The magnitude of Gamma can vary across underlying asset prices. The magnitude of Gamma depends on the moneyness of the option, as detailed in Table 9.7.
Table 9.7 Gamma across underlying asset prices Position Gamma: long,
short, or neutral
Gamma when deep-OTM
Gamma when near-the-money
Gamma when deep-ITM Long
forward
Neutral N/A N/A N/A
Short forward
Neutral N/A N/A N/A
Long call Long Approaches zero Highest Approaches zero
Short call Short Approaches zero Lowest Approaches zero
Long put Long Approaches zero Highest Approaches zero
Short put Short Approaches zero Lowest Approaches zero
Note that the concept of moneyness does not apply to forwards as both counterparties are obligated to transact. Note as well that while technically option Gamma is not exactly zero even when deep-ITM or deep-OTM, the values for Gamma approach very close to zero.
The diagrams in Figure 9.2 illustrate Gamma across the underlying asset price for long and short positions in calls and puts for the same example that is presented in Figure 9.2.
Figure 9.2 Gamma across the underlying asset price for long and short calls and puts Observe the following from Figure 9.2:
Gamma approaches zero when the options are either deep-OTM or deep-ITM.
The magnitude of an option's Gamma is largest near-the-money.
Call and put Gamma are identical to each other.
Let's explore each of these observations in detail.
Why does Gamma approach zero when the options are either deep-OTM or deep-ITM?
Gamma approaches zero when Delta is not sensitive to changes in the underlying asset price, which occurs when deep-OTM and deep-ITM, for the following reasons:
A deep-OTM option is unlikely to be exercised. Hence, change in the underlying asset price barely impacts option value. Therefore, Delta is stable at close to zero, and
Gamma approaches zero.
A deep-ITM option is likely to be exercised, and change in the underlying asset price barely changes this likelihood. Hence, Delta is stable at close to 1 or −1, and Gamma approaches zero.
Why is the magnitude of an option's Gamma largest near-the-money?
Delta is quite sensitive to changes in the underlying asset price when near-the-money.
After all, when near-the-money, a change in the underlying asset price can significantly change the likelihood of the option being exercised and, as a result, the value of the
option. Therefore, Delta is very sensitive to changes in the underlying asset price, and the magnitude of Gamma is relatively large.
Why are the Gammas of a call and put identical to each other?
Long call Gamma is identical to long put Gamma. This is despite the fact that a long call is always long Delta and a long put is always short Delta. The Gammas are identical because both long call and long put Delta increase in identical fashion as the underlying asset price increases, as illustrated in Figure 9.1:
The long call Delta increases from 0 to 1.
The long put Delta increases from −1 to 0.
Similarly, short call Gamma is identical to short put Gamma, as the Delta of both
decreases in an identical fashion as the underlying asset price increases, as illustrated in Figure 9.1.
Knowledge check
Q 9.34: For which positions does Gamma not vary across the underlying asset
price?
Q 9.35: How does long call Gamma vary across the underlying asset price?
Q 9.36: How does short call Gamma vary across the underlying asset price?
Q 9.37: How does long put Gamma vary across the underlying asset price?
Q 9.38: How does short put Gamma vary across the underlying asset price?
Q 9.39: Why does Gamma approach zero when the option positions are either deep-OTM or deep-ITM?
Q 9.40: Why is the magnitude of Gamma largest near-the-money?
Q 9.41: Why are the Gammas of a call and put identical to each other?
KEY POINTS
The expressions “long,” “short,” and “neutral” refer to whether a given Greek is positive, negative, or zero.
Long Delta describes positive value sensitivity to changes in the underlying asset price. Short Delta describes negative sensitivity. Delta neutrality describes zero sensitivity.
Long Gamma describes positive Delta sensitivity to changes in the underlying asset price. Short Gamma describes negative sensitivity. Gamma neutrality describes zero sensitivity.
A long forward is long Delta and Gamma neutral. A long call is long Delta and long Gamma. A long put is short Delta and long Gamma. Short positions have the opposite characteristic to their corresponding long positions due to the zero-sum nature of forwards and options.
A purchasing counterparty is long Delta and a selling counterparty is short Delta.
Option Delta varies across the underlying asset price and forward Delta does not. The Delta of a forward position is constant because its counterparties are both obligated to transact. The Delta of an option is sensitive to the underlying asset price as its long position is not obligated to transact.
An option's Gamma varies across the underlying asset price and a forward's Gamma is always zero.
Chapter 10
Understanding Vega, Rho, and Theta INTRODUCTION
In the previous chapter we learned about position sensitivity as measured by Delta and Gamma. This chapter explores the use of Vega, Rho, and Theta to understand and
describe position sensitivity.
After you read this chapter, you will be able to
Understand that Vega, Rho, and Theta are measures of sensitivity.
Describe each position's sensitivity using Vega, Rho, and Theta.
Understand the Vega, Rho, and Theta characteristics of each position.
Understand the symmetrical sensitivity of forwards and the asymmetric sensitivity of options.
Explain when positions are long Theta and when they are short Theta.