Option Greeks: Gamma and Vega: Practice Problems and Solutions

This is Section 40 of the Study Guide. See Section 1 here. See Section 2 here. See Section 3 here. See Section 4 here. See Section 5 here. See Section 6 here. See Section 7 here. See Section 8 here. See Section 9 here. See Section 10 here. See Section 11 here. See Section 12 here. See Section 13 here. See Section 14 here. See Section 15 here. See Section 16 here. See Section 17 here. See Section 18 here. See Section 19 here. See Section 20 here. See Section 21 here. See Section 22 here. See Section 23 here. See Section 24 here. See Section 25 here. See Section 26 here. See Section 27 here. See Section 28 here. See Section 29 here. See Section 30 here. See Section 31 here. See Section 32 here. See Section 33 here. See Section 34 here. See Section 35 here. See Section 36 here. See Section 37 here. See Section 38 here. See Section 39 here.

The option Greek gamma (Γ) "measures the change in delta when the stock price increases by $1" (McDonald 2006, p. 382).

Whether a call or a put is purchased, gamma is positive. Thus, delta increases as the stock price increases.

"For a call, delta approaches 1 as the stock price increases. For a put, delta approaches 0 as the stock price increases" (McDonald 2006, p. 384).

For a call and a put with the same strike price and time to expiration, gamma is the same.

Deep in-the-money and out-of-the money options both have values of gamma close to 0.

The option Greek vega "measures the change in the option price when there is an increase in volatility of one percentage point" (where one percentage point = 0.01).

R. L. McDonald suggests the following mnemonic device to help memorize the definition of vega: "vega" and "volatility" begin with the same letter.

The higher the volatility of the underlying asset price, the higher the price of both the call and put options on that asset. Thus, vega is positive for purchased calls and puts.

According to R. L. McDonald, "vega tends to be greater for at-the-money options, and greater for options with moderate than with short times to expiration" - but this is not the case with very long-lived options (386).

Due to put-call parity, the vega for calls and puts with the same strike prices and times to expiration is the same.

Source: McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006, Ch. 12, pp. 382-386.

Original Practice Problems and Solutions from the Actuary's Free Study Guide:

Problem OGGV1. The stock of Delta-Gamma Corporation has a price of $506. Certain call options on Delta-Gamma Corporation stock have a delta of 0.4 and trade for $33 per option. These options also have a gamma of 0.03. The stock price suddenly increases to $508. What is the new call option delta?

SolutionOGGV1. A gamma of 0.03 means that delta increases by 0.03 whenever the stock price increases by $1. Since the stock price has increased by $2, delta has increased by 0.06, and the new delta is 0.46.

Problem OGGV2. The stock of Vega Corporation has a price of $567 and a volatility of 0.45. A certain put option on Vega Corporation has a price of $78 and a vega of 0.23. Suddenly, volatility increases to 0.51. Find the new put option price.

Solution OGGV2. A vega of 0.23 means that for every increase in volatility by 0.01, the option price will increase by 0.23. Volatility here has increased by 6 percentage points, to the put option price increased by 6*0.23 = 1.38 and the new put option price is 78 + 1.38 = $79.38

Problem OGGV3. Certain call options on the stock of Simultaneous Co. have a delta of 0.3, a gamma of 0, and a vega of 0.11. The stock currently trades at $34 per share with volatility of 0.4, and a call option trades at $3 per contract. Suddenly, the stock price increases to $36, and volatility falls to 0.2. Find the new call option price.

Solution OGGV3. Since the gamma of this option is 0, a change in the stock price does not affect delta. Thus, the effect of the stock price change is to increase the option price by (36-34)∆ = 2*0.3 = 0.6

But the effect of the volatility decrease is to decrease the option price by (0.4 - 0.2)*vega/0.01 =

20*vega = 20*0.11 = 2.2

Thus, the new option price is 3 + 0.6 - 2.2 = $1.40

Problem OGGV4. The stock of Gamma LLC currently trades for $60 per share. For which of these otherwise equivalent options and strike prices (K) is the gamma the highest?

(a) Call, K = 2
(b) Put, K = 20
(c) Call, K = 45
(d) Put, K = 61
(e) Call, K = 98
(f) Put, K = 102

Solution OGGV4. The farther an option is in-the-money or out-of-the-money, the closer the option's gamma is to 0. Options (a), (c), and (f) are significantly in-the-money, whereas options (b) and (e) and significantly out-of-the-money. (d), however, is extremely close to being right at-the-money. So the gamma for (d) should be the highest, and (d) is the correct answer.

Problem OGGV5. The stock of Vega Corporation has a price of $567. For which of these strike prices (K) and times to expiration (T, in years) is the vega for one of these otherwise equivalent call options most likely to be the highest?

(a) K = 564, T = 0.2
(b) K = 564, T = 1
(c) K = 564, T = 30
(d) K = 598, T = 0.2
(e) K = 598, T = 1
(f) K = 598, T = 30

Solution OGGV5. Vega tends to be greatest for at-the money options with moderately long times to expiration. A call with K = 564 is closer to being at-the-money than a call with K = 598. Furthermore, T = 0.2 is a short time period, whereas T = 1 is a longer time period without being extremely long. For very long-lived options, vega tends to be smaller than the vega for moderately lived options that are otherwise equivalent. Thus, the greatest value of vega most likely pertains to answer (b).

See other sections of The Actuary's Free Study Guide for Exam 3F / Exam MFE.