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## Real Options

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**Real Options**“Life wasn’t designed to be risk-free. The key is not to eliminate, but to estimate it accurately and manage it wisely.” -William Schreyer, former Chair and CEO of Merrill Lynch**Options Are Everywhere!**• Callable Bonds • Convertible Securities • Warrants • Secure Loans • Firm with Debt • Exotics • Compensation • Real Options**Real versus Financial Options**• Transparency • Information • Contract details • Valuation?**What is a real option?**• The holder of an option has the right, butnot the obligation to do something • Some typical capital investment real options • Timing: Can the project occur sometime besides now or never? • Growth: Can you alter capacity? • Abandonment: Can you stop production before the end of economic life of the asset? • Flexibility: Can you alter operations?**Current State of Real Options**• Gaining traction in valuing corporate investments (but the slope is slippery) • About a third of the CFO’s “always” or “almost always” use real options (Graham & Harvey) • 10% to 15% of CFO’s use real options “always” or “often” (Ryan & Ryan) • 9% of senior executives use real options but about a third had stopped using (Bain & Co.)**Advantages of the Option Framework**• Creates a different way to think of how a firm can create shareholder value • Ties strategic decisions directly to maximizing shareholder wealth • May use capital market data • More closely models actual decision process • Encourages optimal growth opportunity investing, i.e. R&D, or infrastructure • Creates metric to better monitor and reward managers • Think less of the “most likely” cash flow and more of the distribution of cash flows**How do we deal with real options?**• Use NPV and assume option value is zero • Estimate expected cash flow • Each cash flow is usually the most likely case or the probability weighted average of the cash flows • Incorporate the riskiness of cash flows into r • Assume the project is a “now or never” opportunity • Use NPV and qualitatively recognize option • Same as above except recognize options exist • Without computing a value make a qualitative appraisal • Financially engineer a project specific model • Sky is the limit!**Another Method: Financial Option Models**• Assumptions which may be problematic for real options • Prices have various distributions • Least restrictive assumption: Brownian motion • No arbitrage: A replicating portfolio must exist • Marketed Asset Disclaimer (MAD): assumes the project without the option is the replicating portfolio • Binomial option pricing model (discrete time) • More flexible, can be difficult to model • Typically better economic description of real options • Black-Scholes variations (continuous time) • Requires stricter assumptions • Better economic description of certain financial options • Simple method for first pass valuation**Call Option Basics**• The buyer of an option has the right, butnot the obligation to buy an asset • The seller has a commitment to sell an asset • Underlying Asset Price (S) • Strike Price (X) • Expiration Date (T) • American: Exercise at any time • European: Only exercise at maturity**Call Option Value Basics**• Option Premium • Option Value • Intrinsic • At the expiration date: • ST > X value = • ST = X value =0 • ST < X value =0 • Time: At expiration date=0 • Premium=Intrinsic value+Time Value • Limited Liability ST - X**Long (Buying) a Call Payoff Diagram**Value of Option at expiration With Premium (P) $0 $-P Value of underlying asset (ST) at expiration $X $X+P**Selling (Writing/Short) a Call**• The seller has a commitment to sell an agreed amount of an asset at an agreed price on or before a specified future date. • Intrinsic value • At the expiration date: • ST > X value = • ST = X value =0 • ST < X value =0 • Unlimited liability -(ST - X)**Short Call Payoff Diagram**Value of Option at expiration $P With Premium (P) $0 Value of underlying asset (ST) at expiration $X $X+P**Buying Versus Selling**• Liability • Obligation versus Commitment**Put Option Basics**• The buyer of a put option has the right, butnot the obligation to sell an agreed amount of an asset at an agreed price on or before a specified future date. • Intrinsic Value • At the expiration date: • Value = Max [0 , X-ST ] • Value = X-ST if ST < X • Value = 0 if ST X • Limited liability**Long (Buying) a Put Payoff Diagram**Value of Option at expiration With Premium (P) $0 $-P Value of underlying asset (ST) at expiration $X-P $X**Selling a Put**• The buyer of a put option has the right, butnot the obligation to sell an asset • The seller has a commitment to buy an asset . • Intrinsic value • At the expiration date: • Value = -(X-ST ) if ST < X • Value = 0 if ST X • Unlimited liability**Short (Selling/Writing) a Put Payoff Diagram**Value of Option at expiration $P $0 With Premium (P) Value of underlying asset (ST) at expiration $X-P $X**How Do We Value?