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Partial Derivatives of Index Straddle w.r.t. Index Constituents

Hi everyone,

I'm an undergraduate Math/Econ student planning on going onto a PhD in Finance. I am trying to replicate the results of the Driessen et al Journal of Finance paper on pricing correlation risk.

In order to delta/vega hedge a short index straddle I need to find the partial derivative of the straddle with respect to the variance of each constituent security, and the partial derivative of the straddle with respect to the constituent stock.

\[ \frac{\partial O_I}{\partial S_i} \text{ and } \frac{\partial O_I}{\partial \phi_i^2} \]

Where \(O_I\) denotes the price of the straddle, \( i \in [1, \ldots, n] \) are constituents of the index, and \( \phi_i^2 \) is the variance of the constituent.

As far as I understand Delta and Vega are linear functions, so can I just use the weighted greek of the straddle? Can I just take \( \omega_i \cdot \Delta_I \) and similar to get Vega?

Thanks for any help you can offer. My knowledge of option pricing is fairly limited, but I am anxious to learn more, so if you can point me in the direction of a book or some papers on the topic it would be great.

As a side note: If anyone is familiar with the OptionMetrics data set and knows what secid should be used for SPY I would love to hear. The one I am currently using lacks the Greeks, which isn't really an issue since I can just calculate them myself, but since I am using the pre-defined values for all other options it would be nice to have uniformity.

Edit: I have been thinking a bit about this and here is what I have so far:

If we suppose that movement in each stock indexed by \( i \in [1,\ldots, n]\) is governed by the following process:
\[ \frac{dS_i}{S_i} = \mu_i dt + \phi_i dW_i \]
(I might be wrong here, but should we have a third term on the right: \(\sum_j \rho_{ij} \phi_j S_j dW_j\) for each \( i \neq j \)?)

Then we have the index as a function:
\[ S_I(t, W_1,\ldots, W_n) = \sum_i S_i \]
I am not quite sure about the application of Ito's Lemma here to get the process for \(dS_I\)... I took a finance class aimed at students looking to grad school in which we briefly covered the Ito's Lemma in the context of BS, but I don't feel confident in applying it to a multi-variate function as in this example. I would assume that we would just get a unit weighted linear combination of applications of ito's lemma w.r.t. each of the constituent processes.

Any tips on how I would proceed? Once I can get the process for \(dS_I\) it seems straightforward to use Ito's Lemma to get the process for call and put options on index, then assuming this still works neatly into the heat equation I think I could get at the delta and vega.

Possibly I am overcomplicating this, and I should just plug \( S_I(t, W_1,\ldots, W_n) = \sum_i S_i \) into the standard Black-Scholes formula. In this case, then it seems my intuition of weighting Delta and Vega by the index weights would be correct.
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