Internet Appendix for A Bayesian Approach to Real Options: The Case of Distinguishing Between Temporary and Permanent Shocks

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1 Intenet Appendix fo A Bayesian Appoach to Real Options: The Case of Distinguishing Between Tempoay and Pemanent Shocks Steven R. Genadie Gaduate School of Business, Stanfod Univesity Andey Malenko Gaduate School of Business, Stanfod Univesity Januay 00

2 Poofs and Deivations Smooth-pasting conditions in Eqs. (7) and (8). Let M denote the set of (G t )-stopping times and M be an element of this set. By de nition, the value of the option G ( (t) ; p (t)) satis es h R G ( (t) ; p (t)) = sup M E e (s t) t p (s) e R s t p(u)du H (s) ds +' +e R (A) t p(s)ds e ( t) (S ( () ; p ()) I) jg t : The ight-hand side of (A) consists of two tems. The st tem coesponds to the payo if the shock evets befoe the m invests. The second tem coesponds to the payo if the m invests befoe the shock eveses. Let Z + D ( (t) ; p (t)) E e (s t t) p (s) e R s t p(u)du H (s) dsjg t ; (A) + ' d (p (t)) Z t t 0 ( + p (s)) ds; (A) whee t 0 is the time when the shock aives. Then, fo any (t) > 0 and p (t) (0; 0 ], we can ewite (A) as e d(p(t)) (G ( (t) ; p (t)) D ( (t) ; p (t))) = sup M E (A4) e d(p()) (S ( () ; p ()) I D ( () ; p ())) jg t Because e d(p()) (S ( () ; p ()) I D ( () ; p ())) is C eveywhee and the dependence of ( (t) ; p (t)) on any initial point (; p) is explicit and smooth, the poblem satis es the smooth- t pinciple (Peski and Shiyaev (006)). Theefoe, at all points (p) ; p the deivatives of e d(p) (G (; p) D (; p)) and e d(p) (S (; p) I D (; p)) On p.5 Peski and Shiyaev (006) pove this esult fo one-dimensional poblems, but state that it also extends to highe dimensions (see p.50). See also p.44, whee they state that the smooth- t pinciple holds in multiple dimensions if the state pocess afte stating at the bounday of the stopping egion entes the inteio of the stopping egion immediately.

3 with espect to and p must be the same. Taking the two deivatives, we obtain G (p) ; p = S (p) ; p ; (A5) G p (p) ; p = Sp (p) ; p : (A6) Poof of the existence of a solution to the xed-point poblem in Eq. (9). Note that a solution G(; p); (p) to Eqs. (5) - (8) will satisfy (9). Fom the above we know that if (p) exists, then it satis es (5) - (8). We thus need to demonstate the existence of a bounday between an execise egion and a continuation egion fo any p (0; 0 ]. Conside state ( (t) ; p (t)), and suppose that the m exogenously leans the type of the outstanding shock in a moment. Then, the value of the investment option just befoe the m leans the type of the shock is equal to ( p (t)) G ( (t) ; 0) + p (t) G ( (t) ; ). Fomally, let (Gs; 0 s t) denote ltation geneated by (B (s) ; M (s) ; N (s)), s t, but unde which the m leans the identity of an outstanding shock in a moment. In othe wods, Gs 0 contains the identity of the shock fo all s > t, but not fo s = t. Notice that while infomation ltations (G s ; s t) and (Gs; 0 s t) ae di eent, the evolution of (s), s t unde the full-infomation ltation is the same. Let G 0 ( (t) ; p (t)) denote the value of the investment option unde (Gs; 0 s t) G 0 ( (t) ; p (t)) sup M 0 E R e s (s) ds e IjG 0 t = ( p (t)) G ( (t) ; 0) + p (t) G ( (t) ; ) ; (A7) whee M 0 is the set of optimal stopping times adapted to (G 0 s; s t). Because (G t ) is a ltation geneated by (B; M; N), fo any s > t, G s G 0 s, and fo s = t, G s and G 0 s coincide. By de nition of G ( (t) ; p (t)), Z G ( (t) ; p (t)) = E e s (s) ds e IjG t ; ()

