1. Slides by: Pamela L. Hall, Western Washington University Francis & IbbotsonChapter 28: Options1 Options Chapter 28

2. Francis & IbbotsonChapter 28: Options 2 Background  Put and call prices are affected by –Price of underlying asset –Option’s exercise price –Length of time until expiration of option –Volatility of underlying asset –Risk-free interest rate –Cash flows such as dividends  Premiums can be derived from the above factors  Investors’ expectations about the direction of the underlying asset’s price change does not impact the value of an option

3. Francis & IbbotsonChapter 28: Options 3 Introduction to Binomial Option Pricing  A simple valuation model is used to determine the price for a call option –Assumes only two possible rates of return over time period The price could either rise or fall –For instance, if a stock’s price is currently $45.45 and it can change by either ±10% over the next period, the possible prices are »$45.45 x 1.10 = $50 »$45.45 x 0.90 = $40.91 Ignores taxes, commissions and margin requirements Assumes investor can gain immediate use of short sale funds Assume no cash flows are paid

4. Francis & IbbotsonChapter 28: Options 4 One-Period Binomial Call Pricing Formula  Intrinsic Value Call = MAX[0, {Stock Price – Exercise Price}] –If the an option has an exercise price of $40 and The stock price was $50 upon expiration the option would be valued at –COP Up = MAX[0,$50 - $40] = $10 The stock price was $40.91 upon expiration the option would be valued at –COP Down = MAX[0, $40.91 - $40] = $0.91

5. Francis & IbbotsonChapter 28: Options 5 One-Period Binomial Call Pricing Formula  If we borrowed the money needed to purchase the optioned security at the risk- free rate –We would not need to invest any money to get started (AKA: self-financing portfolio) If the stock price rose the ending value of the portfolio would be –V Up = Value of stock – (1+ risk-free)(amount borrowed) If the stock price fell the ending value of the portfolio would be –V Down = Value of stock – (1+ risk-free)(amount borrowed)

6. Francis & IbbotsonChapter 28: Options 6 One-Period Binomial Call Pricing Formula  To find the option’s price you must find the values for the amount of stock and borrowed funds that will equate COP Up and COP Down to Value Up and Value Down or –MAX [0, Price 0Up – exercise price] = COP Up = Value Up = Up value of stock – (1+risk-free rate × amount borrowed) –MAX [0, Price 0Down – exercise price] = COP Down = Value Down = Down value of stock – (1+risk- free rate × amount borrowed) Equations can be solved simultaneously to determine the hedge ratio

7. Francis & IbbotsonChapter 28: Options 7 One-Period Binomial Call Pricing Formula  The hedge ratio represents the number of shares of stock costing P 0 financed by borrowing B* dollars –This will duplicate the expiration payoffs from a call option

8. Francis & IbbotsonChapter 28: Options 8 One-Period Binomial Call Pricing Formula  The initial price (COP 0 ) of the call option is –Hedge ratio × P 0 – B* = COP 0  Observations –P 0 is a major determinant of a call option’s initial price –The probability of the price fluctuations do not impact COP 0 –Model is risk-neutral Call has the same value whether investor is risk-averse, risk-neutral or risk-seeking

9. Francis & IbbotsonChapter 28: Options 9 Multi-Period Binomial Call Pricing Formula  One-period model can be used to –Encompass multiple time periods –Value common stock, bonds, mortgages  What if stock currently priced at $45.45 could either rise or fall in value by 10% over each of the next two time periods

10. Francis & IbbotsonChapter 28: Options 10 Multi-Period Binomial Call Pricing Formula

11. Francis & IbbotsonChapter 28: Options 11 Multi-Period Binomial Call Pricing Formula  This concept can be extended to any number of time periods  Can add cash flow payments to the branches  When a large number of small time periods are involved, we obtain Pascal’s triangle

12. Francis & IbbotsonChapter 28: Options 12 Multi-Period Binomial Call Pricing Formula Resembles a normal probability distribution as n increases.

