Binary Backwards

A Twitter exchange leads to a useful discussion on option modeling for Rolf Poulsen. So, social media isn’t all bad…

Anybody who writes exams or performs job interviews knows the value of questions. If they are based on true stories or statements, even better. To my delight this showed up in my Twitter timeline:

(Let us assume @FMTrader1 describes an at-the-money down-binary (or digital) option with one week (five business days; 5/252 years) to expiry.)

Starter for ten, Q1: What is the initial price of the digital option?

The payoff is either 1 or 0, thus 1 is the only case with a positive rate of return, so the price, p, must solve (1–p)/p = 0.7, i.e., p = 0.588.

Going into modeling, Q2: Is that price consistent with the Black–Scholes model? In the Black–Scholes model, the price of this at-the-money down-binary option is:

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