Financial Options

1. Financial Options

2. Topics in Chapter • What is a financial option? • Options Terminology • Profit and Loss Diagrams • Where and How Options Trade? • The Binomial Model • Black-Scholes Option Pricing Model

3. 1. What is a financial option? • An option is a contract which gives its holder the right, but not the obligation, to buy (or sell) an asset at some predetermined price within a specified period of time. • It does not obligate its owner to take any action. It merely gives the owner the right to buy or sell an asset.

4. 2.Option Terminology • A call option gives you the right to buy within a specified time period at a specified price • The owner of the option pays a cash premium to the option seller in exchange for the right to buy

5. 2. Option Terminology • A put option gives you the right to sell within a specified time period at a specified price • It is not necessary to own the asset before acquiring the right to sell it

6. 2. Option Terminology • Strike (or exercise) price: The price stated in the option contract at which the security can be bought or sold. • Expiration date: The last date the option can be exercised.

7. 2. Option Terminology • Exercise value: The value of a call option if it were exercised today = • Max[0, Current stock price - Strike price] • Note: The exercise value is zero if the stock price is less than the strike price. • Option price: The market price of the option contract.

8. 2. Option Terminology • The price of an option has two components: • Intrinsic value: • For a call option equals the stock price minus the striking price • For a put option equals the striking price minus the stock price • Time value equals the option premium minus the intrinsic value

9. 3. Profit and Loss Diagrams • For the Disney JUN 22.50 Call buyer: Maximum profit is unlimited Breakeven Point = $22.75 $0 -$0.25 Maximum loss $22.50

10. 3. Profit and Loss Diagrams • For the Disney JUN 22.50 Call writer: Maximum profit Breakeven Point = $22.75 $0.25 $0 Maximum loss is unlimited $22.50

11. 3. Profit and Loss Diagrams • For the Disney JUN 22.50 Put buyer: Maximum profit = $21.45 Breakeven Point = $21.45 $0 -$1.05 Maximum loss $22.50

12. 3. Profit and Loss DiagramsConsider the following data:

13. 3. Profit and Loss Diagrams Exercise Value of Option

14. 3. Profit and Loss Diagrams Market Price of Option

15. Time Value of Option

16. Call Time Value Diagram Option value 30 25 20 15 10 5 Market price Exercise value 5 10 15 20 25 30 35 40 Stock Price

17. 4. Where and How Options Trade?

18. 4. Where and How Options Trade? • Options trade on four principal exchanges: • Chicago Board Options Exchange (CBOE) • American Stock Exchange (AMEX) • Philadelphia Stock Exchange • Pacific Stock Exchange

19. 4. Where and How Options Trade? • AMEX and Philadelphia Stock Exchange options trade via the specialist system • All orders to buy or sell a particular security pass through a single individual (the specialist) • The specialist: • Keeps an order book with standing orders from investors and maintains the market in a fair and orderly fashion • Executes trades close to the current market price if no buyer or seller is available

20. 4. Where and How Options Trade? • CBOE and Pacific Stock Exchange options trade via the marketmaker system • Competing marketmakers trade in a specific location on the exchange floor near the order book official • Marketmakers compete against one another for the public’s business

21. 4. Where and How Options Trade? • Any given option has two prices at any given time: • The bid price is the highest price anyone is willing to pay for a particular option • The asked price is the lowest price at which anyone is willing to sell a particular option

22. 4. Where and How Options Trade?

