PSTAT 170, QUIZ 2 INTRODUCTION TO MATHEMATICAL FINANCE 2022
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PSTAT 170, QUIZ 2
INTRODUCTION TO MATHEMATICAL FINANCE
FEBRUARY 3, 2022
Problem 1. A stock currently sells for $40.00. A 6-month call option written on the stock with a strike price of $45.00 has a premium of $3.87. Assuming a 4% continuously compounded risk-free rate and a 7% continuous dividend yield, what is the price of a 6-month put option written on the stock with a strike price of $45.00? [Show your solution with details.] Problem 2. Suppose call and put option prices on the same underlying with same expiration are given by:
Strike Call Price Put Price |
45 12 3 |
50 9 6 |
60 5 10 |
(1) (3 points) Show that you can not efect arbitrage only based on call options.
(2) (2 points) Find the convexity violations.
(3) (5 points) What spread would you use to efect arbitrage? Demonstrate that the spread position is an arbitrage.
Problem 3. (10 points) Consider the following two-period model for the stock S: S2 = $144
S1 = $120
S0 = $100
S2 = $108
S1 = $90
S2 = $81
Assume the both interest rate and dividend yield rate are zero. Find the risk neutral probability. [S1 is stock price after one period.]
Problem 4. (10 points) Let S = $100, K = $95, r = 8%, T = 0.5, and 6 = 0. r is the continuously compounded interest rate. Let u = 1.1, d = 0.9, and n = 2. Construct the binomial tree for pricing and hedging a European put option on S . At each node provide the option premium, and B.
1. Solution
Problem 1 . Using standard put-call parity formula
P(45, 0.5) = C(45, 0.5) — e —6T S0 + e —rT 45.
P(45, 0.5) = $3.87 — e —0.07*0.540 + e —0.04*0.545 = $9.354
样
Problem 2. (1) (a) Call option premium decreases as the strike price K increases. (b)
C(K1 ) — C(K2 ) = 3 K2 — K1 = 5
C(K2 ) — C(K3 ) = 4 K3 — K2 = 10
(c)
C(K1 ) — C(K2 ) 3 4 C(K2 ) — C(K3 )
K2 — K1 5 10 K3 — K2
Since all inequality conditions are satisied for the option premium, we can not efect arbitrage only based on call options.
(2)
P(K2 ) — P(K1 ) 3 4 P(K3 ) — P(K2 )
K2 — K1 5 10 K3 — K2 ,
so the put option premium violates the convexity condition.
(3) λ = K3—K2K3—K1 = 23, so consider buy 2 45-strike put options, 1 60-strike put option and sell 3 50-strike put options.
Transaction |
t = 0 |
ST < 45 |
45 ST 50 |
50 ST 60 |
ST > 60 |
Buy 2 45-strike put Sell 3 50-strike put Buy 1 60-strike put |
-6 +18 -10 |
2(45 — ST ) —3(50 — ST ) 60 — ST |
0 —3(50 — ST ) 60 — ST |
0 0 60 — ST |
0 0 0 |
Total |
+2 |
0 |
2ST — 90 0 |
60 — ST 0 |
0 |
From the table above, we can see that we can efect arbitrage with this combi- nation.
Problem 3. By usual formula from Chapter 10 of MacDonald:
P* = (1 + r — d)/(u — d) = (1 — 0.9)/(1.2 — 0.9) = 1/3
Problem 4 .
= e —6h Pu — PdS(u—d)
B = e —rh uPd — dPu
Option price P = · S + B = e —rh (Pu e(r—6)h — du—d + Pd u — e(r—6)hu—d)
Based on the formulas, we can construct the following binomial tree:
S0 = $100
P0 = $2.141
= -0.274
B = 29.518
S1 = $110
Pu = $0
= 0
B = 0
S1 = $90
Pd = $5.475
= -0.778
B = 75.475
S2 = $121
Puu = $0
S2 = $99 Pud = Pdu = $0
S2 = $81
Pdd = $14
2022-08-27