Report

Real Options Dr. Lynn Phillips Kugele FIN 431 Options Review • Mechanics of Option Markets • Properties of Stock Options • Introduction to Binomial Trees • Valuing Stock Options: The Black-Scholes Model • Real Options OPT-2 Mechanics of Options Markets OPT-3 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 OPT-4 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. OPT-5 Options on Common Stock 1. 2. 3. 4. 5. • Identity of the underlying stock Strike or Exercise price Contract size Expiration date or maturity Exercise cycle American or European 6. Delivery or settlement procedure OPT-6 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 OPT-7 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 OPT-8 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 OPT-9 Option Payoffs & Profits Notation: • • • • S0 = current stock price per share ST = 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 OPT-10 Option Payoffs & Profits Call Holder Payoff to Call Holder (S - K) 0 if S >K if S < K = Max (S-K,0) Profit to Call Holder Payoff - Option Premium Profit =Max (S-K, 0) - C OPT-11 Option Payoffs & Profits Call Writer Payoff to Call Writer - (S - K) 0 if S > K = -Max (S-K, 0) if S < K = Min (K-S, 0) Profit to Call Writer Payoff + Option Premium Profit = Min (K-S, 0) + C OPT-12 Payoff & Profit Profiles for Calls Call Payoff and Profit K = $20 c = $5.00 Stock Call Holder Price Payoff Profit 0 $0 -$5 $10 $0 -$5 $20 $0 -$5 $30 $10 $5 $40 $20 $15 Payoff: Profit: Max(S-K,0) Max (S-K,0) – c Call Writer Payoff Profit $0 $5 $0 $5 $0 $5 -$10 -$5 -$20 -$15 -Max(S-K,0) -[Max (S-K, 0)-p] OPT-13 Payoff & Profit Profiles for Calls Call Payoff and Profit $25 $20 Call Holder $15 $10 $$ $5 $0 0 $10 $20 $30 $40 -$5 -$10 -$15 Call Writer -$20 -$25 Stock Price Call Holder Payoff Call Holder Profit Call Writer Payoff Call Writer Profit OPT-14 Payoff & Profit Profiles for Calls Payoff Profit Call Holder Profit 0 Call Writer Profit Stock Price OPT-15 Option Payoffs and Profits Put Holder Payoffs to Put Holder 0 (K - S) if S > K if S < K = Max (K-S, 0) Profit to Put Holder Payoff - Option Premium Profit = Max (K-S, 0) - P OPT-16 Option Payoffs and Profits Put Writer Payoffs to Put Writer 0 if S > K -(K - S) if S < K = -Max (K-S, 0) = Min (S-K, 0) Profits to Put Writer Payoff + Option Premium Profit = Min (S-K, 0) + P OPT-17 Payoff & Profit Profiles for Puts Put Payoff and Profit K = $20 p = $5.00 Stock Put Holder Price Payoff Profit 0 $20 $15 $10 $10 $5 $20 $0 -$5 $30 $0 -$5 $40 $0 -$5 Payoff: Profit: Max(K-S,0) Max (K-S,0) – p Put Writer Payoff Profit -$20 -$15 -$10 -$5 $0 $5 $0 $5 $0 $5 -Max(K-S,0) -[Max (K-S, 0)-p] OPT-18 Payoff & Profit Profiles for Puts Put Payoff and Profit $25 $20 Put Holder $15 $10 $$ $5 $0 0 $10 $20 $30 $40 -$5 -$10 -$15 Put Writer -$20 -$25 Stock Price Put Holder Payof Put Holder Profit Put Writer Payoff Put Writer Profit OPT-19 Payoff & Profit Profiles for Puts Profits Put Writer Profit 0 Put Holder Profit Stock Price OPT-20 Option Payoffs and Profits CALL PUT Holder: Payoff (Long) Profit Max (S-K,0) Max (S-K,0) - C “Bullish” Max (K-S,0) Max (K-S,0) - P “Bearish” Writer: Payoff (Short) Profit Min (K-S,0) Min (K-S,0) + C “Bearish” Min (S-K,0) Min (S-K,0) + P “Bullish” OPT-21 Long Call Long Call Profit = Max(S-K,0) - C Call option premium (C) = $5, Strike price (K) = $100. 