Option Strategist Exercise 2: Exploring the Option Greeks
This exercise is designed to illustrate how the delta, theta, gamma and vega exposures of an options portfolio change as market conditions vary. We will use the following option contract:
Underlying: GBP/USD
Price quoted in USD
Contract size: £1,000
European Option
Options on cash
One tick = 1/10,000 of quoted price
Value of one tick = 0.1 USD
USD Year basis = 360 days
GBP Year basis = 365 days
Set up a 30 day £1,000,000 call (= 1000 contracts), struck at 1.69, with market conditions as follows:
Cable spot rate 1.69
Volatility 20%
US Interest Rates 5%
UK Interest Rates 5%
Note down the option’s premium, as well as its delta and gamma for the following spot rates:
Spot Cable
Premium($)
Delta %
Gamma %
1.4500
100
0.4
0.2
1.5500
2700
6.9
2.3
1.6200
12700
23.8
5.4
1.6900 ATM
38500
51
6.9
1.7600
83700
76.6
5.3
1.8300
143300
91.8
2.5
1.8900
200300
97.2
1
1.9500
259200
99
0.3
2.3000
607600
99.6
0
What is the relationship between premium, delta and gamma? Why is delta not 1 when spot is 2.300, or 0 when spot is 1.4500?
As the underlying increases premium increases, delta approaches 100% and the gamma approaches 0%. As the underlying decreases, premium decreases, delta approaches 0% and gamma also approaches 0%. Delta is not 1 or 0 because there is 30 days to maturity, so there is still a probability of ending up I or out of the money respectively. If you put days elapsed to 27 then the delta will go to 1 or 0.
Draw a rough sketch showing how delta and gamma (include the option price curve) vary with changes in the underlying from the table above:
Delta,
Gamma
Spot Cable
If the delta value was +1.00, how would you expect the call premium to change if the spot rate moved up by one big figure?
By exactly the same amount (e.g 1 cent)
In the following exercise we will analyse the net delta of a position when more than one option is involved. We will set up a short straddle using 20 day ATM puts and calls. Restore the spot rate back to 1.69
Sell £1,000,000 ATM calls (-1000 contracts)
Sell £1,000,000 ATM puts (-1000 contracts)
What are the individual call and put deltas, the overall position delta and the P&L, for the following spot cable rates:
Spot Rate
Call delta %
Put delta%
Position delta%
Position P&L
1.5000
-0.6
99.1
98.5
-126700
1.6700
-40.8
58.9
18.1
-1600
1.6900
-50.8
48.9
-1.9
0
1.7800
-86.9
12.8
-74.1
-37900
1.8800
-98.7
1.1
-97.6
-127300
1.9500
-99.6
0.1
-99.5
-196500
What happens to the position delta as the spot rate increases, and why?
It approaches -1, or -£1m. The short put is deep OTM (delta approaching 0( , the call is deep ITM (delta approaching 1)
Explain the variation in delta as the spot rate moves above and below the ATM strike.
As the underlying trades up, the call delta approaches 1 and the put delta 0, so the net delta exposure approaches -£1m. As the underlying trades down the call delta approaches 0 and the put delta 1, so the net delta approaches +£1m.
d) Restore spot back to 1.6900. In the following table show the delta exposure in %, in Sterling , the required spot Sterling delta hedge and the delta adjustment needed to remain delta neutral:
Spot Rate
Delta Exp %
Delta Exp £
Delta Hedge £
Delta Adjustment
1.6000
74.5
745K
-745k
0
1.6200
61.9
619k
-619k
+126k
1.6400
46.1
461k
-461k
+158k
1.6600
27.9
279k
-279k
+182k
1.6800
8.1
81k
-81k
+198k
1.6600
27.9
279k
-279k
-198k
1.6400
46.1
461k
-461k
-182k
e) Set up a long ATM straddle (strike = 1.6900) and repeat the exercise in part d). What do you observe?
Spot Rate
Delta Exp %
Delta Exp £
Delta Hedge £
Delta Adjustment
1.6000
-74.5
-745K
745k
0
1.6200
-61.9
-619k
619k
-126k
1.6400
-46.1
-461k
461k
-158k
1.6600
-27.9
-279k
279k
-182k
1.6800
-8.1
-81k
81k
-198k
1.6600
-27.9
-279k
279k
198k
1.6400
-46.1
-461k
-61k
182k
f) Restore the short straddle position. What is the value of gamma for the put and call options, and the overall position gamma for the following spot cable rates:
Spot Rate
Call gamma%
Put gamma%
Position gamma%
1.5500
-1.6
-1.6
-3.2
1.6200
-5.8
-5.8
-11.6
1.6500
-7.5
-7.5
-15
1.6900
-8.5
-8.5
-17
1.7500
-6.3
-6.3
-12.6
At what spot rate is the straddle delta most sensitive? Why is this the case?
At 1.6900. This ATM therefore the gamma is highest.
Restore the spot rate back to 1.69. What is the position gamma and P&L (Theta) after the following days have elapsed, other things being equal?:
Days Elapsed
Position gamma%
P&L
Theta
5
-19.6
+8300
+8300
10
-24
+18300
+10000
15
-34
+31300
+13000
19
-76.2
+48700
+17400
The short straddle is short gamma. One way of reducing this exposure is to “buy gamma”. Buy “2 big figure OTM” calls and puts (this is a strangle). The net position is a butterfly. Note the net gamma exposure that you have created. What other combinations of options can be used to set up a butterfly?
2 big figure OTM calls and puts would be a 1.67 strike put and a 1.71 strike call. This reduces the net gamma considerably because we have “bought gamma”, BUT, it also reduces our net theta exposure to 0. This is because the strikes are too close together. This position is called a long butterfly.