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I have a complicated product with knock-out barriers combined with other exotic options. I am curious if there is a fast and loose way to figure out the delta, gamma, rho, theta and possibly vega, with the historical prices of the product.

I mean, I would calculate the greeks by hand since it isnt impossible to write them out in a formula, but it would be a hideous expression, and my basic calculus skills are rusty.

So, is there a relatively simple way to get the sensitivities from historical prices? Can I just simply do $$ \frac{\Delta P}{\Delta S} $$ with the historical underlying and historical option price for the delta, for instance? Would this be similar to what I would get by differentiating the price formula?

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The observed time series of the historical option price would give you a time series of $\Delta P$, however these changes in price is not purely driven by change in spot. You also have the impact of change of vol and theta: $\Delta P \approx \Delta *\Delta S + 0.5 * \Gamma * \Delta S^2 + vega*\Delta \sigma + \theta$

Assuming that you would know the change of spot, rate and vol. You still have only 1 equation with many unknown greeks (so no unique solution).

Furthermore, you have the problem that the answer to your problem depends on the moneyness of your option, distance to the barriers, time to expiry etc.

Would it maybe be a solution for you to approximate your very exotic product by a combination of simpler products? You do an overhedge/underhedge etc.

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