30 Profit ($) 20 10 70 0 -5 80 90 100 Terminal stock price (S) 110 120 130 OPT-22 Short Call Short Call Profit = -[Max(S-K,0)-C] = Min(K-S,0) + C Call option premium (C) = $5, Strike price (K) = $100 Profit ($) 5 0 -10 110 120 130 70 80 90 100 Terminal stock price (S) -20 -30 OPT-23 Long Put Long Put Profit = Max(K-S,0) - P Put option premium (P) = $7, Strike price (K) = $70 30 Profit ($) 20 10 0 -7 Terminal stock price ($) 40 50 60 70 80 90 100 OPT-24 Short Put Short Put Profit = -[Max(K-S,0)-P] = Min(S-K,0) + P Put option premium (P) = $7, Strike price (K) = $70 Profit ($) 7 0 40 50 Terminal stock price ($) 60 70 80 90 100 -10 -20 -30 OPT-25 Properties of Stock Options OPT-26 Notation c p S0 ST K T D r = European call option price (C = American) = European put option price (P = American) = Stock price today =Stock price at option maturity = Strike price = Option maturity in years = Volatility of stock price = Present value of dividends over option’s life = Risk-free rate for maturity T with continuous compounding OPT-27 American vs. European Options An American option is worth at least as much as the corresponding European option Cc Pp OPT-28 Factors Influencing Option Values Effect on Option Value European Call Put 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 Dividend D + ? + + - + ? + + American Call Put + + + + - + + + + OPT-29 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 OPT-30 Option Price Quotes Calls MSFT (MICROSOFT CORP) $ 25.98 July 2008 CALLS Strike Last Sale Bid Ask Vol Open Int 15.00 10.85 10.95 11.10 10 85 17.50 10.54 8.45 8.55 0 33 20.00 6.00 6.00 6.05 4 729 22.50 3.60 3.55 3.65 195 3891 24.00 2.30 2.24 2.27 422 2464 25.00 1.50 1.45 1.48 3190 10472 26.00 0.83 0.83 0.85 2531 15764 27.50 0.31 0.29 0.31 2554 61529 OPT-31 Option Price Quotes Puts MSFT (MICROSOFT CORP) $ 25.98 July 2008 PUTS Strike Last Sale Bid Ask Vol Open Int 15.00 0.01 0.00 0.01 0 2751 17.50 0.01 0.00 0.02 0 2751 20.00 0.01 0.01 0.02 0 5013 22.50 0.03 0.03 0.04 13 4788 24.00 0.11 0.11 0.12 50 25041 25.00 0.25 0.24 0.25 399 7354 26.00 0.45 0.45 0.47 10212 51464 27.50 0.80 0.82 0.84 2299 39324 OPT-32 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 OPT-33 Option Price Quotes MSFT (MICROSOFT CORP) STRIKE = $25.00 CALLS Last Sale July 2008 1.42 August 2008 1.80 October 2008 2.36 January 2009 3.10 Bid 1.45 1.85 2.43 3.15 Ask 1.48 1.87 2.46 3.20 Vol 355 257 41 454 Open Int 10472 927 3309 59244 PUTS July 2008 August 2008 October 2008 January 2009 Bid 0.45 0.80 1.39 2.06 Ask 0.47 0.82 1.41 2.08 Vol 419 401 215 2524 Open Int 51464 1591 25323 155877 Last Sale 0.47 0.81 1.43 2.09 25.98 OPT-34 Effect on Option Values Volatility = σ • 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 OPT-35 Effect on Option Values Risk-free Rate = r • As r : –Investor’s required return increases –The present value of future cash flows decreases = Increases value of calls = Decreases value of puts OPT-36 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 OPT-37 Upper Bound for Options • Call price must be ≤ stock price: c ≤ S0 C ≤ S0 • Put price must be ≤ strike price: p≤K P≤K p ≤ Ke-rT OPT-38 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. OPT-39 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 OPT-40 Lower Bound for European Call Prices Non-dividend-paying Stock c Max(S0 –Ke –rT,0) Portfolio Portfolio A: 1 European call + Ke-rT cash If S > K Stock Option Cash Total A S-K K S B S S Portfolio B: 1 share of stock If S < K Stock Option Cash Total S 0 K K S OPT-41 