How To Value Real Options

Transcription

1 FIN 673 Pricing Real Options Professor Robert B.H. Hauswald Kogod School of Business, AU From Financial to Real Options Option pricing: a reminder messy and intuitive: lattices (trees) elegant and mysterious: Black-Scholes-Merton Option theory in corporate finance? managerial flexibility and projects as options strategy as a collection of options Key concepts: arbitrage ideas risk-adjusted probabilities risk-neutral pricing 1/24/211 Real-Options Pricing Robert B.H. Hauswald 2

2 Having the Cake and Eat It, too Options confer contractual rights on holder: a right to buy (sell) a fixed amount of currency at (over) a specified time (period) in the future at a price specified today Insurance vs. fixed commitment: right to buy or sell at discretion of holder wait and see security: even over time have an opinion while cutting off catastrophes Right means choice: choice means value 1/24/211 Real-Options Pricing Robert B.H. Hauswald 3 A Short Options Menu: Review Style: European or American exercisable at maturity only (e) or any time (a) Type: the right to buy (call) or to sell (put) corporate: growth call, retrenchment put Underlying: financial markets: spot or futures corporate finance: real asset, firm value Parties: buyer (holder), seller (writer) 1/24/211 Real-Options Pricing Robert B.H. Hauswald 4

3 Pricing Terminology: Review Three price elements: current price of underlying asset: spot, futures, forward strike (exercise): price at which transaction occurs (option) premium: the option s price itself Price location: at/in/out-of-the-money options at: current spot strike in: option profitable if exercised immediately out: option could not be profitably exercised Intrinsic value: extent to which an option is in-themoney (profit of immediate exercise) 1/24/211 Real-Options Pricing Robert B.H. Hauswald 5 Pricing Real Options How do I take into account the risk of the real options? Does it matter if the underlying asset is traded in financial markets? How do I go about implementing a real options model? What are the limitations of real options analysis? 1/24/211 Real-Options Pricing Robert B.H. Hauswald 6

4 Compare Projects The insight: all projects or assets with similar risk should have similar returns The challenge: find a twin security or project - and use its rate of return The alternative: use a standard equilibrium theory that relates risk to return (economics) The strategy: In the case of options, need to find a replicating portfolio that has the same risk The concept: no need to appeal to no-arbitrage 1/24/211 Real-Options Pricing Robert B.H. Hauswald 7 Futures Market? If there is a futures market for the underlying product (e.g. oil), then PV is readily computed it is simply today s price of the product (adjusted for a convenience yield ) times the volume. If we don t have a futures market, we need to find the appropriate rate of return for the underlying project (the firm s cost of capital perhaps) risk-neutral valuation 1/24/211 Real-Options Pricing Robert B.H. Hauswald 8

5 Implementing Real Options Analysis There are three different approaches to value real options: formulaic approach (e.g. Black-Scholes) lattice model (e.g. binomial model) Monte-Carlo Simulation Formulas are easy to implement, but they have limited applicability and are very much black boxes the other two approaches are more viable in general 1/24/211 Real-Options Pricing Robert B.H. Hauswald 9 Price Determinants Current spot price, (dividend and) interest rate futures or forward: by C&C from spot price and interest rates foregone revenue in real option: dividend yield Exercise price Time to maturity: length of period to expiration Underlying price process: volatility Type: European or American A right: use probability theory to evaluate contingencies Prerequisite: a model of the underlying asset value Distributional assumption: the spot (forward) price s logarithmic change is normally distributed 1/24/211 Real-Options Pricing Robert B.H. Hauswald 1

6 Pricing European Options: BSM Apply the classics: modify the seminal work of Black, Scholes and Merton to calculate theoretically fair prices Pricing formula: call c d t 1 [ St N( d1) exp{ r( T t) } KN ( d2 )] 2 1 [ log( S K ) + ( r + 2)( T t) ], d d σ T t p t σ 2 1 σ T t [ S N( d ) + { r( T t) } KN( )] Put: t t 1 exp d 2 Interpretation: payoffs S - K and K - S weighted by discount factor: future strike and spot probability of prices realizations: expected values PCP - put-call-parity: fundamental arbitrage equation 1/24/211 Real-Options Pricing Robert B.H. Hauswald 11 Black-Scholes-Merton Example Assumes option is European - exercise only at the option s maturity date e.g. residual value guarantee on a machine - a put option which can be only exercised at T. Value at T (forward price) 48 d1.98 Guarantee level 5 d Maturity (years) 5 N(-d1).461 Volatility of Value at T.4 N(-d2).619 Annual interest rate.5 P (X N(-d2) - F N(-d1)) exp(-rt) /24/211 Real-Options Pricing Robert B.H. Hauswald 12

