The investment game in incomplete markets.

Transcription

1 The investment game in incomplete markets. M. R. Grasselli Mathematics and Statistics McMaster University RIO 27 Buzios, October 24, 27

2 Successes and imitations of Real Options Real options accurately describe the value of flexibility in decision making under uncertainty. According to a recent survey, 26% of COs in North America always or almost always consider the value of real options in projects. This is due to familiarity with the option valuation paradigm in financial markets and its lessons. But most of the literature in Real Options is based on different combinations of the following unrealistic assumptions: (1) infinite time horizon, (2) perfectly correlated spanning asset, (3) absence of competition. Though some problems have long time horizons (3 years or more), most strategic decisions involve much shorter times. The vast majority of underlying projects are not perfectly correlated to any asset traded in financial markets. In general, competition erodes the value of flexibility.

3 Alternatives The use of well known numerical methods (e.g finite differences) allows for finite time horizons. As for the spanning asset assumption, the absence of perfect correlation with a financial asset leads to an incomplete market. Replication arguments can no longer be applied to value managerial opportunities. The most widespread alternative to replication in the decision-making literature is to introduce a risk-adjusted rate of return, which replaces the risk free rate, and use dynamic programming. This approach lacks the intuitive understanding of opportunities as options. inally, competition is generally introduced using game theory. Surprisingly, game theory is almost exclusively combined with real options under the hypothesis of risk-neutrality!

4 A one period investment model Consider a two factor market where the discounted prices for the project V and a correlated traded asset S follow: (us, hv ) with probability p 1, (us (S T, V T ) =, lv ) with probability p 2, (1) (ds, hv ) with probability p 3, (ds, lv ) with probability p 4, where < d < 1 < u and < l < 1 < h, for positive initial values S, V and historical probabilities p 1, p 2, p 3, p 4. et the risk preferences be specified through an exponential utility U(x) = e γx. An investment opportunity is model as an option with discounted payoff C t = (V e rt I ) +, for t =, T.

5 European Indifference Price The indifference price for the option to invest in the final period as the amount π that solves the equation max E[U(x +H(S T S )] = max E[U(x π+h(s T S )] (2) H H Denoting the two possible pay-offs at the terminal time by C h and C l, the European indifference price is explicitly given by π = g(c h, C l ) (3) where, for fixed parameters (u, d, p 1, p 2, p 3, p 4 ) the function g : R R R is defined as g(x 1, x 2 ) = q ( ) γ log p 1 + p 2 p 1 e γx 1 + p2 e γx (4) q ( ) p 3 + p 4 log γ p 3 e γx 1 + p4 e γx, 2 with q = 1 d u d.

6 Early exercise When investment at time t = is allowed, it is clear that immediate exercise of this option will occur whenever its exercise value (V I ) + is larger than its continuation value π C. That is, from the point of view of this agent, the value at time zero for the opportunity to invest in the project either at t = or t = T is given by C = max{(v I ) +, g((hv e rt I ) +, (lv e rt I ) + )}.

7 A multi period model Consider now a continuous-time two factor market of the form ds t = (µ 1 r)s t dt + σ 1 S t dw dv t = (µ 2 r)v t dt + σ 2 V t (ρdw + 1 ρ 2 dz). We want to approximate this market by a discrete time processes (S n, V n ) following the one period dynamics (1). This leads to the following choice of parameters: u = e σ 1 t, h = e σ 2 t, d = e σ 1 t, l = e σ 2 t, p 1 + p 2 = e(µ 1 r) t d, p 1 + p 3 = e(µ 2 r) t l u d h l ρσ 1 σ 2 t = (u d)(h l)[p 1 p 4 p 2 p 3 ], supplemented by the condition p 1 + p 2 + p 3 + p 4 = 1.

8 Numerical Experiments - Act I We now investigate how the exercise threshold varies with the different model parameters. The fixed parameters are I = 1, r =.4, T = 1 µ 1 =.115, σ 1 =.25, S = 1 σ 2 =.2, V = 1 Given these parameters, the CAPM equilibrium expected rate of return on the project for a given correlation ρ is ( ) µ1 r µ 2 = r + ρ σ 2. (5) The difference δ = µ 2 µ 2 is the below equilibrium rate of return shortfall and plays the role of a dividend rate paid by the project, which we fix at δ =.4. σ 1

9 Known Thresholds In the limit ρ ±1 (complete market), the closed form expression for the investment threshold obtained in the case T = gives V DP = 2. This should be contrasted with the NPV criterion (that is, invest whenever the net present value for the project is positive) which in this case gives V NPV = 1. The limit γ in our model corresponds to the McDonald and Siegel (1986) threshold, obtained by assuming that investors are averse to market risk but neutral towards idiosyncratic risk. or our parameters, the adjustment to market risks is accounted by CAPM and this threshold coincides with V DP = 2

10 Dependence on Correlation and Risk Aversion != "=!=2 1.8!=8 1.8 "=.6 "= threshold !1! correlation threshold risk aversion igure: Exercise threshold as a function of correlation and risk aversion.

