1. 1 9.220 Chapter 22 - Options

2. 2 Options §If you have an option, then you have the right to do something. I.e., you can make a decision or take some action. l The option owner has a choice to make. l Usually the choice can be made over time after more information is known. l Having an option is valuable; options never have a negative value, because, at worst, the option owner can discard the option and not take any action.

3. 3 A European Call Option gives the holder the right to buy the underlying asset for a prescribed price (exercise/strike price), on a prescribed date (expiry date). A European Put Option gives the holder the right to sell the underlying asset for a prescribed price (exercise/strike price), on a prescribed date (expiry date). American Options exercise is permitted at any time during the life of the option (call or put).

4. 4 Underlying Asset (S) The specific asset on which an option contract is based (e.g. stock, bond, real-estate, etc.). For traded Stock Options: one call (put) option contract represents the right to buy (sell) 100 shares of the underlying stock. Strike/Exercise Price (E) The specified asset price at which the asset can be bought (sold) by the holder of a call (put) if s/he exercised his/her right. Expiration Date (T) The last day an option exists.

5. 5 Option Writer §The option writer is the person who created (or wrote) the option and then sold it to someone else. As the writer did not previously own the option, the act of writing and selling the option is equivalent to short-selling the option. §The option writer has sold an option or right to the option owner. l Thus the writer has taken on an obligation to act on the instruction of the option owner as specified by the option contract. l When selling the option to the buyer, the writer receives as compensation the “option premium”= the price the buyer pays for the option.

6. 6 Writer: Seller of an option (takes a short position in the option). Holder: Buyer of the option (takes a long position in the option). Elements of an option contract: u type (put or call) u style (American or European) u underlying asset (stock/bond/etc…) u unit of trade u exercise price u expiration date

7. 7 I’m “in the money” §An option is said to be “in the money” if exercising it would produce a positive payoff. §An “at the money” option would generate a zero payoff if exercised. §An “out of the money” option would generate a negative payoff if exercised. (Thus an out of the money option would never be exercised.)

8. 8 Option Payoffs at Expiration §It is useful to examine the payoffs of options when they are about to expire. At this point in time, the owner is forced to make the decision to exercise or to abandon the option. The option owner will exercise if the option is in the money when it is about to expire. §Consider Call and Put Options…

9. 9 Holding A European Call Option Contract- An Example European style IBM corp. September 100 call: entitles the buyer (holder) to purchase 100 shares of IBM common stock at $100 per share (E), at the options expiration date in September (T). At the options expiration date (T): For the Call Option Holder If S T > E=$100: Exercise the call option - pay $100 for an IBM stock with a market value of S T (e.g. S T =$105). Payoff at T: S T - E = $105-$100=$5 > 0. If S T  E=$100: Can buy IBM stocks in the market for S T (e.g. S T =$90). Holder will not choose to exercise (option expires worthless). Payoff at T: $0.

10. 10 Holding a European Call Conclusion: A call option holder will never lose at T (expiration), since his/her payoff is never negative: If S T  E=$100 If S T > E=$100 Call option value at T$0 S T - E = S T - $100 Payoff at T 0 E=$100 STST 45 0

11. 11 For the Call Option Writer (Short Seller) If S T > E = $100: Holder will exercise. Writer will deliver an IBM stock with a market value of S T ($105) to the holder, in return for E dollars ($100). Payoff at T: E-S T = $100-$105= - $5 < 0. If S T  E = $100: Holder will not exercise. Payoff at T: $0.

12. 12 For the Call Option Writer (Short Seller) Conclusion: A call option writer will never gain at T (expiration), since his/her payoff is never positive: If S T  E=$100 If S T > E=$100 Call option value at T$0 E - S T = $100 - S T Payoff at T 0 E=$100 STST 45 0

13. 13 Holding A European Put Option Contract- An Example European style IBM corp. September 100 put entitles the buyer (holder) to sell 100 shares of IBM corp. common stock at $100 per share (E), at the option’s expiration date in September (T). At the options expiration date (T): For the Put Option Holder If S T < E=$100: Exercise the put option - receive $100 for an IBM stock with a market value of S T (e.g. S T =$90). Payoff at T: E - S T = $100-$90=$10 > 0. If S T  E=$100: Can sell IBM stocks in the market for S T (e.g. S T =$105). Holder will not choose to exercise (option expires worthless). Payoff at T: $0.

14. 14 Holding A European Put Conclusion: A put option holder will never lose at T, since his/her payoff is never negative: If S T  E=$100 If S T < E=$100 Put option value at T$0 E - S T = 100 - S T Payoff at T 0STST 45 0 $100 E=$100

15. 15 For the Put Option Writer (Short Seller) If S T <E = $100: Holder will exercise. Writer will pay E dollars ($100) in return for an IBM stock (worth S T = $90). Payoff at T: S T - E = $90-$100 = -$10< 0. If S T  E = $100 : Holder will not exercise. Payoff at T: $0.

