FNCE 445 Problem Set #6
Consider one-period binomial trees (Q1 and Q2):
1. Let S = 100,K = 105, r = 8%, T = 0.5 and δ = 0. Let u = 1.3 and
d = 0.8.
(a) What are ∆, B, and the premium for a European call?
(b) What are ∆, B, and the premium for a European put?
2. Let S = 100,K = 95, r = 8%, T = 0.5 and δ = 0. Let u = 1.3 and
d = 0.8.
(a) What are ∆, B, and the premium for a European put?
(b) Suppose you observe a put price of $8. What is the a
itrage?
(c) Suppose you observe a put price of $6. What is the a
itrage?
Consider multi-period binomial tree models in which n refers the numbe
of periods (Q3 - Q7):
3. Let S = 100,K = 95, r = 0.08, T = 1 and δ = 0. Let u = 1.3, d = 0.8,
and n = 2. Construct the binomial tree for a call option. At each
node provide the premium, ∆, and B.
4. Repeat the option-price calculation in the previous question for stock
prices of 80, 90 and 110. What happens to the initial option ∆ as the
stock price increase?
5. Let K = 95, σ = 0.3, r = 0.08, T = 1 and δ = 0. Let u = 1.3,
d = 0.8, and n = 2. Construct the binomial tree for a put option fo
S = 80, 110, and 130. At each node provide the premium, ∆, and B.
What happens to the initial put ∆ as the stock price increase?
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6. Try the supplementary exercise (posted at D2L) and refer the solution.
Then, repeat the problem assuming that the stock pays a continuous
dividend 8% per year. Calculate the prices of the American and Eu-
opean puts and calls. Which options are early-exercised?
7. Suppose that the exchange rate is $0.92/Euro. The dollar-denominated
interest rate is 4% and the euro-denominated interest rate is 3%.
u = 1.2, d = 0.9, T = 0.75, n = 3, and K = $1.00.
a. What is the price of a 9-month European put?
. What is the price of a 9-month American put?
8. Consider a one-period binomial tree when the underlying stock pays
the continuous dividend with yield δ. The interest rate is r.
a. Suppose u < e(r−δ)h. Show that there is an a
itrage opportunity.
. Suppose d > e(r−δ)h. Show that there is an a
itrage opportunity.
In answering the following problems, refer the standard normal density
table to compute N(d1) and N(d2). It is helpful if you verify your solution
y using the spreadsheets on the CD-ROM from the textbook.
9. Suppose S = 100,K = 95, σ = 30%, r = 0.08, δ = 0.03 and T = 0.75
a. Compute the BS price of a call.
. Compute the BS price of a call for which S = 100e−0.03×0.75,K =
95e−0.08×0.75, σ = 30%, r = 0, δ = 0 and T = 0.75. How does you
answer compare to that for (a)?
c. What is the 9-month forward price for the stock?
d. (optional) Compute that price of a 95-strike 9-month call option
on a futures contract.
e. (optional) What is the relationship between your answer to (d)
and the price you computed in (a) and (b)?
10. The exchange rate is U95/Euro, the yen-denominated interest rate is
1.5%, the euro-denominated interest rate is 3.5%, and the exchange
ate volatility is 10%.
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a. what is the price of a 90-strike yen-denominated euro put with 6
months to expiration?
. what is the price of a 1/90-strike euro-denominated yen call with
6 months to expiration?
c. What is the relation between your answer in (a) and (b)?
11. Consider a bull spread where you buy a 40-strike put and sell a 45-
strike put. Suppose σ = 0.30, r = 0.08, δ = 0, and T = 0.5. Use
the excel spreadsheets when you compute gamma and vega. Use the
normal table when you compute delta.
a. Suppose S = 40. What are delta, gamma, and vega?
. Suppose S = 45. What are delta, gamma, and vega?
c. Consider a bull spread when you buy a 40-strike call and a 45-
strike call. Repeat (a) and (b). You can refer the answer key of
Exercise 11.14 of the text book. Compare the results with the
esult of (a) and (b) above. Which greeks are identical? Why are
they identical?
12. Suppose S = 40, σ = 0.3, r = 0.08, and δ = 0. Suppose you sell a
45-strike call with 91 days to expiration.
(a) What is delta?
(b) If the option is on 100 shares, what investment is required for a
delta-hedged portfolio? What is your overnight profit if the stock
tomo
ow is 39? What if the stock price is 40.50?
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