PSTAT 170, QUIZ 2 INTRODUCTION TO MATHEMATICAL FINANCE
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PSTAT 170, QUIZ 2
INTRODUCTION TO MATHEMATICAL FINANCE
2021
The quiz begins at the scheduled time of 9:30am Paci
c. The submission window closes at 10:40am Paci
c. You have 70 minutes to complete this quiz and successfully upload your solutions to GauchoSpace. Show all work, do not just write an answer. For questions that use a calculator, make sure to write down the formula you are using that is typed into the calculator. Please make your 
nal results easy to read. This quiz consists of 4 questions. Good luck!
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 45 50 60
Call Price 12 9 5
Put Price 3 6 10
(1) (3 points) Show that you can not e
ect arbitrage only based on call options.
(2) (2 points) Find the convexity violations.
(3) (5 points) What spread would you use to e
ect 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 
= 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 
TS0 + e 
rT45.
P(45, 0.5) = $3.87 
e 
0.07
0.540 + e 
0.04
0.545 = $9.354
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 satis
ed for the option premium, we can not e
ect 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) 
= K3
K2
K3
K1 = 2
3, 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 |
50 |
ST > 60 |
|
Buy 2 45-strike put Sell 3 50-strike put Buy 1 60-strike put |
-6 +18 -10 |
2(45 |
0
|
0 0
60 |
0 0 0 |
|
Total |
+2 |
0 |
2ST |
60 |
0 |
From the table above, we can see that we can e
ect 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
= e 
h 
B = e 
rh u 
d
Option price P = 
S + B = e 
rh 
Pu e(r
)h 
d
u
d + Pdu 
e(r
)h
u
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
2022-02-18