The investment game in incomplete markets

The investment game in incomplete markets M. R. Grasselli Mathematics and Statistics McMaster University Pisa, May 23, 2008

Strategic decision making We are interested in assigning monetary values to strategic decisions. Traditional, these include the decision to: create a new firm;

Strategic decision making We are interested in assigning monetary values to strategic decisions. Traditional, these include the decision to: create a new firm; invest in a new project;

Strategic decision making We are interested in assigning monetary values to strategic decisions. Traditional, these include the decision to: create a new firm; invest in a new project; start a real estate development; finance R&D;

Many other non-financial decisions can be addressed in the same framework. For instance, the decision to: enroll in an MBA program;

Many other non-financial decisions can be addressed in the same framework. For instance, the decision to: enroll in an MBA program; get married;

Many other non-financial decisions can be addressed in the same framework. For instance, the decision to: enroll in an MBA program; get married; change the constitution of a country;

Many other non-financial decisions can be addressed in the same framework. For instance, the decision to: enroll in an MBA program; get married; change the constitution of a country; introduce environmental laws;

Elements of Valuation In all of the previous problems, we can identify the following common elements: uncertainty about the future; To account for these elements, we are going to base our decisions on values obtained using the following theoretical tools:

Elements of Valuation In all of the previous problems, we can identify the following common elements: uncertainty about the future; some degree of irreversibility; To account for these elements, we are going to base our decisions on values obtained using the following theoretical tools:

Elements of Valuation In all of the previous problems, we can identify the following common elements: uncertainty about the future; some degree of irreversibility; timing flexibility; interaction with other people s decisions. To account for these elements, we are going to base our decisions on values obtained using the following theoretical tools:

Successes and Limitations of Real Options Real options accurately describe the value of flexibility in decision making under uncertainty.

Successes and Limitations of Real Options Real options accurately describe the value of flexibility in decision making under uncertainty. According to a recent survey, 26% of CFOs 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 (30 years or more), most strategic decisions involve much shorter times.

Alternatives The use of well known numerical methods (e.g finite differences) allows for finite time horizons.

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.

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 0, hv 0 ) with probability p 1, (us (S T, V T ) = 0, lv 0 ) with probability p 2, (1) (ds 0, hv 0 ) with probability p 3, (ds 0, lv 0 ) with probability p 4, where 0 < d < 1 < u and 0 < l < 1 < h, for positive initial values S 0, V 0 and historical probabilities p 1, p 2, p 3, p 4. Let the risk preferences be specified through an exponential utility U(x) = e γx.

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 0 )] = max E[U(x π+h(s T S 0 )+C T ] H H

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 0 )] = max E[U(x π+h(s T S 0 )+C T ] 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 ) (2) 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 (3) 2 + 1 q ( ) p 3 + p 4 log γ p 3 e γx 1 + p4 e γx, 2 with q = 1 d u d.

Early exercise When investment at time t = 0 is allowed, it is clear that immediate exercise of this option will occur whenever its exercise value (V 0 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 = 0 or t = T is given by C 0 = max{(v 0 I ) +, g((hv 0 e rt I ) +, (lv 0 e rt I ) + )}.

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).

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.

Numerical Experiments We now investigate how the exercise threshold varies with the different model parameters.

Numerical Experiments We now investigate how the exercise threshold varies with the different model parameters. The fixed parameters are I = 1, r = 0.04, T = 10 µ 1 = 0.115, σ 1 = 0.25, S 0 = 1 σ 2 = 0.2, V 0 = 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. (4) 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 = 0.04. σ 1

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.

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 0 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.

Dependence with Correlation and Risk Aversion 2 1.9 1.8!=0.5!=2!=8 2 1.9 1.8 "=0 "=0.6 "=0.9 1.7 1.7 1.6 1.6 threshold 1.5 threshold 1.5 1.4 1.4 1.3 1.3 1.2 1.2 1.1 1.1 1!1!0.5 0 0.5 1 correlation 1 0 2 4 6 8 10 risk aversion

Dependence with Volatility 5.5 5.5 5.5 5 4.5!=0!=0.6!=0.9 5 4.5!=0!=0.6!=0.9 5 4.5!=0!=0.6!=0.9 4 4 4 threshold 3.5 3 threshold 3.5 3 threshold 3.5 3 2.5 2.5 2.5 2 2 2 1.5 1.5 1.5 1 0.4 0.6 0.8 volatility 1 0.4 0.6 0.8 volatility 1 0.4 0.6 0.8 volatility

Dependence with Dividend Rate 14 12 10 threshold 8 6 "=0 "=0.6 "=0.9 4 2 0!0.05 0 0.05 0.1 0.15 0.2 0.25 0.3!