4 whee is the optimal stopping time, which is adapted to ltation (G t ). Because is adapted to (G t ), it is also adapted to (G 0 t). Hence, G 0 ( (t) ; p (t)) = sup M 0 E R R e s (s) ds e IjG 0 t E e s (s) ds e IjG 0 t = E R e s (s) ds e IjG t = G ( (t) ; p (t)) : (A9) Combining (A7) with (A9), we obtain ( p) G (; 0) + pg (; ) G (; p) (A0) fo any (; p). Now, conside G (; 0) and G (; ). Thee is no leaning when p f0; g, so the option picing poblems ae standad. The execise bounday fo p = 0 is given by (0) =. To solve fo (), note that in the ange p =, < ( + '), Eq. (5) is solved by G (; ) = C ~ + ( ( + ') I); (A) whee = 4 s + + ( + ) 5 > > : (A) The unknown constant ~ C and the execise bounday () ae given by the bounday conditions (6) - (7). Simplifying, we obtain an implicit solution fo (): + ( )(+') () () = I + ( + ( + ') I): (A) De ne Q(x) = I + x ( ( + ') I) + ( )(+') x : (A4) +

5 We have Q(0) = I > 0; Q(( + ') ) = ' + < 0; Q 00 (x) = ( ) (+') ( I)x > 0: (A5) Hence, thee exists a unique point between 0 and ( + '), at which Q (x) = 0. It satis es the su cient conditions (Dixit and Pindyck (994)), so it is equal to (). Because both (0) and () ae below ( + '), fo any > ( + '), G (; 0) = I and G (; ) = + (+')( ) + I. Thus, fo any > ( + ') and p (0; 0 ], ( p) G (; 0) + pg (; ) = S (; p) I: (A6) By de nition of G (; p), G (; p) S (; p) I: (A7) Combining Eqs. (A0), (A6), and (A7), we nd that fo any p (0; 0 ] and any > ( + '), G (; p) = S (; p) I, and thus it is always optimal to execise the investment option. Also, fo any p (0; 0 ] and any < ( ) I, it is always optimal not to execise the option, as the payo fom the execise is negative in this ange. Theefoe, fo any p (0; 0 ], thee is a bounday between execise and continuation egions. Closed fom solutions fo the model of Section. when > 0 and = 0. While the execise tigge p () is chaacteized by (9), it is not solvable in closedfom, since the value function G(; p) itself is not available in closed-fom. Howeve, fo the special case in which = 0, 0, the closed-fom solution fo the tigge is p () j =0 = I + I(+') I I : (A8) (+')( ) 4

6 The coesponding value of the investment option equals G (; p) j =0 = p I ( + ') I + ( p) p ; (A9) whee (y) = y )t e( y (+')I y +'+ + )t (+ )(+') e( I whee t (z) is a function de ned implicitly by e t + = e t y ; z (+')I ze t I Ie t + I ze t ( )(+') y (A0) : (A) Notice that when = 0, investment does not occu when the jump evets. Theefoe, in this case, the tigge (A8) coincides with (). Deivation of the investment tigge fo the case of multiple shocks in Section 5. Let Y ( (t) ; n (t) ; p (t)) be a continuously di eentiable function, whee (t) is the cuent value of the cash ow pocess, n (t) is the cuent numbe of outstanding shocks, and p (t) = (p 0 (t); p (t) ; :::; p n (t)) is the vecto of cuent beliefs. Applying Itô s lemma fo semimatingales (see Theoem in Potte (004)), we get the dynamics of Y (t) Note that since in () 0 denotes the level of (t) befoe the positive jump occued, we need to use p ( 0 ( + ')) to ensue equivalence. 5

