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Real Options Dr. Lynn Phillips Kugele FIN 431 Slide 2 OPT-2 Options Review Mechanics of Option Markets Properties of Stock Options Introduction to Binomial Trees Valuing Stock Options: The Black-Scholes Model Real Options Slide 3 OPT-3 Mechanics of Options Markets Slide 4 OPT-4 Option Basics Option = derivative security Value derived from the value of the underlying asset Stock Option Contracts Exchange-traded Standardized Facilitates trading and price reporting. Contract = 100 shares of stock Slide 5 OPT-5 Put and Call Options Call option Gives holder the right but not the obligation to buy the underlying asset at a specified price at a specified time. Put option Gives the holder the right but not the obligation to sell the underlying asset at a specified price at a specified time. Slide 6 OPT-6 Options on Common Stock 1.Identity of the underlying stock 2.Strike or Exercise price 3.Contract size 4.Expiration date or maturity 5.Exercise cycle American or European 6.Delivery or settlement procedure Slide 7 OPT-7 Option Exercise American-style Exercisable at any time up to and including the option expiration date Stock options are typically American European-style Exercisable only at the option expiration date Slide 8 OPT-8 Option Positions Call positions: Long call = call holder Hopes/expects asset price will increase Short call = call writer Hopes asset price will stay or decline Put Positions: Long put = put holder Expects asset price to decline Short put = put writer Hopes asset price will stay or increase Slide 9 OPT-9 Option Writing The act of selling an option Option writer = seller of an option contract Call option writer obligated to sell the underlying asset to the call option holder Put option writer obligated to buy the underlying asset from the put option holder Option writer receives the option premium when contract entered Slide 10 OPT-10 Option Payoffs & Profits Notation: S 0 = current stock price per share S T = stock price at expiration K = option exercise or strike price C = American call option premium per share c = European call option premium P = American put option premium per share p = European put option premium r = risk free rate T = time to maturity in years Slide 11 OPT-11 Payoff to Call Holder ( S - K) if S >K 0if S < K Profit to Call Holder Payoff - Option Premium Profit =Max (S-K, 0) - C Option Payoffs & Profits Call Holder = Max (S-K,0) Slide 12 OPT-12 Payoff to Call Writer - (S - K) if S > K = -Max (S-K, 0) 0if S < K= Min (K-S, 0) Profit to Call Writer Payoff + Option Premium Profit = Min (K-S, 0) + C Option Payoffs & Profits Call Writer Slide 13 OPT-13 Payoff & Profit Profiles for Calls Payoff: Max(S-K,0) -Max(S-K,0) Profit: Max (S-K,0) c -[Max (S-K, 0)-p] Slide 14 OPT-14 Payoff & Profit Profiles for Calls Call Holder Call Writer Slide 15 OPT-15 Payoff & Profit Profiles for Calls Profit Stock Price 0 Call Writer Profit Call Holder Profit Payoff Slide 16 OPT-16 Payoffs to Put Holder 0if S > K (K - S) if S < K Profit to Put Holder Payoff - Option Premium Profit = Max (K-S, 0) - P Option Payoffs and Profits Put Holder = Max (K-S, 0) Slide 17 OPT-17 Payoffs to Put Writer 0 if S > K= -Max (K-S, 0) -(K - S) if S < K= Min (S-K, 0) Profits to Put Writer Payoff + Option Premium Profit = Min (S-K, 0) + P Option Payoffs and Profits Put Writer Slide 18 OPT-18 Payoff & Profit Profiles for Puts Payoff: Max(K-S,0) -Max(K-S,0) Profit: Max (K-S,0) p -[Max (K-S, 0)-p] Slide 19 OPT-19 Payoff & Profit Profiles for Puts Put Writer Put Holder Slide 20 OPT-20 Payoff & Profit Profiles for Puts 0 Profits Stock Price Put Writer Profit Put Holder Profit Slide 21 OPT-21 CALL PUT Holder: Payoff Max (S-K,0) Max (K-S,0) (Long) Profit Max (S-K,0) - C Max (K-S,0) - P Bullish Bearish Writer: Payoff Min (K-S,0) Min (S-K,0) (Short) Profit Min (K-S,0) + C Min (S-K,0) + P Bearish Bullish Option Payoffs and Profits Slide 22 OPT-22 Long Call Call option premium (C) = $5, Strike price (K) = $100. 