The Real Option Approach to Plant Valuation: from Spread Options to Optimal Switching

Transcription

1 The Real Option Approach to Plant Valuation: from Spread Options to René 1 and Michael Ludkovski 2 1 Bendheim Center for Finance Department of Operations Research & Financial Engineering Princeton University 2 Department of Mathematics University of Michigan CFC 2007, January 19, 2007

2 Motivation: Valuing a Tolling Agreement Stylized Version Leasing an Energy Asset Fossil Fuel Power Plant Oil Refinery Pipeline Owner of the Agreement Decides when and how to use the asset (e.g. run the power plant) Has someone else do the leg work

3 Classical Power Plant Valuation Real Option Approach Lifetime of the plant [T 1, T 2 ] C capacity of the plant (in MWh) H heat rate of the plant (in MMBtu/MWh) P t price of power on day t G t price of fuel (gas) on day t K fixed Operating Costs Value of the Plant (ORACLE) T 2 C e rt E{(P t HG t K ) + } t=t 1 String of Spark Spread Options

4 Beyond Plant Valuation: Credit Enhancement (Flash Back) The Calpine - Morgan Stanley Deal Calpine needs to refinance USD 8 MM by November 2004 Jan. 2004: Deutsche Bank: no traction on the offering Feb. 2004: The Street thinks Calpine is heading South March 2004: Morgan Stanley offers a (complex) structured deal A strip of spark spread options on 14 Calpine plants A similar bond offering How were the options priced? By Morgan Stanley? By Calpine?

5 Pricing Spreads & European Baskets Enormous literature R.C. - V. Durrleman Tech. Rep. Journal of Computational Finance SIAM Review C. Alexander S. Borovkova M. Dempster

6 Plant Operation Model: the Finite Mode Case Markov process (state of the world) X t = (X (1) t, X (2) t, ) (e.g. X (1) t = P t, X (2) t = G t, X (3) t = O t for a dual plant) Plant characteristics Z M = {0,, M 1} modes of operation of the plant H 0, H 1, H M 1 heat rates {C(i, j)} (i,j) ZM regime switching costs (C(i, j) = C(i, l) + C(l, j)) ψ i (t, x) reward at time t when world in state x, plant in mode i Operation of the plant (control) u = (ξ, T ) where ξ k Z M = {0,, M 1} successive modes 0 τ k 1 τ k T switching times

7 Plant Operation Model: the Finite Mode Case T (horizon) length of the tolling agreement Total reward H(x, i, [0, T ]; u)(ω) = T ψ us (s, X s ) ds C(u τk, u τk ) 0 τ k <T

8 Stochastic Control Problem U(t)) acceptable controls on [t, T ] (adapted càdlàg Z M -valued processes u of a.s. finite variation on [t, T ]) where Problem J(t, x, i) = sup J(t, x, i; u), u U(t) J(t, x, i; u) = E [ H(x, i, [t, T ]; u) X t = x, u t = i ] = E [ T ψ us (s, X s ) ds C(u τk, u τk ) X t = x, u t = i ] 0 τ k <T

9 Iterative Optimal Stopping Consider problem with at most k mode switches U k (t) = {(ξ, T ) U(t): τ l = T for l k + 1} Admissible strategies on [t, T ] with at most k switches [ J k (t, x, i) = T esssup u U k (t) E ψ us (s, X s) ds t t τ k <T ] C(u τk, u τk ) X t = x, u t = i.

10 Alternative Recursive Construction [ J 0 (t, x, i) = T ] E ψ i (s, X s ) ds X t = x, t [ J k (t, x, i) = τ ] sup E ψ i (s, X s ) ds + M k,i (τ, X τ ) X t = x. τ S t t Intervention operator M { } M k,i (t, x) = max j i C i,j + J k 1 (t, x, j). Hamadène - Jeanblanc (M=2)

11 Variational Formulation Notation Assume L X X space-time generator of Markov process X t in R d Mφ(t, x, i) = max j i { C i,j + φ(t, x, j)} intervention operator φ(t, x, i) in C 1,2( ([0, T ] R d ) \D ) C 1,1 (D) D = i { (t, x): φ(t, x, i) = Mφ(t, x, i) } (QVI) for all i Z M : 1 φ Mφ, 2 E x[ T 0 φ Mφ dt ] = 0, 3 L ( X φ(t, x, i) + ψ i (t, x) )( 0, φ(t, x, i) = 0, ) 4 L X φ(t, x, i) + ψ i (t, x) φ(t, x, i) Mφ(t, x, i) Conclusion = 0. φ is the optimal value function for the switching problem

