B. Maddah ENMG 625 Financial Eng g II 7/7/9 Chapter 12 Basic Option Theory (1) Option basics An option is the right, but not the obligation, to sell or buy an asset at specific terms. E.g., the option of purchasing a home in exactly one year for $ K. The one year time in this example is called the exercise time. The $ K is the exercise or the strike price of the option. An option that gives the right to buy is a call option. An option that gives the right to sell is a put option. imilar to futures contracts options are used to hedge risk, or to make money via speculation. Unlike a futures contract, an option has a price or a premium. E.g., you may pay $1 K to obtain the option to buy the $ K home in one year. An option is a derivative security with an underlying asset that can be sold or bought (e.g., a house). The value of an option depends on the underlying asset value at possible exercise times. E.g., the home call option has no value if the home value < $ K after 1 year, and is worth $1 K if it s $3 K. 1
European and American options A European option can be exercised only at a specific date (the expiration date). E.g., a call option to buy 1 GM shares for $5/share in exactly two months. An American option can be exercised any time before and including the expiration date. E.g., a call option to buy 1 GM shares for $5/share within two months. Option trading There are two sides to any option o The party that grants the option who is said to sell or write the option; o The party that obtain the option who is said to buy it. tock options are traded on an exchange where trades are done through a broker. The exchange clearing house guarantees the performance of all parties. An option writer is required to post margin (security deposit). Exchange traded options are listed in the financial press. Prices are quoted on a per-share basis. However, a single option contract is for 1 shares. 2
Option value and profit at expiration The value of a call option with strike price K at expiration is max(, K ), where is the stock price at expiration. The profits of the buyer and the seller of a call option are max(, K ) C, C max(, K). That is buyer makes money if the stock price exceeds the strike price plus premium ( > K+C). The seller makes money otherwise. 3
An example of a call option value and buyer and seller profits are shown in the following figures with K = $5 and C = $1. 1 8 V 6 6 8 1 R buycall R sellcall 6 8 1 imilarly the value, and buyer and seller profits of a put option at expiration with premium P, are respectively max(, K ), max(, K ) P, P max(, K ). 4
An example of put option value, buyer and seller profits are shown in the following figures with K = $5 and P = $1. 1 8 V 6 6 8 1 R buyput R sellput 6 8 1 A call option is in the money, at the money, or out of the money, if > K, = K, or < K, respectively. The reverse terminology applies to put options. 5
Option values at times earlier than expiration A call option which is out of the money before expiration has a value because the stock price could increase at expiration. The value of the option at time zero is the price or the premium of the options. Determining the price of an option is among the most important problems of Financial Engineering. The following figure shows the option price as a function of time to expiration. It is obvious from the figure that the longer the time to expiration the higher the option price (why?). 6
Factors that affect stock option price (Hull 6) There are six factors: (1) The current stock price, ; (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) Expected dividends payments. 7
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Option Combinations Combinations of options and stock can approximate any payoff function by a piece-wise linear function. A common option combination is the butterfly spread. It is constructed by buying two calls with strike prices K 1 and K 3 and by selling two units of a call with strike price K 2, K 1 < K 2 < K 3. The following figure shows the payoff at of a butterfly spread with K 1 =, K 2 = 5, K 3 = 6, C 1 = 14, C 2 = 1, and C 3 = 9, where C i is the price of the option with strike price K i. 1 R buycall1 R buycall3 R sellcall2 R buycall1 + R buycall3 + R sellcall2 6 8 1 The butterfly spread yields a positive profit if the stock price at expiration is close to K 2, and a small loss, otherwise. K 2 is usually close to the current stock price. o, the butterfly spread is used if one believes that the stock price will not vary much. 1 The payoffs at expiration are R buycall1 () = max(, K 1 ) C 1 and R buycall3 () = max(, K 3 ) C 3 and R sellcall2 () = 2 [C 2 max(, K 2 )]. 9
Put-Call Parity For European options, there is a simple relationship between the price of call and put options with the same strike price, K, and expiration time, T. This relationship is found by noting that a combination of buying a call, selling a put, and lending an amount d(, T)K, are equivalent to buying and holding the stock. This fact is best understood graphically. The following figure illustrates this for K = 5. 2 V buycall V sellput V buycall + V sellput + K 1 7 6 5 3 1 1 3 5 6 1 3 5 6 7 8 9 1 Therefore, the put-call parity relationship is as follows. P C d(, T) K= C P+ d(, T) K=, where is the current stock price. K 2 In the figure, the values of buying the call and selling the put are V buycall () = max(, K) and V sellput () = max(, K ). Note that V buycall () + V sellput () = K. Add to this the payoff of the loan K, you get. 1