The option's delta is the rate of change of the price of the option with respect to its underlying security's price. The delta of an option ranges in value from 0 to 1 for calls (0 to -1 for puts) and reflects the increase or decrease in the price of the option in response to a 1 point movement of the underlying asset price.
Far out-of-the-money conventional options tend to have deltas close to zero. A deep in-the-money call tends toward +1; a deep in-the-money put tends toward −1. These signs describe the option itself; a short position reverses them.
Up delta , down delta
As the delta can change even with very tiny movements of the underlying stock price, it may be more practical to know the up delta and down delta values. For instance, the price of a call option with delta of 0.5 may increase by 0.6 point on a 1 point increase in the underlying stock price but decrease by only 0.4 point when the underlying stock price goes down by 1 point. In this case, the up delta is 0.6 and the down delta is 0.4.
Passage of time and its effects on the delta
As expiration approaches with other inputs held fixed, an in-the-money call’s delta tends toward +1 and an in-the-money put’s delta toward −1. Out-of-the-money deltas tend toward zero. A call-only chart should not be read as saying that an in-the-money put’s signed delta rises toward +1.
The chart above illustrates the behaviour of the delta of options at various strikes expiring in 3 months, 6 months and 9 months when the stock is currently trading at $50.
Changes in volatility and its effect on the delta
Changing implied volatility changes delta as well as option value. In the illustrated call example, higher volatility raises the delta of out-of-the-money calls and lowers it for some in-the-money calls. The exact effect depends on moneyness, time, rates, dividends and model conventions; this is not a universal monotonic rule for every call and put.
The chart above depicts the relationship between the option's delta and the volatility of the underlying security which is trading at $50 a share.
The charts illustrate model relationships with selected inputs held fixed. They are not current market quotes or universal curves for every option.
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Using the estimate
Greeks are local model sensitivities, not guaranteed price changes. They change as the stock, time and implied volatility change. Check whether a quoted value is per share or per contract, which volatility increment is used, and whether theta is measured per day or per year. Selling an option reverses the position’s Greek signs.
Changing delta in contract dollars
Three standard calls with delta 0.40 initially have 120 shares of delta exposure. For a small $1 stock rise, the first-order option-value change is about +$120 with other inputs fixed. If gamma per share is 0.05 per $1 move, a $2 rise changes modeled delta toward 0.50 and exposure toward 150 shares. A fixed 120-share estimate misses the changing sensitivity.
Using a second-order approximation for the same move gives 3 × 100 × [0.40 × 2 + ½ × 0.05 × 2²] = $270, versus $240 from delta alone. These are local estimates; repricing can differ when gamma, IV and time also change.
Why delta is not a promised probability
In the basic no-dividend European call model, delta is N(d1), while risk-neutral probability of expiring above strike is N(d2). The two are different. Neither automatically represents the real-world probability of a profitable trade after premium and costs. Probability of touching a strike is another event again.
When combining underlyings, 100 shares of delta in one stock is not directly comparable with 100 in a differently priced stock. State whether a portfolio report uses share delta, dollar delta or beta-adjusted exposure, and identify the assumptions behind any conversion.