30 20 10 0 -5 708090100 110120130 Profit ($) Terminal stock price (S) Long Call Profit = Max(S-K,0) - C Slide 23 OPT-23 Short Call Call option premium (C) = $5, Strike price (K) = $100 -30 -20 -10 0 5 708090100 110120130 Profit ($) Terminal stock price (S) Short Call Profit = -[Max(S-K,0)-C] = Min(K-S,0) + C Slide 24 OPT-24 Long Put Put option premium (P) = $7, Strike price (K) = $70 30 20 10 0 -7 706050408090100 Profit ($) Terminal stock price ($) Long Put Profit = Max(K-S,0) - P Slide 25 OPT-25 Short Put Put option premium (P) = $7, Strike price (K) = $70 -30 -20 -10 7 0 70 605040 8090100 Profit ($) Terminal stock price ($) Short Put Profit = -[Max(K-S,0)-P] = Min(S-K,0) + P Slide 26 OPT-26 Properties of Stock Options Slide 27 OPT-27 Notation c = European call option price ( C = American) p = European put option price ( P = American) S 0 = Stock price today S T =Stock price at option maturity K = Strike price T = Option maturity in years = Volatility of stock price D = Present value of dividends over options life r = Risk-free rate for maturity T with continuous compounding Slide 28 OPT-28 American vs. European Options An American option is worth at least as much as the corresponding European option C c P p Slide 29 OPT-29 Factors Influencing Option Values Slide 30 OPT-30 Effect on Option Values Underlying Stock Price (S) & Strike Price (K) Payoff to call holder: Max (S-K,0) As S , Payoff increases; Value increases As K , Payoff decreases; Value decreases Payoff to Put holder: Max (K-S, 0) As S , Payoff decreases; Value decreases As K , Payoff increases; Value increases Slide 31 OPT-31 Option Price Quotes Calls Slide 32 OPT-32 Option Price Quotes Puts Slide 33 OPT-33 Effect on Option Values Time to Expiration = T For an American Call or Put: The longer the time left to maturity, the greater the potential for the option to end in the money, the grater the value of the option For a European Call or Put: Not always true due to restriction on exercise timing Slide 34 OPT-34 Option Price Quotes Slide 35 OPT-35 Effect on Option Values Volatility = Volatility = a measure of uncertainty about future stock price movements Increased volatility increased upside potential and downside risk Increased volatility is NOT good for the holder of a share of stock Increased volatility is good for an option holder Option holder has no downside risk Greater potential for higher upside payoff Slide 36 OPT-36 Effect on Option Values Risk-free Rate = r As r : Investors required return increases The present value of future cash flows decreases = Increases value of calls = Decreases value of puts Slide 37 OPT-37 Effect on Option Values Dividends = D Dividends reduce the stock price on the ex-div date Decreases the value of a call Increases the value of a put Slide 38 OPT-38 Upper Bound for Options Call price must be stock price: c S 0 C S 0 Put price must be strike price: p K P K p Ke -rT Slide 39 OPT-39 Upper Bound for a Call Option Price Call option price must be stock price A call option is selling for $65; the underlying stock is selling for $60. Arbitrage: Sell the call, Buy the stock. Worst case: Option is exercised; you pocket $5 Best case: Stock price < $65 at expiration, you keep all of the $65. Slide 40 OPT-40 Upper Bound for a Put Option Price Put option price must be strike price Put with a $50 strike price is selling for $60 Arbitrage: Sell the put, Invest the $60 Worse case: Stock price goes to zero You must pay $50 for the stock But, you have $60 from the sale of the put (plus interest) Best case: Stock price $50 at expiration Put expires with zero value You keep the entire $60, plus interest Slide 41 OPT-41 Lower Bound for European Call Prices Non-dividend-paying Stock c Max(S 0 Ke rT,0) Portfolio A: 1 European call + Ke -rT cash Portfolio B: 1 share of stock Slide 42 OPT-42 Lower Bound for European Put Prices Non-dividend-paying Stock p Max(Ke -rT S 0,0) Portfolio C: 1 European put + 1 share of stock Portfolio D: Ke -rT cash Slide 43 OPT-43 9.43 Put-Call Parity No Dividends Portfolio A: European call + Ke -rT in cash Portfolio C: European put + 1 share of stock