The priority option: the value of being a leader in complete and incomplete markets.

Transcription

1 s. a leader in s. Mathematics and Statistics - McMaster University Joint work with Vincent Leclère (École de Ponts) Eidgenössische Technische Hochschule Zürich, December 09, 2010

2 s

3 s.

4 Combining options and s s. Most of the real options approach consider monopolistic decision making. Option value leads to conservative exercise strategies. Intuitively, competition should erode the option value. A systematic application of both real options and theory in strategic decisions has been proposed in the literature (see Smit and Trigeorgis (2004) for a review). The essential idea can be summarized in two rules: 1 whenever the outcome of a given involves a wait and see strategy, its pay-off should be calculated as the value of a real option; 2 whenever the pay-off of a given involves a, its value should calculated as the equilibrium solution to the. In this way, option valuation and theoretical equilibrium become dynamically related.

5 Competition in continuous times s. Consider the model of Grenadier (2000), where two firms contemplating the decision to pay a cost K to invest in a project leading to instantaneous cash flows Y t D Q where dy t Y t = νdt + ηdw t, (1) where Y t is a stochastic demand shock and D Q is the inverse demand function when Q firms are present. Assume both completeness and infinite maturity. More specifically, assume that Y t is perfectly correlated with a traded financial asset dp t P t = µdt + σdw t = rdt + σ(dw t + λdt), λ = µ r σ. (2)

6 Project value s. Let ξ = ν r η and λ = µ r σ be the Sharpe ratios for the project and the spanning asset. After both firms have invested, the value of the project is given by the expected value of all discounted future cash flows, that is [ ] E Q e r(s t) Y s D 2 ds Y t = y = yd 2 δ, t where δ = η(λ ξ). We see that δ plays the role of a dividend rate. Given that the leader has already invested, the value for the follower is then given by F (y) = sup E Q [ e rτ Φ(Y τ )1 {τ< } Y 0 = y ], (3) τ 0 where τ is a stopping time and Φ(y) = D 2 y/δ K.

7 s. The follower value then satisfies η 2 2 y 2 F (y) + (r δ)yf (y) rf (y) 0 F (y) Φ(y) [ ] [F (y) Φ(y)] η 2 2 y 2 F (y) + (r δ)yf (y) rf (y) = 0. (4) supplemented by F (v) 0 and F (0) = 0. The solution to variational inequality is ( ) β K y F (y) = β 1 Y F, Y YF yd(2) δ K, y Y F where Y F = δkβ D 2 (β 1) and β > 1 is a solution of 1 2 η2 β(β 1) + (r δ)β = r.

8 s. F(y) D 2 y/δ-k Kδ/D2 Y F

9 and simultaneous exercise s. After investing, the leader has no more options to exercise. As a result, the value of becoming a leader can be obtained entirely by expected value of future cash flow at a rate Y t D 1 until the process Y reaches Y F and Y t D 2 thereafter. The solution to this simple first passage time problem is ( ) β yd(1) L(y) = δ D 1 D 2 D 2 β K y β 1 Y F K, y < YF yd 2 δ K, y Y F Finally, it is clear that the value obtained from simultaneous exercise is S(y) = yd 2 δ K

10 Threshold for the leader s. It can be shown that there exists a unique point Y L (0, Y F ) such that L(Y ) < F (Y ), Y < Y L In addition L(Y ) = F (Y ), Y = Y L L(Y ) > F (Y ), Y L < Y < Y F L(Y ) = F (Y ), Y Y F S(Y ) < min(l(y ), F (Y ), Y < Y F S(K) = L(Y ) = F (Y ), Y Y F

11 Threshold for the leader s. d 1 (y)=l(y)-f(y) Y L d 2 (y)=s(y)-f(y) Y F -K

12 Randomized strategies s. Denote by x i (t) {0, 1} the state for firm i at time t. A randomized strategy consists of a pair (x i (t), G i t) where G i is an independent enlargement of the filtration F. Denote p i (t) := P(x i (t) = 1 F t ). A strategy is pure at time t if p i (t) {0, 1}. Otherwise it is mixed and represented by the randomization parameter η i (t) as x i (t) = 1 {ηi (t) p i (t)}, η i (t) U[0, 1], η i (t) F t If G 1 t G 2 t = F t, the strategies of the two players are independent. Alternatively, correlated equilibria can be introduced through a communication device γ ij (t) implemented via a third party.

