For those who do not know, the delta is the sensitivity of an option to a movement of the underlying. This can be extended to a portfolio in which the delta reflects the sensitivity of the portfolio to the underlying. It is usually the case that the delta is given to us by the software we use to trade.
The delta in its mathematical form is just the derivative of the option with respect to the underlying. The question to ask of course is how good is this number? if we are dealing with a big portfolio errors in the delta can be damaging in our assessment of exposure to movements of the underlying. The problem with the traditional delta that comes from the black and scholes formula, is that it assumes that trading is continuous and not discrete as it really is.
Discrete Delta
We can evaluate what is known as a “discrete” delta which can give us a better estimate of the true sensitivity. Here is an example. Suppose a stock is traded at $100 and a call option on that stock trades at $5. We note that when the stock goes up to 101, the price of the option goes up to $5.5. The delta in our case would be:
This is the essence of finding a “discrete” derivative. This would prove more accurate than the black scholes formula. This could be extended to even other “greeks” such as the gamma and vega.
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