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The price of a traded security follows a geometric Brownian motion...
The price of a traded security follows a
... [Show More] geometric
Brownian motion with drift 0.06 and volatility 0.4. Its current price
is 40. A brokerage firm is offering, at cost C, an investment that will
pay 100 at the end of 1 year either if the price of the security at 6 months
is at least 42 or if the price of the security at 1 year is at least 5 percent
above its price at 6 months. That is, the payoff occurs if either S(0.5) ≥
42 or S(1) > 1.05 S(0.5). The continuously compounded interest rate
is 0.06.
(a) If this investment is not to give rise to an arbitrage, what is C?
(b) What is the probability the investment makes money for its buyer?
Exercise 7.12 The price of a traded security follows a geometric
Brownian motion with drift 0.04 and volatility 0.2. Its current price
is 40. A brokerage firm is offering, at cost 10, an investment that will
pay 100 at the end of 1 year if S(1)>(1 + x)40. That is, there is a pay-
off of 100 if the price increases by at least 100x percent. Assume that
the continuously compounded interest rate is 0.02, and that the new in-
vestment can be bought or sold.
(a) If this investment is not to give rise to an arbitrage, what is x?
(b) What is the probability that the investment makes money for its
buyer?
A stock price is currently $50. It is known that at the end of 6 months it will be either $45
or $55. The risk-free interest rate is 10% per annum with continuous compounding.
What is the value of a 6-month European put option with a strike price of $50?
13.5. A stock price is currently $100. Over each of the next two 6-month periods it is expected
to go up by 10% or down by 10%. The risk-free interest rate is 8% per annum with
continuous compounding. What is the value of a 1-year European call option with a
strike price of $100?
13.6. For the situation considered in Problem 13.5, what is the value of a 1-year European put
option with a strike price of $100? Verify that the European call and European put prices
satisfy put-call parity.
13.7. What are the formulas for u and d in terms of volatility?
13.8. Consider the situation in which stock price movements during the life of a European
option are governed by a two-step binomial tree. Explain why it is not possible to set up
a position in the stock and the option that remains riskless for the whole of the life of the
option.
13.9. A stock price is currently $50. It is known that at the end of 2 months it will be either $53
or $48. The risk-free interest rate is 10% per annum with continuous compounding.
What is the value of a 2-month European call option with a strike price of $49? Use no-
arbitrage arguments.
13.10. A stock price is currently $80. It is known that at the end of 4 months it will be either $75
or $85. The risk-free interest rate is 5% per annum with continuous compounding. What
is the value of a 4-month European put option with a strike price of $80? Use no-
arbitrage arguments.
13.11. A stock price is currently $40. It is known that at the end of 3 months it will be either $45
or $35. The risk-free rate of interest with quarterly compounding is 8% per annum.
Calculate the value of a 3-month European put option on the stock with an exercise
price of $40. Verify that no-arbitrage arguments and risk-neutral valuation arguments
give the same answers.
13.12. A stock price is currently $50. Over each of the next two 3-month periods it is expected
to go up by 6% or down by 5%. The risk-free interest rate is 5% per annum with
continuous compounding. What is the value of a 6-month European call option with a
strike price of $51?
13.13. For the situation considered in Problem 13.12, what is the value of a 6-month European
put option with a strike price of $51? Verify that the European call and European put
prices satisfy put-call parity. If the put option were American, would it ever be optimal
to exercise it early at any of the nodes on the tree? [Show Less]