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Session 13: Payoff & Profit/Loss DiagramsA.

i) No. of bought (1)/ Bid/Ask Exercise Type of Initialunits sold (-1) price price option payoff

NAB share 1 1 30.03 30.03NAB Jul $29.00 call 1 -1 0.12 31 C -0.12

Total initial payoff 29.91

ii) S* 29 29.5 30 30.5 31 31.5 32 32.5 33Payoff of share 29 29.5 30 30.5 31 31.5 32 32.5 33Payoff of option 0 0 0 0 0 -0.5 -1 -1.5 -2Final payoff of covered call 29 29.5 30 30.5 31 31 31 31 31P/L of covered call -0.91 -0.41 0.09 0.59 1.09 1.09 1.09 1.09 1.09P/L in share -1.03 -0.53 -0.03 0.47 0.97 1.47 1.97 2.47 2.97P/L in option 0.12 0.12 0.12 0.12 0.12 -0.38 -0.88 -1.38 -1.88

iii) The payoff diagram

The Payoff Diag ram for a Covered Call

-5

0

5

10

15

20

25

30

35

29 29.5 30 30.5 31 31.5 32 32.5 33

range of stock pr ices on expiration date

P a y o

f f $

Payoff of share

Payoff of option

Final payoff of covered call

iv) The P/L diagram

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The Profit/Loss Diag ram for a Covered Call

-3

-2

-1

0

1

2

3

4

29 29.5 30 30.5 31 31.5 32 32.5 33

range of stock price S*

p r o

f i t / l o s s

$

P/L in sh are

P/L in opti on

P/L of covered call

v) The call option is currently out of the money since the current stock price of $30.03 is lessthan the option exercise price of $31.

vi) final payoff = S * max{0, S * - X } = 30.03 max(0, 30.03-31) = $30.03

P/L = S * max{0, S * - X } ( S 0 C 0) = 30.03 29.91 = $0.12.

vii) When investors anticipate that there is little change in the price of the stock , they mayform a covered call to earn the premium (the time value to be exact) and result in anoverall positive return despite a zero return on the stock .

viii) To solve for S * when P/L = S * max{0, S * - X } ( S 0 C 0) = 0, S * must be less than X so that S * 0 ( S 0 C 0) = 0

S * = ( S 0 C 0) = 30.03 0.12 = $29.91.Otherwise, S * cancel out in the equation.

ix) If the call I wrote is exercised, I must deliver the share in exchange for the exercise price.When S * >> X , the call is in the money:final payoff = S * max{0, S * - X } = X = $31, andP/L = S * max{0, S * - X } ( S 0 C 0) = X (S 0 C 0) = 31 (30.03 0.12) = $1.09

B. i) Initial payoff = - P L + 2 P M - P H = -0.105 + 2 0.23 - 0.37 = $-0.015

ii) Final payoff = -max{0, X L S*} + 2 max{0, X M S*} - max{0,X H S*} = -max{0, 29 24} + 2 max{0, 29.5 24} - max{0,30 24}= -5 + 11 - 6 = $0

P/L = final payoff initial payoff= 0 -0.015 = $0.015

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iii) Final payoff = -max{0, X L S*} + 2 max{0, X M S*} - max{0,X H S*}

= -max{0, 29 35} + 2 max{0, 29.5 35} - max{0,30 35}= -0 + 0 - 0 = $0

P/L = final payoff initial payoff= 0 -0.015 = $0.015

iv) To solve for S * whenP/L = -max{0, X L S*} + 2 max{0, X M S*} - max{0,X H S*} -0.015 = 0

Note that S * cannot be less than X L nor greater than X H , otherwise, S * cancel out in theequation.

The first breakeven point occurs when X L < S * < X M so thatP/L = -0 + 2 max{0, X M S*} - max{0,X H S*} + 0.015 = 0 S * = 2 X M X

H + 0.015 = 29.015

The second breakeven point occurs when X M < S * < X H so thatP/L = -0 + 2 0 - max{0, X H S*} + 0.015 = 0 S * = X H 0.015 = 29.985

C. If traders believe that (i) the stock price is to fall or (ii) the quoted price of the callis above its model price, they may write naked calls hoping that the call will fall invalue at which time they will buy the same call to close off the position and earn aprofit. Obviously, they have to take into account of transaction costs and thepossibility that their belief may turn out to be wrong.

D. i) Payoff & P/L table on expiry date

S* 25 26 27 28 29 30 31 32 33 34 35

Payoff of share 25 26 27 28 29 30 31 32 33 34 35

Payoff of put 4 3 2 1 0 0 0 0 0 0 0

Payoff of call 0 0 0 0 0 0 0 -1 -2 -3 -4

Final payoff of collar 29 29 29 29 29 30 31 31 31 31 31

P/L of collar -1.045 -1.045 -1.045 -1.045 -1.045 -0.045 0.955 0.955 0.955 0.955 0.955

P/L in share -5.03 -4.03 -3.03 -2.03 -1.03 -0.03 0.97 1.97 2.97 3.97 4.97

P/L in put 3.865 2.865 1.865 0.865 -0.135 -0.135 -0.135 -0.135 -0.135 -0.135 -0.135

P/L in call 0.12 0.12 0.12 0.12 0.12 0.12 0.12 -0.88 -1.88 -2.88 -3.88

ii) Payoff & P/L diagram on expiry date

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The Payoff and P/L Diagram for a Collar

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0

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30

35

25 26 27 28 29 30 31 32 33 34 35

range of stock pr ices on expiration date

P a y o

f f / P / L Payoff

P/L

iii) There is little chance of seeing stock prices rising above $31 as a call is sold togive up the profit potential beyond that price. On the other hand, the userbelieves that there is room for the price to fall below $29 as a put is purchasedto floor the potential loss (see the P/L) or the value (see the payoff) of his stockinvestment.

iv) The put option protects the downside risk of the stock investment. As soon asS * falls below $29, the put becomes in the money and offsets the loss in thestock.

v) The call reduces the cost of insurance (or downside risk protection) associatedwith the purchase of the put. As long as S * does not rise beyond $31, the userkeeps the premium of the call.