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Derivatives Securities and Markets

Problem Set: Properties of Stock Options

Question #a

What is a lower bound for the price of a six-month call option on a non-dividend-paying stock when the stock price is $80, the strike price is $75, and the risk-free interest rate is 10% per annum?

Question #b

What is a lower bound for the price of a two-month European put option on a non-dividend-paying stock when the stock price is $58, the strike price is $65, and the risk-free interest rate is 5% per annum?

Question #c

The price of a European call that expires in six months and has a strike price of $30 is $2. The underlying stock price is $29, and a dividend of $0.50 is expected in two months and again in five months. The term structure is flat, with all risk-free interest rates being 10%. What is the price of a European put option that expires in six months and has a strike price of $30?

Question #d

A European call option and put option on a stock both have a strike price of $20 and an expiration date in three months. Both sell for $3. The risk-free interest rate is 10% per annum, the current stock price is $19. Identify the arbitrage opportunity open to a trader.

Question #e

What is the impact (if any) of negative interest rates on:

i. The put–call parity result for European options

ii. The result that American call options on non-dividend-paying stocks should never be exercised early.

iii. The result that American put options on non-dividend paying stocks should sometimes be exercised early.

Assume that holding cash earning zero interest is not possible.