11 Dependence on Volatility and Dividend Rate !=!=.6!= !=!=.6!=.9 threshold threshold volatility " igure: Exercise threshold as a function of volatility and dividend rate.

12 Dependence on Time to Maturity 2 ow risk aversion!= Higher risk aversion!= threshold "= "=.6 threshold "= "= "= "= time to maturity time to maturity igure: Exercise threshold as a function of time to maturity.

13 Values for the Option to Invest (V!I) +!=!=.99.6 option value V igure: Option value as a function of underlying project value. The threshold for ρ = is and the one for ρ =.99 is

14 Suspension, Reactivation and Scrapping et us denote the value of an idle project by, an active project by 1 and a mothballed project by M. Then = option to invest at cost I 1 = cash flow + option to mothball at cost E M M = cash flow + option to reactivate at cost R + option to scrap at cost E S We obtain its value on the grid using the recursion formula k (i, j) = max{continuation value, possible exercise values}. As before, the decisions to invest, mothball, reactivate and scrap are triggered by the price thresholds P S < P M < P R < P H.

15 Numerical Experiments - Act II We calculate these thresholds by keeping track of three simultaneous grids of option values. The fixed parameters now are µ 1 =.12, σ 1 =.2, S = 1 σ 2 =.2, V = 1 r =.5, δ =.5, T = 3 I = 2, R =.79, E M = E S = C = 1, m =.1 ρ =.9, γ =.1

16 Dependence on Mothballing Sunk Cost 1.8 Thresholds Vs. Mothballing Sunk Cost, Increment Size: Thresholds Mothballing Sunk Cost

17 Dependence on Mothballing Running Cost 2.5 Thresholds Vs Mothballing Running Cost, Increment Size: Thresholds Mothballing Running Cost

18 Dependence on Correlation 1.6 Thresholds Vs Correlation, Increment Size: Thresholds !1!.8!.6!.4!

19 Dependence on Risk Aversion 5 Thresholds Vs Risk Aversion, Increment Size:.5, Rho fixed at Thresholds

20 Combining options and games or a systematic application of both real options and game theory in strategic decisions, we consider the following rules: 1. Outcomes of a given game that involve a wait and see strategy should be calculated by option value arguments. 2. Once the solution for a given game is found on a decision node, its value becomes the pay-off for an option at that node. In this way, option valuation and game theoretical equilibrium become dynamically related in a decision tree.

21 One period expansion option under monopoly Suppose now that a firm faces the decision to expand capacity for a product with uncertain demand: { hy with probability p Y 1 = ly with probability 1 p, (6) correlated with a traded asset The expansion requires a sunk cost I. The state of the firm after the investment decision at time t k is { 1 if the firm invests at time tk x(k) = (7) if the does not invest at time t k The cash flow per unit demand for the firm is denoted by D x(k).

22 The NPV solution If no expansion occurs at time t, then the value of the project at time t is v out = D Y + g(d hy, D ly ) = D Y + π (D Y 1 ). If expansion occurs, then the value of the project at time t is v in = (D 1 Y I ) + g(d 1 hy, D 1 ly ) = D 1 Y + π (D 1 Y 1 ). textcolorredif the decision needs to be taken at time t, then according to NPV the firm should expand provided v in v out, that is, if the sunk cost I is smaller then I NPV = (D 1 D )Y + (π (D 1 Y 1 ) π (D Y 1 )). (8)

23 The RO solution By contrast, if the decision to invest can be postponed until time t 1, then the value of the project when no investment occurs at time t is v wait = D Y + π (C 1 ), where C 1 denotes the random variable C 1 = C 1 (Y 1 ) = max{d Y 1, D 1 Y 1 I } D Y 1. Accordingly, the firm should invest at time t provided v in v wait, that is, if the sunk cost is smaller than I RO = (D 1 D )Y + (π (D 1 Y 1 ) π (C 1 )). (9) Since the function g is non-decreasing in each of its arguments, I NPV I RO = π (C 1 ) π (D Y 1 ). (1) That is, according to RO, the firm is less likely to expand at time t.

24 A multi-period investment game Consider two firms and each operating a project with an option to re-invest at cost I and increase cash flow according to an uncertain demand dy t = µ(t, Y t )dt + σ(t, Y t )dw. Suppose that the option to re-invest has maturity T, let t m, m =,..., M be a partition of the interval [, T ] and denote by (x (t m ), x (t m ) {(, ), (, 1), (1, ), (1, 1)} the possible states of the firms after a decision has been at time t m. et D xi (t m)x j (t m) denote the cash flow per unit of demand of firm i. Assume that D 1 > D 11 > D > D 1. We say that there is MA is (D 1 D ) > (D 11 D 1 ) and that there is SMA otherwise.