13. Francis & IbbotsonChapter 28: Options 13 Multi-Period Binomial Call Pricing Formula  Pascal’s triangle in tree form

14. Francis & IbbotsonChapter 28: Options 14 Black and Scholes Call Option Pricing Model  Black & Scholes (B&S) developed a formula to price call options –Assume normally distributed rates of return

15. Francis & IbbotsonChapter 28: Options 15 B&S Call Valuation Formula  Use a self-financing portfolio –COP 0 = (P 0 h – B)  Assume a hedge ratio of N(x)  Borrowings equal XP[e (-RFR)d ]N(y)  B&S equation –COP 0 = P 0 N(x)- XP[e (-RFR)d ]N(y) where () [] d σ d0.5VAR(r)RFR  XP P ln x 0.5 0 ++ = Fraction of year until call expires Values of x and y have no intuitive meaning.

16. Francis & IbbotsonChapter 28: Options 16 B&S Call Valuation Formula  N(x) is a cumulative normal-density function of x –Gives the probability that a value less than x will occur in a normal probability distribution  To use the B&S model you need –Table of natural logarithms (or a calculator) –Table of cumulative normal distribution probabilities

17. Francis & IbbotsonChapter 28: Options 17 Example  Given the following information, calculate the value of the call –P 0 = $60 –XP (strike or exercise price) = $50 –d (time to expiration) = 4 months or 1/3 of a year –Risk-free rate = 7% –Variance (returns) = 14.4%

18. Francis & IbbotsonChapter 28: Options 18 Example  Substituting the values for N(x) and N(y) into the COP 0 equation –COP 0 = $60(0.853) - $50(0.977)(0.796) = $51.18 - $38.89 = $12.29 Looking this value up in the table yields an N(x) of 0.853. Looking this value up in the table yields an N(y) of 0.796.

19. Francis & IbbotsonChapter 28: Options 19 The Hedge Ratio  Represents the fraction of a change in an option’s premium caused by a $1 change in the price of the underlying asset –AKA delta, neutral hedge ratio, elasticity, equivalence ratio –Calls have a hedge ratio between 0 and 1  Hedgers would like a hedge ratio that will completely eliminate changes in their hedged portfolio –Is presented as N(x) in the B&S equation If x has a value of 1.65, N(x) has a value of 0.9505 –Means that 95.05 shares of a stock should be sold short to establish a perfect hedge against 100 shares in an offsetting position

20. Francis & IbbotsonChapter 28: Options 20 Risk Statistics and Option Values  Investor normally estimates an asset’s standard deviation of returns and uses it as an input into the B&S model –However, can insert the call’s current price into the model and compute the implied volatility of the underlying asset Risk statistics change over time

21. Francis & IbbotsonChapter 28: Options 21 Put-Call Parity Formula  Formula represents an arbitrage-free relationship between put and call prices on the same underlying asset –If the two options have identical strike prices and times to maturity  Consider the following: Portfolio of 3 positions in same stock Position Value When Options Expire If P < XPIf P > XP 1) Long position on underlying stockPP 2) Short position in call option (sell call)0XP – P 3) Long position in put option (buy put)XP – P0 Portfolio’s two total valuesXP Values are the same whether the stock is in or out of the money when options expire—thus portfolio is perfectly hedged.

22. Francis & IbbotsonChapter 28: Options 22 Put-Call Parity Formula  This portfolio is worth the present value of the option’s exercise price or –XP  (1+RFR) d under either outcome  The portfolio must also be worth –P + POP – COP  This leads to the Put-Call Parity equation –P + POP – COP = XP  (1+RFR) d

23. Francis & IbbotsonChapter 28: Options 23 Pricing Put Options  We can use put-call parity to value a put after the value of a call on the same security has been determined –POP= COP + (XP  (1+RFR) d ) – P Example: Calculate the price of a put option on a stock with a current price of $60, a strike price of $50, 4 months remaining until expiration, a risk-free rate of 7% and a variance of 14.4% with a call valued at $12.29 –POP = $12.29 + (50  (1.07) 0.333 )-$60 = $1.18

24. Francis & IbbotsonChapter 28: Options 24 Checking Alignment of Put and Call Prices  When prices for both puts and calls on the same underlying stock are available –Put-call parity can be used to determine if the prices are properly aligned If not, arbitrage profits can be earned

