BE332 - Class 10 - Options
1) What does it mean to assert that the delta of a call option is 0.7? How can a short position in 1,000
options be made delta neutral when the delta of each option is 0.7?
Suggested solution:
A delta of 0.7 means that, when the price of the stock increases by a small amount, the price of the option
increases by 70% of this amount. Similarly, when the price of the stock decreases by a small amount, the
price of the option decreases by 70% of this amount. A short position in 1,000 options has a delta of −700
and can be made delta neutral with the purchase of 700 shares.
2) Calculate the delta of an at-the-money six-month European call option on a non-dividend-paying
stock when the risk-free interest rate is 10% per annum and the stock price volatility is 25% per
annum.
Suggested solution:
In this case, S0 = K , r = 01,  = 025 , and T = 05 . Also,
ln(S  K ) + (01+ 0252  2)05
d1 =
0 = 03712
025
The delta of the option is N (d1) or 0.64.
3) What does it mean to assert that the theta of an option position is –0.1 when time is measured in
years? If a trader feels that neither a stock price nor its implied volatility will change, what type of
option position is appropriate?
Suggested solution:
A theta of −01 means that if t
units of time pass with no change in either the stock price or its volatility,
the value of the option declines by 01t . A trader who feels that neither the stock price nor its implied
volatility will change should write an option with as high a negative theta as possible. Relatively short-life
at-the-money options have the most negative thetas.
4) What is meant by the gamma of an option position? What are the risks in the situation where the
gamma of a position is large and negative and the delta is zero?
Suggested solution:
The gamma of an option position is the rate of change of the delta of the position with respect to the asset
price. For example, a gamma of 0.1 would indicate that when the asset price increases by a certain small
amount delta increases by 0.1 of this amount. When the gamma of an option writer’s position is large and
negative and the delta is zero, the option writer will lose significant amounts of money if there is a large
movement (either an increase or a decrease) in the asset price.
5) A financial institution has the following portfolio of over-the-counter options on sterling:
Type Position Delta of Option Gamma of Option Vega of Option
Call −1,000 0.5 2.2 1.8
Call −500 0.8 0.6 0.2
Put −2,000 -0.40 1.3 0.7
Call −500 0.70 1.8 1.4
A traded option is available with a delta of 0.6, a gamma of 1.5, and a vega of 0.8.
(a) What position in the traded option and in sterling would make the portfolio both gamma neutral and
delta neutral?
(b) What position in the traded option and in sterling would make the portfolio both vega neutral and delta
neutral?
Suggested solution:
The delta of the portfolio is
−1 000  050 − 500  080 − 2 000  (−040) − 500  070 = −450
The gamma of the portfolio is
−1 000  22 − 500  06 − 2 000 13 − 500 18 = −6 000
The vega of the portfolio is
−1 000 18 − 500  02 − 2 000  07 − 500 14 = −4 000
(a) A long position in 4,000 traded options will give a gamma-neutral portfolio since the long position has a
gamma of 4 000 15 = +6 000 . The delta of the whole portfolio (including traded options) is then:
4 000  06 − 450 = 1 950
Hence, in addition to the 4,000 traded options, a short position of 1,950 in sterling is necessary so that
the portfolio is both gamma and delta neutral.
(b) A long position in 5,000 traded options will give a vega-neutral portfolio since the long position has a
vega of 5 000  08 = +4 000 . The delta of the whole portfolio (including traded options) is then
5 000  06 − 450 = 2 550
Hence, in addition to the 5,000 traded options, a short position of 2,550 in sterling is necessary so that
the portfolio is both vega and delta neutral.
6) Consider again the situation in Q6. Suppose that a second traded option with a delta of 0.1, a gamma
of 0.5, and a vega of 0.6 is available. How could the portfolio be made delta, gamma, and vega
neutral?
Suggested solution:
Let w1 be the position in the first traded option and w2 be the position in the second traded option. We
require:
6 000 = 15w1 + 05w2
4 000 = 08w1 + 06w2
The solution to these equations can easily be seen to be w1 = 3 200 ,
has a delta of
w2 = 2 400 . The whole portfolio then
−450 + 3 200  06 + 2 400  01 = 1 710
Therefore the portfolio can be made delta, gamma and vega neutral by taking a long position in 3,200 of the
first traded option, a long position in 2,400 of the second traded option and a short position of 1,710 in
sterling.