LECTURE 1 : Introduction and Review of Option Payoffs

Transcription

1 AALTO UNIVERSITY Derivatives LECTURE 1 : Introduction and Review of Option Payoffs Matti Suominen

2 I. INTRODUCTION QUESTIONS THAT WE ADDRESS: What are options and futures and swaps? How to value options Warrants Convertibles Exotic Options Empirical results on option pricing Executive Options How and why should a firm do risk management? How to value flexibility in investments? What is financial engineering? DEFINITION Derivative = an asset whose payoff is derived from another asset s price. There are several different types of derivatives securities. We will focus our attention mainly on options. 2

3 SOME STATISTICS SIZE OF GLOBAL FINANCIAL MARKETS (27 trillion in 212): HIGH TURNOVER: Turnover in US equity market in 213 was 96% of market capitalization, close to 2 trillion USD. Globally I estimate the equity market turnover at roughly 6-8% of market capitalization, around 4- trillion USD in 213. Global daily foreign exchange turnover in 213 was $.3 trillion. In three days foreign exchange turnover is sufficient to cover world trade in a year. Sources: World Bank, Credit Suisse, World Federation of Exchanges, The Economist 3

4 Global Derivative Markets Source:BIS Notional amounts outstanding at year- end, in trillions of USD OTC Instruments Foreign exchange contracts Forwards and forex swaps Currency swaps 2, Options 3,7 13 Interest rate contracts Forward rate agreements, Interest rate swaps Options 8, Equity- linked contracts 1, 7, 6,9 Forwards and swaps,1 1,8 2,4 Options 1,3,7 4, Commodity contracts,4 7,1 2,2 Gold,2,6,3 Other commodities,2 6, 1,9 Forwards and swaps,1 2,8 1,3 Options,1 3,7,6 Credit default swaps Single- name instruments Multi- name instruments ,6 of which index products - 7,9 Unallocated/OTC 43 2 Exchange- traded instruments Futures 8, Interest rate 8, 24 2 Currency,,2,2 Equity index,3 1 1,6 Options, Interest rate 4, Currency,,1,1 Equity index,9 6,6 4

5 Turnover of financial Derivatives traded on organized exchanges Notional amounts in trillions of US dollars Source:BIS Interest rate futures Interest rate options Currency futures 2, Currency options, 1,1 3 Equity index futures 18, Equity index options 14, Total in North America in Europe in Asia and Pacific in Other Markets 3,4 1 26

6 Why are derivatives important? 1) Derivatives market is large relative to the underlying market (stock, bonds etc.) 2) Firms use derivatives to hedge (i.e., to take protection or insurance) against risks Research indicates that use of derivatives increases firm value by 4-12% (see Stulz, 24). 6

7 II. OPTIONS Why are options so important? Options are popular tool in risk management. Many capital investments have options built into them (option to expand production, option to abandon the project at a future date, etc.). These are known as Real Options. Most corporate securities such as common stock, risky debt, convertible securities, warrants are or contain options. They are common in executive compensation! Same techniques (no arbitrage conditions) are used to price other assets such as government bonds. The Chicago Board Options Exchange was founded in 1973 as the first organized options exchange. Today options are traded in most industrialized countries and on several underlying securities such as: stocks, currencies, commodities, futures, government bonds. 7

8 1. BASIC STOCK OPTIONS Two kinds of options: CALL OPTIONS: right (but not the obligation) to buy a stock - at a predetermined price (exercise or strike price) - before or at a given date (expiration date) PUT OPTIONS: right (but not the obligation) to sell a stock - at a predetermined price (exercise or strike price) - before or at a given date (expiration date) European options: may be exercised only at expiration. American options: may be exercised anytime before expiration. 8

