Why Historical Option Prices Cannot Identify Greeks on Their Own

Summary

The document considers whether an exotic option’s Greeks can be estimated by dividing historical option price changes by historical changes in the underlying. It explains that an option’s observed price movement is generally influenced by several factors at once: changes in spot, curvature with respect to spot, changes in implied volatility, and the passage of time. A local approximation includes delta, gamma, vega, and theta contributions, so a single observed price change does not isolate any one sensitivity.

Even if changes in market inputs are observed, one equation with multiple unknown Greeks does not determine a unique answer. Sensitivities also vary with the option’s moneyness, proximity to knock-out barriers, and time to expiry, making a historical ratio especially unreliable for a path-dependent exotic. The response suggests approximating the complicated payoff with simpler instruments and managing the resulting hedge mismatch. It does not give a calibrated estimation procedure or discuss how to account for noisy prices, changing exposure, or market data quality.

Key ideas

• An option’s price change can reflect several Greeks and changing market inputs simultaneously.
• A historical price change divided by an underlying move does not generally identify delta.
• One observed change cannot uniquely recover multiple unknown sensitivities.
• Exotic option sensitivities depend on state variables such as barrier distance and time to expiry.
• A simpler instrument portfolio may approximate an exotic payoff, with residual overhedging or underhedging risk.

Tags

# Finding the delta and gamma with historical data


# Finding the delta and gamma with historical data












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?

## Answer by mbison (score 1, accepted)

https://quant.stackexchange.com/a/33828

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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.