25 Derivation of project values (1) et V (x i (t m 1 ),x j (t m 1 )) i (t m, y) denote the project value for firm i at time t m and demand level y. Denote by v (x i (t m),x j (t m)) i (t m, y) the continuation values: v (1,1) (1,1) g(v i (t m+1, y u ), (V (1,1) i (t m+1, y d )) i (t m, y) = D 11 y t + e r t v (1,) (1,) g(v (t m+1, y u ), (V (1,) (t m+1, y d )) (t m, y) = D 1 y t + e r t v (,1) (,1) g(v (t m+1, y u ), (V (,1) (t m+1, y d )) (t m, y) = D 1 y t + e r t v (1,) (1,) g(v (t m+1, y u ), (V (1,) (t m+1, y d )) (t m, y) = D 1 y t + e r t v (,1) (,1) g(v (t m+1, y u ), (V (,1) (t m+1, y d )) (t m, y) = D 1 y t + e r t v (,) (,) g(v i (t m+1, y u ), (V (,) i (t m+1, y d )) i (t m, y) = D y t + e r t

26 Derivation of project values (2) or fully invested firms, the project values are simply given by V (1,1) i (t m, y) = v (1,1) i (t m, y). Now consider the project value for firm when has already invested and hasn t: V (1,) (t m, y) = max{v (1,1) (t m, y) I, v (1,) (t m, y)}. Similarly, the project value for when has invested and hasn t is V (,1) (t m, y) = max{v (1,1) (t m, y) I, v (,1) (t m, y)}.

27 Derivation of project values (3) Next consider the project value for when it has already invest and hasn t: { V (1,) v (1,1) (t m, y) = (t m, y) if v (1,1) (t m, y) I > v (1,) (t m, y), (t m, y) otherwise. v (1,) Similarly, the project value for when it has already invest and hasn t is { V (,1) v (1,1) (t m, y) = (t m, y) if v (1,1) (t m, y) I > v (,1) (t m, y), (t m, y) otherwise. v (,)

28 Derivation of project values (4) inally, the project values V (,) i are obtained as a Nash equilibrium, since both firms still have the option to invest. The pay-off matrix for the game is irm Invest Wait Invest (v (1,1) irm I, v (1,1) I ) (v (1,) I, v (1,) ) Wait (v (,1), v (,1) I ) (v (,), v (,) )

29 MA: dependence on risk aversion. 2 Gamma =.1 Rho =.5 2 Gamma =.1 Rho = !5!1!15 V V V11!I V1!I V1 5!5!1!15 V V V11!I V1!I V1! Demand! Demand 2 Gamma = 1 Rho =.5 2 Gamma = 2 Rho = !5!1!15 V V V11!I V1!I V1 5!5!1!15 V V V11!I V1!I V1! Demand! Demand igure: Project values in MA case for different risk aversions.

30 MA: dependence on correlation. 9 V Gamma = 1 V Gamma=1 Rho =.5 9 V Gamma = 1 V Gamma = 2 Rho = V V 2!.8!.6!.4! Correlation 3 V V 2!.8!.6!.4! Correlation 9 V Gamma = 2 V Gamma = 1 Rho = V V 2!.8!.6!.4! Correlation igure: Project values in MA case as function of correlation.

31 SMA: dependence on risk aversion 2 SMA Gamma=.1 2 SMA Gamma= !5!1!15 V V V11!I V1!I V1 5!5!1!15 V V V11!I V1!I V1! Demand! Demand 2 SMA Gamma=1 2 SMA Gamma = !5!1!15 V V V11!I V1!I V1! Demand 5!5!1!15 V V V11!I V1!I V1! Demand igure: Project values in SMA case for different risk aversions.

32 SMA: dependence on correlation. 2 V Gamma = 1 V Gamma = 1 Y=15 2 V Gamma = 1 V Gamma = 2 Y= V V!.8!.6!.4! Correlation V V!.8!.6!.4! Correlation 2 V Gamma = 2 V Gamma = 1 Y= V V!.8!.6!.4! Correlation igure: Project values in SMA case as function of correlation.

33 SMA x MA 2 MA/SMA Gamma=.1 2 MA/SMA Gamma= !5!1!15 V MA V MA V11!I V1 V SMA V SMA 5!5!1!15 V MA V MA V11!I V1 V SMA V SMA! Demand! Demand 2 MA/SMA Gamma=1 2 MA/SMA Gamma= !5!1!15 V MA V MA V11!I V1 V SMA V SMA 5!5!1!15 V MA V MA V11!I V1 V SMA V SMA! Demand! Demand igure: Project values for MA and SMA.