**• Option Value=Intrinsic+Time • Focus on Time Component • Time till Maturity (T) • What if the call option is out of the money? • Near the money? • In the money? • Exercise before expiration? • S0-PV(X)**Option Value**Value of a Call B Upper bound Lower bound C A Exercise price Share price**What are the determinants of value?**• Exercise price (X) • Time till expiration (T) • Underlying asset (S) • Interest rate (rf) • Variability of the value of the underlying asset…Volatility (sS)**Option Pricing Model**• Characterize the underlying asset price • Simple Binomial World: • 1 period (time t) • Two possible prices: uS0 and dS0 St=uS0=150 S0=100 St=dS0=50**Call Option Intrinsic Value**• Call Option (C) • Strike Price (X) 100 • Expires at t St=uS0=150 Ct=50 S0=100 C0=? St=dS0=50 Ct=0**What is the value of the call today?**• Assumption: No Arbitrage Opportunities • Establish an arbitrage portfolio of the option and underlying asset • This portfolio has a constant return • What is that return? • Portfolio • Short Call • Long stock**Portfolio Composition**• Portfolio: Short call and long stock • What proportions? • Say we short ONE call, now how many stocks do we buy (h)? • Remember the goal is constant payoff • Payoff in up state=payoff in down state • 150h-50=50h-0 h=0.5 • Generalize h for one period model: huS0=h150 1Ct=-50 S0=100 hdS0=h50 1Ct=0**Portfolio’s Value**• Self-financing portfolio • Total cash flows must equal zero • What’s the valuation equation? huS0=0.5(150)=75 1Ct=-50 Pt=25 S0=100 C0=? P0=? hdS0=0.5(50)=25 1Ct=0 Pt=25**Call Price**• Assume r=10% • hS0-C=PV(payoff) • hS0-C=[huS0-Cu]/(1+r) • (0.5)100-C=[75-50]/1.1 • (0.5)100-C=25/1.1 • C=50-22.73 • C=27.27**Generalization of the One Step Model**Substitute in the following for h: Implications:**Important Characteristics**• Backward induction method • Rollback values one period at a time • Arbitrage portfolio guarantees that the payoff is identical in all states of nature • No longer concerned which path the price on the underlying asset takes • Implies that you are not concerned about state dependent risk • Implies that all investors are risk neutral • Used in all derivatives pricing • Simplifying assumption • What is the appropriate r? • The risk free rate!**Risk Neutral Valuation**• Investors are risk-neutral (the probability of underlying asset moves don’t matter), they require 10% • The share price can increase/decrease by 50% • What is the probability of an increase/decrease? • p=[(1+r)-d]/(u-d)=[1.1-0.5]/(1.5-0.5)=0.6 • Intuitively: • 1+0.1 = P(increase) × 1.5 + P(decrease) × 0.5 • 1.1 = P(increase) × 1.5 + P(1- increase) × 0.5 • P(increase) =60% and P(decrease)=40% (Risk-neutral probabilities) • Given the call prices for the two scenarios: • [1/1.1] x [pCu+(1-p)Cd] • [1/1.1] x [0.60 × $50 + 0.40 × $0] = $27.27**Two Ways to Value**• No-Arbitrage Valuation: To value an option, you can take a levered position in the underlying asset that replicates the payoffs of the option. So you have to estimate the price of the replicating portfolio and the option. • Risk-Neutral Valuation: Assume investors do not care about risk, so that the expected return on the underlying asset is equal to the risk-free interest rate. Calculate the expected future value of the option then discount it to time 0. Only have to estimate stock and option prices (no probabilities!).**Two Step Binomial Model**Assume: u=1+0.5, d=1-0.5, r=0.10, t=1, S=100 and X=100 First: Build the stock prices from today forward by u and d. uuS0=225 C=? uS0=150 C=? udS0=75 C=? S0=100 C=? dS0=50 C=? ddS0=25 C=?