When the price of the underlying falls, the call option loses value, but by how much? This discussion brings us to today's topic of option delta risk.

Delta is an indicator used to describe the relationship between changes in the price of the underlying and changes in the value of the option. Specifically, delta refers to the number of units by which the option value should theoretically increase (decrease) when the price of the underlying rises (decreases) by one unit. For example, when delta is 0.5, the price of the option rises by 0.5 units when the price of the underlying rises by 1 unit, while when delta is -0.5, the option price rises by -0.5 units when the price of the underlying rises by 1 unit, i.e. It falls by 0.5 units. Therefore, without taking into account volatility and time changes, an option with a larger absolute delta value will increase in value the more the underlying price changes in a favorable direction, while an option with a smaller absolute delta value will decrease in value the less the underlying price changes in an unfavorable direction.

What about the different options' delta values? Firstly, it is obvious that the delta of a call option must be greater than 0, while the delta of a put option must be less than 0. This is because the price of a call option rises when the underlying rises, while the price of a put option falls. Secondly, the delta will vary according to the real and imaginary value of the option. In simple terms, as the option moves from the imaginary value to the real value, the probability of exercise at expiry also moves from low to high, and because the absolute value of delta is positively correlated with the probability of exercise at expiry, the absolute value of delta also moves from low to high until it approaches 1. For a flat option, the absolute value of delta is around 0.5.

The probability of an option being exercised at expiry will vary depending on the remaining time to expiry, and so will the delta. Obviously, the probability of exercise is related to the real and imaginary values and therefore the impact of expiry on delta varies depending on the real and imaginary values. The closer the expiry date, the more reassured the real option holder will be and the higher the absolute delta will be; the more the false option has one foot off the edge of the cliff, the closer the absolute delta will be to zero; the flat option has a 50/50 chance of becoming real and false, so the delta will always be around 0.5. The graph on the right shows the relationship between the underlying strike price of a 2000-point call option, delta, and the underlying price and remaining time to expiry. The interested reader can try plotting the delta of the put option.

There is also a relationship between implied volatility and delta. We can find that for a dummy call option, implied volatility and delta are positively correlated. For real call options, implied volatility and delta are inversely correlated. This is instructive because, when you are a seller of a vanilla call option and the market jumps up, you are likely to make more profit than your delta position can explain. This is because the instantaneous rise in implied volatility pushes up the delta of the position and therefore creates an accelerating money-making effect.

Some people would think that delta is quite a complicated thing, but in reality, it is the opposite. We learn delta to simplify our understanding of the function of options and to give us an idea of how a portfolio works under different conditions. In a complex options strategy, instead of analyzing the profitability characteristics of each investment separately, we can simply add up the delta and know how to capture the returns and manage the risk, which is the unique delta appeal of options.