My question is about the following (from Maxime de Bellefroid, Ch. 5 The Greeks): enter image description here

  1. From my understanding $\Delta_2$ is the sensitive of the option (on the first instrument with underlying $S_1$) with respect to the price of the second instrument $S_2$, I guess this value can only be estimated and there are no closed formulas?

  2. How is the value of $\frac{\partial S_2}{\partial S_1}$ calculated?


1 Answer 1


Look carefully. $\frac{\partial S_2}{\partial S_1}$ is explicitly given as $\frac{\partial S_2}{\partial S_1} = \frac{\rho_{12}\sigma_2}{\sigma_1}\frac{S_2}{S_1}$

If you know the correlation and the standard deviations of daily returns for asset 1 and asset 2 you can use this formula. But if not you can just do a least squares regression of the daily price changes (not returns) of asset 2 on the daily price changes of asset 1 and essentially come up with the same result (remeber the formula for the slope on a bivariate regression, it is the correlation times the ratio of the two standard deviations).

Then the closed formula for finding $\Delta_2$ is $\Delta_2 = \Delta / \frac{\partial S_2}{\partial S_1}$, a simple division (let us hope the denominator is not zero).

  • $\begingroup$ When you say '$\frac{\partial{S_2}}{\partial{S_1}}$ is explicitly given' you mean that it is defined that way? If so why? $\endgroup$
    – Mattiatore
    Feb 1 at 16:40
  • 1
    $\begingroup$ It is a poor explanation if you ask me, very confusing. He was able to write $\Delta_2 \frac{\partial S_2}{\partial S_1} = \Delta_2 \rho_{12} \frac{\sigma_2}{\sigma_1}\frac{S_2}{S_1}$ because he calculated for you what $\frac{\partial S_2}{\partial S_1}$ is equal to, but he does not show the details. So it is implicit, not explicit, I suppose. $\endgroup$
    – nbbo2
    Feb 1 at 16:53

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