Quantifying Flexibility Real Options Calculus

Transcription

1 MPRA Munch Personal RePEc Archve Quantfyng Flexblty Real Optons Calculus V. G. Makhankov and M. A. Aguero-Granados Unversdad Autonoma del Estado de Mexco, Los Alamos Natonal Laboratory 7. July 010 Onlne at MPRA Paper No. 4419, posted 15. August :45 UTC

2 Quantfyng Flexblty Real Optons Calculus V. G. Makhankov (*), M. Agüero (**) (*) Chaparron Place, Santa Fe, NM. USA (**) Unversdad Autonoma del Estado de Mexco, Toluca, Mexco, and Theoretcal Dvson, Los Alamos Natonal Laboratory, NM USA Lecture Notes Abstract We expose a real optons theory as a tool for quantfyng the value of the operatng flexblty of real assets. Addtonally, we have ponted out that ths theory s an approprated methodology for determnng optmal operatng polces, and provde an example of successful applcaton of our approach to power ndustres, specfcally to valuate the power plant of electrcty. In partcular by ncreasng the volatlty of prces wll eventually lead to hgher assets values.

3 I. Introducton As s well known not a long tme has passed from the date when Black and Scholes had publshed ther breakng paper [1]. But ther model has had a sgnfcant mpact on the developments of dervatves market. After that much nterest n opton prcng has been generated from the development of new optons markets. The rapd development of theory and consequently ts dverse applcaton to the opton prcng problem occurred after that. Snce throughout the paper we wll deal wth the concept of call optons, let us frst say somethng fundamental about the term Call Optons. We wll need throughout the whole paper to use ths term n many dfferent modes. Let us assume that there are two partes. These partes need to sgn an agreement that s a fnancal contract between them. So, we have the buyer and the seller that have n common an opton. It s an opton to buy shares of stock at a specfed tme n the future. Often t s smple called an opton. The buyer has the rght but not the oblgaton to buy a gven quantty of stock at a gven prce on or before a gven date from the seller of the opton of a partcular commodty or fnancal nstrument. The seller or also called as wrter s oblgated to sell the commodty or fnancal nstrument should the buyer so decde. The buyer pays the fee, named also premum for ths rght. There are markets n call optons on stocks, commodtes, currences, stock ndexes, futures and nterest rates. Specfc optons are prced dfferently but ther common features can be restrcted to the followng defnton. The gven quantty s fxed and s usually ether 100 unts or 1,000 unts. The gven prce s known as the exercse prce or strke prce (K). The gven date s known as the expry date (T). Often the stock underlyng the opton s referred to as the underlyng wth the prce S(t). The opton prce (value) we denote as C(t). Exchange traded stock optons are lsted wth three, sx and nne months of lfe and varous strke prces. Accordng ther features of the end transactons we wll have the followng dfferent possbltes. In-the-money s an opton whose strke prce s below the current stock prce. At-the-money s an opton whose strke prce s close to the current stock prce. Out-of-the-money an opton whose strke prce s above the current stock prce. Most exchanges contnually lst optons and there are all three types of optons for each expry cycle. Amercan opton s that defned earler, can be exercsed on or before maturty. European opton can be only exercsed on the maturty date. Usually ther prces only slghtly dffer. Intrnsc value s the dfference between the current stock prce and the strke prce. Tme value s the dfference between the opton prce and the current stock prce and s the money the nvestor has at rsk f the stock prce stays constant.

4 Trees and Black-Scholes Approach Let us dscuss now the so-called tree approach to opton prcng and ts connecton to contnuous Black-Scholes one. Later on we assume that the nterest rates r 0. Due to asymmetry and no-arbtrage one can see that max( S K,0) C( t) S( t) that s presented at the graph. 1 C Fg.1 K S The parameters necessary to calculate the opton prce C are K, the strke prce; S(0), r S(T) = S ds, (spot prce) wth no probablty needed for umpng up or down. Black-Scholes prcng formula through Cox, Ross & Rubnsten tree. For better explanng ths approach let us consder a smple example of the bnomal tree for stock dynamcs wth r 0.08, K $10. So, the dagram for obtanng the value of called opton C s S=100, C=? q 1- q S 180, C max( S K,0) 60 S 60, C max( S K,0) 0

5 Next we can buld a portfolo: Long N shares at $100 (current prce). Borrow $B amount of money at r = 0.5. The net out-of-pocket cost s NS B. We consder ths portfolo as a replca of the opton C: C ( NS B) Ths portfolo gves the same return as the call opton at the cycle end. So now we have the tree NS (1 r) B C C NS B NS (1 r) B C As can be easly calculated the solutons at the end of the cycle are C C 60 0 N 0.5 S S SC SC 1 NS C 0,5.600 B. 4 S S 1r 1r 1.5 The replca of the call on the cycle s long N = 0.5 shares at S = $100 and borrowng B = $4 at r = 0.5, then we have obtaned pc (1 p) C C NS B r 1 r C f wth (1 rs ) S p S S 0.54 Beng a rsk-neutral probablty (due to no-arbtrage). Denotng, S us S ds

