The magnitude of Delta can vary across the underlying asset price. Table 9.5 presents the value of Delta across the underlying asset price for each position.
Table 9.5 Delta across the underlying asset price Position Delta:
long or short
Delta range
Delta when deep- OTM
Delta when deep-ITM
Delta when underlying asset price is much lower than the forward price/strike price
Delta when underlying asset price is much higher than the
forward price/strike price
Long forward
Long Always 1 N/A N/A 1 1
Short forward
Short Always
−1
N/A N/A −1 −1
Long call Long Between 0 and 1
Approaches 0
Approaches 1
Approaches 0 Approaches 1
Short call Short Between
−1 and 0
Approaches 0
Approaches -1
Approaches 0 Approaches −1 Long put Short Between
−1 and 0
Approaches 0
Approaches -1
Approaches −1 Approaches 0 Short put Long Between
1 and 0
Approaches 0
Approaches 1
Approaches 1 Approaches 0 Some notes in relation to Table 9.5 are as follows:
Deltas of option positions vary across the underlying asset price, while Deltas of forward positions do not.
The concept of moneyness does not apply to forwards, as both parties face obligations and neither party gets to choose whether to transact.
A long call is deep-OTM when the underlying asset price is much lower than the strike price, and deep-ITM when the underlying asset price is much higher than the strike price.
A long put is deep-OTM when the underlying asset price is much higher than the strike, and deep-ITM when the underlying asset price is much lower than the strike price.
Technically, option Delta is not exactly equal to 0, 1, and −1, even when deep-ITM or deep-OTM, though the values for Delta approach very close to these values.
The ranges of values are for forwards and European-style options where the
underlying asset pays no income. The ranges for other variations are similar but not identical.
Because long forward Delta never changes, a long forward is known as a “one Delta” or
“Delta one” position. It is described this way as its Delta is universally the same number across the underlying asset price.
The diagrams in Figure 9.1 illustrate Delta across underlying asset prices for long and short positions in calls and puts for the following example:
Figure 9.1 Delta across the underlying asset price for long and short calls and puts Strike price = $95
Time to expiration = 1 year
Underlying asset volatility = 20%
Risk-free interest rate = 2%
Why does the Delta of a forward position remain constant, while the Delta of an option position does not? The explanation is provided next.
9.3.1 Why Delta of a forward position is constant
The Delta of a forward position is constant because the counterparties to a forward are both obligated to transact. Since the transaction will certainly take place, the
counterparties fully experience gains or losses as the underlying asset price changes. After all, the long forward will, with certainty, eventually receive the asset while the short
forward will, with certainty, eventually deliver the asset. Hence, each position is fully sensitive to the change in the underlying asset price. Therefore, the long forward Delta is 1 and the short forward Delta is −1.
9.3.2 Why Delta of an option position varies across the underlying asset price
The Delta of an option position varies across the underlying asset price as the long position to an option is not obligated to transact. Therefore:
When the option is deep-OTM, it doesn't matter very much that the underlying asset price has changed, as there is low likelihood the transaction will occur. Hence, there is little value sensitivity and Delta is negligible.
When the option is deep-ITM, changes in the underlying asset price impact the
counterparties, as there is high likelihood that the transaction will occur. The option positions are fully sensitive and Delta is 1 for the purchasing counterparty and −1 for the selling counterparty.
When the option is near-the-money, changes in the underlying asset price are
somewhat impactful, as there is some likelihood that the transaction will occur. The option positions are therefore somewhat sensitive, and Delta will be between 0 and 1 for the purchasing counterparty and between 0 and −1 for the selling counterparty.
To illustrate, consider a call option that has a strike price of $100. Delta across the underlying asset price is as follows:
When the underlying asset price is $20, this option is deep-OTM. It is highly unlikely that this option will be exercised. After all, the long call will not rationally exercise its right to purchase the underlying asset for $100 when the asset is worth only $20.
Should the underlying asset price increase to $21, it is not impactful: Just as the long call will not exercise its right to purchase for $100 when the underlying asset price is
$20, so too the long call will not exercise when the underlying asset price is $21.
Hence, both the long and short calls' Deltas are negligible.
When the underlying asset price is $150, this option is deep-ITM. It is highly likely that this option will be exercised. Should the underlying asset price then increase to
$151, it is very impactful: Since the long call will almost certainly exercise, the long call benefits fully from the increase in the underlying asset price. Hence, Delta will approach 1 for the long position and −1 for the short call.
When the underlying asset price is $100, this option is ATM. There is some likelihood of it being exercised. Therefore, should the underlying asset price increase to $101, it is somewhat impactful. Hence, Delta will be between 0 and 1 for the long call and between 0 and −1 for the short call.
We see that Delta close to 0 indicates that the option will very likely not be exercised, while a Delta of close to either 1 or −1 indicates that the option will very likely be exercised.
The absolute value of Delta approaches 1 when an option is deep-ITM and approaches 0 when deep-OTM. Hence the absolute value of Delta can be perceived as a measure of moneyness. When perceiving Delta from this perspective, Delta is presented in absolute terms and without the use of a decimal. For example, an absolute Delta of 75 indicates that the position is ITM, while an absolute Delta of 25 indicates that the position is OTM.
Knowledge check
Q 9.19: For which positions does Delta not vary across the underlying asset price?
Q 9.20: How does long call Delta vary across the underlying asset price?
Q 9.21: How does short call Delta vary across the underlying asset price?
Q 9.22: How does long put Delta vary across the underlying asset price?
Q 9.23: How does short put Delta vary across the underlying asset price?
Q 9.24: What is a “one Delta” position?
Q 9.25: Why are forward positions one Delta?
Q 9.26: Why does the Delta of an option position vary across the underlying asset price?