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]:

enter image description here

can you explain how to obtain the following result: enter image description here

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

  • $\begingroup$ Also, the authors mention a set of lecture notes from Bruno Dupire form his 2019 continuous time finance course where Dupire shows an arbitrage strategy that can be applied to incorrect binary forecasts. I could not find these notes. If anyone knows where they are I would appreciative it. $\endgroup$ – roz Aug 9 '19 at 16:17
  • $\begingroup$ I believe that he is using the reflection principle but I am not exactly sure how. $\endgroup$ – roz Aug 9 '19 at 16:47

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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    $\begingroup$ Can you please explain more clearly what you mean in your second paragraph (and correct me if you think what I say is wrong). I understand that by imposing bounds on the price of a binary we also get bounds on the possible volatility of the forecast. I also understand that the crux of the argument is basically that if the volatility of the forecast is too high or too low we can generate returns via arbitrage. I just do not exactly understand what you mean by "roof it to -inf, +inf". Could you possibly be more concrete? $\endgroup$ – roz Aug 9 '19 at 20:17
  • $\begingroup$ hi, sure. I don't think that price bounds do constrain volatility, in NNT's twisted logic; and that's precisely his "conceit". If probability was not bounded [0,1], it could randomly-walk [-inf,+fin] or [0,+inf] like any other option on anytinh. But allow the same random walk with the same vol, but bounded thus, then suddenly you get a free lunch "arb'ing" those bounds. Except there's no hint given how to infer "sigma" here. There is no proposal for a level of vol that is too high (or too low) to arb. NNT is (again) full of hot air and ego... $\endgroup$ – demully Aug 9 '19 at 22:36

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