The Pricing of Binary Options

As you are trading your binary options did you ever stop and ask yourself how are binary options priced? Well, for the most part their value is calculated based most of the time on the Black–Scholes model. This mathematical model is based on a derivatives market which will give the price of a European style option. Independent tests of the model have shown that the model produces quite close to actual quotes with some discrepancies known as the “option smile.”

The Black Scholes model is a partial differential equation that describes the price of the option vs. time. The key concept is to flawlessly “hedge” the option by buying and selling the underlying asset that cancels risk. This strategy is named delta hedging and is the basis for many other trading strategies. As such the formula computes that there is one true price on the option which is calculated by the Black Scholes formula.

The value of a call option for a non-dividend-paying underlying stock in terms of the Black–Scholes parameters is:

The price of a corresponding put option based on put-call parity is:

For both, as above:

One of the most important components of the equation, as mention earlier above, is the delta. The binary call option delta measures the variance in the price of the call option based on the change in the underlying asset’s price and is the angle of the slope of the binary options price profile versus the underlying assets. The option pricing formula uses Greek symbols, and from all these symbols, the binary option’s delta is regarded as the most practical tool because it indicates the status of the underlying asset. For example, a binary call option with a delta of 0.5 implies that if the underlying share price goes up 1¢ then the binary call will also increase by ½¢. Another example shows that a short 400 contract position in S&P500 binary calls with a delta of 0.25 equals a short position in 100 S&P500 short futures. Remember though that the delta is always changing because of the change in the underlying asset, and any other change in other variables will cause the delta to change. So if any or all of the variables in the equation adjust which include underlying price, time to expiry, implied volatility changes, then the binary option will not necessarily always increase in value by ½¢ or the above mentioned example of S&P position short futures. Yet for all it’s worth, the utility of the delta- from all the Greek symbols used in the formula- is the most implemented part of tool used in trading.