Assume the following unrealistic assumptions The interest rate is equal to zero Investors are risk-neutral (they don’t care about risk) • That is the price of assets that will give $100 with a probability of 50% and $0 with the probability of 50% will be... • Now assume that a stock is traded today (28/02) for $9. • What will be the price of a call option with an exercise price of $10 that expires at the end of May (notation C10May), if the price of the stock (underlining asset) can increase or decrease by $2 until the expiration date with equal probabilities? • Will the value of the call option increase or decrease if the price of the stock can increase or decrease by $3 until the expiration date?

A trader holds a call option with a strike price of $50, and the current market price of the underlying asset is $60. If the option expires, what is the potential profit (payoff) from exercising the call option?

Consider a world in which some stock, S, can either go up by 25% or down by 20% in one year and noother outcomes are possible. The continuously compounded risk-free interest, r, is 5.5% and the current priceof the stock, S0, is $100.1. What are the possible stock values in one year’s time, ST ?2. What are the possible payoffs of a European call option written on stock S with a strike price, X, of$100 and time-to-expiration of 1 year, T = 1 ?3. Suppose you want to form a portfolio, P , consisting of short on one call option and long on somenumber, ∆, of the stock, such that the portfolio value in one year’s time, PT , does not depend on thevalue of the stock, ST . What would be the appropriate value of ∆, also called the hedge ratio or delta?4. What would be the (certain) portfolio value in one year’s time, PT ?5. What is the arbitrage-free value of the portfolio today, P0 ?6. What is the premium of the call option today, c0, if there is no arbitrage opportunity?7. Define p = (erT − d) /(u − d), and call this the risk-neutral probability that the stock price increases.What is the value of p ?8. What is the expected value of the stock in one year’s time, E (ST ), under the risk-neutral probabilities?9. At what continuous rate would the stock price have to grow to end up at the expected value?10. What would be the expected value of the call option in one year’s time, E (cT ), under the risk-neutralprobabilities?11. At what continuous rate would the call price have to grow to end up at the expected value?

Consider an option on a non-dividend-paying stock when the stock price is $30, the exercise price is $29, the risk-free interest rate is 5% per annum, the volatility is 25% per annum, and the time to maturity is four months. (a) What is the price of the option if it is a European call? (1 mark) (b) What is the price of the option if it is an American call? (1 mark) (c) What is the price of the option if it is a European put? (1 mark) (d) Verify that put–call parity holds. (1 mark)

When ABC was trading at $52 per share, you paid $6.40 for a call option (for one share) on the stock of ABC with a strike price of $50, and six months until maturity. After six months, the share price of ABC is $54.10.What is the value of the call option at expiration? Do not include the $ sign and answer to the nearest $0.01.

The no-arbitrage condition is violated at option expiry when the value of an in-the-money:A.put option is below its exercise price.B.put option is below its exercise value.C.call option is below the price of its underlying.