CoachA digital option (binary option) pays a fixed amount if the underlying ends above (call) or below (put) the strike, and nothing otherwise. Under Black–Scholes a cash digital call paying 1 is worth .
Examples
Example 15.2 (A three-month digital on chapter 9’s surface)
Scale chapter 9’s surface to an index at 5 000. A three-month at-the-money digital call on 10 million is worth 4.84 million with Black–Scholes at the at-the-money volatility of 16.4%. With the smile it is worth 5.77 million: the vega term adds 0.93 million, almost a fifth of the price.