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Posted: November 29th, 2022

Name _________________________________________________________
ID # __________________________________________________________
ISE 563 Mid-Term
July 13th, 2022
Notes:
 Do all problems.
 Open book, open notes, …
 Show work and be clear.
 No consulting with others
1. Given the following parameters for a 2-period binomial tree :
a) What is the price of an European Put option?
b) What is the price of an American Put option?

Time 0 Step 1 Step 2
Time 0 0.50 0.75
Dt 0.50 0.25
Volatility (s) 40% 30%
risk-free rate 2% 3%
Dividend paid @ end of period 2 0
Stock 100
Srike 100
2. Given the following zero risk-free rates with annual coupons and
compounding:
a) What are the discount factors?
b) What are the Par rates?
c) What are the forward rates?
d) What is the value of a 10-year swap which receives 5% fixed and pays
floating?
e) What is the par swap rate which starts in 5 years for 4 years?

Time Zero DF Par Fwd
0 1
1 1.75%
2 2.00%
3 2.25%
4 2.50%
5 2.75%
6 3.00%
7 3.25%
8 3.50%
9 3.75%
10 4.00%
3. Given the real-world drift of a stock as
𝑑𝑆
𝑆
= 𝜇𝑆𝑑𝑡 + 𝜎𝑆𝑑𝑧
With a risk-free rate of “r”.
For a function of the stock G(S,t) defined as
𝐺(𝑆,𝑡) = 𝑆

a) What process does dG(S,t) follow?
b) What is the 1% (lower) confidence interval of G?
c) What is the value of a call option on G with strike of K and maturity T?
Do all work symbolically.

4. Given a set of 10 trials with 3 annual steps for a stock process. The strike is
100. The resulting stock prices and zero risk-free rates are given below.
What is the value of a 3-year call option with the following modifications?
If before t=3, the stock goes above 120, receive an immediate $20 payoff and the
option ends.
If before t=3, the stock goes below 80, pay an immediate $20 and the option
continues

Trial t0 t=1 t=2 t=3
1 100 84 112 110
2 100 107 138 132
3 100 115 123 116
4 100 124 132 128
5 100 155 126 110
6 100 102 75 129
7 100 93 88 88
8 100 129 113 86
9 100 78 86 118
10 100 95 81 124
Risk-free Rate 2% 3% 5%
5. Use call options (Any strike), stocks, or bonds to create the following
terminal payoff structure?

-150
-100
-50
0
50
100
150
200
0 50 100 150 200 250 Payoff at maturity
Stock price at Maturity
Payoff at Maturity
6. You own one butterfly spread with strikes at 75, 100, and 125. The options
are all European exercise, 1-year maturity, and the initial stock price is $100.
The 1-year volatility surface is given below. Risk-free rate is 0% and a $2
dividend is paid in 18 months. Use call options to create the butterfly
spread.
a) What is the value of the butterfly spread?
b) What stock position would be needed to delta hedge the butterfly spread?
c) What is the forward volatility (for ATM) between years 2 and 3?
d) What is the same forward volatility if the stock dropped 20% immediately
and the volatility surface remained the same?
Skew
Maturity 75 100 125
1 30% 27% 24%
2 28% 25% 22%
3 26% 23% 20%

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