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## Real Options

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**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**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**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.**Options on Common Stock**Identity of the underlying stock Strike or Exercise price Contract size Expiration date or maturity Exercise cycle American or European Delivery or settlement procedure**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**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**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**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**Payoff to Call Holder**(S- K) if S >K 0 if S< K Profit to Call Holder Payoff - Option Premium Profit =Max (S-K, 0) - C Option Payoffs & ProfitsCall Holder = Max (S-K,0)**Payoff to Call Writer**- (S - K) if S > K = -Max (S-K, 0) 0 if S < K = Min (K-S, 0) Profit to Call Writer Payoff + Option Premium Profit = Min (K-S, 0) + C Option Payoffs & ProfitsCall Writer**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]**Payoff & Profit Profiles for Calls**Call Holder Call Writer**Payoff & Profit Profiles for Calls**Payoff Call Holder Profit Profit 0 Call Writer Profit Stock Price**Payoffs to Put Holder**0 if 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)**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**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]**Payoff & Profit Profiles for Puts**Put Holder Put Writer**Payoff & Profit Profiles for Puts**Profits Put Writer Profit 0 Put Holder Profit Stock Price**Option Payoffs and Profits**CALLPUT 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”**Long Call**Call option premium (C) = $5, Strike price (K) = $100. Profit ($) 30 20 10 Terminal stock price (S) 70 80 90 100 0 110 120 130 -5 Long Call Profit = Max(S-K,0) - C**Short Call**Call option premium (C) = $5, Strike price (K) = $100 Profit ($) 110 120 130 5 0 70 80 90 100 Terminal stock price (S) -10 -20 -30 Short Call Profit = -[Max(S-K,0)-C] = Min(K-S,0) + C**Long Put**Put option premium (P) = $7, Strike price (K) = $70 Profit ($) 30 20 10 Terminal stock price ($) 0 40 50 60 70 80 90 100 -7 Long Put Profit = Max(K-S,0) - P**Short Put**Put option premium (P) = $7, Strike price (K) = $70 Profit ($) Terminal stock price ($) 7 40 50 60 0 70 80 90 100 -10 -20 -30 Short Put Profit = -[Max(K-S,0)-P] = Min(S-K,0) + P**Notation**c= European call option price (C = American) p= European put option price (P = American) S0 = Stock price today ST=Stock price at option maturity K= Strike price T= Option maturity in years = Volatility of stock price D = Present value of dividends over option’s life r=Risk-free rate for maturity Twith continuous compounding**American vs. European Options**An American option is worth at least as much as the corresponding European option Cc Pp**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**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**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**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**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**Upper Bound for Options**• Call price must be ≤ stock price: c ≤ S0 C ≤ S0 • Put price must be ≤ strike price: p ≤ KP ≤ K p ≤ Ke-rT**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.**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**Lower Bound for European Call PricesNon-dividend-paying**Stock cMax(S0 –Ke –rT,0) Portfolio A: 1 European call + Ke-rT cash Portfolio B: 1 share of stock**Lower Bound for European Put PricesNon-dividend-paying Stock**pMax(Ke -rT–S0,0) Portfolio C: 1 European put + 1 share of stock Portfolio D: Ke-rT cash**Put-Call ParityNo 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 9.43**Put-Call ParityAmerican Options**• Put-Call Parity holds only for European options. • For American options with no dividends:**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**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**Consider the Portfolio: Long D shares Short 1 call**option Portfolio is riskless when: 22D – 1 = 18D or D = 0.25 Setting Up a Riskless Portfolio 22D – 1 18D**Valuing the PortfolioRisk-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