16. 16 Conclusion: A put option writer will never gain at T, since his/her payoff is never positive: If S T  E=$100 If S T < E=$100 Put option value at T$0 S T - E = S T - 100 Payoff at T 0 E=$100 STST 45 0 - $100 For the Put Option Writer (Short Seller)

17. 17 Combinations of Options You purchase a BCE stock, and simultaneously write (short sell) the July $85 European call option. Your payoff diagram at expiration in July (T): Payoff at T 0 STST 45 0 Buy Stock Payoff at T 0 STST Short Sell Call Payoff at T 0 STST 45 0 Combination E=$85 Same as Short Sell a Put and Buy A T-bill $85

18. 18 The Put-Call Parity Relationship* You purchase the BCE stock, the July $85 put option, and short sell the July $85 call option (both options are European). Your payoff at expiration in July (T): If S T = $100If S T = $80 Stock (S) Put (P) Call(C)__ Total (Certain) Payoff: To calculate the PV of the certain payoff ($85=E) today, we use the risk-free rate: *Only for European Options

19. 19 You purchase the BCE stock, the July $85 European put option, and short sell the July $85 European call option Your payoff at expiration in July (T): 45 0 The Put-Call Parity Portfolio Payoff at T 0 STST 45 0 Buy Stock STST Short Sell Call STST 45 0 Combination E=$85 A Certain Payoff of E=$85 E=$85 Buy Put E=$85 $85

20. 20 Using Put-Call-Parity (PCP) to Replicate Securities A synthetic security l Definition - A portfolio of other securities which will pay the same future cash flows as the security being replicated. l Since payoffs at expiration (cash flows) are the same for the synthetic security and the original security under all states of the world, their current prices must be identical. l Otherwise, if one is currently cheaper than the other, an arbitrage opportunity will exist: buy (long) the cheaper security today for the lower price, and simultaneously short sell the expensive security for the higher price. This results in a positive initial cash flow. l This positive cash flow is an arbitrage profit (“free lunch”), since at expiration, the cash flows from both positions will offset each other, and the total cash flow at expiration will be zero.

21. 21 uThe Put-Call-Parity (PCP) Relationship: This is a risk free T-bill that pays E dollars in T years uRecall that the PCP portfolio was created by: Long one stock (+S), Long one put (+P), and Short one call (-C) uWe saw that this is equivalent to: Long a T-bill (+Ee - Tr f ) uThus, we replicated a long position in a T-bill with: long stock, long put and short call. uFor security replication purposes, use PCP with the following rule: Long is “+” Short is “-” How Do We Replicate Securities?

22. 22 A Synthetic Stock u We first rearrange the PCP equation to isolate S: u According to the above replication rule: Long one stock (+S) = Long a T-bill (+Ee - Tr f ) & Long one call (+C) & Short one put (-P), u The payoff (cash flow) at maturity: S T E Long T-bill E E Long Call0S T - E Short Put-(E - S T )$0 Total Replicated Payoff: +S T +S T u Conclusion - holding the replicated portfolio is the same as holding the stock

23. 23 A Synthetic Call u We first rearrange the PCP equation to isolate C: u According to the above replication rule: Long one call (+C) = Long one stock (+S) & Long one put (+P) & Short a T-bill (-Ee - Tr f ) u The payoff (cash flow) at maturity: S T E Long Stock Long Put Short T-bill Total Replicated Payoff: Conclusion - holding the replicated portfolio is the same as holding a call

24. 24 A Synthetic Put u We first rearrange the PCP equation to isolate P: u According to the above replication rule: Long one put (+P) = Long a T-bill (+Ee - Tr f ) & Long one call (+C) & Short one stock (-S) u The payoff (cash flow) at maturity: S T E Long T-bill Long Call Short Stock Total Replicated Payoff: u Conclusion - holding the replicated portfolio is the same as holding a put

25. 25 Bounding The Value of An American Call The value of an American call can never be: ubelow the difference b/w the stock price (S) and the exercise price (E). u If C < S - E: investors will pocket an arbitrage profit. u Example: S = $100, E = $90, C = $8=>C = 8 < 10 = S – E u Arbitrage Strategy: uBuy the call for $8, and exercise it immediately by paying the exercise price ($90)to get the stock (worth $100). This results in an immediate arbitrage profit (i.e. “free lunch”) of: -8-90+100= $2 uExcess demand will force C to rise to $10 u As long as there is time to expiration, we will have C > $10 = S - E uAbove the value of the underlying stock (S) u If it is, buy the stock directly Boundary Conditions Payoff at t 0 E=$90 StSt 45 0 value of the American call will be here

26. 26 For an American Call: C = C (S, E, T, , r) (+) (-) (+) (+) (+) S - The higher the share price now, the higher the profit from exercising. Thus the higher the option price will be. E - The higher the exercise price now, the more it needs to be paid on exercise. Thus, the lower the option value will be. T - The more time there is to expiration, the higher the chance that the stock price will be higher at T, and the higher the option value will be.  - The larger the volatility, the more probable a profitable outcome, thus the higher C is. r - The higher the interest rate, the P.V. of the future exercise price decreases. The call price will increase. For an American Put: P = P(S, E, T, , r) (-) (+) (+) (+) (-) Determinants of American Option Pricing

27. 27 Determinants of American Option Pricing Determinants ofRelation toRelation to Option PricingCall OptionPut Option Stock pricePositiveNegative Strike price NegativePositive Risk-free rate PositiveNegative Volatility of the stockPositivePositive Time to expiration datePositivePositive