Both are worth max( S T, K ) at maturity They must therefore be worth the same today: c + Ke -rT = p + S 0 Slide 44 OPT-44 Put-Call Parity American Options Put-Call Parity holds only for European options. For American options with no dividends: Slide 45 OPT-45 Introduction to Binomial Trees Slide 46 OPT-46 A Simple Binomial Model (Cox, Ross, Rubenstein, 1979) A stock price is currently $20 In three months it will be either $22 or $18 Stock Price = $22 Stock Price = $18 Stock price = $20 Slide 47 OPT-47 Stock Price = $22 Option Price = $1 Stock Price = $18 Option Price = $0 Stock price = $20 Option Price=? A Call Option A 3-month European call option on the stock has a strike price of $21. Slide 48 OPT-48 Consider the Portfolio:Long shares Short 1 call option Portfolio is riskless when: 22 1 = 18 or = 0.25 22 1 18 Setting Up a Riskless Portfolio Slide 49 OPT-49 Valuing the Portfolio Risk-Free Rate = 12% Assuming no arbitrage, a riskless portfolio must earn the risk-free rate. The riskless portfolio is: Long 0.25 shares Short 1 call option The value of the portfolio in 3 months is 22 0.25 1 = 4.50 or 18 x 0.25 = 4.50 The value of the portfolio today is 4.5e 0.12 0.25 = 4.3670 Slide 50 OPT-50 Valuing the Option DescriptionValue PortfolioLong 0.25 shares Short 1 call $4.367 Shares=0.25 x $20$5.000 Call option= $5.000 4.367$0.633 Slide 51 OPT-51 Generalization 1-Step Tree S0u uS0u u S0d dS0d d S0S0 At t=0After move up After move down Stock price S0S0 S0uS0uS0dS0d Option price ffufu fdfd u> 1 d < 1 Slide 52 OPT-52 Generalization (continued) Consider the portfolio that is long shares and short 1 derivative The portfolio is riskless when: S 0 u u = S 0 d d or S 0 u u S 0 d d Slide 53 OPT-53 Generalization (continued) Value of the portfolio at time T is: S 0 u u Cost to set up the portfolio today: S 0 f = Value of the portfolio today today: S 0 f = (S 0 u u )e rT Hence = S 0 ue -rT )+ u e rT Slide 54 OPT-54 Generalization (continued) Substituting for we obtain = e rT [ p u + (1 p ) d ] where Slide 55 OPT-55 Risk-Neutral Valuation p and (1 p ) can be interpreted as the risk-neutral probabilities of up and down movements The value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate. Expected payoff from option: Slide 56 OPT-56 Irrelevance of Stocks Expected Return Expected return on the underlying stock is irrelevant in pricing the option Critical point in ultimate development of option pricing formulas Not valuing option in absolute terms Option value = f(underlying stock price) Slide 57 OPT-57 Original Example Revisited Risk-Neutral = No Arbitrage Since p is a risk-neutral probability 20 e 0.12 0.25 = 22p + 18(1 p ); p = 0.6523 Alternatively, use the formula S 0 u = 22 u = 1 S 0 d = 18 d = 0 S0 S0 p (1 p ) Slide 58 OPT-58 Valuing the Option Value of the option: = e 0.12 x 0.25 [0.6523 1 + 0.3477 0] = 0.633 S 0 u = 22 u = 1 S 0 d = 18 d = 0 S0S0 0.6523 0.3477 Slide 59 OPT-59 Relevance of Binomial model Stock price only having 2 future price choices appears unrealistic Consider: Over a small time period, a stocks price can only move up or down one tick size (1 cent) As the length of each time period approaches 0, the Binomial Model converges to the Black- Scholes Option Pricing Model. Slide 60 OPT-60 Valuing Stock Options: The Black-Scholes Model Slide 61 OPT-61 BSOPM Black-Scholes (-Merton) Option Pricing Model BS = Fischer Black and Myron Scholes With important contributions by Robert Merton BSOPM published in 1973 Nobel Prize in Economics in 1997 Values European options on non-dividend paying stock Slide 62 OPT-62 BSOPM Assumptions = expected return on the stock = volatility of the stock price Therefore in time t: t= mean of the return = standard deviation and: Slide 63 OPT-63 The Lognormal Property Assumptions ln S T is normally distributed with mean: and standard deviation : Because the logarithm of S T is normal, S T is lognormally distributed Slide 64 OPT-64 The Lognormal Property continued where m, v ] is a normal distribution with mean m and variance v Slide 65 OPT-65 The Lognormal Distribution