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Lecture

# Assignment 2.docx

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York University

Administrative Studies

ADMS 4503

Nabil Tahani

Winter

Description

ANSWER 1.
a. Fair price of 6 month call on GS:
p≥ max ( D + Ke – S , 0)
0
S0= $150.53 , T = 0.5 , K = $165,p = $18, r = 0.04
D = PV ( all dividends expected to be paid within 6 months)
-0.04 xx 4/12
= 0.50e = 0.50 x 0.987084 = $0.49
-0.04 x 0.5
Fair price of GS = 0.49 + 165e – 150.53
= 0.49 + 165 (0.980199) – 150.53
= 0.49 + 161.73 – 150.53
= $11.69
b. Current market price of 6 month call is $7 , then the put call parity will show as follows :
c + D + Ke = p + S
0
taking the Left hand side (LHS)
-rT
c + D + Ke = $7 + $0.49 + 161.73 = $169.22
taking the right hand side (RHS)
p + S0= 18 + 150.53 = 168.53
since the LHS is higher, we shall short a call and buy a put and stock
so that :
At time 0 After 6 months
Short call $ 7
Buy a put -$ 18
Buy a stock -$150.53
Dividend $0.49
Borrow at 4% $149.04 -$149.04e 0.04 x = $152.05
Total 0
If S goes above $165, you exercise the call and sell the stock making a profit ( after paying off
T
the loan)
$165 - $152.05 = $12.95
If T goes below $165 , you exercise put to sell stock and make an arbitrage profit
$165 - $152.05 = $12.95
c. While on one hand, American options give holders the opportunity to exercise it before maturity,
it can however be used for arbitrage purposes by following its put call parity equation of :
-rT
S0− D − K ≤ C − P ≤ S 0 Ke
150.53 – 0.49 – 165 ≤ C – P ≤ 150.53 – 161.73
-14.96 ≤ C-P ≤ -11.2
-14.96 + P ≤ C –P + P≤ -11.2 + P
-14.96 + 18 ≤ C ≤ -11.2 + 18 3.04 ≤ C ≤ 6.8
Since the call option price do not fall within the bounds, the holder can easily take advantage
of an arbitrage .
The holder can follow the same step as would be followed for European option ( as
explained in answer 1 b ) , however , the characteristic of an American option makes it
possible to be excercised early.
The option holder should not exercise it if –
- There is dividend expected during the life of option
- The investor would be able to earn some interest prior to maturity
- There may be favourable fluctuations in the stock price later on ( and closer to maturity
) and as such should not be excercised early.
ANSWER 2
a. We can price strategy at once by using the total pay off at the terminal nodes. The bull call
spread price to be paid is $35.6353 - $25.5473 = $10.088 . See the trees below.
u = exp ( σ x √δt ) = exp ( 30% x √ 1/12 ) = 1.0905; d = 1/u = 0.9170
a;d 0.0863
a = exp ( r x δt ) = exp ( 4% x 1/12) = 1.0033 ; p = u;d =0.1735 = 0.4974
b. The break-even point is $200 + $10.088 = $ 210.088 . The Maximum loss is the premium paid
which is : $10.088 and the maximum profit is $20 – $10.088 = $9.912.
c. With the help of the put-call parity for each strike price, the bear put spread premium ( paid) is
$ 9.5160. Here are the details:
p = c S + K x exp ( -rT)
1 1 – 0 1
p2= c 2 – 0 x 2xp ( -rT)
p2–p 1 c –2c +1(K – 2 ) x1exp ( -rT)
p2–p 1 - 10.088 + 20 x exp ( -4% x 0.5)
= - 10.088 + 20 ( 0.980199)
= $9.5160
( the binomial trees are shown in the next pages)
For EQIX call option with the strike price of $200
At each node:
Upper value = Underlying Asset Price
Lower value = Option Price
Values in red are a result of early exercise.
442.0545 Strike price = 200 242.0545
Discount factor per step = 0.9934 391.0982
Time step, dt = 0.1667 years, 60.83 days 192.4271
Growth factor per step, a = 1.0067 346.0157 346.0157
148.6647
Probability of up move, p = 0.4967 146.0157
Up step size, u = 1.1303 306.13 306.13
110.0902 107.4589
Down step size, d = 0.8847
270.8419 270.8419 270.8419
78.23333
73.49087 70.8419
239.6215 239.6215 239.6215
53.61995 47.83899 40.95044
212 212 212 212
30.04588 23.1634
35.63527 12
187.5624 187.5624 187.5624
5.920306
18.3629 12.88827
165.9418 165.9418 165.9418
7.07905
2.920836 0
146.8135 146.8135
1.44102 0
129.8901 129.8901
0
0
114.9174
0
101.6707
0
Node Time:
0.0000 0.1667 0.3333 0.5000 0.6667 0.8333 1.0000
For the EQIX call option with the strike price of $220
At each node:
Upper value = Underlying Asset Price
Lower value = Option Price
Values in red are a result of early exercise.
442.0545
Strike price = 220 222.0545 Discount factor per step = 0.

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