7 Black-Scholes with Dividends Dividends are a form of asset leakage if dividend are paid repeatedly, adjust B-S-M to allow for constant proportional dividends: c t Se S δ where and -δt N(d N(d S δ ) Ke ) Ke Se δt rt ln (S d rt δ N(d N(d / 2 ]t and δ is a constant dividend 1/24/211 Real-Options Pricing Robert B.H. Hauswald 13 3 σ 3 σ σ /K) + [r + σ t t ) t ) 2 yield Perpetual Options: Infinite-Horizon Consider an opportunity to develop a piece of land: V X P γ 1 P* γ γ, P* X, γ γ 1 function( σ, r, δ ) Value of developed land 1 Gamma Cost of development 1 P* Annual Volatility.2 Annual interest rate.6 Option Value 27.7 Annual "Dividend Yield".45 1/24/211 Real-Options Pricing Robert B.H. Hauswald 14

8 Lattice Methods: Trees Most common is the binomial model one up or down movement at a time workhorse of the financial industry: pricing American options Solve by starting at the end and working backwards time honored principle: dynamic programming (engineering), backward induction Probabilities in the lattice have been adjusted to reflect risk of underlying variable; discount at risk-free rate pricing theory For example, an option to invest in a project 1/24/211 Real-Options Pricing Robert B.H. Hauswald 15 Three Period Binomial Option Pricing Example: Review There is no reason to stop with just two periods: generalize to three, four, periods The principles are the same: find q construct the underlying asset value lattice working forward construct the option value working backward Find the value of a three-period at-the-money call option written on a $25 stock that can go up or down 15 percent each period when the risk-free rate is 5% 1/24/211 Real-Options Pricing Robert B.H. Hauswald 16

9 Stock Price Lattice $ 25. (1.15) /3 $25 1/ $ 25. (1.15) $ 25. (1.15) 1/3 1/24/211 Real-Options Pricing Robert B.H. Hauswald /3 $ 25. (1.15) $ 25. (1.15)(1.15) 1/ /3 $ 25. (1. 15) 1/ /3 2 $25. (1.15) (1.15) 1/3 2/ $ 25. (1.15) (1.15) 1/ /3 $ 25. (1.15) 3 2 Risk-Neutral Probabilities: Review S(), V() q 1- q S(U), V(U) V () S(D), V(D) q V ( U ) + (1 q) V ( D) (1 + ) r f The key to finding q is to note that it is already impounded into an observable security price: the value of S() q S( U ) + (1 q) S( D) S() (1 + r f ) (1 + rf ) S() S( D) A minor bit of algebra yields: q S( U ) S( D) 1/24/211 Real-Options Pricing Robert B.H. Hauswald 18

10 Call Option Lattices C3( U, U, U) max[$38.2 $25,] $ (1 3) $3.1 C2( U, U ) 2/ (1.5) C1( U ) C3( D, U, U ) $ (1 3) $1.97 C3( U, D, U ) C3( U, U, D) 2/ (1.5) 1/3 max[$28.1 $25,] C ( 2 U, D) C2( D, U ) /3 2 3 $3.1 + (1 3) $ 6.5 2/3 3.1 C1( D) 1/3 (1.5) C (,, ) $25 3 U D D 2 3 $ (1 3) $ C3( D, U, D) C3( D, D, U ) 4.52 (1.5) 2/ /3 1/3 max[$2.77 $25,] C2( D, D) $ + (1 3) $ /3 (1.5) 1/3 C,, ) ( D D D 2 3 $6.5 + (1 3) $1.25 max[$15.35 $25,] C 1/3 (1.5) /24/211 Real-Options Pricing Robert B.H. Hauswald 19 Risk-Neutral Valuation in Practice S(), V() q 1- q r (1 + r d e f f ) d u d u d S(U), V(U) S(D), V(D) Use observed volatility to determine size of up and down steps and generate value lattice: fit model to observed uncertainty! Some more algebra yields (note continuous compounding!): (1 + rf ) S() S( D) (1 + rf ) S() ds() q S( U ) S( D) us() ds() σ u e S ( U ) us σ t ( ) q S( U ) + (1 q) S( D) S() (1 + ) ( D) ds ( ) S 1 σ t, d e ; u d? u 1/24/211 Real-Options Pricing Robert B.H. Hauswald 2 r f

