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Bounded Forecasts, Binary Options, and Volatility Arguments

Article Quant Q&A · Author: roz
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Summary

The document concerns an argument connecting bounded probability forecasts, modeled as an arithmetic Brownian motion, with arbitrage restrictions on binary forecasts. The response frames a forecast bet as similar to a cash-or-nothing binary option: its payoff is bounded, while its current value reflects an expected probability. It suggests that an option-pricing expression involving the normal cumulative distribution may underlie the formula being discussed.

However, the response does not derive the claimed bound on the Brownian motion or explain why the maximum of the expression on the right-hand side bounds it. It explicitly presents its formula interpretation as a guess, then criticizes assumptions about normally distributed forecasts and the meaning of volatility. The material is best treated as a partial intuition and a record of unresolved questions, not as a proof or a reliable trading rule. The claims about selling volatility are not supported by calculations or empirical evidence in the document.

Key ideas

  • The response interprets a bounded probability bet as resembling a cash-or-nothing binary option.
  • It suggests that a Black–Scholes binary call formula may be related to the expression under discussion.
  • The explanation does not derive the Brownian motion bound or justify the maximum used to bound it.
  • The response questions whether normal forecast assumptions and volatility comparisons are appropriate.
  • The trading interpretation is speculative and is not backed by empirical evidence in the document.

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# Why is the value of the Brownian motion bounded by the maximum value of this square difference?


# Why is the value of the Brownian motion bounded by the maximum value of this square difference?












This comes from Taleb and Madeka's paper (https://www.academia.edu/39998351/All_Roads_Lead_to_Quantitative_Finance_Response_to_Clayton_?auto=download) regarding arbitrage restrictions on binary forecasts.

If we have an arithmetic Brownian motion bounded between [L, H]:

can you explain how to obtain the following result:

and how the authors know that the max of the RHS bounds the value of Bt?

## Answer by demully (score 1)

https://quant.stackexchange.com/a/47050

Apologies upfront if my Finance is better than my grasp on the finer points of advanced calculus. I know the argument(s) he's making; and just hope someone more Quant than I can land the point home.

He's arguing that a "simple" bet and a binary option, which is an "exotic" option, are one and the same thing! Which is true. Both end up worth 0 or 1, versus a prior cost = expected probability of p.

The conceit here is that instead of looking at the risk-reward of expected p vs p priced (classic Kelly stuff), he's looking at it in option terms, ie sigma, essentially in vol of price = vol of forecast terms. And making the not-too-exotic point if p is bounded [0,1], then it cannot roof it to [-inf,+inf]. In which case, if IV>RP, selling vol generates returns, irrespective of actual p.

Shock horror, all that before; albeit maybe put in this precise context dressed up exactly this way ;-)

Where I run short on answers is the precise formula. The exposition he's using is the European cash-or-nothing call segment of Black Scholes. Which is N(d2) bit of the classic formulae. My guess is that the integral of that ends up in the formula above.

Which leads to the oh-so revolutionary concept of selling vol when RV>IV. Never heard that one before ;- Even if the basis for the fair value of sigma is gloriously unclear. Even if the assumption that forecast probabilities are normally distributed is news to me, and evidently flawed. How can a move from a 49.5% to a 50.5% expectation be equivalent to 1%/99% going to 0%/100%. Sorry, but that's just mega-bonkers as models go. Even if all of Taleb's prior work screams arrogance on a level I'd struggle to match with anyone historically. We've had warrior-poets; philosopher-kings; warrior-poet-kings etc. The closest I can come to a warrior-poet-philosopher-king to match NNT is Marcus Aurelius. Trouble is, the basis of his philosophy was modesty and how little we could know... Taleb wins ;-)

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