FE HW

Homework - Options: general properties and pricing in discrete time (Chapters 10-11 and

13 in Hull)

FE-620

June 26, 2020

Problem 3.1 Consider a forward contract to purchase one share of stock in 3 months for a forward price of $25, and an European call option on the same stock with strike K = 25$. In both cases we end up paying $25 at maturity in return for one share of stock (assuming that the option is exercised). Describe three ways in which these contracts are different.

Hint: Consider for example the risk profile of each contract: how much can we gain/lose in each case.

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Problem 3.2 Suppose that c1, c2 and c3 are the prices of European call options with strike prices K1, K2 and K3, respectively, where K3 > K2 > K1 and K3 − K2 = K2 −K1. All option have the same maturity. Show that

c2 ≤ 0.5(c1 + c3)

This inequality means that the option price c(K) is a convex function of strike K.

Hint: Consider a portfolio that is long one option with strike K1, long one option with strike K3, and short two options with strike price K2. It helps to draw the diagram of the payoff of this portfolio as function of the stock price at maturity S(T ).

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Problem 3.3

1. What is the lower bound for the price of a 3-month European call option on a non-dividend paying stock when the stock price is $100, the strike price is $90, and the risk-free interest rate is 1% per annum?

2. What is the lower bound on the price of a 3-month European put option on a non-dividend paying stock when the stock price is $100, the strike price is $110, and the risk-free interest rate is 1% per annum?

3. Assume that the price of a 3-month European call option on a non- dividend paying stock with stock price $100 and strike price $90 is $11.67. The risk-free interest rate is 1% per annum. What is the price of the European put option on the same stock with the same strike and maturity?

In all cases the risk-free interest rate is quoted with continuous-time com- pounding.

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Problem 3.4 The current price of a non-dividend paying stock is $120 with a volatility of 25%. The risk-free rate is 2% with continuous time compounding.

For a 3-month time step in a binomial tree model compute the following:

1. The percentage of the up stock price movement.

2. The percentage of the down stock price movement.

3. The probability of an up movement in a risk-neutral world.

4. The probability of a down movement in a risk-neutral world.

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Problem 3.5 A stock price is currently $50. It is known that at the end of 3 months it will be either $60 or $42. The risk-free rate of interest with continuous compounding is 1% per annum. Calculate the value of a 3-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.

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