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Real Options Dr. Lynn Phillips Kugele FIN 431

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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

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OPT-3 Mechanics of Options Markets

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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

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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.

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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

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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

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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

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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

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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

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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)

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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

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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]

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OPT-14 Payoff & Profit Profiles for Calls Call Holder Call Writer

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OPT-15 Payoff & Profit Profiles for Calls Profit Stock Price 0 Call Writer Profit Call Holder Profit Payoff

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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)

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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

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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]

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OPT-19 Payoff & Profit Profiles for Puts Put Writer Put Holder

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OPT-20 Payoff & Profit Profiles for Puts 0 Profits Stock Price Put Writer Profit Put Holder Profit

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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

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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

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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

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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

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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

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OPT-26 Properties of Stock Options

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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 option’s life r = Risk-free rate for maturity T with continuous compounding

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OPT-28 American vs. European Options An American option is worth at least as much as the corresponding European option C c P p

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OPT-29 Factors Influencing Option Values

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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

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OPT-31 Option Price Quotes Calls

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OPT-32 Option Price Quotes Puts

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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

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OPT-34 Option Price Quotes

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OPT-35 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

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OPT-36 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

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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

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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

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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.

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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

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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

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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

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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

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OPT-44 Put-Call Parity American Options Put-Call Parity holds only for European options. For American options with no dividends:

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OPT-45 Introduction to Binomial Trees

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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

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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.

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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

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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

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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

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OPT-51 Generalization – 1-Step Tree S0u ƒuS0u ƒu S0d ƒdS0d ƒd S0ƒS0ƒ At t=0After move up After move down Stock price S0S0 S0uS0uS0dS0d Option price ffufu fdfd u> 1 d < 1

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OPT-52 Generalization (continued) Consider the portfolio that is long shares and short 1 derivative The portfolio is riskless when: S 0 u – ƒ u = S 0 d – ƒ d or S 0 u – ƒ u S 0 d – ƒ d

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OPT-53 Generalization (continued) Value of the portfolio at time T is: S 0 u – ƒ u Cost to set up the portfolio today: S 0 – f = Value of the portfolio today today: S 0 – f = (S 0 u – ƒ u )e –rT Hence ƒ = S 0 ue -rT )+ƒ u e –rT

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OPT-54 Generalization (continued) Substituting for we obtain ƒ = e –rT [ p ƒ u + (1 – p )ƒ d ] where

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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:

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OPT-56 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)

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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 ƒ u = 1 S 0 d = 18 ƒ d = 0 S0 ƒS0 ƒ p (1 – p )

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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 ƒ u = 1 S 0 d = 18 ƒ d = 0 S0ƒS0ƒ 0.6523 0.3477

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OPT-59 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 Black- Scholes Option Pricing Model.

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OPT-60 Valuing Stock Options: The Black-Scholes Model

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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

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OPT-62 BSOPM Assumptions = expected return on the stock = volatility of the stock price Therefore in time Δt: μ Δt= mean of the return = standard deviation and:

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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

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OPT-64 The Lognormal Property continued where m, v ] is a normal distribution with mean m and variance v

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OPT-65 The Lognormal Distribution Restricted to positive values

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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 Δt Geometric mean of returns – 2 /2

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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

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OPT-68 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.

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OPT-69 Notation c and p = European option prices (premiums) S 0 = stock price K = strike or exercise price r = risk-free rate σ = volatility of the stock price T = time to maturity in years

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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 )

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OPT-71 The Black-Scholes Formulas

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OPT-72 BSOPM Example Given: S 0 = $42r = 10%σ = 20% K = $40T = 0.5

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OPT-73 BSOPM Call Price Example d 1 = 0.7693N(0.7693) = 0.7791 d 2 = 0.6278N(0.6278) = 0.7349

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OPT-74 BSOPM Put Price Example d 1 = 0.7693N(-0.7693) = 0.2209 d 2 = 0.6278N(-0.6278) = 0.2651

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OPT-75 BSOPM in Excel N(d 1 ): =NORMSDIST(d 1 ) Note the “S” in the function “S” denotes “standard normal” ~ Φ(0,1) =NORMDIST() → Normal distribution Mean and variance must be specified

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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)

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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)

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Real Options

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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

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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 (σ) –Interest rate is 10% per annum Continuously compounded = ln(1.1) = 9.53% (r)

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OPT-81 Options Approach

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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?

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OPT-83 Multiple Options

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OPT-84

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OPT-85 If Expansion and S 1 = 375

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OPT-86 If No Expansion and S 1 = 375

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OPT-87 If Expansion and S 1 = 240

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OPT-88 If No Expansion and S 1 = 240

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OPT-89 Real Options Recap

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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

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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

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OPT-92 Probability of Up Movement

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Fundamentals of Futures and Options Markets, 5 th Edition, Copyright © John C. Hull 2004 11.1 Introduction to Binomial Trees Chapter 11.

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