Why Vega and Theta Are Not Model-Free Option Greeks
Summary
The discussion asks whether the risk-neutral probability density inferred from option prices can provide vega and theta in the same nearly model-independent way sometimes used to describe delta and gamma. The answer is that vega depends on how volatility and option prices are represented. Local volatility, stochastic volatility, jump models, and Black–Scholes implied volatility can assign different meanings and values to vega; volatility itself is not directly observable in the way spot is.
Theta is also model-dependent because its behavior can reflect interactions among gamma, vega, and other sensitivities. The answer notes that Black–Scholes vega remains definable when market prices are represented with strike-specific implied volatilities. These are conceptual arguments rather than a derivation or empirical comparison. The document does not give a general calculation procedure, and its comments about model-independent delta and gamma are qualified by restrictions on the model class.
Key ideas
- Vega depends on the model or volatility convention used to express option prices.
- A risk-neutral density does not determine a unique model-free vega.
- Theta can reflect interactions among several option sensitivities, making it model-dependent.
- Black–Scholes vega can be defined using implied volatility, including strike-specific implied volatility.
- Claims about model-independent delta and gamma rely on particular assumptions.
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# Can the risk neutral pdf derived from Breeden-Litzenberger Method be used to calculate vega and theta? # Can the risk neutral pdf derived from Breeden-Litzenberger Method be used to calculate vega and theta? I have been researching volatility smoothing techniques and risk-neutral pdf. I noticed one interesting post in Does the risk neutral pdf that is derived using Litzenberger-Breeden Method correspond to gamma and it's integral correspond to delta? Kevin, the answerer of the post, pointed out a property to derive spot delta based on dual delta and spot gamma based on dual gamma. I am wondering is there a similar way to derive vega, theta, and rho in a (almost) model-free approach? Any suggestions and comments are highly appreciated! ## Answer by Frido (score 5, accepted) https://quant.stackexchange.com/a/76237 No, vega cannot be derived in a model-free manner. The reason for this is because in contrast to delta and gamma, there are multiple definitions of vega, and an even deeper underlying reason may be that spot volatility is not an observable. The consequence of this is that 'vega' starts to depend on the quoting mechanism; i.e. do you express the market price of an option in terms of a local vol model, a stochastic vol model, a jump diffusion, or Black Scholes with an implied volatility? All these different quoting mechanisms will lead to a different concept and value of 'vega'. As an example; suppose you are able to fit a pure jump model to the market price of options for a particular time to maturity. What is vega in a pure jump model? However you can (always) translate your pure jump model prices to Black-Scholes prices with implied vols that depend on strike. Then you do have vega. Hope this example makes it clearer. Regarding theta: here too there is no model-free quantity. Recall that in SV models the theta of an option balances the gamma, vega, vanna, volga. As only gamma potentially is model-free, you cannot hope to have model-free theta. As regards delta and gamma: Here under some circumstances (for the class of so-called homogeneous stochastic vol models) there is a model free definition, and this is (partly) because the spot price is an observable and therefore has to occur in any function you use to quote the market price by. Last remark: as option market prices can always be expressed in terms of Black-Scholes prices with strike dependent IV, BS vega is always well-defined. ## Answer by Arshdeep (score 0) https://quant.stackexchange.com/a/76323 Calculating vega in a model free manner is an oxymoron. Vega can be seen as the way to jump from one model's price to another. For example, a model which specifies vol as 10 bps and a model which specifies vol as 11 bps have price difference equal to vega at 10 bps (roughly). If we're saying vega is independent of models (='what vol is'), then prices must be linear in vol. Which doesn't work out because when vol is infinite, price is finite.
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