内容文本文稿damianov

1、ECON41415 Derivative MarketsTopic 6b. Introduction to Options1Learning Outcomes Defining the payoffs of call and put options. Differentiating between European and American options. Deriving upper and lower bounds on the prices of call and put options. Understanding the put-call parity relationship.2

2、 There are two basic types of options. A call option gives the holder of the option the right to buy an asset by a certain date for a certain price. A put option gives the holder the right to sell an asset by a certain date for a certain price.3 Options can be either American or European, a distinct

3、ion that has nothing to do with geographical location.4Call Options Consider the situation of an investor who buys a European call option with a strike price of $100 to purchase 100 shares of a certain stock. Suppose that the current stock price is $98, the expiration date of the option is in 4 mont

4、hs, and the price of an option to purchase one share is $5. The initial investment is $500.5 Because the option is European, the investor can exercise only on the expiration date. If the stock price on this date is less than $100, the investor will clearly choose not to exercise. (There is no point

5、in buying for $100 a share that has a market value of less than $100.) In these circumstances, the investor loses the whole of the initial investment of $500. If the stock price is above $100 on the expiration date, the option will be exercised.6 Suppose, for example, that the stock price is $115. B

6、y exercising the option, the investor is able to buy 100 shares for $100 per share. If the shares are sold immediately, the investor makes a gain of $15 per share, or $1,500, ignoring transactions costs. When the initial cost of the option is taken into account, the net profit to the investor is $1,

7、000.7Figure 1:Profit from buying a European call option on one share of a stock. Option price = $5; strike price = $100.8Put Options Whereas the purchaser of a call option is hoping that the stock price will increase, the purchaser of a put option is hoping that it will decrease. Consider an investo

8、r who buys a European put option with a strike price of $70 to sell 100 shares of a certain stock. Suppose that the current stock price is $65, the expiration date of the option is in 3 months, and the price of an option to sell one share is $7. The initial investment is $700.9 Because the option is

9、 European, it will be exercised only if the stock price is below $70 on the expiration date. Suppose that the stock price is $55 on this date. The investor can buy 100 shares for $55 per share and, under the terms of the put option, sell the same shares for $70 to realize a gain of $15 per share, or

10、 $1,500. When the $700 initial cost of the option is taken into account, the investors net profit is $800. There is no guarantee that the investor will make a gain. If the final stock price is above $70, the put option expires worthless, and the investor loses $700.10Figure 2:Profit from buying a Eu

11、ropean put option on one share of a stock. Option price = $7; strike price = $7011Option Positions There are two sides to every option contract. On one side is the investor who has taken the long position (i.e., has bought the option). On the other side is the investor who has taken a short position

12、 (i.e., has sold or written the option).12 Figures 3 and 4 show the variation of the profit or loss with the final stock price for writers of the options considered in Figures 1 and 2.13Figure 3:Profit from writing a European call option on one share of a stock. Option price = $5; strike price = $10

13、014Figure 4:Profit from writing a European put option on one share of a stock. Option price = $7; strike price = $7015 There are four types of option positions: (a) A long position in a call option(b)A long position in a put option(c) A short position in a call option(d)A short position in a put opt

14、ion16 It is often useful to characterize a European option in terms of its payoff to the purchaser of the option. The initial cost of the option is then not included in the calculation. If K is the strike price and ST is the final price of the underlying asset:17(a) The payoff from a long position i

15、n a European call option is max(ST - K, 0) This reflects the fact that the option will be exercised if ST K and will not be exercised if ST K.18(b) The payoff to the holder of a short position in the European call option is - max(ST - K, 0) = min(K - ST, 0)19(c) The payoff to the holder of a long po

16、sition in a European put option is max(K - ST, 0)20(d) The payoff from a short position in a European put option is - max(K - ST, 0) = min(ST - K, 0)21 Figure 5 illustrates these payoffs.22Figure 5:Payoffs from positions in European options23Arbitrage Arguments Arbitrage arguments can be used to exp

17、lore the relationships between European option prices, American option prices, and the underlying stock price. The most important of these relationships is put-call parity, which is a relationship between the price of a European call option, the price of a European put option, and the underlying sto

18、ck price.24 We will use the following notation:S0Current stock priceKStrike price of optionTTime to expiration of optionSTStock price on the expiration daterContinuously compounded risk-free rate of interest for an investment maturing in time TCValue of American call option to buy one sharePValue of

19、 American put option to sell one sharecValue of European call option to buy one sharepValue of European put option to sell one share25Upper Bounds for Calls and Puts The price of a call option on a stock must always be worth less than the price of the stock itself. Similarly, the price of a put opti

20、on on a stock must always be worth less than the options strike price.26 An American or European call option gives the holder the right to buy one share of a stock for a certain price. No matter what happens, the option can never be worth more than the stock. Hence, the stock price is an upper bound

21、 to the option price:c S0 and C S027 An American or European put option gives the holder the right to sell one share of a stock for K. No matter how low the stock price becomes, the option can never be worth more than K. Hence,p K and P K28 For European options, we know that at maturity the option c

22、annot be worth more than K. It follows that it cannot be worth more than the present value of K today:p Ke-rT29Lower Bound for European Calls on Non-Dividend-Paying Stocks A lower bound for the price of a European call option on a non-dividend-paying stock isS0 Ke-rT30Example Suppose that S0 = $20,

23、K = $18, r =10% per annum, and T = 1 year. In this case, S0 Ke-rT = 20 - 18e-0.l = 3.71 31 Consider the situation where the European call price is $3.00, which is less than the theoretical minimum of $3.71. An arbitrageur can short the stock and buy the call to provide a cash inflow of $20.00 - $3.0