9 Call option payoff at expiration: EX= 2, C() = 8 Payoff S - 9

10 Put option payoff at expiration: EX= 2, P() = 6 Payoff 2 1 S

11 Writing a call option; payoff at expiration: EX= 2, C() = 8 Payoff S

12 Writing a put option; payoff at expiration: EX= 2, P() = 6 Payoff S

13 Payoff Payoff from buying a stock; S() =2 S Payoff from shorting the stock; S() = 2 Payoff 2 S

14 Options are very flexible instruments: Example 1: Payoff Straddle: Buy one call and put; EX=2, C() = 4, P() = 3 S

15 Example 2: Payoff Short Straddle: Write one call and put; EX=2, C() = 4, P() = 3 S

16 Pricing options: PUT-CALL PARITY Evaluate the terminal payoff from two portfolios: PORTFOLIO 1: 1 call option + 1 bond paying EX at expiration Payoff EX EX S Cost of this portfolio today: 16

17 PORTFOLIO 2: 1 share + 1 put option (same exercise price) Payoff EX EX S Cost of this portfolio today: The two portfolios must be equally expensive: Price of a call + PV(EX) = S + price of a put ð P = C + PV(EX) - S 17

18 Example 3: How to buy shares without buying shares: Synthetic shares Payoff Payoff from buying a call, writing a put and investing PV(EX) in bonds; EX =2 S How expensive are synthetic shares? 18

19 GENERAL PROPERTIES OF CALL OPTIONS The price of a call option must satisfy the following relations: 1. C S ; 2. C S - PV(EX); 3. If S =, then C = ; 4. For large S, C S - PV(EX). Payoff 4 Bounds on the value of a call option; EX = S 19

20 Other properties of call options: Keeping other things equal: 1. The price of a call option, C, increases with the current stock price, S. (That is, if S increases, C increases); 2. C decreases with the strike price, EX; 3. C increases with the interest rate; 4. C increases with the volatility of underlying stock;. C increases with time to expiration. 2

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22 Review questions for lecture 1 (answers are given below): Q1: A European call option on IBM gives you the right to purchase 1 share of IBM at a predetermined exercise price, EX, at some predefined date in the future (the options expiration date). What is the optimal exercise policy and the payoff at maturity for a European call option? Q2: A European put option on IBM gives you the right to sell 1 share of IBM at a predetermined exercise price, EX, at some predefined date in the future (the options expiration date). What is the optimal exercise policy and the payoff at maturity for a European put option? Q3: What is put-call parity? Q4: Knowing the final payoff for a European call option, can you provide some bounds on the value of the call option (for instance can you argue why the price of the call option should be less than the price of the stock?) Q: True or false? The price of the call option is: a) Increasing with the exercise price. b) Decreasing with the interest rate. c) Increasing with price volatility. 22

23 Answers: Q1: A European call option on IBM gives you the right to purchase 1 share of IBM at a predetermined exercise price, EX, at some predefined date in the future (the options expiration date). What is the optimal exercise policy and the payoff at maturity for a European call option? Let S = share price at option s expiration Let EX = exercise price Optimal exercise policy and payoff: Excercise if S > EX ð payoff is S-EX (you can buy stock which is worth S at a lower price EX). Do not exercise if EX S ð payoff is zero. We can write the payoff conveniently as: max(s-ex;), that is, the payoff is the maximum of S-EX and. Q2: A European put option on IBM gives you the right to sell 1 share of IBM at a predetermined exercise price, EX, at some predefined date in the future (the options expiration date). What is the optimal exercise policy and the payoff at maturity for a European put option? In case of a put option the optimal exercise policy is: Excercise if EX > S ð payoff is EX-S (you can sell stock which is worth S at a higher price EX). Do not exercise if S EX ð payoff is zero. We can write the payoff conveniently as: max(ex-s;). Q3: What is put-call parity? Put-call parity states that the prices of put and call options are related in a deterministic manner. Put-call parity states that: C = S + P - PV(EX) where C, S and P are the current market prices of call option, share and put option, respectively, and PV(EX) the present value of the exercise price. We can understand this relation as follows: 23

