Price of a Binary Option

Consider an underlying price of 918. You are presented with the choice of buying or selling a European binary call option with 1 day till expiration and a strike price of 918 (i.e. at the money). The 30 day historical volatility of the underlying, as measured by standard deviation, is 5%, and the risk free rate is 1.50%.

Since this is a binary option, the bet is about whether the underlying price is going up or down tomorrow. So how should this option be priced?  If the price of the underlying is more than 918 in a day, the binary pays off 1, if the price is lower than 918 in a day, the binary pays off 0. Is this a 50/50 bet?  An amateur would say the option should be worth 0.50 since there is an equal chance of each event happening. But the price of this binary call is actually worth slightly more than 0.50. How so?

To value the option scenario described above, we need to consider the (1) time value of money and (2) the volatility of the underlying. If we use a black-scholes model, with the variables listed above, we come up with a call price of 0.506.

Try out various calculations on this example, check out the CME binary calculator link, here.

Using the same variables, but changing the time to expiration, comes up with the following values.