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16 Option Valuation 1 What is an Option Worth? • At expiration, an option is worth its intrinsic value. • Before expiration, put-call parity allows us to price options. But, • To calculate the price of a call, we need to know the put price. • To calculate the price of a put, we need to know the call price. • So, we want to know the value of a call option: • Before expiration, and • Without knowing the price of the put 16-2 A Simple Model to Value Options Before Expiration, I. • Suppose we want to know the price of a call option with • • • • One year to maturity (T = 1.0) A $110 exercise price (K = 110) The current stock price is $108 (S0 =108) The one-year risk-free rate is 10 percent (r = 10%) 16-3 A Simple Model to Value Options Before Expiration, I. • We know what the stock price will be in one year. • S1 = $130 or $115 (but no other values). • S1 is still uncertain. • We do not know the probabilities of these two values. • Therefore, the call option value at expiration will be: • $130 – $110 = $20 or • $115 - $110 = $5 • This call option is certain to finish in the money. • A similar put option is certain to finish out of the money. 16-4 A Simple Model to Value Options Before Expiration, II. • If you know the price of a similar put, you can use put-call parity to price a call option before it expires. C - P S0 - K/(1 r)T C - 0 $108 $110/(1.10 ) C $108 - $100 $8. • The chosen pair of stock prices guarantees that the call option finishes in the money. • Suppose, however, we want to allow the call option to expire in the money OR out of the money. 16-5 The One-Period Binomial Option Pricing Model—The Assumptions • S = the stock price today; the stock pays no dividends. • Assume that the stock price in one period is either S × u or S × d, where: • u (for “up” factor) is > 1 • and d (for “down” factor) < 1 • Suppose S = $100, u = 1.1 and d = .95 • S1 = $100 × 1.1 = $110 or S1 = $100 × .95 = $95 • What is the call price today, if: • K = 100 • R = 3% 16-6 The One-Period Binomial Option Pricing Model—The Setup • Consider the following portfolio: • Buy a fractional share of the underlying asset• The Greek letter D (Delta) = this fraction • Sell one call option • Finance the difference by borrowing $(DS – C) • Key Question: What is the value of this portfolio, today and at option expiration? 16-7 The Value of this Portfolio (long D Shares and short one call) is: • Important: DS is NOT the change in S. Rather, it is a dollar amount, DS = Δ * S. DS×u - Cu DS - C Cu and Cd : The intrinsic value of the call if the stock price increases to S×u or decreases to S×d. DS×d - Cd Portfolio Value Today Portfolio Value At Expiration 16-8 To Calculate Today’s Call Price, C: • There is one combination of a fractional share and one call that makes this portfolio risk-less. • The portfolio will have the same value when the underlying asset increases as it does when the underlying asset decreases in value. • The portfolio is riskless if: DSu – Cu = DSd – Cd • We know all values in this equation, except D. • S = $100; Su = $110; Sd = $95 • Cu = MAX(Su – K, 0) = MAX($110 – 100,0) = $10 • Cd = MAX(Sd – K, 0) = MAX($95 – 100,0) = $0 16-9 First Step: Calculate D To make the portfolio riskless: DSu – Cu = DSd – Cd DSu – DSd = Cu – Cd D(Su – Sd) = Cu – Cd, We can calculate D: D = (Cu – Cd) / (Su – Sd) D = (10 – 0) / 110 – 95 D = 10 / 15 D=2/3 16-10 Sidebar: What is D? • D, delta, is the riskless hedge ratio. • D, delta, is the fractional share amount needed to hedge one call. • Therefore, the number of calls to hedge one share is 1/D 16-11 The One-Period Binomial Option Pricing Model—The Formula • A riskless portfolio today should be worth (DS – C)(1+r) in one period. • So: (DS – C)(1+r) = DSu – Cu (which equals DSd – Cd because we chose the “correct” D). • Solve the equation above for C. 16-12 The One-Period Binomial Option Pricing Model—The Formula • Solving the equation above for C: ( DS - C)(1 r) DSu - Cu ΔS(1 r) C(1 r) ΔSu Cu ΔS(1 r) ΔSu C(1 r) Cu ΔS(1 r u) C(1 r) Cu ΔS(1 r u) Cu C(1 r) ΔS(1 r u) Cu C 1 r 16-13 Calculate the Call Price, C. C C C C ΔS(1 r u) C u 1 r (2/3)($100 )(1 .03 1.10) $10 1 .03 ($200/3)( .07) $10 1 .03 $5.33 $5.18 1.03 16-14 Now We Can Calculate the Call Price, C. If C= $5.18, what is the price of a similar put? Using Put-Call Parity: P S C K/(1 r) P $100 $5.18 $100/1.03 P $5.18 $100/1.03 100 P $2.27 16-15 The Two-Period Binomial Option Pricing Model • Suppose there are two periods to expiration instead of one. • Repeat much of the process used in the one-period binomial option pricing model. • This method can be used to price: • European call options. • European put options. • American calls and puts (with a modification to allow for early exercise). • An exotic array of options (with the appropriate modifications). 