# Option Strategist Exercise Exploring the Option Greeks

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```					   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

1)   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%
a) Note down the option’s premium, as well as its delta and
gamma for the following spot rates:

1.4500
1.5500
1.6200
1.6900 ATM
1.7600
1.8300
1.8900
1.9500
2.3000

b) 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?
c) 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?
2)    In the following exercise we will analyse the net delta of a
Spot Rate    Call delta    Put delta    Position delta     Position P&L

1.5000
1.6700

1.6900
1.7800

1.8800

1.9500

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)
a) What are the individual call and put deltas, the overall
position delta and the P&L, for the following spot cable
rates:

b) What happens to the position delta as the spot rate
increases, and why?
c) 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
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
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

g) At what spot rate is the straddle delta most sensitive? Why is
this the case?

3)       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          P&L              Theta
gamma
5
10

15

19

4)        The short straddle is short gamma. One way of reducing
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?

4)        Set up a short strangle (spot cable = 1.6900) with a call
strike of 1.7400 and a put strike of 1.6400. Vary the spot
rate and assess the riskiness of this position against that of