7 unde (F t ): dy = Y + Y n p k (k +Y x db (t) + [Y (( + ') ; n + ; ^p (p)) h + Y ; n ; ~p (p) Y +' i= p ii) dt Y ] (dm (t) + dm (t)) i dn (t) ; (A) whee, p k, and Y denote (t), p k (t), and Y (t), espectively, and ^p (p) and ~p (p) ae the updated vectos of beliefs de ned by (8) - (9). In (A), the intensities of M (t) and M (t) ae and if n (t) < N and zeo if n (t) = N. By the law of iteated expectations, E [dn (t) jg t ] = E [E [dn (t) jf t ] G t ] = E [k (t) dtjg t ] = n kp k dt: k= (A) Theefoe, the instantaneous conditional expected change in Y (; n; p) is equal to E h dy (;n;p) dt i n jg t = Y Y p k (k k= + n< N [Y (( + ') ; n + ; ^p (p)) Y ] ( + ) h i n + Y ; n ; ~p (p) Y +' kp k ; k= i= p ii) (A4) whee n< N is an indicato function taking the value if n < N and 0 othewise. If Y (; n; p) denotes the value of a contingent claim that continuously pays a cash ow of y (; n; p) and the discount ate is equal to, then it must be the case that 6

8 h i E dy (;n;p)+y(;n;p)dt jg dt t = Y (; n; p). Hence, Y = Y + Y n p k (k i= p ii) + n< N [Y (( + ') ; n + ; ^p (p)) Y ] ( + ) h i n Y ; n ; ~p (p) Y +' kp k + y (; n; p) : k= (A5) Simplifying: k= n p k k Y = Y + Y k= n p k (k n i= p ii) + n< N ( + ) Y (( + ') ; n + ; ^p(p)) n + p k k Y ; n ; ~p(p) + y(; n; p): +' k= (A6) Let S (; n; p) denote the value of the undelying poject. It is the expected discounted value of cash ows that the m gets if it immediately execises the investment option. S(; n; p) is thus a special case of Y (; n; p), with y(; n; p) =. Thus S (; n; p) must satisfy: k= n p k k S = S + S k= n p k (k n i= p ii) + n< N ( + ) S (( + ') ; n + ; ^p(p)) n + p k k S ; n ; ~p(p) + : +' k= The solution can be witten as S (; n; p) = a n 0 + (A7) n p k (a n k a n 0) ; (A8) k= 7

9 whee constants a n k, k = 0; ; :::; n, n = 0; ; :::; N ae de ned late. Let G (; n; p) denote the value of the investment option. Befoe the investment occus, G(; n; p) is a special case of Y (; n; p), with y(; n; p) = 0. Thus G (; n; p) must satisfy: n k= + + p k (k n p k k G = G + G k= i= p ii) + n< N ( + ) G (( + ') ; n + ; ^p(p)) n + p k k G ; n +' k= ; ~p(p) : (A9) The optimal investment decision can be descibed by a tigge function (n; p). Eq. (A9) is solved subject to the following value-matching and smooth-pasting conditions: k= G (n; p) ; n; p = S (n; p) ; n; p I; G (n; p) ; n; p = S (n; p) ; n; p (n;p);n;p) P p k (k n i= p ii) = (n;p);n;p) (A0) The st equation is the value-matching condition. The second and thid equations ae the smooth-pasting conditions (with espect to and t, espectively). Combining (A7), (A9) and (A0) gives us: (n; p) I = (n; p) G (n; p) ; n; p + n< N ( + ) G (n; p) ( + ') ; n + ; ^p (p) + I S (n; p) ( + ') ; n + ; ^p (p) + ( k= p kk) h G (n;p) +' ; n ; ~p (p) + I S (n;p) +' ; n ; ~p (p) i : (A) When is vey close to the tigge (n; p), the aival of a new positive shock will esult Note that since S(; n; p) is linea in, S = 0. 8