13 strategies s. We focus on independent strategies only. Because of the Markovian structure of the, it is enough to consider p i (t) p i (Y t ). Assume that the is played successively until one of the firms exercises. For y Y F we have that p (y) = p 1 (y) = p 2 (y) = 1 is a Nash equilibrium. For y Y L we have that p (y) = p 1 (y) = p 2 (y) = 0 is a Nash equilibrium. The interesting region is Y L < y < Y F.

14 strategies (continued) s. For Y L < y < Y F, the pay-off for firm i is V i = [p i (1 p j )L + p i p j S + (1 p i )p j F ] = p i(1 p j )L + p i p j S + (1 p i )p j F 1 (1 p i )(1 p j ) [(1 p i )(1 p j )] k k=0 Maximizing this expression with respect to p i and using symmetry leads to p (y) = p 1 (y) = p 2 (y) = L(y) F (y) L(y) S(y), which can be shown to be a Nash equilibrium.

15 Expected payoff s. Observe that the expected payoff for each firm is F (y), y < Y L V (y) = (1 p S ) F (y)+l(y) 2 + p S S(y), y (Y L, Y F ), S(y), y > Y F where p S (y) is the probability of simultaneous exercise. Using he expression for p (y) we find p S (y) = p (y) 2 1 (1 p (y)) 2 = L F L + F 2S This gives V (y) = F (y) for all y!

16 Predetermined roles s. Define L π (Y ) as the project value for a firm that has been predetermined as the Leader. Following the same reasoning as before, this value is given by L π (y) = sup E Q [ e rτ L(Y τ )1 {τ< } Y 0 = y ]. (5) τ 0 Observe that D 1 L (y) = δ (D ( ) 1 D 2 )β y β 1 if y < Y F δ Y F, D 2 if y Y F δ so L(y) is not differentiable at Y F. Moreover, L (y) < 0 for 0 y < Y F.

17 Obstacle problem for the leader s. D 2 y/δ L π (y) F(y) Y L Y 1 Y 2 Y F Y 3 L(y) D 2 y/δ-k -K

18 Variational inequality for the leader s. Formally, the value for a predetermined leader satisfies η 2 2 y 2 (L π ) + (r δ)y(l π ) rl π 0 L π (y) L(y) (6) [ ] η [L π 2 L] 2 y 2 (L π ) + (r δ)y(l π ) rl π = 0, supplemented by the conditions L π (y) 0 and L(0) = 0. Since the obstacle is not differentiable, there is not guarantee that this can be formulated in strong sense.

19 Variational inequality in weak sense s. Define M(y) = L(y) D 2y δ + K, χ(y) = L(y) D 2y δ and f (y) = rk D 2 y. Then the weak formulation of (??) is b(m, M M) (f, M M) κ, M K, M K. Here b(, ) is the bilinear form b(g, g) = [ yg (y) η 2 1 y 2 (κ 1) 0 0 ] 1 + y 2 (r δ) g (y) g (y)y 2 η 2 ω(y)dy K g(y)ω(y)dy rg(y) g(y)ω(y)dy on the Sobolev space H 1 κ(0, ) and (, ) κ is a weighted inner product on the Hilbert space L 2 κ(0, ) with weight function ω(y) = 1/(1 + y 2 ) κ. Finally K = {g H 1 g χ, g(0) = K}.

