FIN 5216 Homework 1
Fall 2023
Due Sep 20th Wednesday 23:59 pm
Instructions
a. You can either write down your assignment or type it in a Word file.
b. Please show all the steps of your work. Please keep 4 decimal places for intermediate
steps and 2 decimal places for final answers.
c. Please submit your homework electronically through Canvas. Please include your full
name in the file name. If you write down the assignment, you can take a picture and
upload it. Make sure everything is complete and clear before you upload the
assignment.
d. No late submissions will be accepted, except with prior permission.
Q1. An investor buys a European put on a share for $3. The stock price is $42 and the strike
price is $40.
1) Under what circumstances will the option be exercised?
2) Under what circumstances does the investor make a profit?
3) Draw a diagram showing the variation of the investor’s profit with the stock price at the
maturity of the option.
4) What is the break-even stock price at expiration?
Q2. Use the software DerivaGem to plot the graphs that describe the effect of changes in
volatility and risk-free rate on option prices when S0 = 50, K = 50, r = 5%, σ = 30%, and T = 1.
Copy and paste the screenshot of your graphs here.
1)
2)
3)
4)
The effect of changes in volatility (0.01% - 60%) on European call option prices
The effect of changes in volatility (0.01% - 60%) on European put option prices
The effect of changes in risk-free interest rate (0% - 10%) on European call option prices
The effect of changes in risk-free interest rate (0% - 10%) on European put option prices
Q3. What is a lower bound for the price of a 4-month call option on a non-dividend-paying stock
when the stock price is $28, the strike price is $25, and the risk-free interest rate is 8% per
annum?
Q4. Keep working on the question from Lecture 1 slide 28. Suppose that c = 3, S0 = 31, T = 0.25, r
= 10%, K = 30, D = 0.
1) What should be the proper European put option price under the put-call parity
relationship?
2) What is the arbitrage possibility when p = 1? Use a table to show the positions you
would hold at time 0 and the profit you will get at time T.
Q5. Put-call parity states that c + Ke-rT = p + S0
However, this holds only for European options. How about American options?
Consider:
A. A portfolio consisting of a short position in an American call a long position in one share
of the stock;
B. A portfolio consisting of a short position in a European put option plus a long position in
a zero-coupon bond that pays K at time T (or lending)
Use notations: C for American call value, K for strike price (both American call and European
put), r for continuously compounded annualized risk-free interest rate, T for expiration date
(both American call and European put), p for European put value, S0 for stock price at time 0, St
for stock value at time t, and ST for stock value at time T.
If the American call option is exercised early at time t (t < T), complete the table below:
Cost at Time 0
Portfolio A
Portfolio B
- C + S0
Payoff at time t
Payoff at time T
- (St – K) + St = K
No need to fill here
Assume that portfolio A’s payoff at time t will earn interest till time T.
1) Which portfolio has a larger value at T for sure? Note that -max(a, b) = min(-a, -b).
2) Write down the relationship between the portfolio costs at time 0.
It must hold that portfolio A’s current price at time 0 is larger than portfolio b.
3) Incorporate American put option value (P ≥ p) into the relationship from 2). Find out the
upper bound for C – P.
If the American call option is not exercised early, complete the table below:
Cost at Time 0
Portfolio A
Payoff at time T
- C + S0
Portfolio B
4) Verify that both portfolios deliver the same payoff at time T. Note that -max(a, b) =
min(-a, -b).