Risk Reversals, Volatility Skew, and Digital Option Prices
Summary
The discussion examines an apparent conflict between a more negative risk reversal and the price of a European digital put. A steeper downside volatility skew is sometimes read as evidence that a decline is more likely, yet changing the skew while holding at-the-money volatility fixed can reduce the digital put price. The replies clarify that higher downside implied volatility does not, by itself, mean a higher probability of a down move; it can reflect expectations of larger or more violent declines when they occur.
The note also distinguishes a digital option, which pays only if spot finishes beyond its strike at expiry, from a one-touch option, which pays if spot reaches a level during the option's life. The latter may respond differently to a skew change. These are qualitative explanations rather than a full pricing derivation, and the discussion does not establish that skew has one universal interpretation across markets or models.
Key ideas
- Downside implied volatility being higher does not alone imply a greater probability of a down move.
- A volatility skew can reflect the expected size or severity of moves as well as market pricing of risk.
- A European digital pays according to where spot finishes at expiry.
- A one-touch option depends on whether spot reaches a level before expiry, so its price can react differently to skew.
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Full text
# A PARADOX? - relationship between risk reversal (slope of vol smile) and digital price # A PARADOX? - relationship between risk reversal (slope of vol smile) and digital price 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. ## Answer by Lliane (score 3) https://quant.stackexchange.com/a/51829 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%. ## Answer by joe smith (score 0) https://quant.stackexchange.com/a/61765 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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