# Consider a european call option and a european (answer attached)

1. Consider a European call option and a European put option on a nondividend-paying stock. You are given:
(i) The current price of the stock is 60.
(ii) The call option currently sells for 0.15 more than the put option.
(iii) Both the call option and put option will expire in 4 years.
(iv) Both the call option and put option have a strike price of 70.
Calculate the continuously compounded risk-free interest rate.
(A) 0.039
(B) 0.049
(C) 0.059
(D) 0.069
(E) 0.079

2. Near market closing time on a given day, you lose access to stock prices, but some
European call and put prices for a stock are available as follows:
Strike Price Call Price Put Price
\$40 \$11 \$3
\$50 \$6 \$8
\$55 \$3 \$11
All six options have the same expiration date.
After reviewing the information above, John tells Mary and Peter that no arbitrage
opportunities can arise from these prices.
Mary disagrees with John. She argues that one could use the following portfolio to
obtain arbitrage profit: Long one call option with strike price 40; short three call
options with strike price 50; lend \$1; and long some calls with strike price 55.
Peter also disagrees with John. He claims that the following portfolio, which is
different from Mary’s, can produce arbitrage profit: Long 2 calls and short 2 puts
with strike price 55; long 1 call and short 1 put with strike price 40; lend \$2; and
short some calls and long the same number of puts with strike price 50.
Which of the following statements is true?
(A) Only John is correct.
(B) Only Mary is correct.
(C) Only Peter is correct.
(D) Both Mary and Peter are correct.
(E) None of them is correct.

3-An insurance company sells single premium deferred annuity contracts with return
linked to a stock index, the time-t value of one unit of which is denoted by S(t). The
contracts offer a minimum guarantee return rate of g%. At time 0, a single premium
of amount  is paid by the policyholder, and π  y% is deducted by the insurance
company. Thus, at the contract maturity date, T, the insurance company will pay the
policyholder
π  (1 y%)  Max[S(T)/S(0), (1 + g%)T].
You are given the following information:
(i) The contract will mature in one year.
(ii) The minimum guarantee rate of return, g%, is 3%.
(iii) Dividends are incorporated in the stock index. That is, the stock index is
constructed with all stock dividends reinvested.
(iv) S(0)  100.
(v) The price of a one-year European put option, with strike price of \$103, on the
stock index is \$15.21.
Determine y%, so that the insurance company does not make or lose money on this
contract.
4. For a two-period binomial model, you are given:
(i) Each period is one year.
(ii) The current price for a nondividend-paying stock is 20.
(iii) u  1.2840, where u is one plus the rate of capital gain on the stock per period if
the stock price goes up.
(iv) d  0.8607, where d is one plus the rate of capital loss on the stock per period if
the stock price goes down.
(v) The continuously compounded risk-free interest rate is 5%.
Calculate the price of an American call option on the stock with a strike price of 22.
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4

5. Consider a 9-month dollar-denominated American put option on British pounds.
You are given that:
(i) The current exchange rate is 1.43 US dollars per pound.
(ii) The strike price of the put is 1.56 US dollars per pound.
(iii) The volatility of the exchange rate is   0.3.
(iv) The US dollar continuously compounded risk-free interest rate is 8%.
(v) The British pound continuously compounded risk-free interest rate is 9%.
Using a three-period binomial model, calculate the price of the put.

6. You are considering the purchase of 100 units of a 3-month 25-strike European call
option on a stock.
You are given:
(i) The Black-Scholes framework holds.
(ii) The stock is currently selling for 20.
(iii) The stock’s volatility is 24%.
(iv) The stock pays dividends continuously at a rate proportional to its price. The
dividend yield is 3%.
(v) The continuously compounded risk-free interest rate is 5%.
Calculate the price of the block of 100 options.
(A) 0.04
(B) 1.93
(C) 3.63
(D) 4.22
(E) 5.09

7. Company A is a U.S. international company, and Company B is a Japanese local
company. Company A is negotiating with Company B to sell its operation in
Tokyo to Company B. The deal will be settled in Japanese yen. To avoid a loss at
the time when the deal is closed due to a sudden devaluation of yen relative to
dollar, Company A has decided to buy at-the-money dollar-denominated yen put of
the European type to hedge this risk.
You are given the following information:
(i) The deal will be closed 3 months from now.
(ii) The sale price of the Tokyo operation has been settled at 120 billion Japanese
yen.
(iii) The continuously compounded risk-free interest rate in the U.S. is 3.5%.
(iv) The continuously compounded risk-free interest rate in Japan is 1.5%.
(v) The current exchange rate is 1 U.S. dollar = 120 Japanese yen.
(vi) The natural logarithm of the yen per dollar exchange rate is an arithmetic
Brownian motion with daily volatility 0.261712%.
(vii) 1 year = 365 days; 3 months = ¼ year.
Calculate Company A’s option cost.

8. You are considering the purchase of a 3-month 41.5-strike American call option on
a nondividend-paying stock.
You are given:
(i) The Black-Scholes framework holds.
(ii) The stock is currently selling for 40.
(iii) The stock’s volatility is 30%.
(iv) The current call option delta is 0.5.
Determine the current price of the option.
(A) 20 – 20.453  
0.15  / 2d
2
e x x
(B) 20 – 16.138  
0.15  / 2d
2
e x x
(C) 20 – 40.453  
0.15  / 2d
2
e x x
(D) 16.138 d 20.453 0.15 2 / 2
  
e x x
(E)  
40.453 0.15  / 2d
2
e x x – 20.453

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