11 Binomial Real Options 1. Calculate PV of project s value (net cash flows) taking future financial strategy (WACC) as given: V S 2. Find appropriate risk-free interest rate: r 3. Determine the required investment amount(s): K 4. Model current asset value (cash flow) uncertainty ~ 2 r d r ~ r t σ t 1, σ : r e, u e, d q u u d 5. Build cash flow and associated option value lattices 6. Recover object of interest (c, V, K); extend model 1/24/211 Real-Options Pricing Robert B.H. Hauswald 21 Binomial Model: Example Parameter Inputs Project value 1 Exercise Price 1 Maturity (years) 2 Annual Volatility.3 Annual interest rate.7 Number of Periods 4 Step Size (T/N).5 Annual lost revenues.4 Exercise Price at Maturity 1 Risk-Neutral Probabilities u d rhat dhat q ((rhat/dhat) - d)/(u-d) Lattice for the Underlying Project Value Date Jun-97 Jun-98 Jun-99 Downs/Period Lattice for the Option Value Date Jun-97 Jun-98 Jun-99 Downs/Period /24/211 Real-Options Pricing Robert B.H. Hauswald 22

12 From Option to Project Valuation Project Variable Call Option Required expenditure X, K Strike, exercise price Operating value of assets S, F Price of underlying asset (spot, futures, forward) Length of time to final decision Riskiness of operating CFs 2 σ t, T-t Time to expiration ~ r d u, d q u d Variance of underlying asset s return, price, etc. Time value of money r Default risk-free rate of return 1/24/211 Real-Options Pricing Robert B.H. Hauswald 23 Monte-Carlo Simulation Some applications involve options that are pathdependent their values depend on the particular path of cash flows (not just the lattice node at some point in time) Compound options: options on options feasibility study to build prototype with new technology Monte-Carlo simulation: somewhat similar to scenario analysis can properly account for path probabilities and risk works in a forward, rather than backward, fashion 1/24/211 Real-Options Pricing Robert B.H. Hauswald 24

13 Implementation Challenges Multiple sources of uncertainty model the interaction of risks Modeling resolution of uncertainty what is learned when: from decision to value trees Estimating inputs volatility distribution of underlying cost of capital on underlying dividend yields: lost revenue (in %) 1/24/211 Real-Options Pricing Robert B.H. Hauswald 25 Benefits of Option Analysis Plausibility check: initial outlay, PV of project, later investment(s) given any two, back out the third Pricing business and financial strategy warm and fuzzy becomes cold and hard Corporate finance: ANPV PNPV +π why overpay? paying a premium now amounts to what? applications: resource extraction, growth, synergy, R&D, governance (abandonment, cash out ) options 1/24/211 Real-Options Pricing Robert B.H. Hauswald 26

14 Who is Using Real Options? A survey of 4 CFOs reported that Twenty seven percent of CFOs said that they always use real options to analyze large projects. (Graham and Harvey, The theory and practice of corporate finance, Journal of Financial Economics 6, May/June 21: ) Industries applying real-options analysis (no order): pharmaceuticals, petrochemicals, aerospace, power generation, mineral extraction, finance, real estate, electronics, forest products, telecommunications, metallurgy, oil and gas, etc. 1/24/211 Real-Options Pricing Robert B.H. Hauswald 27 Capital Budgeting Techniques How freqently does your firm use the following techniques when deciding which project or acquisition to pursue? Source: Graham Harvey JFE 21 n 392 IRR NPV Hurdle rate Payback Evaluation technique Sensitivity analysis P/E multiple Discounted payback Real options Book rate of return Simulation analysis Profitability index APV.% 1.% 2.% 3.% 4.% 5.% 6.% 7.% 8.% % always or almost always 1/24/211 Real-Options Pricing Robert B.H. Hauswald 28

15 Binomial Real Options: Appendix The four most common types of real options 1. The opportunity to make follow-up investments. 2. The opportunity to abandon a project 3. The opportunity to wait and invest later. 4. The opportunity to vary the firm s output or production methods. Recall the relationship between active and passive NPV: Value Real Option NPV with option - NPV without option 1/24/211 Real-Options Pricing Robert B.H. Hauswald 29 Option to Wait Intrinsic Value identifies it as what type of option? Option Price Asset Price 1/24/211 Real-Options Pricing Robert B.H. Hauswald 3