16-16 The Method We can find binomial option prices for two (or more) periods by using the following five steps: 1. Build a price “tree” for stock prices through time. 2. Use the intrinsic value formula to calculate the possible option values at expiration. 3. Calculate the fractional share needed to form each riskless portfolio at the next-to-last date. 4. Calculate all possible option prices at the next-tolast date. 5. Repeat this process by working back to today. 16-17 The Binomial Option Pricing Model with Many Periods 16-18 What Happens When the Number of Periods Gets Really, Really Big? • For European options on non-dividend paying stocks, the binomial method converges to the Black-Scholes option pricing formula. • To calculate the prices of many other types of options, however, we still need to use a computer (and methods similar in spirit to the binomial method). 16-19 The Black-Scholes Option Pricing Model • The BSOPM calculates the price of a call option before maturity • • • • • Dates from the early 1970s Professors Fischer Black and Myron Scholes Facilitated option pricing CBOE was launched soon after BSOPM appeared 1997 Nobel Prize in Economics • Important contributions by professor Robert Merton • The Black-Scholes-Merton option pricing model 16-20 Black-Scholes Option Valuation Variables C P S K r T ln e N(d) = Current call option value = Current put option value = Current stock price = Option strike or exercise price = Risk-free interest rate Stock price volatility = time to maturity of the option in years = Natural log function; ln(x) = 2.71828, the base of the natural log; exp() = probability that a random draw from a normal distribution will be less than d Solution Variables Input Variables Functions 16-21 Black-Scholes Option Valuation C SN(d1) Ke P Ke rT rT N (d 2 ) N(d 2 ) SN(d1) ln(S / K ) (r 2 / 2)T d1 T d 2 d1 T 16-22 Formula Functions • e-rt = exp(-rt) = natural exponent of the value of –rt (a discount factor) • ln(S/K) = natural log of the "moneyness" term, S/K • N(d1) and N(d2) denotes the standard normal probability for the values of d1 and d2. • Formula makes use of the fact that: N(-d1) = 1 - N(d1) 16-23 Example: Computing Prices for Call and Put Options Suppose you are given the following inputs: S = $50 K = $45 T = 3 months (or 0.25 years) = 25% (stock volatility) r = 6% 16-24 Step 1: Calculating d1 and d2 ln(S K (r σ 2 2 T d1 σ T 2 ( ( ln 50 45 0.06 0.25 2 0.25 0.25 0.25 0.10536 0.09125 0.25 1.02538 0.125 d2 d1 σ T 1.02538 0.25 0.25 0.90038 16-25 Step 2: Excel’s “=NORMSDIST(x)” Function =NORMSDIST(1.02538) = 0.84741 = N(d1) =NORMSDIST(0.90038) = 0.81604 = N(d2) N(-d1) = 1 - N(d1): N(-1.02538) = 1 – N(1.02538) = 1 – 0.84741 = 0.15259 = N(-d1) N(-0.90038) = 1 – N(0.90038) = 1 – 0.81604 = 0.18396 = N(-d2) 16-26 Step 3a: The Call Price C = SN(d1) – Ke–rTN(d2) = $50(0.84741) – 45(e-(.06)(.25))(0.81604) = 50(0.84741) – 45(0.98511)(0.81604) = $6.195 16-27 Step 3b: The Put Price: P = –rT Ke N(–d 2) – SN(–d1) = $45(e-(.06)(.25))(0.18396) – 50(0.15259) = 45(0.98511)(0.18396) – 50(0.15259) = $0.525 16-28 Verify Our Results Using Put-Call Parity Note: Options must be European-style C P S Ke rT $5.985 $0.565 50 - 45e (0.060.25) $5.42 $50.00 $44.33 $5.42 16-29 Valuing the Options Using Excel Stock Price: Strike Price: Volatility (%): Time (in years): Riskless Rate (%): 50.00 45.00 25.00 0.2500 6.00 Stock: Discounted Strike: 50.00 44.33 d(1): N(d1): 1.02538 0.84741 N(-d1): 0.15259 d(2): N(d2): 0.90038 0.81604 N(-d2): 0.18396 Call Price: $ 6.195 Put Price: $ 0.525 16-30 Using a Web-based Option Calculator • www.numa.com. 16-31 Varying the Option Price Input Values • An important goal of this chapter is to show how an option price changes when only one of the five inputs changes. • The table below summarizes these effects. 16-32 Factors Influencing Option Values Effect on Common Option Value Name Call Put Table 16.3 Input Factor Underlying stock price S Strike price of option contract K Time remaining to expiration T Volatility of the underlying stock price σ Risk-free interest rate r + + + + + + + - Delta 16-33 Varying the Underlying Stock Price • Changes in the stock price have a big effect on option prices. 