20 with priority s. Using a penalty approximation technique, we can show that M is smoother than the obstacle and satisfies a variational inequality in strong sense. It then follows that Ay β if 0 y < Y 1 L(y) if Y 1 y Y 2 L π (y) = By β + Cy β 1 if Y 2 < y < Y 3 D 2 y K if y Y 3, δ Observe that Y L < Y 1, so the priority option delays investment. The value of the priority option is then given by π(y) = L π (y) F (y). (7)

21 Priority option value s.

22 s s. Suppose now that the stochastic demand Y t is correlated with the portfolio P t as follows: { dy Y = νdt + ηdw t = µdt + σdb, t dp P where W t and B t have instantaneous correlation ρ. For simplicity, take r = 0. According to CAPM, if Y could be traded its equilibrium rate of return ν would satisfy ν η = ρµ σ We then define δ(ρ) := ν ν = η(ρλ ξ) as the below equilibrium shortfall rate, which plays the role of a dividend yield in this case.

23 Utility problem s. As before, we calculate the project value when both firms have already invested as [ ] E e ν(s t) Y s D 2 ds Y t = y = yd 2 ν ν = yd 2 δ(ρ). t For a utility function U(x) = e γx, define [ F (x, y) = sup E (τ,θ) Here U(x) = e γx and e λ2 τ 2 U (X θ τ + ( ) )] D2 Y + τ δ(ρ) K, dx θ t = θ dp t P t = θσ(λdt + dw t ). (8)

24 function s. Using Henderson (2007), let β(ρ) = 1 + 2δ(ρ) η 2 > 1 and define Y F (ρ) as the solution to D 2 Y F (ρ) δ(ρ) Then F (x, y) = K = [ 1 γ(1 ρ 2 ) log 1 + γ(1 ] ρ2 )D 2 Y F (ρ), β(ρ)δ(ρ) ( ) e [1 γx γ(1 ρ 2 ( ) ] 1 )D 2 Y β F y 1 ρ 2 δβ+γ(1 ρ 2, )D 2 Y Y ( ) F F D2 e γx e γ y δ K, y Y F 0 y<y F

25 function s. As before, the value for the leader can be found by expected discounted cash flows assuming that the follower exercises optimally: L(x, y) = [ e γ [ e γ x+ D(1) ( D(2) D(1) δ y+ δ ) ( ) α ] y Y F K Y F, 0 y Y F x+ D(2) δ y K ], y Y F, ( ) ( ) 2 where α = 1 2 ν + 1 η 2 2 ν η + 2 ν 2 η 2 Similarly, the value for simultaneous exercise is S(x, y) = e γ [ x+ D(2) δ y K ]

26 Leader threshold s. We can again show that, for each fixed x, there exists a unique point Y L (0, Y F ) such that In addition L(x, y) < F (x, y), y < Y L L(x, y) = F (x, y), y = Y L L(x, y) > F (x, y), Y L < y < Y F L(x, y) = F (x, y), y Y F S(x, y) < min(l(x, y), F (x, y), y < Y F S(x, y) = L(x, y) = F (x, y), y Y F

27 strategies s. Following the same arguments as before, we have that: For y Y F, p (x, y) = p 1 (x.y) = p 2 (x, y) = 1. For y Y L, p (x, y) = p 1 (x, y) = p 2 (x, y) = 0. For Y L < y < Y F. p (x, y) = p 1 (x, y) = p 2 (x, y) = L(x, y) F (x, y) L(x, y) S(x, y).

28 option s. Define L π (Y ) as the expected utility for a firm that has been given a priority option for choosing to be the Leader. Formally, this has the same type of two-interval solution as in the complete, but a rigorous proof is still open. The value for the priority option can then be obtained by an indifference value argument comparing L π (X, Y ) and the equilibrium value V i without the priority option.

29 Obstacle problem in s s. Y L Y 1 Y 2 Y F Y 3 L π (x,y) y L(x,y) F(x,y)

30 s. Real options and theory can be combined in a dynamic framework for decision making under uncertainty and competition. For a complete, we found the leader and follower values as well as the equilibrium strategies for symmetric firms competing for an investment opportunity. Comparing this with the solution of a gives the priority option value. The effects of ness and risk aversion can be incorporated using the concept of indifference pricing. We again found the leader and follower values and equilibrium strategies. We characterize a candidate solution for the leader value with priority. Much more work is necessary for a large number of firms. Danke!