10 in immediate investment. This implies: G (n; p) ( + ') ; n + ; ^p (p) = S (n; p) ( + ') ; n + ; ^p (p) I: (A) Thus, the second tem on the ight-hand side of (A) is zeo, so (n; p) satis es P h (n; p) = ( n k= p (n;p) kk) G ; n ; ~p (p) (n;p) i + I S ; n ; ~p (p) +' +' +I + (n; p) G (n; p) ; n; p ; (A) Deivation of constants a n k in Eq. (A8). Plugging (A8) in (A7), using the de nitions of ^p (p) and ~p (p), and simplifying, we obtain P ( n k= p kk) a n 0 + ( ) k= p k (a n k a n 0) = ( + ') a n+ 0 + a n+ + k= p k a n+ k a n+ 0 + a n+ k+ a n+ 0 k= p kka n k + ; + +' (A4) fo n = 0; ; :::; N. Fo n = N, S (; n; p) satis es (A7) without the tems with and. Hence, P N + k= p kk ( + ) P N k= p k a N k a N 0 = +' N a 0 + P N (A5) k= p N kka k + : Eqs. (A4) and (A5) must hold fo all p. Hence, the coe cients in constant tems and p k, k = ; :::; n on both sides of each equation must be equal. Thus, we get the following equations: fo n = 0; ; :::; N : ( + + ) a n 0 = ( + ') a n+ 0 + a n+ + ; (A6) 9

11 fo k = ; :::; n and n = ; :::; N : k a n 0 + ( ) (a n k a n 0) (A7) = ( + ') a n+ k a n+ 0 + a n+ k+ a n+ + + ' an k ; fo k = ; :::; N: N k a 0 + ( + ) a Nk a N0 = + ' a N k : (A8) nally, matching the constant tem in the equation fo n = N: ( ) a N 0 = : (A9) This gives us N + N + linea equations that fully detemine N + N + constants a n k, k = 0; :::; n, n = 0; :::; N. Numeical Pocedues Numeical pocedue fo computing (p) in Section. To compute the tigge functions we use a vaiation of the least-squaes method developed by Longsta and Schwatz (00). Note that when p = 0, the model becomes standad, so (0) =. Also, note that p (T ) 0 as T, whee T is the time that passes afte the aival of the shock. Because of that, we can appoximate (p (T )) fo a lage T by (0). Afte that, we take a small and compute p (T ) fom (). Then, we use the least squaes method of Longsta and Schwatz (00) to estimate the second deivative of the conditional expected payo fom waiting at time T until time T. Then, we use this estimate and (9) to compute (p (T )). We epeat this N times fo a su ciently lage N such that p (T N) > 0. Moe speci cally, at any step n: 0

12 . Use (p (T k)), k = 0; ; :::; n and the least squaes method to estimate the second deivative of the conditional expected payo fom waiting at time T n.. Use this estimate as input in (9) to compute (p (T n)). Numeical pocedue fo computing (n; p) in Section 5. We conside the following set of paamete values: N =, = 0:04, = = 0, ' = 0:, = 0:, =, =, I = 5. Fist, we compute S (; n; p). Fom (A6) - (A9), when N =, we get 6 linea equations that detemine 6 unknowns a 0 0, a 0, a, a 0, a, and a. Solving this system of equations, we obtain a 0 0 = 9:49, a 0 = 7:9, a = 6:8, a 0 = 5, a = 4:766, a = 4:98. Second, we compute (n; p). Because the option value is a elatively at function of the execise theshold aound the optimal theshold, computing the option values by simulations yields elatively pecise option values but elatively impecise execise theshold. To ovecome this poblem, we compute the option values by simulations and use them as inputs in Eq. (5) to compute the execise thesholds: (0; p) = I; (A40) (; p) (; p) (; p) = p G + ' ; 0; + I S (; p) = (p + p ) G (; p) ; ; ~p (p) + ' + ' ; 0; + I; (A4) (; p) + I S ; ; ~p (p) +I: (A4) + ' Speci cally, ou numeical pocedue is set up in the following way:. Fo given thesholds (; p ; p ) and (; p p ; p ; p ), expessed as polynomials of p and (p ; p ), espectively, calculate the option value G (; 0; ) by simulations. We use the fouth-ode polynomials and 0; 000 simulations.. Maximize the option value G (; 0; ) with espect to coe cients of polynomials.. Using the computed thesholds, calculate option values G (; 0; ) and G (; ; p ; p ).

13 4. Use option values G (; 0; ) and G (; ; p ; p ) fom step to obtain the execise thesholds (; p) and (; p) fom (A4) and (A4). The ode of the polynomials used in step can be expanded, and steps and 4 can be iteated until convegence with vey little e ect on the values of the execise thesholds.