Solutions to Homework 6

FM 5021 Mathematical Theory Applied to Finance

8.1 An investor buys a European put on a share for $3. The stock price is$42 and the strike price is $40. Under what circumstances does the investormake a profit? Under what circumstances will the option be exercised? Drawa diagram showing the variation of the investor’s profit with the stock price atthe maturity of the option.The investor makes a profit if the stock price on the expiration date is less than $37, be-cause the gain from exercising the option is greater than $3. Taking into account the initialcost of the option ($3), the profit will be positive. The option will be exercised if the stockprice is less than $40 on the expiration date. Below is the graph of the investor’s profit asa function of the stock price at the maturity of the option.

8.2 An investor sells a European call on a share for $4. The stock price is$47 and the strike price is $50. Under what circumstances does the investormake a profit? Under what circumstances will the option be exercised? Drawa diagram showing the variation of the investor’s profit with the stock price atthe maturity of the option.The investor makes a profit if the price of the stock is below $54 on the expiration date.If the stock price is below $50, the option will not be exercised, and the investor makes aprofit of $4 (the price that he received for the option). The option is exercised when thestock price is greater than $50 on the expiration date. If the stock price is between $50and $54, the option is exercised and the investor makes a profit between $0 and $4. Belowis the graph of the investor’s profit as a function of the stock price at the maturity of theoption.

1

8.4 Explain why brokers require margins when the clients write options butnot when they buy options.When an investor buys an option, he must pay it upfront. There are no future liabilitiesand, therefore, no need for a margin account. When an investor writes an option, thereare potential future liabilities. To protect against the risk of a default, margins are required.

8.11 Describe the terminal value of the following portfolio: a newly entered-into long forward contract on an asset and a long position in a European putoption on the asset with the same maturity as the forward contract and a strikeprice that is equal to the forward price of the asset at the time the portfolio isset up. Show that the European put option has the same value as a Europeancall option with the same strike price and maturity.The terminal value of the long forward contract is ST − F0, where ST is the price of theasset at maturity and F0 is the price of the asset at the time the portfolio is set up. Theterminal value of the European put option is max(F0 − S, 0).The terminal value of the portfolio is, therefore,

ST − F0 + max(F0 − S, 0) = max(0, ST − F0).

Note that this is the same as the terminal value of a European call option with the samematurity as the forward contract and an exercise price equal to F0. Thus, the forwardcontract plus the put option is worth the same as a call option with the same strike priceand time to maturity as the put option. Since the forward contract is worth zero at thetime the portfolio is set up, the European put option is worth the same as the Europeancall option at the time the portfolio is set up.

8.13 Explain why an American option is always worth at least as much asa European option on the same asset with the same strike price and exercisedate.The holder of an American option has all the rights the holder of a European option doesand even more (exercise prior to maturity). Therefore, the American option must be worth

2

at least as much as the European option.

8.14 Explain why an American option is always worth at least as much asits intrinsic value.The holder of an American option has the right to exercise it immediately. The Americanoption must thus be worth at least as much as its intrinsic value. If it were not, an arbi-trageur could lock in a profit by buying the option and exercising it immediately.

8.21 Explain why the market maker’s bid-offer spread represents a real costto options investors.A ”fair” price for the option can reasonably be assumed to be half way between the bid andoffer price quoted by a market maker. An investor typically buys at the market maker’soffer and sells at the market maker’s bid. Each time he makes this there is a hidden costequal to half the bid-offer spread.

9.1 List the six factors that affect stock option prices.The six factors affecting the price of a stock option are:1. The current stock price, S0.2. The strike price, K.3. The time to expiration, T .4. The volatility of the stock price, σ.5. The risk-free interest rate, r.6. The dividends expected during the life of the option.

9.2 What is a lower bound for the price of a 4-month call option on a non-dividend-paying stock when the stock price is $28, the strike price is $25, andthe risk-free interest rate is 8% per annum?A lower bound for the option’s price is

$28− $25e−0.08× 412 = $3.66.

9.3 What is a lower bound for the price of a 1-month European put optionon a non-dividend-paying stock when the stock price is $12, the strike price is$15, and the risk-free interest rate is 6% per annum?A lower bound for the option’s price is

$15e−0.06× 112 − $12 = $2.93.

9.8 Explain why the arguments leading to put-call parity for European op-tions cannot be used to give a similar result for American options.When early exercise is not possible, two portfolios that are worth the same at time T mustbe worth the same at earlier times. When early exercise is possible, the same argumentdoes not hold. Suppose, for example, that P + S > C + Ke−rT . This does not necessarilylead to arbitrage opportunities. If we buy the call, short the put, and short the stock, we

3

cannot be sure of the result because we do not know when the put will be exercised.

9.13 Give an intuitive explanation of why the early exercise of an Americanput becomes more attractive as the risk-free rate increases and volatility de-creases.The early exercise of an American put option is attractive when the interest earned onthe strike price is greater than the insurance element lost. When interest rates increase,the value of the interest earned on the strike price increases making early exercise moreattractive. When volatility decreases, the insurance element is less valueable. This makesearly exercise more attractive.

9.18 Prove the result in equation (9.4). (Hint: For the first part of the rela-tionship, consider (a) a portfolio consisting of a European call plus an amountof cash equal to K, and (b) a portfolio consisting of an American put optionplus one share.)Since P ≥ p, from the put-call parity

p + S0 = c + Ke−rT

we obtain thatP ≥ c + Ke−rT − S0

and since c = C (because it is never optimal to exercise an American call option on anondividend paying stock prior to expiry),

P ≥ C + Ke−rT − S0,

i.e.C − P ≥ S0 −Ke−rT .

Consider now the following two portfolios:Portfolio (a): One European call option plus and amount of cash equal to K.Portfolio (b): One American put option plus one share.Both options have the same exercise price and expiration date. Assume that the cash inportfolio (a) is invested at the risk-free interest rate. If the put option is not exercised earlyportfolio (b) is worth

ST + max(K − ST , 0) = max(K, ST )

at time T . Portfolio (a) is worth

max(ST −K, 0) + Ke−rT = max(ST , K)−K + Ke−rT

at time T . Therefore, portfolio (a) is worth more than portfolio (b). Suppose next that theput option in portfolio (b) is exercised early, say at time Tearly. This means that portfolio(b) is worth K at time Tearly. However, portfolio (a) is worth at least Ke−rTearly at timeTearly. Thus, portfolio (a) is worth at least as much as portfolio (b) in all circumstances.Hence

c + K ≥ P + S0.

4

Since c = C,C + K ≥ P + S0,

i.e.,C − P ≥ S0 −K.

Therefore, we have established (9.4)

S0 −K ≤ C − P ≤ S0 −Ke−rT .

5