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how do we resolve this seeming paradox? lets take GBPUSD now: it has a negative risk reversal, ie putvols > call vols , because traders expect spot to fall, so they are buying puts, pushing their vols up. and if the risk reversal becomes MORE negative , then based on the above, you'd say theres even MORE expectation of the spot falling. HOWEVER, if you actually go ahead and request a quote in the market for betting on the spot falling , eg a 3month european digital put (which are priced as leveraged put spreads, ie taking into account slope of vol smile), you'll see that when the RR becomes more negative (ie if you change your volsmile SLOPE to achieve this , without moving the atm), the digital price (which represents traders view on the probabilty of the payout occuring) goes DOWN , despite that we established above that traders think that change in RR means the prob goes UP!

also, perhaps a simpler way to look at this is , if you price an ATM digital put , you'd think it would be worth more than 50%, but it's less.

Now i understand WHY this occurs - i understand the maths here of digital pricing - but what i want to understand is how can one logically explain it, given my above explanation of what causes the vol skew slope.

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I think you have a misunderstanding here, the fact that the vol is higher on the downside doesn't mean the probability of the price going down is higher, it means that the magnitude of down moves (if they happen) is higher than the magnitude of up moves. It doesn't mean traders expect a down move to be more likely, it means traders expect a down move would be more violent (and imply more subsequent volatility) that an up move.

You can very well have 40% probability of down moves (with a 20% down move) and 60% of up move (with a 13% move up), in that case the puts may be more valuable than the calls, but the price of a digital put should be 40%.

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  • $\begingroup$ i understand what you say , but it does not fully add up,. perhaps i can prove you are wrong by contradiction: i submit to you that its impossible to have a vol smile that gives you a 60% prob of down move, where low strike vols are above high strike vols , ie where by your explanation, market thinks down moves will be bigger (higher vol) than up moves ! @Lliane $\endgroup$
    – Randor
    Mar 26, 2020 at 15:04
  • $\begingroup$ I don't understand your reply, by definition no arbitrage you can't have both a higher probability of down moves and larger moves on the downside. If it was the case you can just short the asset and have a positive expected payoff. $\endgroup$
    – Lliane
    Mar 30, 2020 at 3:36
  • $\begingroup$ Nevertheless , i think that such scenarios can occur in reality of for example 55% prob of down move of 5% vs 45% prob of upmove of 2% , eg in the markets today i would think! I dont think youre right that this is an arbitrage, as arbitrage works always, not just in expectation $\endgroup$
    – Randor
    Mar 30, 2020 at 20:25
  • $\begingroup$ That's your personal expectation, but that's not what is implied in the market probabilities. $\endgroup$
    – Lliane
    Mar 31, 2020 at 4:32
  • $\begingroup$ when we talk about downside vol and upside vol , is there some mathematical description you can give of that so that we can verify mathematically that these vols are in fact as you describe? eg how can we mathematically show that the vol at a low strike is the standard deviation of the underlyings movements if it is going down in the vicinity of that strike ? that would be very interesting . as it is confusing a bit, since the volsmile shows implied vol for different STRIKES. not for different values of the underlying. $\endgroup$
    – Randor
    Apr 2, 2020 at 20:44
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A relevant point here is that the european digital pays out only at expiry. Intuitively a decrease in the risk reversal (more negative) implies that spot will be more volatile at the strike level. If spot does in fact cross the strike level then the vega of the option will have flipped where higher vols will mean lower price. If you check the price the of a one touch with the same strike then the more negative RR will show a higher price of the option.

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