Lower Bound for European Put Prices Non-dividend-paying Stock p Max(Ke -rT–S0,0) Portfolio Portfolio C: 1 European put + 1 share of stock Portfolio D: Ke-rT cash If S > K Stock Put Option Cash Total If S < K Stock Put Option Cash Total C S 0 D K K S S K-S K K S OPT-42 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(ST , K ) at maturity • They must therefore be worth the same today: c + Ke -rT = p + S0 OPT-43 9.43 Put-Call Parity American Options • Put-Call Parity holds only for European options. • For American options with no dividends: S0 K C P S0 Ke rT OPT-44 Introduction to Binomial Trees OPT-45 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 = $20 Stock Price = $18 OPT-46 A Call Option A 3-month European call option on the stock has a strike price of $21. Stock Price = $22 Option Price = $1 Stock price = $20 Option Price=? Stock Price = $18 Option Price = $0 OPT-47 Setting Up a Riskless Portfolio • Consider the Portfolio: Long D shares Short 1 call option 22D – 1 18D • Portfolio is riskless when: 22D – 1 = 18D or D = 0.25 OPT-48 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 – 0.120.25 = 4.3670 OPT-49 Valuing the Option Description Value Portfolio Long 0.25 shares Short 1 call $4.367 Shares =0.25 x $20 $5.000 Call option = $5.000 – 4.367 $0.633 OPT-50 Generalization – 1-Step Tree S0u ƒu S 0 S0d ƒd ƒ Stock price Option price u> 1 d<1 At t=0 After move up After move down S0 f S0u fu S0d fd OPT-51 Generalization (continued) • Consider the portfolio that is long D shares and short 1 derivative S0uD – ƒu S0dD – ƒd • The portfolio is riskless when: S0uD – ƒu = S0d D – ƒd or ƒu f d D S0 u S0 d OPT-52 Generalization (continued) • Value of the portfolio at time T is: S0u D – ƒu • Cost to set up the portfolio today: S0D – f = Value of the portfolio today today: S0 D – f = (S0u D – ƒu )e–rT • Hence ƒ = S0 D(1ue-rT)+ƒu e–rT OPT-53 Generalization (continued) • Substituting for D we obtain ƒ = e–rT[ p ƒu + (1 – p )ƒd ] where e d p u d rT OPT-54 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: pfu ( 1 p ) f d OPT-55 Irrelevance of Stock’s 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) OPT-56 Original Example Revisited Risk-Neutral = No Arbitrage S0 ƒ S0u = 22 ƒu = 1 S0d = 18 ƒd = 0 • Since p is a risk-neutral probability 20e0.12 0.25 = 22p + 18(1 – p ); p = 0.6523 • Alternatively, use the formula e rT d e 0.120.25 0.9 p 0.6523 ud 1.1 0.9 OPT-57 Valuing the Option S0 ƒ S0u = 22 ƒu = 1 S0d = 18 ƒd = 0 Value of the option: = e–0.12 x 0.25 [0.65231 + 0.34770] = 0.633 OPT-58 Relevance of Binomial model • Stock price only having 2 future price choices appears unrealistic • Consider: – Over a small time period, a stock’s 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 BlackScholes Option Pricing Model. OPT-59 Valuing Stock Options: The Black-Scholes Model OPT-60 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 OPT-61 BSOPM Assumptions m = expected return on the stock = volatility of the stock price Therefore in time Δt: μ Δt = mean of the return Dt = standard deviation and: DS ~ ( mDt , 2 Dt ) S OPT-62 The Lognormal Property • Assumptions → ln ST is normally distributed with mean: lnS0 ( m 2 / 2 )T and standard deviation: T • Because the logarithm of ST is normal, ST is lognormally distributed OPT-63 The Lognormal Property continued l nST ~ l nS0 ( m 2 2 )T , 2T or ST ln ~ ( m 2 2 )T , 2T S0 where m,v] is a normal distribution with mean m and