12 Reflected Backward SDE s Assume X 0 = x & (Y x, Z x, A) adapted to (F X and E [ sup Yt x t T Y x t = Yt x T 0 T t T 0 t ) Z x t 2 dt + A T 2] < ψ i (s, X x s ) ds + A T A t M k,i (t, X x t ), (Y x t Conclusion: if Y x 0 = Jk (0, x, i) then T t Z s dw s, M k,i (t, X x t )) da t = 0, A 0 = 0. Y x t = J k (t, X x t, i)

13 System of Reflected Backward SDE s QVI for optimal switching: coupled system of reflected BSDE s for (Y i ) i ZM, Y i t = Y i T t ψ i (s, X s ) ds + A i T Ai t t max{ C i,j + Y j t }. j i T t Z i s dw s, Existence and uniqueness Directly for M > 2? M = 2, Hamadène - Jeanblanc use difference process Y 1 Y 2.

14 Discrete Time Dynamic Programming DPP Time Step t = T /M Time grid S = {m t, m = 0, 1,..., M } Switches are allowed in S For t 1 = m t, t 2 = (m + 1) t consecutive times J k (t 1, X t1, i) = max (E [ t 2 ( ψ i (t 1, X t1 ) t + E [ J k (t 2, X t2, i) F t1 ] ) Tsitsiklis - van Roy ψ i (s, X s) ds + J k ] ) (t 2, X t2, i) F t1, M k,i (t 1, X t1 ) t 1 ( { Ci,j + J k 1 (t 1, X t1, j) }). max j i (1)

15 Longstaff-Schwartz Version Recall [ τ k J k (m t, x, i) = E ψ i (j t, X j t ) t + M k,i (τ k t, X τ k t) ] X m t = x. Analogue for τ k : j=m τ k (m t, x l m t, i) = { τ k ((m + 1) t, x l (m+1) t, i), no switch; m, switch, and the set of paths on which we switch is given by {l: ĵ l (m t; i) i} with ( ĵ l (t 1 ; i) = arg max C i,j + J k 1 (t 1, xt l j 1, j), ψ i (t 1, xt l [ 1 ) t + Êt 1 J k (t 2,, i) ] ) (xt l 1 ). (3) The full recursive pathwise construction for J k is { J k (m t, xm t, l ψ i (m t, x l i) = m t) t + J k ((m + 1) t, x(m+1) t, l i), no switch; C i,j + J k 1 (m t, xm t, l j), switch to j. (4) (2)

16 Remarks Spread Options Regression used solely to update the optimal stopping times τ k Regressed values never stored Helps to eliminate potential biases from the regression step.

17 Algorithm Spread Options 1 Select a set of basis functions (B j ) and algorithm parameters t, M, N p, K, δ. 2 Generate N p paths of the driving process: {x l m t, m = 0, 1,..., M, l = 1, 2,..., N p } with fixed initial condition x l 0 = x 0. 3 Initialize the value functions and switching times J k (T, x l T, i) = 0, τ k (T, x l T, i) = M i, k. 4 Moving backward in time with t = m t, m = M,..., 0 repeat the Loop: Compute inductively the layers k = 0, 1,..., K (evaluate E [ J k (m t + t,, i) F m t ] by linear regression of {J k (m t + t, x l m t+ t, i)} against {B j (x l m t)} NB j=1, then add the reward ψ i (m t, x l m t) t) Update the switching times and value functions 5 end Loop. 6 Check whether K switches are enough by comparing J K and J K 1 (they should be equal). Observe that during the main loop we only need to store the buffer J(t, ),..., J(t + δ, ); and τ(t, ),, τ(t + δ, ).

18 Convergence Spread Options Bouchard - Touzi Gobet - Lemor - Warin

19 Example 1 Spread Options dx t = 2(10 X t ) dt + 2 dw t, X 0 = 10, Horizon T = 2, Switch separation δ = Two regimes Reward rates ψ 0 (X t ) = 0 and ψ 1 (X t ) = 10(X t 10) Switching cost C = 0.3.

