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Options. Chapter 28. 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

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  1. Options Chapter 28 Chapter 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 Chapter 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 Chapter 28: Options

  4. One-Period Binomial Call Pricing Formula • Intrinsic ValueCall = 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 • COPUp = MAX[0,$50 - $40] = $10 • The stock price was $40.91 upon expiration the option would be valued at • COPDown = MAX[0, $40.91 - $40] = $0.91 Chapter 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 • VUp = Value of stock – (1+ risk-free)(amount borrowed) • If the stock price fell the ending value of the portfolio would be • VDown = Value of stock – (1+ risk-free)(amount borrowed) Chapter 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 COPUp and COPDown to ValueUp and ValueDown or • MAX [0, Price0Up – exercise price] = COPUp = ValueUp = Up value of stock – (1+risk-free rate × amount borrowed) • MAX [0, Price0Down – exercise price] = COPDown = ValueDown = Down value of stock – (1+risk-free rate × amount borrowed) • Equations can be solved simultaneously to determine the hedge ratio Chapter 28: Options

  7. One-Period Binomial Call Pricing Formula • The hedge ratio represents the number of shares of stock costing P0 financed by borrowing B* dollars • This will duplicate the expiration payoffs from a call option Chapter 28: Options

  8. One-Period Binomial Call Pricing Formula • The initial price (COP0) of the call option is • Hedge ratio × P0 – B* = COP0 • Observations • P0 is a major determinant of a call option’s initial price • The probability of the price fluctuations do not impact COP0 • Model is risk-neutral • Call has the same value whether investor is risk-averse, risk-neutral or risk-seeking Chapter 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 Chapter 28: Options

  10. Multi-Period Binomial Call Pricing Formula Chapter 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 Chapter 28: Options

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

  13. Multi-Period Binomial Call Pricing Formula • Pascal’s triangle in tree form Chapter 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 Chapter 28: Options

  15. [ ] ( ) + + ln RFR 0.5VAR(r) d XP P = 0 x 0.5 σ d B&S Call Valuation Formula • Use a self-financing portfolio • COP0 = (P0h – B) • Assume a hedge ratio of N(x) • Borrowings equal XP[e(-RFR)d]N(y) • B&S equation • COP0 = P0 N(x)- XP[e(-RFR)d]N(y) • where Fraction of year until call expires Values of x and y have no intuitive meaning. Chapter 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 Chapter 28: Options

  17. Example • Given the following information, calculate the value of the call • P0 = $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% Chapter 28: Options

  18. Example • Substituting the values for N(x) and N(y) into the COP0 equation • COP0 = $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. Chapter 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 Chapter 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 Chapter 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: Values are the same whether the stock is in or out of the money when options expire—thus portfolio is perfectly hedged. Chapter 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 Chapter 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 Chapter 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 Chapter 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 + 60/1.060.0833 – 57 • 4.125  4.3344 • Either puts were under priced by 21¢ or calls were over priced by 21¢ • Ignores transaction costs Chapter 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 Chapter 28: Options

  27. The Effects of Cash Dividend Payments • Impacts the value of an American call option 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). On the ex-dividend date the stock price drops from Pd to Pe. Option prices usually do not drop by the same amount because the slope of the price curve < +1. Chapter 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 • COPe = [P0 – Div/(1+RFR)]N(x) – XP[e(-RFR)d]N(y) • Example • P0 = $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 • COPe = [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. Chapter 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 Chapter 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 Chapter 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 Chapter 28: Options

  32. Example • Contrasting the actual and synthetic long positions 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). Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 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 Chapter 28: Options

  43. Strangles • Involves a put and call with same expiration date but different strike prices • Involves smaller total outlay than a straddle Chapter 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 Chapter 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 Chapter 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 Chapter 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 XPB • Short butterfly spread • Profitable if optioned asset experiences large up and/or down price fluctuations Chapter 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 Chapter 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 Chapter 28: Options

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