23. 5. The Binomial Model

24. 5. The Binomial Model • The value of the replicating portfolio at time T, with stock price ST, is ΔST + (1+r)T*B where Δ is the # of share s and B is cash invested • At the prices ST = Sou and ST = Sod, a replicating portfolio will satisfy (Δ * Sou ) + (B * (1+r)T )= Cu (Δ * Sod ) + (B * (1+r)T )= Cd Solving for Δ and B Δ = (Cu - Cd) / (So(u –d) ) B = (Cu - (Δ * Sou) )/ (1+r)T and substitute in C = ΔSo + B to get the option price

25. 5. The Binomial Model • Stock assumptions: • Current price: So = $27 • In next 6 months, stock can either • Go up by factor of u = 1.41 • Go down by factor of 0.71 • Call option assumptions • Expires in t = 6 months = 0.5 years • Exercise price: X = $25 • Risk-free rate: 6% a year a day (r = 0.06/365 a day)

26. Ending "up" price = fu = Sou = $38.07 Option payoff: Cu = MAX[0, fu−X] = $13.07 Currentstock priceSo = $27 Ending “down" price = = fd = Sod = $19.17 Option payoff: Cd = MAX[0, fd −X] = $0.00 u = 1.41d = 0.71X = $25 5. The Binomial Model

27. Ending "up" price = fu = $38.07 Ending "up" stock value = Δ*fu = $26.33Option payoff: Cu = MAX[0,fu −X] = $13.07Portfolio's net payoff = Δ*fu - Cu = $13.26 Currentstock priceSo = $27 Ending “down" stock price = fd = $19.17 Ending “down" stock value = Δ*fd = $13.26Option payoff: Cd = MAX[0,fd −X] = $0.00Portfolio's net payoff = Δ*fd - Cd = $13.26 u = 1.41d = 0.71X = $25Δ = 0.6915 5. Riskless Portfolio’s Payoffs at Call’s Expiration: $13.26

28. 5. Create portfolio by writing 1 option and buying Δ shares of stock. • Portfolio payoffs: • Stock is up: Δ * Sou − Cu • Stock is down: Δ * Sod − Cd

29. 5. The Hedge Portfolio with a Riskless Payoff • Set payoffs for up and down equal, solve for number of shares: • Δ = (Cu− Cd) / (So(u− d)) • B = (Cu − (Δ * Sou) )/ (1+r)T = - (Portfolio's net payoff )/ (1+r)T • In our example: • Δ = ($13.07 − $0) / $27(1.41− 0.71) =0.6915 • B = - $13.26/(1 + 0.06 / 365 )365*0.5 = -$12.87

30. 5. The Hedge Portfolio with a Riskless Payoff • Option Price C =ΔSo + B =0.6915($27) + (-$12.87) = $18.67 − $12.87 = $5.80 If the call option’s price is not the same as the cost of the replicating portfolio, then there will be an opportunity for arbitrage. Option Premium C + (Commission)

31. 5. Arbitrage Example • Suppose the option sells for $6. • You can write option, receiving $6. • Create replicating portfolio for $5.80, netting $6.00 −$5.80 = $0.20. • Arbitrage: • You invested none of your own money. • You have no risk (the replicating portfolio’s payoffs exactly equal the payoffs you will owe because you wrote the option. • You have cash ($0.20) in your pocket.

32. 5. Arbitrage and Equilibrium Prices • If you could make a sure arbitrage profit, you would want to repeat it (and so would other investors). • With so many trying to write (sell) options, the extra “supply” would drive the option’s price down until it reached $5.80 and there were no more arbitrage profits available. • The opposite would occur if the option sold for less than $5.80.

33. 5. Multi-Period Binomial Pricing • If you divided time into smaller periods and allowed the stock price to go up or down each period, you would have a more reasonable outcome of possible stock prices when the option expires. • This type of problem can be solved with a binomial lattice. • As time periods get smaller, the binomial option price converges to the Black-Scholes price, which we discuss in later slides.

34. 6. Black-Scholes • Fischer Black and Myron Scholes published the derivation and equation of the Black-Scholes Option Pricing Model in 1973 under several assumptions. • Bottom line: options of traded stocks are fundamentally priced

35. 6. Black-Scholes Option Pricing Model • The stock underlying the call option provides no dividends during the call option’s life. • There are no transactions costs for the sale/purchase of either the stock or the option. • Risk-free rate, rRF, is known and constant during the option’s life. (More...)

