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Option Greeks, Fixed Parameters, and Model Uncertainty

Article Quant Q&A · Author: Mining
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Summary

The document examines whether probability belongs in explanations of option Greeks. Its response distinguishes uncertainty in future price paths, given fixed model parameters, from uncertainty in the parameters themselves. Under the fixed-parameter mathematical model, Greeks are sensitivities of option value to inputs; probability-based descriptions of delta or vega offer intuition about how payoff outcomes shift as inputs change.

The answer argues that estimated parameters add another layer of uncertainty that basic models such as Black–Scholes do not capture when estimates are inserted as if known. It uses the difference between prediction intervals and confidence intervals to illustrate how ignoring estimation uncertainty can make a forecast distribution too narrow. This is a conceptual discussion rather than a derivation of Greeks or a calibrated comparison, and its claim about known parameters describes the model framework rather than the practical certainty of market inputs.

Key ideas

  • Greeks are mathematical sensitivities of option value to model inputs.
  • Probability language can help explain how changes in inputs affect the distribution of possible payoffs.
  • Fixed-parameter models account for price-path uncertainty but omit uncertainty in estimated parameters.
  • Treating parameter estimates as known can understate the range of possible outcomes.
  • Prediction intervals and confidence intervals describe different sources of uncertainty.

Tags

OptionsDerivatives pricingVolatilityStatistics
Full text
# No Probability in Greeks


# No Probability in Greeks












In an interview, I was once told that I should not consider probability when talking about option greeks since from a mathematical point of view greeks have nothing to do with probability. That is of course true, from a mathematical point of view, but the way greeks are taught always takes into account the probability reasoning.

For example, in this website Volatility it says this:

> A rise in the implied volatility of a call will decrease the delta for an in-the-money option, because it has a greater chance of going out-of-the-money, whereas for an out-of-the-money option, a higher implied volatility will increase the delta, since it will have a greater probability of finishing in the money.

Again, it is taking into account probability to explain vega. I understand that because I was also taught greeks through probability reasoning. Therefore, how can I understand greeks functioning without taking into account the probability reasoning? How would you explain the relationship vega vs delta without considering "chances of ending in the money"?

## Answer by Dave Harris (score 4, accepted)

https://quant.stackexchange.com/a/68467

There is a really technical issue here. Both Ito's and Stratonovich's methods presume that all parameters are known. It is necessary for the math to work out correctly. If you drop that assumption and rework the rules of math, you get very different models because you have added parameter uncertainty as well.

There is no probability in the Greeks because there is no parameter uncertainty possible. If you were to calculate Apple's option price and various derivatives, it must come to a fixed number based on the parameters you know must be true and that you have not estimated. Of course, that is the rub. You cannot put estimators into Black-Scholes or similar models. That would produce a different result and a different equation for that matter. Of course, that is what everybody does, but without the adjustments.

The probability/certainty issue is a core element of the problems with these models.

Also, "chances of ending in the money" is a multi-faceted issue. If the parameters are known, then there is still the uncertainty created by possible realizations of the price path. If the parameters are not known, then every possible parameterization (of which there are an infinite number) has an infinite number of paths.

To understand the potential differences, consider the differences between a prediction interval and a confidence interval. If you choose a fixed point as your parameter estimate, then you will have too little uncertainty. Your distribution of possible realizations will be too narrow. You have only accounted for the uncertainty in the price process, not the uncertainty in the sample you used to create the estimate.

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.