Why Option Return Correlation Can Differ from Underlying Stock Correlation
Summary
Underlying stock return correlation does not directly determine the correlation between two option positions. The document illustrates this with perfectly correlated stocks and options whose exposures point in opposite directions: a deep in-the-money call on one stock and a deep in-the-money put on the other. As both stocks rise, the call gains while the put loses, demonstrating how option type and directional sensitivity can change the sign of position-level co-movement.
For a variance-covariance Value at Risk calculation, the discussion cautions that this approach can be unreliable for nonlinear instruments. It suggests historical simulation as an alternative, without presenting a comparison or empirical test. If retaining a variance-covariance method, it proposes approximating each option with its delta exposure to the underlying, then applying the covariance calculation to those signed exposures. This is a local linear approximation and does not capture nonlinear effects such as changing delta or volatility sensitivity.
Key ideas
- Option position correlation can differ from the correlation of the underlying stocks.
- Option type and directional exposure can reverse the sign of co-movement between positions.
- Variance-covariance VaR may be unsuitable for nonlinear option exposures.
- A delta-based approximation treats each option as a signed exposure to its underlying stock.
- Historical simulation is suggested as an alternative VaR approach.
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# What is the correlation of stock options?
# What is the correlation of stock options?
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?
## Answer by mbison (score 2)
https://quant.stackexchange.com/a/20634
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.