I want to calculate the VaR of two correlated option positions, and I know the correlation between stock price returns. I want to separately calculate $Var_1$,$Var_2$ for option 1 and 2, and then use $$\sqrt{(w_1var1)^2+(w_2var2)^2+2*corr*w_1*w_2*var_1*var_2}$$ to calculate the portfolio var, but I don't know if the option correlation is the same with its underlying stock returns correlation?
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I do not think that the correlation for stock1 and stock2 equals the correlation for option1 and option2. For example, assume stock1 and stock2 are perfectly correlated. Now assume option1 is a deep in the money call on stock1, assume option2 is a deep in the money put on stock2.
Now stock1 goes up. Since correlation is perfect stock2 goes up. Option1 is deep in the money call and therefore also goes up with a delta close to 1. However, Option2 is a put. Since stock2 went up as well, Option2 will drop in value. As you see the sign flipped on the correlation. So from this example I would think you can not use the stock correlation directly for your options.
It looks like you want to calculate the Value at Risk using a normal variance/covariance approach. I think most text books will say that this method is not the best when dealing with non linear products. I would advise on using a different approach (maybe historical sim).
See link for description of other available methods. https://web.stanford.edu/class/msande444/2012/MS&E444_2012_Group2a.pdf
Btw, if you insist on using the variance/covariance method. I guess you could always use only a delta approximation. Then you simply take the delta of option1 and the delta of option2. This approx your option position as if were just positions in the underlying. With the deltas as weights. My example with the put and the call would give that you have delta of 1 in stock1 and a delta of -1 for stock2. So you d be long stock1 and short stock2.