24、0 =$17.00. If invested for 1 year at 10% per annum, the $17.00 grows to 17e0.l = $18.79. At the end of the year, the option expires.32 For a more formal argument, we consider the following two portfolios: Portfolio A: one European call option plus an amount of cash equal to Ke-rTPortfolio B: one sha

25、re33 In portfolio A, the cash, if it is invested at the risk-free interest rate, will grow to K in time T. If ST K, the call option is exercised at maturity and portfolio A is worth ST. If ST K, the call option expires worthless and the portfolio is worth K. Hence, at time T, portfolio A is worth ma

26、x(ST, K) Portfolio B is worth ST at time T.34 Hence, portfolio A is always worth as much as, and can be worth more than, portfolio B at the options maturity. It follows that in the absence of arbitrage opportunities this must also be true today. Hence, c + Ke-rT S0 or c S0 - Ke-rT35 Because the wors

27、t that can happen to a call option is that it expires worthless, its value cannot be negative. This means that c 0 and therefore c max(S0 - Ke-rT, 0)36Lower Bound for European Puts on Non-Dividend-Paying Stocks For a European put option on a non-dividend-paying stock, a lower bound for the price is

28、Ke-rT - S037Example Suppose that S0 = $37, K = $40, r = 5% per annum, and T = 0.5 years. In this case, Ke-rT - S0 = 40e-0.05x0.5 - 37 = $2.0138 Consider the situation where the European put price is $1.00, which is less than the theoretical minimum of $2.01. An arbitrageur can borrow $38.00 for 6 mo

29、nths to buy both the put and the stock. At the end of the 6months, the arbitrageur will be required to repay 38e0.05x0.5 = $38.96. 39 For a more formal argument, we consider the following two portfolios: Portfolio C: one European put option plus one sharePortfolio D: an amount of cash equal to Ke-rT

30、40 If ST K, then the put option expires worthless, and the portfolio is worth ST at this time. Hence, portfolio C is worth max(ST, K) in time T. Assuming the cash is invested at the risk-free interest rate, portfolio D is worth K in time T. 41 Hence, portfolio C is always worth as much as, and can s

31、ometimes be worth more than, portfolio D in time T. It follows that in the absence of arbitrage opportunities portfolio C must be worth at least as much as portfolio D today. Hence, p + S0 Ke-rT or p Ke-rT - S042 Because the worst that can happen to a put option is that it expires worthless, its val

32、ue cannot be negative. This means that p max(Ke-rT - S0, 0)43Put-Call Parity We now derive art important relationship between p and c. Consider the following two portfolios that were used in the previous section: Portfolio A: one European call option plus an amount of cash equal to Ke-rTPortfolio C:

33、 one European put option plus one share Both are worth max(ST, K) at expiration of the options.44 Because the options are European, they cannot be exercised prior to the expiration date. The portfolios must therefore have identical values today. This means that c + Ke-rT = p + S045 This relationship

34、 is known as put-call parity. It shows that the value of a European call with a certain strike price and exercise date can be deduced from the value of a European put with the same strike price and exercise date, and vice versa. If this does not hold, there are arbitrage opportunities.46Example Supp

35、ose that the stock price is $31, the strike price is $30, the risk-free interest rate is 10% per annum, the price of a 3-month European call option is $3, and the price of a three-month European put option is $2.25.47 In this case, c + Ke-rT = 3 + 30e-0.1x3/12 = $32.26 and p + S0 = 2.25 + 31 =$33.25

36、 48Portfolio C is overpriced relative to portfolio A.An arbitrageur can buy the securities in portfolio A and short the securities in portfolio C.The strategy involves buying the call and shorting both the put and the stock, generating a positive cash flow of -3 + 2.25 + 31 = $30.25 up front.When in

37、vested at the risk-free interest rate, this amount grows to 30.25e0.1 x0.25 = $31.02 in 3 months;If the stock price at expiration of the option is greater than $30, the call will be exercised; and if it is less than $30, the put will be exercised.In either case, the arbitrageur ends up buying one sh

38、are for $30. This share can be used to close out the short position.The net profit is therefore $31.02 - $30.00 = $1.0249 Put-call parity does not hold for American options. However, it is possible to use arbitrage arguments to obtain upper and lower bounds for the difference between the price of an

39、 American call and the price of an American put. It can be shown that, when there are no dividends, S0 K C P S0 - Ke-rT50Example An American call option on a non-dividend-paying stock with strike price $20.00 and maturity in 5 months is worth $1.50. Suppose that the current stock price is $19.00 and

40、 the risk-free interest rate is 10% per annum.5119 20 C P 19 - 20e-0.1x5/12 or 1 P C 0.18 showing that P- C lies between $1.00 and $0.18. With C at $1.50, P must lie between $1.68 and $2.50. In other words, upper and lower bounds for the price of an American put with the same strike price and expira

41、tion date as the American call are $2.50 and $1.68.52Early Exercise: Calls on a Non-Dividend-Paying Stock It is never optimal to exercise an American call option on a non-dividend-paying stock before the expiration date.53 There are two reasons an American call on a non-dividend-paying stock should

42、not be exercised early. One relates to the insurance that it provides. A call option, when held instead of the stock itself, in effect insures the holder against the stock price falling below the strike price. Once the option has been exercised and the strike price has been exchanged for the stock p

43、rice, this insurance vanishes. The other reason concerns the time value of money. From the perspective of the option holder, the later the strike price is paid out, the better.54Early Exercise: Puts on a Non-Dividend-Paying Stock It can be optimal to exercise an American put option on a non-dividend

44、-paying stock early. Indeed, at any given time during its life, a put option should always be exercised early if it is sufficiently deep in the money. It is always optimal to exercise an American put immediately when the stock price is sufficiently low. When early exercise is optimal, the value of the option is K S0.55The Effect of Dividends We will use D to denote the present value of the dividends during the life o