20 Value Functions Spread Options 9 8 Value Function for successive k Years to maturity J k (t, x, 0) as a function of t

21 Exercise Boundaries Switching boundary Switching boundary Time Units Time Units k = 2 (left) k = 7 (right) NB: Decreasing boundary around t = 0 is an artifact of the Monte Carlo.

22 One Sample Spread Options 12 State process and boundaries Cumulative wealth Time Units

23 Example 2: Comparisons Spark spread X t = (P t, G t ) { log(p t ) OU(κ = 2, θ = log(10), σ = 0.8) log(g t ) OU(κ = 1, θ = log(10), σ = 0.4) P 0 = 10, G 0 = 10, ρ = 0.7 Agreement Duration [0, 0.5] Reward functions ψ 0 (X t ) = 0 ψ 1 (X t ) = 10(P t G t ) ψ 2 (X t ) = 20(P t 1.1 G t ) Switching costs C i,j = 0.25 i j

24 Numerical Comparison Method Mean Std. Dev Time (m) Explicit FD LS Regression TvR Regression Kernel Quantization Table: Benchmark results for Example 2.

25 Example 3: Dual Plant & Delay log(p t ) OU(κ = 2, θ = log(10), σ = 0.8), log(g t ) OU(κ = 1, θ = log(10), σ = 0.4), log(o t ) OU(κ = 1, θ = log(10), σ = 0.4),. P 0 = G 0 = O 0 = 10, ρ pg = 0.5,ρ po = 0.3, ρ go = 0 Agreement Duration T = 1 Reward functions ψ 0 (X t) 0 ψ 1 (X t) = 5 (P t G t) ψ 2 (X t) = 5 (P t O t), ψ 3 (X t) = 5 (3P t 4G t) ψ 4 (X t) = 5 (3P t 4O t). Switching costs C i,j 0.5 Delay δ = 0, 0.01, 0.03 (up to ten days)

26 Numerical Results Setting No Delay δ = 0.01 δ = 0.03 Base Case Jumps in P t Regimes 0-3 only Regimes 0-2 only Gas only: 0, 1, Table: LS scheme with 400 steps and paths. Remarks High δ lowers profitability by over 20%. Removal of regimes: without regimes 3 and 4 expected profit drops from to 9.21.

27 Example 4: Exhaustible Resources Include I t current level of resources left (I t non-increasing process). [ τ ] J(t, x, c, i) = sup E ψ i (s, X s) ds + J(τ, X τ, I τ, j) C i,j X t = x, I t = c. τ,j t (5) Resource depletion (boundary condition) J(t, x, 0, i) 0. Not really a control problem I t can be computed on the fly Mining example of Brennan and Schwartz varying the initial copper price X 0 Method/ X BS PDE FD RMC

28 Extension to Gas Storage & Hydro Plants Can accomodate outages Can include switch separation as a form of delay Can be extended (R.C. - M. Ludkovski) to treat Gas Storage Hydro Plants

29 What Remains to be Done Need to improve delays Need convergence analysis Need better analysis of exercise boundaries Need to implement duality upper bounds we have approximate value functions we have approximate exercise boundaries so we have lower bounds need to extend Meinshausen-Hambly to optimal switching set-up

30 Financial Hedging Spread Options Extending the Analysis Adding Access to a Financial Market Porchet-Touzi Same (Markov) factor process X t = (X (1) t, X (2) t, ) as before Same plant characteristics as before Same operation control u = (ξ, T ) as before Same maturity T (end of tolling agreement) as before Reward for operating the plant H(x, i, T ; u)(ω) = T ψ us (s, X s ) ds C(u τk, u τk ) 0 τ k <T

31 Hedging/Investing in Financial Market Access to a financial market (possibly incomplete) y initial wealth π t investment portfolio Y y,π T corresponding terminal wealth from investment Utility function U(y) = e γy Maximum expected utility v(y) = sup E{U(Y y,π T )} π

32 Indifference Pricing With the power plant (tolling contract) INDIFFERENCE PRICING Analysis of V (x, i, y) = sup E{U(Y y,π T + H(x, i, T ; u))} u,π p = p(x, i, y) = sup{p 0; V (x, i, y) v(y)} BSDE formulation PDE formulation