24 Consider the payoff (at the options expiration) to holding a portfolio where you borrow the present value of EX, buy 1 stock and buy 1 put. This portfolio will cost you today S + P - PV(EX). The payoff from this portfolio (at the option s expiration) is: S + max(ex-s;) - EX Two things are possible at the option s expiration: 1) the put option is in the money and EX S. In this case the payoff from the portfolio is S+EX-S-EX, which is equal to zero. 2) the put option is out of the money and S EX. In this case the payoff from the portfolio is S + - EX = S-EX. This implies that the payoff from a portfolio consisting of one stock, one put and the borrowing of PV(EX) can be written as max(s-ex;), and is thus identical to the payoff from a call option. Because the terminal payoffs are the same, it better be the case that these two portfolios also cost the same today. This implies the put-call parity relation stated above. Q4: Knowing the final payoff for a European call option, can you provide some bounds on the value of the call option (for instance can you argue why the price of the call option should be less than the price of the stock?) In class we gave four bounds on call options: 1. C S 2. C S - PV(EX) 3. If S =, then C = 4. If S is much larger than EX, then C S - PV(EX) We can understand these bounds as follows: 1. The payoff from an option is never negative, but can be positive. So its value should also always be positive (although sometimes it may be very close to zero). Similarly, the terminal value of the option is never higher than S-EX, and thus always smaller than the payoff from a share (that is, payoff from selling a share at the maturity of the option), which is S. 2. Look at the put-call parity, which states that 24

25 C = S + P - PV(EX) But, as is true for any option, the value of a put option is positive, meaning that C > S - PV(EX) Alternatively we can understand it as follows: Buying a stock, but borrowing PV(EX), to finance part of your purchase, costs you S - PV(EX) today. The value of this portfolio at the option s expiration is S-EX. But the option s payoff is max(s-ex;) and so its payoff is identical to that of the above portfolio if the option is in the money, and higher than that if the option is out of the money. Thus the option gives you a higher payoff than you would get by purchasing stock and borrowing PV(EX). Because of this, the market price of the option, C, must also be higher than the cost of this alternative portfolio, which is S - PV(EX). 3. If the share price is today zero, it means that the company is bankrupt and there is no hope that the share price will have any positive value in the future. Because of this also the call option is worthless. 4. Look again at the put call parity, which states that C = S + P - PV(EX) When the share price is very large as compared with the exercise price, the probability that the put option will have a positive payoff is very close to zero. So its value will also be very close to zero. Because of this C S - PV(EX) Alternatively we can understand this as follows: When the stock price is much higher than the exercise price, the probability of exercising the call option is very high. So, effectively, when purchasing such deep in the money call option, we are purchasing stock, but paying part of the price (EX) only later. The value from buying a call option should thus be approximately equal to S - PV(EX), which is what you have to pay if you bought the stock, but borrowed PV(EX) to finance part of your purchase. Q: True or false? The price of the call option is: a) Increasing with the exercise price. b) Decreasing with the interest rate. c) Increasing with price volatility a) False. The payoff of a call option when exercised is S - EX. Lower exercise price means bigger payoff at maturity (you have to pay less to obtain the shares). Thus the price of a call option is decreasing with the exercise price. b) False. The price of a call option (relative to the stock price) is increasing with the interest rate. By buying shares through options you pay part of your purchase later (the exercise 2

26 price). When interest rates are high, the value of this deferred payment is high and thus the value of a call option is high relative to the share price. c) True. Price volatility increases the value of a call option. Consider two stocks A and B, both selling at, and 3-month call options with exercise price on these two stocks. Stock A has low volatility and is either 9 or 11, with equal probabilities, in 3 months time when the option expires. The option payoff is thus either or 1. Stock B, on the other hand, has high volatility and is either or 1 in three months time when the option expires. Thus the payoff to company B s option is either or. Clearly other things equal, the expected payoff on company B s option is higher, and its option should be more valuable. Finally, the fact that C increases with time to expiration can also be understood using the two previous results. First note that the interest savings due to ability to defer payment of the exercise price and the volatility of the final share price both increase with time to expiration. Given this, the value of a call option should also increase with time to expiration.j 26