Dependence with Time to Maturity 1.5 Low risk aversion!=0.5 1.5 Higher risk aversion!=4 1.45 1.4 1.45 1.4 "=0 "=0.6 "=0.9 1.35 1.35 threshold 1.3 1.25 "=0 "=0.6 "=0.9 threshold 1.3 1.25 1.2 1.2 1.15 1.15 1.1 10 20 30 40 50 time to maturity 1.1 10 20 30 40 50 time to maturity

Values for the option to invest 0.5 0.4!=0!=0.99 (V!I) + option value 0.3 0.2 0.1 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 V Figure: Option value as a function of underlying project value. The threshold for ρ = 0 is 1.1972 and the one for ρ = 0.99 is 1.7507.

Combining options and games For a systematic application of both real options and game theory in strategic decisions, we consider the following rules:

Combining options and games For 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.

One period expansion option under monopoly Suppose now that a firm faces the decision to expand capacity for a product with uncertain demand: { hy0 with probability p Y 1 = ly 0 with probability 1 p, (5) 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) = (6) 0 if the does not invest at time t k

The NPV solution If no expansion occurs at time t 0, then the value of the project at time t 0 is v out = D 0 Y 0 + g(d 0 hy 0, D 0 ly 0 ) = D 0 Y 0 + π 0 (D 0 Y 1 ).

The NPV solution If no expansion occurs at time t 0, then the value of the project at time t 0 is v out = D 0 Y 0 + g(d 0 hy 0, D 0 ly 0 ) = D 0 Y 0 + π 0 (D 0 Y 1 ). If expansion occurs, then the value of the project at time t 0 is v in = (D 1 Y 0 I ) + g(d 1 hy 0, D 1 ly 0 ) = D 1 Y 0 + π 0 (D 1 Y 1 ). textcolorredif the decision needs to be taken at time t 0, 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 0 )Y 0 + (π 0 (D 1 Y 1 ) π 0 (D 0 Y 1 )). (7)

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 0 is v wait = D 0 Y 0 + π 0 (C 1 ), where C 1 denotes the random variable C 1 = C 1 (Y 1 ) = max{d 0 Y 1, D 1 Y 1 I } D 0 Y 1. Accordingly, the firm should invest at time t 0 provided v in v wait, that is, if the sunk cost is smaller than I RO = (D 1 D 0 )Y 0 + (π 0 (D 1 Y 1 ) π 0 (C 1 )). (8)

One period duopoly Consider two firms L and F each operating a project with an option to re-invest at cost I and increase cash flow according to the uncertain demand Y.

One period duopoly Consider two firms L and F each operating a project with an option to re-invest at cost I and increase cash flow according to the uncertain demand Y. Let D xi (t m)x j (t m) denote the cash flow per unit of demand of firm i. Assume that D 10 > D 11 > D 00 > D 01.

Equilibrium strategies Lemma Under FMA: 1. If I < IF h, then the equilibrium strategy is (1, 1) for high and low demand. 2. If IF l < I < I F h and I < I L l, then the equilibrium strategy is (1, 1) for high demand and (1, 0) for low demand. 3. If IF h < I < I L l, then the equilibrium strategy is (1, 0) for high and low demand. 4. If IF l < I < I F h and I L l < I, then the equilibrium strategy is (1, 1) for high demand and (0, 0) for low demand. 5. If IL l < I < I L h and I F h < I, then the equilibrium strategy is (1, 0) for high demand and (0, 0) for low demand. 6. If I > I h F, then the equilibrium strategy is (0, 0) for high and low demand.

A multi-period investment game Consider two firms L and F 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 = 0,..., M be a partition of the interval [0, T ] and denote by (x L (t m ), x F (t m ) {(0, 0), (0, 1), (1, 0), (1, 1)} the possible states of the firms after a decision has been at time t m. Let D xi (t m)x j (t m) denote the cash flow per unit of demand of firm i.

Derivation of project values (1) Let 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.

Derivation of project values (1) Let 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 (1,0) (1,0) g(v L (t m+1, y u ), (V (1,0) L (t m+1, y d )) L (t m, y) = D 10 y t + e r t v (0,1) (0,1) g(v L (t m+1, y u ), (V (0,1) L (t m+1, y d )) L (t m, y) = D 01 y t + e r t v (1,0) (1,0) g(v F (t m+1, y u ), (V (1,0) F (t m+1, y d )) F (t m, y) = D 01 y t + e r t v (0,1) (0,1) g(v F (t m+1, y u ), (V (0,1) F (t m+1, y d )) F (t m, y) = D 10 y t + e r t v (0,0) (0,0) g(v i (t m+1, y u ), (V (0,0) i (t m+1, y d )) i (t m, y) = D 00 y t + e r t

Derivation of project values (2) For fully invested firms, the project values are simply given by V (1,1) i (t m, y) = v (1,1) i (t m, y).

Derivation of project values (2) For 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 F when L has already invested and F hasn t: V (1,0) F (t m, y) = max{v (1,1) F (t m, y) I, v (1,0) F (t m, y)}. Similarly, the project value for L when F has invested and L hasn t is V (0,1) L (t m, y) = max{v (1,1) L (t m, y) I, v (0,1) L (t m, y)}.

