How Volatility Skew Changes a Digital Call’s Risk-Neutral Price
Summary
A digital call pays when the underlying finishes above its strike, so its discounted price equals the risk-neutral probability of finishing in the money. The document connects this price to the strike derivative of an ordinary call price. When ordinary call prices are expressed using Black–Scholes implied volatility that varies with strike, differentiating introduces an extra term: Black–Scholes vega multiplied by the strike slope of implied volatility.
This relation explains why a volatility smile or skew affects digital prices beyond the price implied by volatility at the digital’s strike. The answer also gives an intuitive approximation: a digital can be viewed as a narrow call spread. With negative skew, the higher-strike call has lower implied volatility and loses relative value, raising the call spread’s value. The explanation assumes a differentiable call-price curve and uses the Black–Scholes implied-volatility representation; the digital price remains tied to the full risk-neutral distribution, not to a standalone probability inferred from one volatility quote.
Key ideas
- A digital call price is the discounted risk-neutral probability that the underlying finishes above its strike.
- The digital price equals the negative strike derivative of the corresponding ordinary call price.
- Strike-dependent implied volatility adds a vega-times-skew correction to the Black–Scholes digital value.
- A narrow call spread approximates a digital and makes the effect of skew easier to see.
- Negative skew raises the call-spread value in the described setup.
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# Effect of vol smile on risk neutral probability of ITM
# Effect of vol smile on risk neutral probability of ITM
I was asked in an interview about how the vol smile affect the price of a binary option, which is essentially the Prob(ITM) under risk neutral measure. My thought is that the implied vol at spot which makes the option OTM is high, that means the prob(ITM) at that region is higher and so the price of binary option. Please correct me or give a more rigorous extension to my thoughts.
## Answer by AFK (score 3, accepted)
https://quant.stackexchange.com/a/17120
First note that the price of binary call is related to the price of an ordinary call in any model by $$ BinC(T,K) = e^{-rT}\mathbb{E}^{\mathbb{Q}}[1_{S_T>K}] = - \frac{\partial}{\partial K}e^{-rT}\mathbb{E}^{\mathbb{Q}}[(S_T-K)_+] = - \frac{\partial}{\partial K}C(T,K) $$ Now the volatility smile is implicitly defined by $$ C(T,K) = C_{BS}(T,K,\Sigma(T,K)) $$ So, by taking minus the derivative wrt $K$, we get $$ BinC(T,K) = BinC_{BS}(T,K,\Sigma) - \partial_\sigma C_{BS}(T,K,\Sigma )\partial_K\Sigma(T,K) $$ In other words, the volatility smile leads to a corrective term to the price of a binary call which is the BS vega times the skew.
Conversely this formula can be used to calculate the skew based on the price of digitals.
To get a less formal intuition, replace your digital call by a small call spread $\frac{1}{\Delta K}(C(T,K)-C(T,K+\Delta K))$. In BS the vol at $K+\Delta K$ is the same than at $K$. If the volatility decreases with the strike (negative skew) then the volatility at $K+\Delta K$ is lower than the volatility at $K$. So the second call loses value and the call spread's value increases. Since a digital is nothing but the limit of the call spread when $\Delta K \to 0$, you see that its price increases when the skew becomes more negative.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.