36. 6. Black-Scholes Option Pricing Model • Security buyers may borrow any fraction of the purchase price at the short-term risk-free rate. • No penalty for short selling and sellers receive immediately full cash proceeds at today’s price. • Call option can be exercised only on its expiration date. • Security trading takes place in continuous time, and stock prices move randomly in continuous time.

37. 6. Black Scholes Equation • Integration of statistical and mathematical models • For example in the standard Black-Scholes model, the stock price evolves as • dS = μ(t)Sdt + σ(t)SdWt. • where μ is the drift parameter and σ is the implied volatility • To sample a path following this distribution from time 0 to T, we divide the time interval • into M units of length δt, and approximate the Brownian motion over the interval dt • by a single normal variable of mean 0 and variance δt. • The price f of any derivative (or option) of the stock S is a solution of the • following partial-differential equation:

38. 6. Black-Scholes Option Pricing Model • The Black-Scholes OPM:

39. 6. Black-Scholes Option Pricing Model • Variable definitions: • C = theoretical call premium • S = current stock price • t = time in years until option expiration • K = option striking price • R = risk-free interest rate

40. 6. Black-Scholes Option Pricing Model • Variable definitions (cont’d): • = standard deviation of stock returns • N(x) = probability that a value less than “x” will occur in a standard normal distribution • ln = natural logarithm • e = base of natural logarithm (2.7183)

41. 6. Black-Scholes Option Pricing Model Example Stock ABC currently trades for $30. A call option on ABC stock has a striking price of $25 and expires in three months. The current risk-free rate is 5%, and ABC stock has a standard deviation of 0.45. According to the Black-Scholes OPM, but should be the call option premium for this option?

42. 6. Black-Scholes Option Pricing Model Example (cont’d) Solution: We must first determine d1 and d2:

43. 6. Black-Scholes Option Pricing Model Example (cont’d) Solution (cont’d):

44. 6. What is the value of the following call option according to the OPM? • Assume: • N = P = $27 • X = $25 • rRF = 6% • t = 0.5 years • σ = 0.49

45. 6. First, find d1 and d2. d1 = {ln($27/$25) + [(0.06 + 0.492/2)](0.5)} ÷ {(0.49)(0.7071)} d1 = 0.4819 d2 = 0.4819 - (0.49)(0.7071) d2 = 0.1355

46. 6. Second, find N(d1) and N(d2) • N(d1) = N(0.4819) = 0.6851 • N(d2) = N(0.1355) = 0.5539 • Note: Values obtained from Excel using NORMSDIST function. For example: • N(d1) = NORMSDIST(0.4819)

47. 6. Third, find value of option. VC = $27(0.6851) - $25e-(0.06)(0.5)(0.5539) = $19.3536 - $25(0.97045)(0.6327) = $5.06

48. What impact do the following parameters have on a call option’s value? • Current stock price: Call option value increases as the current stock price increases. • Strike price: As the exercise price increases, a call option’s value decreases.

49. 6. Impact on Call Value • Option period: As the expiration date is lengthened, a call option’s value increases (more chance of becoming in the money.) • Risk-free rate: Call option’s value tends to increase as rRF increases (reduces the PV of the exercise price). • Stock return variance: Option value increases with variance of the underlying stock (more chance of becoming in the money).

50. 6. Negative Affects • Myron Scholes was a partner of Long-Term Capital Management (LTCM) established in 1994. • LTCM would use models and experienced investing knowledge to pick differences in short and long securities, profiting from the small margin. • In 1998 the firm had $5B in equity and $125B in liability • August 17, 1998 Russia devalues the rouble and suspended $13.5B of it’s Treasury Bonds, causing LTCM’s equity to decrease drastically, ever increasing it’s leverage ratio • In order to prevent a stock market crash due to the outflux of investors and banks from LTCM’s funds the Federal Reserve organized a $3.5 B rescue effort to take over LTCM’s management • Had LTCM lost leverage, all banks and creditors would pull out of other firms – including Salomon Brothers, Merrill Lynch.