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Chapter 17 Principles of Options and Option Pricing Prof. Rushen Chahal Prof. Rushen Chahal 1 We sent the first draft of our paper to the Journal of Political Economy and promptly got back a rejection letter. We then sent it to the Review of Economics and Statistics, where it was also rejected. Merton Miller and Eugene Fama…then took an interest in the paper and gave us extensive comments on it. They suggested to the JPE that perhaps the paper was worth more serious consideration. The journal then accepted the paper. - Fischer Black Prof. Rushen Chahal 2 Outline • Introduction • Option principles • Option pricing Prof. Rushen Chahal 3 Introduction • Innovations in stock options have been among the most important developments in finance in the last 20 years • The cornerstone of option pricing is the Black- Scholes Option Pricing Model (OPM) – Delta is the most important OPM progeny to the portfolio manager Prof. Rushen Chahal 4 Option Principles • Why options are a good idea • What options are • Standardized option characteristics • Where options come from • Where and how options trade • The option premium • Sources of profits and losses with options Prof. Rushen Chahal 5 Why Options Are A Good Idea • Options: – Give the marketplace opportunities to adjust risk or alter income streams that would otherwise not be available – Provide financial leverage – Can be used to generate additional income from investment portfolios Prof. Rushen Chahal 6 Why Options Are A Good Idea (cont’d) • The investment process is dynamic: – The portfolio managers needs to constantly reassess and adjust portfolios with the arrival of new information • Options are more convenient and less expensive than wholesale purchases or sales of stock Prof. Rushen Chahal 7 What Options Are • Call options • Put options Prof. Rushen Chahal 8 Call Options • 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 Prof. Rushen Chahal 9 Practical Example of A Call Option Prof. Rushen Chahal 10 Put Options • 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 Prof. Rushen Chahal 11 Standardized Option Characteristics • All exchange-traded options have standardized expiration dates – The Saturday following the third Friday of designated months for most options – Investors typically view the third Friday of the month as the expiration date Prof. Rushen Chahal 12 Standardized Option Characteristics (cont’d) • The striking price of an option is the predetermined transaction price – In multiples of $2.50 (for stocks priced $25.00 or below) or $5.00 (for stocks priced higher than $25.00) – There is usually at least one striking price above and one below the current stock price Prof. Rushen Chahal 13 Standardized Option Characteristics (cont’d) • Puts and calls are based on 100 shares of the underlying security – The underlying security is the security that the option gives you the right to buy or sell – It is not possible to buy or sell odd lots of options Prof. Rushen Chahal 14 Where Options Come From • Introduction • Opening and closing transactions • Role of the Options Clearing Corporation Prof. Rushen Chahal 15 Introduction • If you buy an option, someone has to sell it to you • No set number of put or call options exists – The number of options in existence changes every day – Option can be created and destroyed Prof. Rushen Chahal 16 Opening and Closing Transactions • The first trade someone makes in a particular option is an opening transaction – An opening transaction that is the sale of an option is called writing an option Prof. Rushen Chahal 17 Opening and Closing Transactions (cont’d) • The trade that terminates a position by closing it out is a closing transaction – Options have fungibility • Market participants can reverse their positions by making offsetting trades • E.g., the writer of an option can close out the position by buying a similar one Prof. Rushen Chahal 18 Opening and Closing Transactions (cont’d) • The owner of an option will ultimately: – Sell it to someone else – Let it expire or – Exercise it Prof. Rushen Chahal 19 Role of the Options Clearing Corporation • The Options Clearing Corporation (OCC): – Positions itself between every buyer and seller – Acts as a guarantor of all option trades – Regulates the trading activity of members of the various options exchanges – Sets minimum capital requirements – Provides for the efficient transfer of funds among members as gains or losses occur Prof. Rushen Chahal 20 OCC-Related Information on the Web Prof. Rushen Chahal 21 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 Prof. Rushen Chahal 22 Where and How Options Trade (cont’d) • 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 Prof. Rushen Chahal 23 Where and How Options Trade (cont’d) • 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 Prof. Rushen Chahal 24 Where and How Options Trade (cont’d) • 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 Prof. Rushen Chahal 25 The Option Premium • Intrinsic value and time value • The financial page listing Prof. Rushen Chahal 26 Intrinsic Value and Time Value • 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 Prof. Rushen Chahal 27 Intrinsic Value and Time Value (cont’d) • An option with no intrinsic value is out of the money • An option with intrinsic value is in the money • If