25. Francis & IbbotsonChapter 28: Options 25 Example  Given information –On 7/12/2000 KO’s stock was selling for $57 –Call options with a strike price of $60 and one month until expiration were selling for $1.625 –Puts were selling for $4.125 –3-month T-bills were yielding 6%  Plugging data into the put-call parity equation –4.125  1.625 + 60/1.06 0.0833 – 57 –4.125  4.3344 Either puts were under priced by 21¢ or calls were over priced by 21¢  Ignores transaction costs

26. Francis & IbbotsonChapter 28: Options 26 The Effects of Cash Dividend Payments  Ex-dividend date –First trading day after the cash dividend is paid –Stock trades at a reduced price Reduced by the amount of the cash dividend –Stockholders are no longer entitled to the dividend, therefore they should not pay for it  The ex-dividend stock price drop-off –Reduces value of call options –Increases value of put options

27. Francis & IbbotsonChapter 28: Options 27 The Effects of Cash Dividend Payments  Impacts the value of an American call option On the ex-dividend date the stock price drops from P d to P e. Option prices usually do not drop by the same amount because the slope of the price curve < +1. Price curve reflects the option’s price if it is not exercised and not expired (alive). If the option’s live value before ex- dividend > value ex dividend by more than dividend, call should be exercised before it trades ex- dividend to capture cash dividend (while embedded in stock’s price).

28. Francis & IbbotsonChapter 28: Options 28 The Effects of Cash Dividend Payments  The present value of the cash dividend payment should be considered in the B&S option pricing model –COP e = [P 0 – Div/(1+RFR)]N(x) – XP[e (-RFR)d ]N(y) –Example P 0 = $60 XP (strike or exercise price) = $50 d (time to expiration) = 4 months or 1/3 of a year Risk-free rate = 7% Variance (returns) = 14.4% Expected cash dividend of $2 in one year –Present value of dividend = $2/1.07 = $1.869 –COP e = [60 – 1.869]0.853 –50[0.977]0.796 = $10.69 The addition of the cash dividend has lowered the call value by $1.60.

29. Francis & IbbotsonChapter 28: Options 29 Options Markets  Chicago Board Options Exchange (CBOE) –Founded in 1973 but is now the largest options exchange in world  American Stock Exchange –Second largest options exchange  Many options transactions are cleared through –Options Clearing Corporation (OCC)  International Securities Exchange (ISE) –Opened in 2000 –Electronic exchange Competes with CBOE, AMEX, PHIX, PSE

30. Francis & IbbotsonChapter 28: Options 30 Synthetic Positions Can Be Created From Options  Buying a call and selling a put on the same security –Creates the same position as a buy-and- hold position in the security AKA synthetic long position

31. Francis & IbbotsonChapter 28: Options 31 Example  Given information –Phelps’ stock is currently trading for $40 a share You buy a six-month call with a $40 exercise price for a $5 cost You write an 8-month put with an exercise price of $40 for $5 in premium income

32. Francis & IbbotsonChapter 28: Options 32 Example  Contrasting the actual and synthetic long positions Possible Price of stock at option expiration Results from 6-month call with $50 exercise price Results from $5 put written with $50 exercise price Result from combined option positions Result from long position in stock (100 shares) $30-500 -1,000 $35-5000 $40-500+50000 $450+500 $50+500 +1,000 $55-1,000+500+1,500 If the call and put prices , the synthetic position  the actual position. Put-call parity shows that the price of a put must be < the price of a similar call. Thus, to make the put price = call price, put had to have a longer time to expiration (8 months vs. 6 months).

33. Francis & IbbotsonChapter 28: Options 33 Synthetic Positions Can Be Created From Options  Some investors prefer a synthetic long position to an actual long position –Requires smaller initial investment Creates more financial leverage –Owner of a synthetic long position does not collect cash dividends or coupon interest from underlying securities as they do not actually own those securities –Also, when options expire additional premiums must be paid to re-establish position

34. Francis & IbbotsonChapter 28: Options 34 Synthetic Short Position  Can create a synthetic short position by –Selling (writing) a call and simultaneously buying a put with a similar exercise price on the same underlying stock  Superior to an actual short position in the stock –The premium income from selling the call should be > premium paid to buy the put –Requires a smaller initial investment than an actual short sell –Does not have to pay cash dividends on the optioned stock  Disadvantages of a synthetic short position –After expiration of option more money would have to be spent to re-establish position –Could accumulate unlimited losses if the stock price rose high enough