16 Intrinsic Value + Speculative Value Option Value Speculative (time) Value Value of being able to wait Option Price Option to Wait Asset Price 1/24/211 Real-Options Pricing Robert B.H. Hauswald 31 More time More value Option to Wait Option Price Asset Price 1/24/211 Real-Options Pricing Robert B.H. Hauswald 32

17 Option to Abandon: Put Option Dalby Airways Ltd is considering the purchase of a turboprop aeroplane for its business. If the business fails, an option exists to sell the aeroplane for $5,; current value of plane is $553, Risk-free rate: 5% Given the following decision tree of possible outcomes what is the value of the offer (i.e. the put option) and what is the most Dalby Airways should pay for the option? Difference between decision tree and valuation lattice? 1/24/211 Real-Options Pricing Robert B.H. Hauswald 33 Decision Tree: Not a Valuation Lattice, yet Year Month 6 Month (22.6%) 679 (22.6%) (18.4%) PV (22.6%) 451 (18.4%) 368 (18.4%) 1/24/211 Real-Options Pricing Robert B.H. Hauswald 34

18 Intrinsic Value After 12 months Option value exercise price - asset value Example: (or $132 ) Year Month 6 Month () PV () (132) 1/24/211 Real-Options Pricing Robert B.H. Hauswald 35 Valuation Lattice: Extract RN Q After 6 months: Probability of an up-movement Note: movements expressed in rates of change, not our usual up, down and interest factors (same expression, though) q ( interest rate - downside change) ( upside change - downside change) Example: ( (-18. 4) ) ( (-18. 4) ) q. 51 1/24/211 Real-Options Pricing Robert B.H. Hauswald 36

19 Option Value: 6M After 6 months Example: If firm value in month 6 is $451, the option value is: (.51)() + (.49)(132) $65 Value at month 6: discount back 65/1.25 $63 Year Month 6 Month () 832 () NPV () 451(63) 368 (132) 1/24/211 Real-Options Pricing Robert B.H. Hauswald 37 Now Option Value: 12M Expected return (.51)() + (.49)(63) $31 Value today: discount back 31/1.25 $3 Year Month 6 Month () 679 () NPV 553 (3) 553 () 451(63) 368 (132) 1/24/211 Real-Options Pricing Robert B.H. Hauswald 38

20 Corporate Options Decision trees for valuing real options in a corporate setting cannot be practically done by hand. Introduce binomial & B-S-M models Calibrate parameters to observed quantities investment projects corporate strategies synergies from M&A or corporate cooperation 1/24/211 Real-Options Pricing Robert B.H. Hauswald 39 Binomial Pricing 1 + upside change u e σ h Where: σ 1 + downside change d h time as a fraction of a year 1 u standard deviation of annual returns on asset ( t) 1/24/211 Real-Options Pricing Robert B.H. Hauswald 4

21 Binomial Example Investment Project Price 36 σ.4 t 3 days Exercise Price 4 r 1% Maturity 9 days (.4) u e d /24/211 Real-Options Pricing Robert B.H. Hauswald 41 Binomial Pricing 4.37 V u Vu /24/211 Real-Options Pricing Robert B.H. Hauswald 42

22 Binomial Pricing 4.37 V u Vu V d Vd /24/211 Real-Options Pricing Robert B.H. Hauswald 43 Binomial Pricing 5.78 price V t u Vt /24/211 Real-Options Pricing Robert B.H. Hauswald 44

23 Binomial Pricing 5.78 price 1.78 intrinsic value /24/211 Real-Options Pricing Robert B.H. Hauswald 45 The greater of Binomial Pricing price intrinsic value qu Vut qd Vdt+ 1 with VuT IVuT 1 + r ( ) + ( ) /24/ Real-Options Pricing Robert B.H. Hauswald 46

24 V q V q V 1 + r u sut d sdt+ 1 st Binomial Pricing price intrinsic value /24/211 Real-Options Pricing Robert B.H. Hauswald 47 Binomial vs. Black-Scholes Expanding the binomial model to allow more possible price changes 1 step 2 steps 4 steps (2 outcomes) (3 outcomes) (5 outcomes) 1/24/211 Real-Options Pricing Robert B.H. Hauswald 48

25 Binomial vs. Black-Scholes How estimated call price changes as number of binomial steps increases No. of steps Estimated value Black-Scholes 4.5 1/24/211 Real-Options Pricing Robert B.H. Hauswald 49