16-34 Varying the Time Remaining Until Option Expiration 16-35 Varying the Volatility of the Stock Price 16-36 Varying the Interest Rate 16-37 Calculating Delta • Delta measures the dollar impact of a change in the underlying stock price on the value of a stock option. Call option delta = N(d1) > 0 Put option delta = –N(–d1) < 0 • A $1 change in the stock price causes an option price to change by approximately delta dollars. 16-38 Example: Calculating Delta Stock Price: Strike Price: Volatility (%): Time (in years): Riskless Rate (%): 50.00 45.00 25.00 0.2500 6.00 Stock: Discounted Strike: 50.00 44.33 d(1): N(d1): 1.02538 0.84741 N(-d1): 0.15259 d(2): N(d2): 0.90038 0.81604 N(-d2): 0.18396 Call Price: $ 6.195 Put Price: $ 0.525 Call option delta = Put option delta = 0.84741 -0.15259 16-39 The Call "Delta" Prediction: • Call delta = 0.84741 • if the stock price increases by $1, the call option price will increase by about $0.85 • From the previous example: • Stock price = $50 • Call option price = $6.195 • If the stock price is $51: • Call option value = $7.060 • Increase of about $0.868 16-40 The Put "Delta" Prediction: • Put delta value = -0.15259 • if the stock price increases by $1, the put option price will decrease by $0.15. • From the previous example: • Stock price = $50 • Put option price = $0.525 • If the stock price is $51: • Put option value = $0.390 • Decrease of about $0.14 16-41 Hedging with Stock Options • You own 1,000 shares of XYZ stock and you want protection from a price decline. • Let’s use stock and option information from before— in particular, the “delta prediction” to help us hedge. • You want changes in the value of your XYZ shares to be offset by the value of your options position. That is: Change in stock price shares Change in option price # options Change in stock price shares Option Delta # options 16-42 Hedging Using Call Options— The Prediction • Delta = 0.8474; stock price declines by $1: Change in stock price shares Delta # options - 1 1,000 0.8474 # options # options - 1,000 / 0.8474 - 1,180.08 - 1,180.08 / 100 - 12. Write 12 call options with a $45 strike to hedge your stock. 16-43 Hedging with Calls - Results • Call option gain nearly offsets your loss of $1,000. • Why is it not exact? • Call Delta falls when the stock price falls. • Therefore, you did not sell quite enough call options. Stock Price Portfolio Value Call Price Call Position Value $50 $50,000 $6.200 -$7,440 $49 $49,000 $5.370 -$6,444 -$1 ($1,000) -$0.830 $996 16-44 Hedging Using Put Options—The Prediction • Delta = -0.1526; stock price declines by $1: Change in stock price shares Delta # options - 1 1,000 - 0.1526 # options # options - 1,000 / - 0.1526 6,553.08 6,553.08 / 100 66. Buy 66 put options with a strike of $45 to hedge your stock. 16-45 Hedging Using Put Options: Results • Put option gain more than offsets $1,000 loss • Why is it not exact? • Put Delta also falls (gets more negative) when the stock price falls. • Therefore, you bought too many put options—this error is more severe the lower the value of the put delta. • To get closer: Use a put with a strike closer to at-themoney. Stock Price Portfolio Value Put Price $50 $50,000 $0.530 Put Position Value $3,498 $49 $49,000 $0.700 $4,620 -$1 ($1,000) $0.170 $1,122 16-46 Hedging a Portfolio with Index Options • Many institutional money managers use stock index options to hedge equity portfolios Number of option contracts Portfolio beta Portfolio value Option delta Option contract value • Regular rebalancing needed to maintain an effective hedge • Underlying Value Changes • Option Delta Changes • Portfolio Value Changes • Portfolio Beta Changes 16-47 Hedging with Stock Index Options You manage a $10 million stock portfolio. You attempt to maintain a portfolio beta of 1.00 You decide to hedge your position by buying index put options with a contract value of $125,000 and a delta of 0.579. How many contracts do you need to buy? Portfolio beta Portfolio value Number of option contracts Option delta Option contract value 16-48 Hedging with Stock Index Options Portfolio value = $10 million Portfolio beta = Index contract value = $150,800 Number of option contracts Number of option contracts 1.00 Option delta = 0.579 Portfolio beta Portfolio value Option delta Option contract value Portfolio Beta Portfolio Value Option Delta Underlying Value 100 1.00 10,000,000 115 0.579 1508 100 Sell 115 call options 16-49 Implied Standard Deviations • Implied standard deviation (ISD) • = Implied volatility (IVOL) • Stock price volatility estimated from an option price • Of the six BSOPM input factors, only stock price volatility is not directly observable • Calculating an implied volatility requires: • All 5 other input factors • Either a call or put option price 16-50 CBOE Published Implied Volatilities for Stock Indexes • CBOE publishes 3 volatility indexes: • S&P 500 Index Option Volatility (VIX) • S&P 100 Index Option Volatility (VXO) • Nasdaq 100 Index Option Volatility (VXN) • Each volatility index calculated using ISDs from eight options: • 4 calls and 4 puts, each with two maturity dates: • 2 slightly out of the money • 2 slightly in the money • Provides investors with information about market volatility in the coming months. 16-51 Employee Stock Options, ESOs • ESO = call option that a firm grants (gives) to employees. • ESO life generally 10 years • Cannot be sold • “Vesting” period of about 3 years • If employee leaves the company before “vested," the ESOs are lost • Once vested, ESOs can be exercised any time over its remaining life 16-52 52 Why are ESOs Granted? • Motivates employees to make decisions that help the stock price increase • Powerful motivator • Payoffs can be large. • High stock prices: ESO holders gain and shareholders gain. • No upfront costs to the company • Can act as a substitute for ordinary wages • Helpful in recruiting employees 16-53 53 ESO Repricing • Generally issued exactly “at the money” • Intrinsic value = zero. • No value from immediate exercise. • Still valuable • “Underwater” Options • Stock price falls after the ESO is granted • “Restriking” or “Repricing.” • Companies lower the strike prices • Controversial practice 16-54 54 ESO Repricing Controversy • PRO: Once an ESO is “underwater,” it loses its ability to motivate employees. • Small chance for payoff • Employees may leave to get “fresh” options. • CON: Lowering strike price = reward for failing • Decisions by employees made the stock price fall. • If employees know that ESOs will be repriced, the ESOs loose their ability to motivate employees. 16-55 55 ESOs Today • Most companies award ESOs on a regular basis. • Quarterly • Annually • Therefore, employees will always have some “at the money” options. • Regular grants of ESOs mean that employees always have some “unvested” ESOs—giving them the added incentive to remain with the company. 16-56 Valuing Employee Stock Options • Companies must report estimates of ESO values (FASB 123) • BSOPM modified by Merton widely used for this purpose • For example: • In December 2002, the Coca-Cola Company granted ESOs with a stated life of 15 years. • ESOs often exercised before maturity, so CocaCola also used a life of 6 years to value these ESOs. 16-57 Employee Stock Option Valuation The Black-Scholes-Merton Option Pricing Model incorporates the possibility that the underlying stock will pay a dividend during the life of the option: “y” = the annual dividend yield for the underlying stock C Se yT N( d1 ) Ke rT N( d 2 ) ln( S / K ) ( r y 2 / 2 )T d1 T d 2 d1 T “S” equals the stock’s price at the grant date 16-58 Example: Valuing Coca-Cola ESOs Using Excel Stock Price: Discounted Stock: 44.55 35.10 Stock Price: Discounted Stock: 44.55 40.23 Strike Price: Discounted Strike: 44.66 19.13 Strike Price: Discounted Strike: 44.66 36.41 Volatility (%): Time (in years): Riskless Rate (%): Dividend Yield (%): 25.53 15 5.65 1.59 Volatility (%): Time (in years): Riskless Rate (%): Dividend Yield (%): 30.20 6 3.40 1.70 d(1): N(d1): 1.10792 0.86605 d(1): N(d1): 0.50458 0.69307 d(2): N(d2): 0.11915 0.54742 d(2): N(d2): -0.23517 0.40704 Call Price: $ 13.06 Call Price: $ 19.92 16-59 Summary: Coca-Cola Employee Stock Options 16-60 Useful Websites • • • • • • • • www.jeresearch.com (information on option formulas) www.option-price.com (for a free option price calculator) www.numa.com (for “everything about options”) www.wsj.com/free (option price quotes) www.ino.com (Web Center for Futures and Options) www.optionetics.com (Optionetics) www.pmpublishing.com (free daily volatility summaries) www.ivolatility.com (for applications of implied volatility) 16-61