Binary Options: Pricing, Replication and Skew Sensitivity

The final payoff of a binary (or digital) call option is either one if the asset price is above the strike price at the expiry of the option or nothing.
It can be replicated as the limit of a call spread.

If we assume that the only changing variable is the strike price, the price of the binary call is equal to the opposite of the derivatives of the call price with respect to the strike price which can be approximated by finite central difference.

In the Black-Scholes framework there is a closed-form solution for the price of the binary call option, it is equal to the probability that the option will be exercised times the discount factor.

But what if the implied volatility is not constant?
If we assume that the implied volatility is no more constant and is a function of the strike price, we can approximate the price of the binary call option as the sum of two terms.
The first one corresponds to the Black-Scholes price of the binary option, assuming the implied volatility is constant.
The second term is a correction to the Black-Scholes price as the implied volatility is actually not constant. It is the opposite of the product of the vega of a call option and the derivatives of the implied volatility with respect to the strike price, also named volatility skew.

In the equity market we observe in general a downward sloping volatility as a function of the strike price, meaning a negative skew.
Lower strikes tend to have a higher volatility than higher strikes making the price of the call spread which is long on the lower strike and short on the higher strike more expensive.
The price of the digital option increases when the skew becomes more negative. The impact is far from being negligible.

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