Options and Contingent Claim

1. Chapter 15 Options and Contingent Claim

2. Chapter Outline • Option Terminology • How options work? • Asymmetric Gains and Loss of Call and Put options • Investing with options • Put-Call Parity Relation

3. Option Terminology • Buying an option is like buying a ticket to a concern • Call: Right to buy a specific asset at expiration date at a predetermined price. Exercis if S1>E. • Put: . Exercis if S1<E. • Strike or Exercise price (E) • Expiration date • Option premium • American Option: the right to exercise the option at any time between the date of writing and the expiration date. • European Option: Only on the expiration date • Trading : Exchange- traded options or over-the –counter options

4. How options work? Ex1: Listing of IBM option prices Wall street Journal published on May 29, 1998 Underlying stock price: 120. 1/16 Thursday May 28, 1998

5. How options work? • The price of 115 option is (7) contains two values: • (1) Intrinsic value • (2) The time value; the difference (7 - 5.0 625= 1. 9375). • If the intrinsic value =0 ( strike price = stock price)at the money • If the intrinsic value <0 the option called out of money. The IBM 115 put is out of money (115-120.1/16)< 0. • If the intrinsic value > 0 the option called in the money. At any time if a call option in the money, the corresponding put must be out of money.

6. Example : Options payoffs

7. Example : Options payoffs • Suppose you buy a call 50 (contracts) January 30 call contract. • Each contract has 100 shares . You spend 50x 100 ($4) = $20 000 and wait. • In expiration date, you have the right to buy a share at $30. • Assume the stock is selling at $50 per share. • Payoff per share = $20. Per contract = $20(100)=$2000. • Total payoffs = 50 ($2000) = $100 000. • Net profit = $100 000 - $20 000(invested) = $80 000. • If you have bought the stock with $20000 instead of option. • # of shares = $20 000/30.69= 651.68 shares. We can compare the payoffs of these two scenarios:

8. Example : Options payoffs Option magnifies losses and gains S=$30,69

9. Asymmetric Gains and Loss of Call and Put options • Two products( call & put) and two strategies (buy & sell) • Define : • S1= stock price at expiration(1 Period). • S0= Stock Price today • C1= value of the call option on the expiration date (1 period) • C0= Value of the call today. Premium • E = Exercise (striking) price.

10. Asymmetric Gains and Loss of Call and Put options • 1- Buyer of a call option • The payoffs on the expiration date for the buyer of a call option is depending on E & S1 . • Gains if (S1>E + C0 ) . Max lose : premium • Ex: E = $0.585 C0= $.005 S1= $.595 • Profits = S1– (E + C0) • = $0.595 – ( $0.585 + $.005) = $0.005 • Break even : $0.590 • Limited loss & Unlimited gains. • 2- Writer (seller) of a call option • The opposite of the above • Profits = C0 –( S1 – E) = $0.005 – ( $0.595 - $.585) = -$0.005 • Limited gains & Unlimited loss.

11. Asymmetric Gains and Loss of Call and Put options • 3-Buyer of a put • Assume S1 = $. 575 E = $0.585 C0 = $.005 • Profits = E – (S1+C0) = $0.585–( $0.575 +$.005) = $0.005 • Unlimited profits & limited loss • 4- Writers of a put • 0ppossit of (3) • Profits = C0 – (E – S1)= $0.005–( $0.585 + $.575)= -$0.005. • Limited profits and unlimited loss

12. Investing with options • You have $100 000 to invest in 1 year. RF= 5%. No dividends. E= S0 =$100. C0= $10. Four different investment strategies. • Strategy 1: Invest all the money in the stock. • Buy 1000 shares. If S1 > 100 your rate of return increase by 1% for every one $ increase in stock price. Break even = 100 • Strategy 2: Invest all money in calls. • The break even is 110. Buy call ( 10 000 shares). • You can lose your entire money if S1< E. All amount a premium. • Strategy 3. Invest $10 000 in call and the rest in risk-free. • If S1> 110, you gain, same slope as strategy 1 • If S1<110, you loss is the 10 000. • Your Return = (94 500 –100 000) /100 000 = - 5.5% • Thus, options can provide min grantee rate-of return and upside slope = investing in underlying assets

13. Strategy ( 4): buy a share + a put option) Same payoff as (3). Ex. Assume E = $100 .Which strategy is the best? Depends on : S1and tolerance to risk. Investing with options

14. The following are your payoffs: • Rate of return in % • 140 strategy 2(100%Opt.) • 120 Strategy 1(100%Sh) • 100 • 80 • 60 Str. 3 (10% in Opt.) • 20 • 0 • -10 grantee in losses • -20 • -40 Aktiekurs vid lösentidpunkt • -100 50 60 70 80 90 100 110 120 130 140 S1

15. Put-Call Parity Relation • If both strategies give the same payoffs, they should have the same price ( the law of one price); • Put-Call Parity (1) • (2) • (3) • Equation 3 says that: • A call option = buying a stock, borrowing part of the money to do so, buying insurance against downside risk (the Put). • If we have the value of 3 , solve for the fourth.

16. Usefulness of Put-Call Parity • 1-Judging arbitrage possibility. • Ex: if S0 = $100, E =$100, T= 1y. RF = 8% P0=$10 • Correct Price: C0 = $100 – (100/1.08) + 10 = $17.41 • If in the market C0 = 18  arbitrage. Sell the Call & buy the equivalent replicating strategy. The payoff is

17. Usefulness of Put-Call Parity • 2- To determine the relation between Putt- and Call prices. • Rewrite Eq.(1) as: (4) • If the stock price = the PV of E , then C = P. • If the stock price > the PV of E , then C > P. • If the stock price < the PV of E , then P > C. • Ex: if S0 = $110, E =$105, T= 1y. C0 ( a call) =17 P0 = $5 , RF =5% • EQ. (4) : 17-5 = 110 – 105/(1.05) • 12 = 110-100 • 12 ≠10 There is an arbitrage possiblity