Problem Set 10 Solution PDF

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FINS2624 PROBLEM SET 10 SOLUTION Question 1. a) Using the notation introduced in the lecture, we have: U = 1.25 D = 0.8 S0 = 100 Therefore, uS0 = $100 x 1.25 = $125 = S T u S0 dS0 = $100 x 0.8 = $80 = S T d

b) The payoff of a European call option is: 𝐺 = max(𝑆𝑇 − 𝑋, 0) Now, 𝑆𝑇 = �𝑆𝑇𝑢 , 𝑆𝑇𝑑 � Hence, if the market goes up, then: 𝐺𝑇𝑢 = max(𝑆𝑇𝑢 − 𝑋, 0) = max($125 − $100,0) = $25 If the market goes down, then: 𝐺𝑇𝑑 = max�𝑆𝑇𝑑 − 𝑋, 0� = max($80 − $100,0) = $0

c) We want to hold 𝛥 units of stock such that:

𝛥𝑆𝑇𝑢 − max(𝑆𝑇𝑢 − 𝑋, 0) = 𝛥𝑆𝑇𝑑 − max�𝑆𝑇𝑑 − 𝑋, 0� 𝛥$125 − $25 = 𝛥$80

𝛥=

25 ≈ 0.5556 125 − 80

d) Substituting 𝛥 = 0.5556 into either 𝛥$80 or 𝛥$125 − $25 gives us: 𝑃𝑇 = $44.444

e) 𝑃0 = 𝑃𝑇 𝑒 −𝑟 = $44.444𝑒 −0.055 ≈ $42.07

f) At time T, we know that 𝑃𝑇 = 𝛥𝑆𝑇 − 𝐶𝑇 This implies that at time 0, 𝑃0 = 𝛥𝑆0 − 𝐶0

𝐶0 = 𝛥𝑆0 − 𝑃0 𝐶0 = 0.5556 × $100 − $42.07 𝐶0 = $13.49

g) 𝑝= 𝑝=

𝑒 𝑟𝑇 − 𝑑 𝑢−𝑑

𝑒 0.055 − 0.8 ≈ 0.57 1.25 − 0.8

h) 𝐸(𝑆𝑇 ) = 0.57 × $125 + (1 − 0.57) × $80 = $105.65

i) Want to find rs such that: 𝑆0 𝑒 𝑟𝑠 = 𝐸(𝑆𝑇 )

𝐸(𝑆𝑇 ) � ≈ 0.055 𝑟𝑠 = log � 100

j) 𝐸(𝐶𝑇 ) = 0.57 × $25 + (1 − 0.57) × $0 = $14.25

k) Want to find rC such that: 𝑐0 𝑒 𝑟𝑐 = 𝐸(𝐶𝑇 )

𝐸(𝐶𝑇 ) � ≈ 0.055 𝑟𝑐 = log � 𝑐0

End of Chapter Questions BKM Chapter 21 7. Exercise Price 120 110 100 90

Hedge Ratio 0/30 = 0.000 10/30 = 0.333 20/30 = 0.667 30/30 = 1.000

As the option becomes more in the money, the hedge ratio increases to a maximum of 1.0.

9.

a.

uS 0 = 130 ⇒ Pu = 0 dS 0 = 80 ⇒ Pd = 30 The hedge ratio is: H =

Pu − Pd 0 − 30 3 = =− 5 uS0 − dS0 130 − 80

b. Riskless Portfolio Buy 3 shares Buy 5 puts Total

ST = 80

ST = 130

240 150

390 0

390

390

Present value = $390/1.10 = $354.545 c.

The portfolio cost is: 3S + 5P = 300 + 5P The value of the portfolio is: $354.545 Therefore: 300 + 5P = $354.545  P = $54.545/5 = $10.91

10.

The hedge ratio for the call is: H = Riskless Portfolio Buy 2 shares Write 5 calls Total

Cu − C d 20 − 0 2 = = uS0 − dS0 130 − 80 5

S = 80

S = 130

160 0 160

260 -100 160

Present value = $160/1.10 = $145.455 The portfolio cost is: 2S – 5C = $200 – 5C The value of the portfolio is $145.455 Therefore: C = $54.545/5 = $10.91 Does P = C + PV(X) – S? 10.91 = 10.91 + 110/1.10 – 100 = 10.91...