Restricted to positive values Slide 66 OPT-66 The Expected Return Expected value of the stock price S0eTS0eT Expected return on the stock with continuous compounding Arithmetic mean of the returns over short periods of length t Geometric mean of returns 2 /2 Slide 67 OPT-67 Concepts Underlying Black-Scholes Option price and stock price depend on same underlying source of uncertainty A portfolio consisting of the stock and the option can be formed which eliminates this source of uncertainty (riskless). The portfolio is instantaneously riskless Must instantaneously earn the risk-free rate Slide 68 OPT-68 Assumptions Underlying BSOPM 1.Stock price behavior corresponds to the lognormal model with and constant. 2.No transactions costs or taxes. All securities are perfectly divisible. 3.No dividends on stocks during the life of the option. 4.No riskless arbitrage opportunties. 5.Security trading is continuous. 6.Investors can borrow & lend at the risk-free rate. 7.The short-term rate of interest, r, is constant. Slide 69 OPT-69 Notation c and p = European option prices (premiums) S 0 = stock price K = strike or exercise price r = risk-free rate = volatility of the stock price T = time to maturity in years Slide 70 OPT-70 Formula Functions ln(S/K) = natural log of the "moneyness" term N(x) = the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x N(d1) and N(d2) denote the standard normal probability for the values of d1 and d2. Formula makes use of the fact that: N(-d 1 ) = 1 - N(d 1 ) Slide 71 OPT-71 The Black-Scholes Formulas Slide 72 OPT-72 BSOPM Example Given: S 0 = $42r = 10% = 20% K = $40T = 0.5 Slide 73 OPT-73 BSOPM Call Price Example d 1 = 0.7693N(0.7693) = 0.7791 d 2 = 0.6278N(0.6278) = 0.7349 Slide 74 OPT-74 BSOPM Put Price Example d 1 = 0.7693N(-0.7693) = 0.2209 d 2 = 0.6278N(-0.6278) = 0.2651 Slide 75 OPT-75 BSOPM in Excel N(d 1 ): =NORMSDIST(d 1 ) Note the S in the function S denotes standard normal ~ (0,1) =NORMDIST() Normal distribution Mean and variance must be specified Slide 76 OPT-76 Properties of Black-Scholes Formula As S 0 Call Fwd with delivery = K Almost certain to be exercised d 1 and d 2 very large N(d 1 ) and N(d 2 ) 1.0 N(-d 1 ) and N(-d 2 ) 0 c S 0 Ke -rT p 0 c = max(S-K,0) p = max(K-S,0) Slide 77 OPT-77 Properties of Black-Scholes Formula As S 0 0d 1 and d 2 very large & negative N(d 1 ) and N(d 2 ) 0 N(-d 1 ) and N(-d 2 ) 1.0 c 0 p Ke -rT S 0 c = max(S-K,0) p = max(K-S,0) Slide 78 Real Options Slide 79 OPT-79 Real Options Examples Option to vary output / production Option to delay investment Option to expand / contract Option to abandon Use the same option valuation approach for non-financial assets Assume underlying asset is traded Price as any financial asset Slide 80 OPT-80 Example 1 year lease on a gold mine (T) Extract up to 10,000 oz Cost of extraction is $270 per oz (K) Current market price of gold is $300 per oz (S) Volatility of gold prices is 22.3% per annum () Interest rate is 10% per annum Continuously compounded = ln(1.1) = 9.53% (r) Slide 81 OPT-81 Options Approach Slide 82 OPT-82 Option to Expand At t=1, we can expand production for t=2. Up-front capital investment (at t=1) of $150k With the new investment, we can mine up to 12,500 oz per year, at a per unit cost of $280 per oz. How much would you pay at t=0 for this option? Slide 83 OPT-83 Multiple Options Slide 84 OPT-84 Slide 85 OPT-85 If Expansion and S 1 = 375 Slide 86 OPT-86 If No Expansion and S 1 = 375 Slide 87 OPT-87 If Expansion and S 1 = 240 Slide 88 OPT-88 If No Expansion and S 1 = 240 Slide 89 OPT-89 Real Options Recap Slide 90 OPT-90 Conclusions If S 1 = 375: Value of option to expand = $217,089 Subtracting cost of expansion and discounting to t=0 Value = $60,991 If S 1 = 240, net value is negative Slide 91 OPT-91 Probability of Up Movement Know that u = and d = 1/u In our example u=1.25 and d=.8, thus p = 0.677 Option value =.667*$60,991 = $40,680.75 Total lease value = 2-yr without expansion + value of option to expand: $1,688,588 + $40,681 = $1,729,269 Slide 92 OPT-92 Probability of Up Movement