**Second**At expiration, compute the intrinsic value of the option then one period at a time, recursively solve to time 0 (today) using the risk neutral probability (p). uuS0=225 C=125 uS0=150 C=68.18 S0=100 C=37.19 udS0=75 C=0 dS0=50 C=0 ddS0=25 C=0**Valuing a Put**• Remember: Intrinsic value of a put is Max[0,X-ST] • Intuitively, what is different for a put • Underlying stock price? • No difference • At maturity payoff (intrinsic value)? • Max [0,X-ST] instead of Max[0,ST-X] • Recursive solution • Risk neutral probability (p)? • No difference • Put computation (C=)? • No difference**Two Step Binomial Model: Put**Same assumptions as the call: u=1+0.5, d=1-0.5, r=0.10, t=1, S=100 and X=100 First: Build the stock prices from today forward by u and d. NO DIFFERENCE FROM THE CALL! uuS0=225 P=? uS0=150 P=? udS0=75 P=? S0=100 P=? dS0=50 P=? ddS0=25 P=?**Second: Put**At expiration, compute the intrinsic value of the option then one period at a time, recursively solve to time 0 (today) using the risk neutral probability (p). uuS0=225 P=0 uS0=150 P=9.09 S0=100 P=19.83 udS0=75 P=25 dS0=50 P=40.91 ddS0=25 P=75**Valuing an American Option**• Remember an American option can be exercised any time before (or at) maturity • European option can only be exercised at maturity • Intuitively, what is the difference? • Underlying stock price? • No difference • At maturity payoff? • No difference • Backward induction • Risk neutral probability (p)? • No difference • Put or Call computation • Partial difference: The same except at each node, take the higher of the option value or the exercise price (C=S-X, P=X-S)**Real Data**• Usually has many periods • Estimating r, t, S0 and X is relatively easy. • Estimating u and d are more difficult • Usually based on the volatility of the underlying asset. • Typical estimation problems: period, data frequency, etc • Assume Dt is the length of one period in the binomial model , s is the historical volatility over that same period.**Option Delta**• How many shares are needed to replicate an option? • Option Delta = Spread of Possible Option Prices / Spread of Possible Share Prices • Delta = (125 - 0) / (225 - 25)= 5/8 • Delta of a Put = Delta of a Call with the same exercise price minus 1**Black-Scholes versus Binomial Model**• Continuous versus Discrete Time • Assume stock price can be characterized by a continuous process • Special limiting case of Binomial as the length of the period approaches zero • Computationally easier • More restrictive assumptions • Several variations to account for different types of underlying assets**Black-Scholes Model**• European option on non-dividend paying asset • N(d) = cumulative normal density function • X = exercise price • t = number of periods to the expiration date • S = current stock price (underlying asset) • = standard deviation per period of the rate of return on the underlying asset (continuously compounded)**BS Example**• Value a 100 strike price call on ABC • 3 months (0.25 years) till expiration • 0.5 standard deviation • risk free rate/year is 0.04 • ABC is trading at $101 • Value a 100 strike price put on ABC**Option Delta**• First derivative of the option value with respect to the price of the asset (S) • Interpretation: If the price of the asset increases by $1 how much will the value of the option change?**Black-Scholes Variations**• American options on non-dividend paying assets • European options on assets that have • discrete (point in time before expiration) dividend • continuous proportion dividend such as index futures • European options on foreign currency • European options on futures and forwards • Etc.**DCF to Real Options**• Exercise Price (X)? • Initial investment (investment required to acquire the assets) • Underlying asset (S)? • PV (CF) excluding initial investment (value of the operating assets to be acquired) • Time to expiration (t)? • Length of time the choice is available • Volatility (s)? • Riskiness of underlying operating assets • Risk free rate? • Time value of money: If replicating portfolio exists the risk free rate. If not, the risk free rate will provide the upper bound.**DCF versus Real Options**• One method does not completely dominate • Both methods used correctly will give identical values • Each method frames the question differently • DCF • Works well for “assets in place” type of projects • Lower uncertainty, less decision nodes • Real Options • Works well for “growth option” type of projects • Greater span of possible cash flows, many decision nodes