2. Both processes remain correlated Brownian motions driven lognormals, so each has a closed form of geometric Brownian type with the stated volatility.

Follow-up: For a quanto forward on X settled in the Y currency, which of these two adjustments appears in the price?

Key concepts: Ito product rule, correlated Brownian motions, quanto adjustment, effective volatility.