Binary Call Option Delta
Introduction to Binary Call Option Delta
Binary call option delta measures the change in the price of a binary call option owing to a change in the underlying asset price and is the gradient of the slope of the binary options price profile versus the underlying asset price (the underlying).
Of all the Greeks, the binary call option delta could probably be considered the most useful in that it can also be interpreted as the equivalent position in the underlying, i. e. the delta translates options, whether individual options or a portfolio of options, into an equivalent position of the underlying.
A binary call option with a delta of 0.5 means that if the underlying share price goes up 1¢ then the binary call will increase in value by ½¢. Another interpretation would be a short 400 contract position in S&P500 binary calls with a delta of 0.25 which would be equivalent to being short 100 S&P500 futures.
It is important to realise that the delta is dynamically changing as a function of many variables, including a change in the underlying price, and that a change in any of those variables will most likely cause a change in the delta. Therefore, if any or all of the variables, including the underlying price, time to expiry and implied volatility, change then the above option will not necessarily have a delta of 0.5 and increase in value by ½¢ or the equivalent S&P position be short 100 S&P500 futures.
This practicality and simplicity of concept contributes to deltas, out of all the Greeks, being the most utilised amongst traders, especially market-makers.
The following provides an analysis of:
the finite difference method to evaluate deltas,
examples of using the delta to hedge with,
comparisons of conventional call options delta with binary call options delta, and finally
a closed-form formula for the binary call options delta.
Binary Call Option Delta and Finite Delta
The delta Δ of any option is defined by:
Δ = δP / δS
P = price of the option
S = price of the underlying