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
1.5500
1.6200
1.6900 ATM
1.7600
1.8300
1.8900
1.9500
2.3000
• 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?
• 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
d) 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?
• 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
1.6700
1.6900
1.7800
1.8800
1.9500
• What happens to the position delta as the spot rate increases, and why?
• Explain the variation in delta as the spot rate moves above and below the ATM strike.
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
1.6200
1.6400
1.6600
1.6800
1.6600
1.6400
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
1.6200
1.6400
1.6600
1.6800
1.6600
1.6400
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
1.6200
1.6500
1.6900
1.7500
• At what spot rate is the straddle delta most sensitive? Why is this the case?
• 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
10
15
19
• 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?