How Option Prices Relate to Future Price Distributions
Summary
The document distinguishes an option-price surface, organized by strike and expiry, from a surface describing the probability distribution of future underlying prices. It asks whether one can be derived from the other and whether option-pricing models therefore amount to forecasting prices. The response explains that an option’s value can be expressed as the discounted expectation of its payoff under an appropriate pricing framework. Given a future-price distribution, this relationship can be used to calculate option values across strikes and maturities.
The response also says that machine-learning forecasts could inform option pricing, while usefulness depends on the quality and purpose of the model. Black–Scholes is characterized as a model of the underlying price process, with volatility needing to be estimated to produce a distribution. The discussion is brief and does not address risk-neutral versus real-world probabilities, discounting details, calibration, or the non-uniqueness of recovering a full distribution from a finite set of option prices. It therefore introduces the connection without presenting a complete pricing or inference method.
Key ideas
- An option price is related to the discounted expected payoff at expiry.
- A modeled future-price distribution can be used to value options across strikes and maturities.
- Black–Scholes models a price process and requires a volatility input to describe its distribution.
- Machine-learning forecasts may be used in option pricing, but their usefulness is not established by the document.
- Recovering a distribution from observed options requires qualifications not covered in the brief response.
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Full text
# Is option surface same as future price probability surface? # Is option surface same as future price probability surface? Let's consider the Option Chain for the Stock. There are two 3D surfaces representing the probability of the future stock price and the option prices. I wonder if they are representing the same thing? 3D Option Surface: there's the set of PUTs and CALLs options, with different expiration, strikes, and premium. This set of options form 3D surface, with x - time, y - strike, z - premium. 3D Future Price Probability Surface: probability surface of the future stock prices. With x - time, y - possible price and z - density (probability of the given price at a given time). And it seems those surfaces are "equal": - It's possible to compute Option Surface from Future Price Surface. By running Monte Carlo over Future Price Probability Surface and computing the premium for every PUT and CALL option strike. - It's possible to compute Future Price Probability Surface from the Option Surface. By trying all the possible Future Price Probability Surfaces, until we find the one that if used for computing the option prices - give prices closest to our Option Surface. Does that means that option pricing methods are kinda the same as methods for future price prediction? And thus the ML technics like RNN for future price prediction are also could be used in Option Pricing. And technics like Black-Scholes are just a fancy way to predict stock prices? P.S. I'm not talking about the Volatility Surface - because as far as I know (maybe I'm mistaken) the Volatility Surface - is abstract surface representing some parameters in Black-Scholes and not real options. I'm talking here about the real surface formed by real option attributes. ## Answer by CABLE (score 2) https://quant.stackexchange.com/a/54554 Q: Does that means that option pricing methods are kinda the same as methods for future price prediction? A: Yes. Option price is equal to the expectation of discounted payoff. Q: And thus the ML technics like RNN for future price prediction are also could be used in Option Pricing. A: Yes. But whether it is useful is another story. Q: And technics like Black-Scholes are just a fancy way to predict stock prices? A: Black-Scholes is a way to model the stock price process. If one would like to predict stock price distribution using Black-Scholes, one needs to have an estimation of the volatility parameter.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.