Derivation of project values (3) Next consider the project value for L when it has already invest and F hasn t: { V (1,0) v (1,1) L (t m, y) = L (t m, y) if v (1,1) F (t m, y) I > v (1,0) F (t m, y), (t m, y) otherwise. v (1,0) L Similarly, the project value for F when it has already invest and L hasn t is { V (0,1) v (1,1) F (t m, y) = F (t m, y) if v (1,1) L (t m, y) I > v (0,1) L (t m, y), (t m, y) otherwise. v (0,0) F

Derivation of project values (4) Finally, the project values V (0,0) i are obtained as a Nash equilibrium, since both firms still have the option to invest.

Derivation of project values (4) Finally, the project values V (0,0) i are obtained as a Nash equilibrium, since both firms still have the option to invest. The pay-off matrix for the game is Firm F Invest Wait Invest (v (1,1) Firm L L I, v (1,1) F I ) (v (1,0) L I, v (1,0) F ) Wait (v (0,1) L, v (0,1) F I ) (v (0,0) L, v (0,0) F )

FMA: dependence on risk aversion. 200 Gamma = 0.01 Rho = 0.5 200 Gamma = 0.1 Rho = 0.5 150 150 100 100 Project Value 50 0!50!100!150 V00F V00L V11L!I V10L!I V01L Project Value 50 0!50!100!150 V00F V00L V11L!I V10L!I V01L!200 0 50 100 150 200 Demand!200 0 50 100 150 200 Demand 200 Gamma = 1 Rho = 0.5 200 Gamma = 2 Rho = 0.5 150 150 100 100 Project Value 50 0!50!100!150 V00F V00L V11L!I V10L!I V01L Project Value 50 0!50!100!150 V00F V00L V11L!I V10L!I V01L!200 0 50 100 150 200 Demand!200 0 50 100 150 200 250 300 Demand Figure: Project values in FMA case for different risk aversions.

FMA: dependence on correlation. 90 VL Gamma = 1 VF Gamma=1 Rho = 0.5 90 VL Gamma = 1 VF Gamma = 2 Rho = 0.5 80 80 70 70 Project Value 60 50 Project Value 60 50 40 40 30 V00F V00L 20!0.8!0.6!0.4!0.2 0 0.2 0.4 0.6 0.8 Correlation 30 V00F V00L 20!0.8!0.6!0.4!0.2 0 0.2 0.4 0.6 0.8 Correlation 90 VL Gamma = 2 VF Gamma = 1 Rho = 0.5 80 70 Project Value 60 50 40 30 V00F V00L 20!0.8!0.6!0.4!0.2 0 0.2 0.4 0.6 0.8 Correlation Figure: Project values in FMA case as function of correlation.

SMA: dependence on risk aversion 200 SMA Gamma=0.01 200 SMA Gamma=0.1 150 150 100 100 Project Value 50 0!50!100!150 V00F V00L V11L!I V10L!I V01L Project Value 50 0!50!100!150 V00F V00L V11L!I V10L!I V01L!200 0 20 40 60 80 100 120 Demand!200 0 20 40 60 80 100 120 Demand 200 SMA Gamma=1 200 SMA Gamma = 2 150 150 100 100 Project Value 50 0!50!100!150 V00F V00L V11L!I V10L!I V01L!200 0 20 40 60 80 100 120 Demand Project Value 50 0!50!100!150 V00F V00L V11L!I V10L!I V01L!200 0 20 40 60 80 100 120 140 160 180 Demand Figure: Project values in SMA case for different risk aversions.

SMA: dependence on correlation. 200 VL Gamma = 1 VF Gamma = 1 Y0=105 200 VL Gamma = 1 VF Gamma = 2 Y0=105 150 150 Project Value 100 Project Value 100 50 50 0 V00F V00L!0.8!0.6!0.4!0.2 0 0.2 0.4 0.6 0.8 Correlation 0 V00F V00L!0.8!0.6!0.4!0.2 0 0.2 0.4 0.6 0.8 Correlation 200 VL Gamma = 2 VF Gamma = 1 Y0=105 150 Project Value 100 50 0 V00F V00L!0.8!0.6!0.4!0.2 0 0.2 0.4 0.6 0.8 Correlation Figure: Project values in SMA case as function of correlation.

SMA x FMA 200 FMA/SMA Gamma=0.01 200 FMA/SMA Gamma=0.1 150 150 100 100 Project Value 50 0!50!100!150 V00F FMA V00L FMA V11L!I V01L V00L SMA V00F SMA Project Value 50 0!50!100!150 V00F FMA V00L FMA V11L!I V01L V00L SMA V00F SMA!200 0 50 100 150 200 Demand!200 0 50 100 150 200 Demand 200 FMA/SMA Gamma=1 200 FMA/SMA Gamma=2 150 150 100 100 Project Value 50 0!50!100!150 V00F FMA V00L FMA V11L!I V01L V00L SMA V00F SMA Project Value 50 0!50!100!150 V00F FMA V00L FMA V11L!I V01L V00L SMA V00F SMA!200 0 50 100 150 200 Demand!200 0 50 100 150 200 Demand Figure: Project values for FMA and SMA.