**Consider the situation in which stock price movements during the life of a European option are**

**Problem 12.8.**

*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.*

**Problem 12.9.**

*A stock price is currently $50. It is known that at the end of two 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 two-month European call option with a strikeprice of $49? Use no-arbitrage arguments.*

**Problem 12.10.**

*A stock price is currently $80. It is known that at the end of four 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 four-month European put option with a strikeprice of $80? Use no-arbitrage arguments.*

**Problem 12.11.**

*A stock price is currently $40. It is known that at the end of three 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 three-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.*

**Problem 12.12.**

*A stock price is currently $50. Over each of the next two three-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 six-month European call option with a strike price of $51?*

**Problem 12.13.**

*For the situation considered in Problem 12.12, what is the value of a six-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?*

**Problem 12.14.**

*A stock price is currently $25. It is known that at the end of two months it will be either $23 or $27. The risk-free interest rate is 10% per annum with continuous compounding. Suppose ** is the stock price at the end of two months. What is the value of a derivative that pays off ** at this time?*

**Problem 12.15.**

*Calculate **, **, and ** when a binomial tree is constructed to value an option on a foreign currency. The tree step size is one month, the domestic interest rate is 5% per annum, the foreign interest rate is 8% per annum, and the volatility is 12% per annum.*

**Problem 12.16.**

*A stock price is currently $50. It is known that at the end of six months it will be either $60 or $42. The risk-free rate of interest with continuous compounding is 12% per annum. Calculate the value of a six-month European call option on the stock with an exercise price of $48. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers.*

**Problem 12.17.**

*A stock price is currently $40. Over each of the next two three-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 12% per annum with continuous compounding. *

*What is the value of a six-month European put option with a strike price of $42?**What is the value of a six-month American put option with a strike price of $42?*

**Problem 12.18**.

*Using a “trial-and-error” approach, estimate how high the strike price has to be in Problem 12.17 for it to be optimal to exercise the option immediately.*

**Problem 12.19.**

*A stock price is currently $30. During each two-month period for the next four months it is expected to increase by 8% or reduce by 10%. The risk-free interest rate is 5%. Use a two-step tree to calculate the value of a derivative that pays off ** where ** is the stock price in four months? If the derivative is American-style, should it be exercised early?*

**Problem 12.20.**

*Consider a European call option on a non-dividend-paying stock where the stock price is $40, the strike price is $40, the risk-free rate is 4% per annum, the volatility is 30% per annum, and the time to maturity is six months. *

*Calculate**,**, and**for a two step tree**Value the option using a two step tree.**Verify that DerivaGem gives the same answer**Use DerivaGem to value the option with 5, 50, 100, and 500 time steps.*

**Problem 12.21.**

*Repeat Problem 12.20 for an American put option on a futures contract. The strike price and the futures price are $50, the risk-free rate is 10%, the time to maturity is six months, and the volatility is 40% per annum.*

**Category:**Accounting, General Accounting

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