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Exchanges will usually list greeks along with the option, for example this SPX call on the CBOE: SPX241220C04000000

But how does the exchange calculate these greeks? I can think of a few ways to do so:

  • Use the "instantaneous" values of the most recent trades (minimum number to numerically compute the derivative)
  • Fit a hyperplane through a predefined amount of samples (say, minutes? hours? days? of time)
  • Fit the entire trade history to a model (e.g., B-S), and derive the greeks from the model

Presently I'm primarily interested in delta and theta, and I'm wondering how much I can "trust" the greeks to predict the long-term (~months) behaviour of the option price.
If the greeks are "instantaneous" values, then they would be extremely noisy and useless for my purposes. But if they are derived from the long-term history of the option, then it would be much more reliable for me

小太郎
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    Greeks do not predict the behaviour of an option. – amdopt Sep 10 '23 at 13:26
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    These greeks, for sure, are not based on any long- or short-term history of anything. Perhaps you are confusing them with some historical calculations - correlations (also labeled rho), volalility, beta, etc. – Dimitri Vulis Sep 10 '23 at 13:52
  • Doesn't delta measure how much the price changes against changes in the underlying asset, and theta by the passage of time (all else held constant)? – 小太郎 Sep 10 '23 at 14:07
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    They don't measure anything based on history. A good exercise for you may be to build a P&L explain, to get a better feel for what these number can actually do. – Dimitri Vulis Sep 10 '23 at 14:25
  • If it's not built from historical values, what is it built from? That's actually what the question is asking. From knowing how the values are derived in practice, I can understand what they can be useful for – 小太郎 Sep 10 '23 at 15:35
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    Does this answer your question? Black Scholes Theta Finite difference. Wikipedia should answer the rest. With American options it will either be bump and reprice, but for delta, gamma and theta most likely directly from the nodes in CRR (recombining binomial tree) or the grid of FD solver. – AKdemy Sep 10 '23 at 16:02
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    It seems like OP doesn't know anything about greeks in Black-Scholes model so I'm voting to close it as a basic financial question. – Hasek Sep 10 '23 at 16:05
  • I understand that greeks are theoretically calculated from the (e.g., B-S) model, but the model itself has parameters that are unknown. Specifically for B-S, the dividend yield, risk-free interest rate, and volatility/stddev. So to use the model, those parameters need to be estimated. Volatility can be taken from implied volatility, but what about the others? – 小太郎 Sep 10 '23 at 16:25
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    Overview: (0) A model is chosen, such as the Black Scholes Merton model, (1) the unknown parameter(s) in the model, in this case $\sigma$ (volatility) is/are adjusted until the model price matches the market price (2) the Delta, Gamma, Theta etc produced by the model for this value of $\sigma$ are displayed on the CBOE web site. – nbbo2 Sep 10 '23 at 16:26
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    The interest rate $r$ can be found in the interest rate markets, the $r-d$ and therefore $d$ can be found from the pricing of futures/forwards on SPX in various ways... (for ex. the implied forward price is roughly speaking the strike level where thr Put and the Call are equally priced). – nbbo2 Sep 10 '23 at 16:33
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    If the answer and Wikipedia link I provided isn't enough, there is money SE for basic financial questions but if you just read any option book you should understand how Greeks are computed. The risk free rate comes from swap curves, dividend yield can also be derived (from dividend schedules consisting of declared dividends with dates and amounts and forecasts afterwards for example), and IV is simply what you solve the market price for. – AKdemy Sep 10 '23 at 16:34
  • Thanks - that answers my question and was the key missing part of my understanding. The unknown parameters of the model are not all derived from the options price itself and the underlying asset, but from other parts of the market too – 小太郎 Sep 10 '23 at 16:43
  • The sensis do predict, in a very limited sense, the future value of the instrument, but only conditionally and ceteris paribus. E.g., some delta figure says that if the price of the underlying changes, and nothing else changes, then, ignoring the gamma, the option premium will change by dela $\times$ underlying price change. Interpreting this as a "prediction" doesn't sound very helpful. – Dimitri Vulis Sep 10 '23 at 17:35
  • My intention was to use the greeks to solve for the B-S model parameters, and use the model to... model the behaviour of options. I asked the question in a round-about way because I didn't want to assume the use of any specific model, but also incorrectly assumed that all parameters were derived from the options pricing itself like IV is – 小太郎 Sep 10 '23 at 17:51
  • Related questions to add to the linked questions list: https://quant.stackexchange.com/questions/30177/how-much-can-be-said-about-the-greeks-without-picking-a-model https://quant.stackexchange.com/questions/21299/can-i-get-black-scholes-option-price-from-greeks – 小太郎 Sep 10 '23 at 18:05

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