variance v OPT-64 The Lognormal Distribution E ( ST ) S0 e mT var( ST ) S0 e 2 2 mT (e 2T 1) Restricted to positive values OPT-65 The Expected Return Expected value of the stock price S0emT Expected return on the stock with continuous compounding m – 2/2 Arithmetic mean of the returns over short periods of length Δt Geometric mean of returns m m – 2/2 OPT-66 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 OPT-67 Assumptions Underlying BSOPM 1. Stock price behavior corresponds to the lognormal model with μ and σ 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. OPT-68 Notation • c and p = European option prices (premiums) • S0 = stock price • K = strike or exercise price • r = risk-free rate • σ = volatility of the stock price • T = time to maturity in years OPT-69 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(-d1) = 1 - N(d1) OPT-70 The Black-Scholes Formulas c S0 N ( d 1 ) K e rT N( d 2 ) p K e rT N ( d 2 ) S0 N ( d 1 ) w here: d1 ln(S0 / K ) ( r 2 / 2 )T T d 2 d1 T OPT-71 BSOPM Example Given: S0 = $42 K = $40 d1 r = 10% T = 0.5 σ = 20% ln(S0 / K ) ( r 2 / 2 )T T ln(42 40 ) ( 0.10 0.20 2 2 ) 0.5 d1 0.7693 0.20 0.50 d 2 d1 T d 2 0.7693 0.20 0.50 0.6278 OPT-72 BSOPM Call Price Example d1 = 0.7693 N(0.7693) = 0.7791 d2 = 0.6278 N(0.6278) = 0.7349 c S0 N ( d 1 ) K e rT N ( d 2 ) c 40(0.7791) - 42e c $4.76 -.10.5 (0.7349) OPT-73 BSOPM Put Price Example d1 = 0.7693 N(-0.7693) = 0.2209 d2 = 0.6278 N(-0.6278) = 0.2651 pKe rT p 42 e N ( d 2 ) S0 N ( d 1 ) .10 .50 ( 0.2651 ) 40 ( 0.2209 ) p $0.81 OPT-74 BSOPM in Excel • N(d1): =NORMSDIST(d1) Note the “S” in the function “S” denotes “standard normal” ~ Φ(0,1) =NORMDIST() → Normal distribution Mean and variance must be specified OPT-75 Properties of Black-Scholes Formula • As S0 → Call Fwd with delivery = K Almost certain to be exercised d1 and d2 → very large N(d1) and N(d2) → 1.0 N(-d1) and N(-d2) → 0 c = max(S-K,0) c S0 N( d 1 ) K e rT N( d 2 ) p = max(K-S,0) p K e rT N( d 2 ) S0 N( d 1 ) c → S0 – Ke-rT p→0 OPT-76 Properties of Black-Scholes Formula • As S0 →0 d1 and d2 → very large & negative N(d1) and N(d2) → 0 N(-d1) and N(-d2) → 1.0 c = max(S-K,0) c S0 N( d 1 ) K e rT N( d 2 ) p = max(K-S,0) p K e rT N( d 2 ) S0 N( d 1 ) c→0 p → Ke-rT – S0 OPT-77 Real Options 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 OPT-79 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 (σ) – Interest rate is 10% per annum • Continuously compounded = ln(1.1) = 9.53% (r) OPT-80 Options Approach T K S r σ 1 $270.00 $300.00 9.53% 22.3% c S0 N ( d 1 ) Ke rT N ( d 2 ) p Ke rT N ( d 2 ) S0 N ( d 1 ) where : S/K ln(S/K) σ^2 Sqrt(T) 1.1111 0.1054 0.0497 1.0000 d1 (num) d1 (den) d1 N(d1) 0.2255 0.2230 1.0113 0.8441 d2 N(d2) 0.7883 0.7847 d1 ln( S0 / K ) ( r 2 / 2 )T T d 2 d1 T European Call c (component 1) = c (component 2)= c (per oz)= Option Value (c * 10,000) $ $ 253.22 $192.62 60.599 $605,993 OPT-81 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? OPT-82 Multiple Options T K S r σ 2 $270.00 $300.00 9.53% 22.3% c S0 N ( d 1 ) Ke rT N ( d 2 ) p Ke rT N ( d 2 ) S0 N ( d 1 ) where : S/K ln(S/K) σ^2 Sqrt(T) 1.1111 0.1054 0.0497 1.4142 d1 (num) d1 (den) d1 N(d1) 0.3457 0.3154 1.0961 0.8635 d2 N(d2) 0.7808 0.7825 d1 ln( S0 / K ) ( r 2 / 2 )T T d 2 d1 T European Call c (component 1) = c (component 2)= c (per oz)= Option Value (c * 20,000) $ $ 259.05 $174.62 84.429 $1,688,588 OPT-83 σ= t= 22.3% 1 Dt e u= d = 1/u = = 1.250 0.80 $375 x 1.25 = $469 $300 x 1.25 = $375 $375 x 0.80 = $300 $300 $375 x 1.25 = $469 $300 x 0.80 = $240 $375 x 0.80 = $300 OPT-84 If Expansion and S1 = 375 T K S(1) r σ 1 $280.00 $375.00 9.53% 22.3% c S0 N ( d 1 ) Ke rT N ( d 2 ) p Ke rT N ( d 2 ) S0 N ( d 1 ) where : S/K ln(S/K) σ^2 Sqrt(T) 1.3393 0.2921 0.0497 1.0000 d1 (num) d1 (den) d1 N(d1) 0.4123 0.2230 1.8489 0.9678 d2 N(d2) 1.6259 0.9480 d1 ln( S0 / K ) ( r 2 / 2 )T T d 2 d1 T European Call c (component 1) = c (component 2)= c (per oz)= Option Value (c * 12,500) $ $ 362.91 $241.31 121.596 $1,519,953 OPT-85 If No Expansion and S1 = 375 T K S(1) r σ 1 $270.00 $375.00 9.53% 22.3% c S0 N ( d 1 ) Ke rT N ( d 2 ) p Ke rT N ( d 2 ) S0 N ( d 1 ) where : S/K ln(S/K) σ^2 Sqrt(T) 1.3889 0.3285 0.0497 1.0000 d1 (num) d1 (den) d1 N(d1) 0.4487 0.2230 2.0120 0.9779 d2 N(d2) 1.7890 0.9632 d1 ln( S0 / K ) ( r 2 / 2 )T T d 2 d1 T European Call c (component 1) = c (component 2)= c (per oz)= Option Value (c * 10,000) $ $ 366.71 $236.42 130.286 $1,302,864 OPT-86 If Expansion and S1 = 240 T K S(1) r σ 1 $280.00 $240.00 9.53% 22.3% c S0 N ( d 1 ) Ke rT N ( d 2 ) p Ke rT N ( d 2 ) S0 N ( d 1 ) where : S/K ln(S/K) σ^2 Sqrt(T) 0.8571 -0.1542 0.0497 1.0000 d1 (num) d1 (den) d1 N(d1) -0.0340 0.2230 -0.1524 0.4394 d2 N(d2) -0.3754 0.3537 d1 ln( S0 / K ) ( r 2 / 2 )T T d 2 d1 T European Call c (component 1) = c (component 2)= c (per oz)= Option Value (c * 12,500) $ $ 105.46 $90.03 15.436 $192,945 OPT-87 If No Expansion and S1 = 240 T K S(1) r σ 1 $270.00 $240.00 9.53% 22.3% c S0 N ( d 1 ) Ke rT N ( d 2 ) p Ke rT N ( d 2 ) S0 N ( d 1 ) where : S/K ln(S/K) σ^2 Sqrt(T) 0.8889 -0.1178 0.0497 1.0000 d1 (num) d1 (den) d1 N(d1) 0.0024 0.2230 0.0107 0.5043 d2 N(d2) -0.2123 0.4159 d1 ln( S0 / K ) ( r 2 / 2 )T T d 2 d1 T European Call c (component 1) = c (component 2)= c (per oz)= Option Value (c * 10,000) $ $ 121.02 $102.09 18.930 $189,299 OPT-88 Real Options Recap S = price/oz K = extraction cost/oz Time Ounces per year Option Value Tab REAL OPTIONS RECAP S = $300 S = $375 $300 $300 375 375 $270 $270 280 270 1 2 1 1 10,000 20,000 12,500 10,000 $605,993 RO 1 $1,688,588 $1,519,953 $1,302,864 RO 2 RO 3 RO 3 S = $240 240 280 1 12,500 240 270 1 10,000 $192,945 $189,299 RO 4 RO 4 OPT-89 Conclusions S = price/oz K = extraction cost/oz Time Ounces per year Option Value REAL OPTIONS RECAP S = $300 S = $375 $300 $300 375 375 $270 $270 280 270 1 2 1 1 10,000 20,000 12,500 10,000 $605,993 Tab RO 1 $1,688,588 $1,519,953 $1,302,864 RO 2 RO 3 PV = $217,089 ($150,000) $67,089 $60,991 Value of Option to Expand Minus cost to expand S = $240 240 280 1 12,500 240 270 1 10,000 $192,945 $189,299 RO 4 RO 4 RO 3 $3,646 ($150,000) ($146,354) • If S1 = 375: – Value of option to expand = $217,089 – Subtracting cost of expansion and discounting to t=0 – Value = $60,991 • If S1 = 240, net value is negative OPT-90 Probability of Up Movement Dt • Know that u = e and d = 1/u p (1r f )d u d • 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 OPT-91 Probability of Up Movement REAL OPTIONS RECAP S = $300 S = price/oz $300 $300 K = extraction cost/oz $270 $270 Time 1 2 Ounces per year 10,000 20,000 Option Value $605,993 Dt d = 1/u p (1r f )d = = = 1.250 0.80 0.667 375 280 1 12,500 375 270 1 10,000 $1,688,588 $1,519,953 $1,302,864 Value of Option to Expand Minus cost to expand u=e S = $375 PV = $217,089 ($150,000) $67,089 $60,991 PV * p $40,680.75 2-yr + Option $1,729,269 u d OPT-92