1
$\begingroup$

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?

Improve this question
$\endgroup$
0

1 Answer 1

Reset to default
2
$\begingroup$

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).

Improve this answer
$\endgroup$
2
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.