an option’s striking price equals the stock price, the option is at the money Prof. Rushen Chahal 28 The Financial Page Listing • The following slide shows an example from the online edition of the Wall Street Journal: – The current price for a share of Disney stock is $21.95 – Striking prices from $20 to $25 are available – The expiration month is in the second column – The option premiums are provided in the “Last” column Prof. Rushen Chahal 29 The Financial Page Listing Prof. Rushen Chahal 30 The Financial Page Listing (cont’d) • Investors identify an option by company, expiration, striking price, and type of option: Disney JUN 22.50 Call Company Expiration Striking Type Price Prof. Rushen Chahal 31 The Financial Page Listing (cont’d) • The Disney JUN 22.50 Call is out of the money – The striking price is greater than the stock price – The time value is $0.25 • The Disney JUN 22.50 Put is in the money – The striking price is greater than the stock price – The intrinsic value is $22.50 - $21.95 = $0.55 – The time value is $1.05 - $0.55 = $0.50 Prof. Rushen Chahal 32 The Financial Page Listing (cont’d) • As an option moves closer to expiration, its time value decreases – Time value decay • An option is a wasting asset – Everything else being equal, the value of an option declines over time Prof. Rushen Chahal 33 Sources of Profits and Losses With Options • Option exercise • Exercise procedures Prof. Rushen Chahal 34 Option Exercise • An American option can be exercised at any time prior to option expiration – It is typically not advantageous to exercise prior to expiration since this amount to foregoing time value • European options can be exercised only at expiration Prof. Rushen Chahal 35 Exercise Procedures • The owner of an option who decides to exercise the option: – Calls her broker – Must put up the full contract amount for the option • The premium is not a downpayment on the option terms Prof. Rushen Chahal 36 Exercise Procedures (cont’d) • The option writer: – Must be prepared to sell the necessary shares to the call option owner – Must be prepared to buy shares of stock from the put option owner Prof. Rushen Chahal 37 Exercise Procedures (cont’d) • In general, you should not buy an option with the intent of exercising it: – Requires two commissions – Selling the option captures the full value contained in an option Prof. Rushen Chahal 38 Profit and Loss Diagrams • For the Disney JUN 22.50 Call buyer: Maximum profit Breakeven Point = $22.75 is unlimited $0 -$0.25 Maximum loss $22.50 Prof. Rushen Chahal 39 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 Prof. Rushen Chahal 40 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 Prof. Rushen Chahal 41 Profit and Loss Diagrams • For the Disney JUN 22.50 Put writer: Maximum profit Breakeven Point = $21.45 $1.05 $0 Maximum loss = $21.45 $22.50 Prof. Rushen Chahal 42 Option Pricing • Determinants of the option premium • Black-Scholes Option Pricing Model • Development and Assumptions of the model • Insights into the Black-Scholes Model • Delta • Theory of put/call parity • Stock index options Prof. Rushen Chahal 43 Determinants of the Option Premium • Market factors • Accounting factors Prof. Rushen Chahal 44 Market Factors • Striking price – For a call option, the lower the striking price, the higher the option premium • Time to expiration – For both calls and puts, the longer the time to expiration, the higher the option premium Prof. Rushen Chahal 45 Market Factors (cont’d) • Current stock price – The higher the stock price, the higher the call option premium and the lower the put option premium • Volatility of the underlying stock – The great the volatility, the higher the call and put option premium Prof. Rushen Chahal 46 Market Factors (cont’d) • Dividend yield on the underlying stock – Companies with high dividend yields have a smaller call option premium than companies with low dividend yields • Risk-free interest rate – The higher the risk-free rate, the higher the call option premium Prof. Rushen Chahal 47 Accounting Factors • Stock splits: – The OCC will make the following adjustments: • The striking price is reduced by the split ratio • The number of options is increased by the split ratio – For odd-lot generating splits: • The striking price is reduced by the split ratio • The number of shares covered by your options is increased by the split ratio Prof. Rushen Chahal 48 Black-Scholes Option Pricing Model • The Black-Scholes OPM: C S N (d1 ) Ke rt N (d 2 ) ln( S / K ) R ( 2 / 2) t d1 t d 2 d1 t Prof. Rushen Chahal 49 Black-Scholes Option Pricing Model (cont’d) • 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 Prof. Rushen Chahal 50 Black-Scholes Option Pricing Model (cont’d) • 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) Prof. Rushen Chahal 51 Black-Scholes Option Pricing Model (cont’d) 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? Prof. Rushen Chahal 52 Black-Scholes Option Pricing Model (cont’d) Example (cont’d) Solution: We must first determine d1 and d2: ln( S / K ) R ( 2 / 2) t d1 t ln(30 / 25) 0.05 (0.452 / 2) 0.25 0.45 0.25 0.1823 0.0378 0.978 0.225 Prof. Rushen Chahal 53 Black-Scholes Option Pricing Model (cont’d) Example (cont’d) Solution (cont’d): d 2 d1 t 0.978 (0.45) 0.25 0.978 0.225 0.753 Prof. Rushen Chahal 54 Black-Scholes Option Pricing Model (cont’d) Example (cont’d) Solution (cont’d): The next step is to find the normal probability values for d1 and d2. Using Excel’s NORMSDIST function yields: N (d1 ) 0.836 N (d2 ) 0.774 Prof. Rushen Chahal 55 Black-Scholes Option Pricing Model (cont’d) Example (cont’d) Solution (cont’d): The final step is to calculate the option premium: C S N (d1 ) Ke rt N (d 2 ) $30 0.836 $25e (0.05)(0.25) 0.774 $25.08 $19.11 $5.97 Prof. Rushen Chahal 56 Using Excel’s NORMSDIST Function • The Excel portion below shows the input and the result of the function: Prof. Rushen Chahal 57 Development and Assumptions of the Model • Introduction • The stock pays no dividends during the option’s life • European exercise terms • Markets are efficient • No commissions • Constant interest rates • Lognormal returns Prof. Rushen Chahal 58 Introduction • Many of the steps used in building the Black- Scholes OPM come from: – Physics – Mathematical shortcuts – Arbitrage arguments • The actual development of the OPM is complicated Prof. Rushen Chahal 59 The Stock Pays no Dividends During the Option’s Life • The OPM assumes that the underlying security pays no dividends • Valuing securities with different dividend yields using the OPM will result in the same price Prof. Rushen Chahal 60 The Stock Pays no Dividends During the Option’s Life • The OPM can be adjusted for dividends: – Discount the future dividend assuming continuous compounding – Subtract the present value of the dividend from the stock price in the OPM – Compute the premium using the OPM with the adjusted stock price Prof. Rushen Chahal 61 European Exercise Terms • The OPM assumes that the option is European • Not a major consideration since very few calls are ever exercised prior to expiration Prof. Rushen Chahal 62 Markets Are Efficient • The OPM assumes markets are informationally efficient – People cannot predict the direction of the market or of an individual stock Prof. Rushen Chahal 63 No Commissions • The OPM assumes market participants do not have to pay any commissions to buy or sell • Commissions paid by individual can significantly affect the true cost of an option – Trading fee differentials cause slightly different effective option prices for different market participants Prof. Rushen Chahal 64 Constant Interest Rates • The OPM assumes that the interest rate R in the model is known and constant • It is common use to use the discount rate on a U.S. Treasury bill that has a maturity approximately equal to the remaining life of the option – This interest rate can change Prof. Rushen Chahal 65 Lognormal Returns • The OPM assumes that the logarithms of returns of the underlying security are normally distributed • A reasonable assumption for most assets on which options are available Prof. Rushen Chahal 66 Insights Into the Black-Scholes Model • Divide the OPM into two parts: C S N (d1 ) Kert N (d2 ) Part A Part B Prof. Rushen Chahal 67 Insights Into the Black-Scholes Model (cont’d) • Part A is the expected benefit from acquiring the stock: – S is the current stock price and the discounted value of the expected stock price at any future point – N(d1) is a pseudo-probability • It is the probability of the option being in the money at expiration, adjusted for the depth the option is in the money Prof. Rushen Chahal 68 Insights Into the Black-Scholes Model (cont’d) • Part B is the present value of the exercise price on the expiration day: – N(d2) is the actual probability the option will be in the money on expiration day Prof. Rushen Chahal 69 Insights Into the Black-Scholes Model (cont’d) • The value of a call option is the difference between the expected benefit from acquiring the stock and paying the exercise price on expiration day Prof. Rushen Chahal 70 Delta • Delta is the change in option premium expected from a small change in the stock price, all other things being equal: C S C where the first partial derivative of the call premium S with respect to the stock price Prof. Rushen Chahal 71 Delta (cont’d) • Delta allows us to determine how many options are needed to mimic the returns of the underlying stock • Delta is exactly equal to N(d1) – E.g., if N(d1) is 0.836, a $1 change in the price of the underlying stock price leads to a change in the option premium of 84 cents Prof. Rushen Chahal 72 Theory of Put/Call Parity • The following variables form an interrelated securities complex: – Price of a put – Price of a call – The value of the underlying stock – The riskless rate of interest • If put/call parity does not hold, arbitrage is possible Prof. Rushen Chahal 73 Theory of Put/Call Parity (cont’d) • The put/call parity relationship: K CPS (1 R )T where C price of a call P price of a put K option striking price R risk-free interest rate T time until expiration in years Prof. Rushen Chahal 74 Stock Index Options • A stock index option is the option exchanges most successful innovation – E.g., the S&P 100 index option • Index options have no delivery mechanism – All settlements are in cash Prof. Rushen Chahal 75 Stock Index Options (cont’d) • The owner of an in-the-money index call receives the difference between the closing index level and the striking price • The owner of an in-the-money index put receives the difference between the striking price and the index level Prof. Rushen Chahal 76

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