35. Francis & IbbotsonChapter 28: Options 35 Writing Covered Calls  Covered call –Writing a call option against securities you already own Cover the writer’s exposure to potential loss  If call owner exercises the option –Option-writer delivers the already owned securities without having to buy them in the market  Not all covered call positions are profitable –If stock price falls Long position in underlying stock decreases However, receive call premium income

36. Francis & IbbotsonChapter 28: Options 36 Writing Covered Calls  Naked call writing –Occurs when call writer does not own the underlying security Risky if the price of the underlying security increases  Initial margin of 15% or more required –Whereas a covered option writer does not have to put up extra margin to write a covered call

37. Francis & IbbotsonChapter 28: Options 37 Writing Covered Calls  Covered call writers –Gain the most when stock price remains at exercise price and option expired unexercised Receive premium income and get to keep the stock –If stock price increases significantly would have been better off not having written the option Will have to give security to exerciser

38. Francis & IbbotsonChapter 28: Options 38 Straddles  Straddle occurs when –Equal number of puts and calls are bought on the same underlying asset Must have same maturity and strike price  Long straddle position –Profit if optioned asset either Experiences a large increase in price Experiences a large decrease in price Experiences large increases and decreases in price  Useful for a stock experiencing great deal of volatility

39. Francis & IbbotsonChapter 28: Options 39 Long Straddle Position  Infinite number of break-even points for a long straddle position –Downside limit Sum of put and call prices –Upside limit Sum of put and call prices  Believe the underlying stock has potential for enough price movements to make the straddle profitable before expiration –Only a small probability of losing the aggregate premium outlay

40. Francis & IbbotsonChapter 28: Options 40 Short Straddle Position  Symmetrically opposite to long straddle position  Believe stock price will not vary significantly before options expire –Probability that straddle will keep 100% of premium income is small

41. Francis & IbbotsonChapter 28: Options 41 Spreads  The purchase of one option and sale of a similar but different option –Can be either puts or calls but not puts and calls –Spread can occur based on Different strike prices (vertical spreads) Different expirations (horizontal spreads) Time spreads, calendar spreads

42. Francis & IbbotsonChapter 28: Options 42 Spreads  Diagonal spreads combine vertical and horizontal spreads  Credit spreads –Generate premium income exceeding related costs  Debit spreads –Generate an initial cash outflow

43. Francis & IbbotsonChapter 28: Options 43 Strangles  Involves a put and call with same expiration date but different strike prices –Involves smaller total outlay than a straddle Price of underlying asset Payoff from put Payoff from call Strangle’s total payoff XP p  P XP p – P0 XP C > P > XP p 000 P > XP C 0P – XP C

44. Francis & IbbotsonChapter 28: Options 44 Strangles  Long strangle –Debit transaction No premiums from writing options are received  Short strangle –Credit transaction No outlays Small premiums received but also small chance options will be exercised against writer

45. Francis & IbbotsonChapter 28: Options 45 Bull Spread  Vertical spread involving two calls with same expiration date –Debit transaction –Used if believe price of underlying asset will rise, but not significantly

46. Francis & IbbotsonChapter 28: Options 46 Bear Spread  Vertical spread involving two puts with same expiration date but different strike prices –Are profitable only if asset price declines between the two exercise prices Losses are limited if expectations are incorrect

47. Francis & IbbotsonChapter 28: Options 47 Butterfly Spreads  Combination of a bull and bear spread on the same underlying security –Long butterfly spread Will maximize profit if underlying asset’s price does not fluctuate from XP B –Short butterfly spread Profitable if optioned asset experiences large up and/or down price fluctuations

48. Francis & IbbotsonChapter 28: Options 48 The Bottom Line  Binomial option pricing model –Mathematically simple  B&S Option Pricing Model –First closed-form option pricing model Binomial option pricing model is equivalent to B&S if there are an infinite number of tiny time periods  Put prices can be determined using put- call parity formula

49. Francis & IbbotsonChapter 28: Options 49 The Bottom Line  Ex-dividend stock price drop-off decreases (increases) value of a call (put) option  Puts and calls can be assembled to build more complex investing positions –Can build a position that will allow investor to benefit if price of underlying asset Rises Falls Fluctuates up and down Never changes  Options allow us to analyze securities in ways we might not have originally realized