6 Then the followng relatons R u 1, R d 1 are the returns, whle the rskneutral evaluaton s R pr (1 p) R r That s also wrtten as a followng relaton (1 r) d p u d All are pretendng to be n the rsk-neutral world and the rsk s rrelevant. Multple perods If ths s the case, let us subdvde the tme to expry T t nto n equal subntervals, h / n and then the expected termnal opton value s 0 C Eˆ max( SK,0) n n! n n p (1 p) max( u d SK, 0)!( n )! (1 r) n n The factor [ n!/!( n )!] p (1 p) s the bnomal probablty that the stock wll take upward umps n n steps, each wth (rsk-neutral) probablty p. The second factor n max( ud S K,0) gves the call opton value at expry condtonal on the stock followng ups and n- downs. Let C be n-the-money opton for m ups then and wth m nm u d S K K C S[ m, n, p] [ m, n, p] () (1 r) n n [ mnp,, ] ( n!/!( n )!) p(1 p) m n the bnomal DF (probablty at least m ups out of n steps) and u p p 1 r

7 For h / n 0 we have [] N[]. Let us denote as usual [3] 1 u h d u p h r exp( ), 1 /, {1 ( / ) }, ln / After some evaluaton we fnally come to the B-S formula r CSNx ( ) Ke Nx ( ) S B Wth the followng defnton of x ln( S/ K(1 r) ) / The symbol that appears here s due to the dynamcal contnuous rehedgng. As s known the B-S equaton wthout dvdends reads 1 C C C S rs rc S S wth the termnal condton CS (, 0, K) max( S K,0) 0 (1) and the boundary condtons C(0,, K) 0, C( S,, K) S, as S Now let us obtan some concluson from all these approaches. If a constant contnuous compound dvdend yeld s present ( D S Se 1 C C C S ( r D) S rc 0 S S And consequently ) then we have D r CS (,, r,, KD, ) Se Nx ( ) Ke Nx ( ) x{ln( S/ K) ( rd / ) }/

8 Mult-stock descrpton Consder a certan dervatve securty that depends on n state varables and tme, t. We make the assumpton that there a total of at least n+1 traded securtes (ncludng the one under consderaton) whose prces depend on some or all of the n state varables. In practce ths s not unduly restrctve. The traded securtes may be optons wth dfferent strke prces and exercse dates, forward contracts, futures contracts, bonds, stocks, and so on. We assume that no dvdends or other ncome s pad by n+1 traded securtes. 1. The short sellng of securtes wth full use of proceeds s permtted.. There are no transactons costs and taxes 3. All securtes are perfectly dvsble 4. There are no rskless arbtrage opportuntes 5. Securty tradng s contnuous The n state varables are assumed to follow contnuous-tme Ito dffuson processes. We denote the th state varable by (1 n) and suppose that d mdt sdz () Where dz s a Wener process and the parameters m and s are the expected growth rate n. The m and s can be functons of any of the n state varables and tme. Ths s not restrctve. A non dvdend payng securty by renvestng the dvdends n the securty. Other notaton used as follows : Correlaton between dz and dz (1, k n) k f : Prce of the -th traded securty (1 k, n) r: Instantaneous (.e. very short term) rsk free rate k One of the f s the prce of the securty under consderaton. The short-term rsk-free rate, r, may be one of the n state varables. Snce the n+1 traded securtes are all dependent on the t follows from Ito s lemma n Appendx 1A that the f follow dffuson processes: df f dt f dz (3) Where f f f 1 f m kss k k t, k k (4) f f s (5) In these equatons s the nstantaneous mean rate of return provded by f and s the component of the nstantaneous standard devaton of the rate of return provded by f, whch may be attrbuted to the

9 Snce there are n+1 traded securtes and n Wener processes n Equaton (3), t s possble to form an nstantaneously rskless portfolo, P, usng the securtes. Defne k as the amount of the th securty n the portfolo, so that k f (6) The k must be chosen so that the stochastc components of the returns from the securtes are elmnated. From Eq. (3) ths means that k f 0 (7) For 1 n. The return from the portfolo s the gven by d k f dt The cost of settng up the portfolo s k f. If there are no arbtrage opportuntes, the portfolo must earn the rsk-free nterest rate, so that Or k f r k f (8) k f ( r) 0 (9) Eq.(7) and (9) can be regarded as n+1 lnear equatons n the k s. The k s are not all zero. From a well known theorem n lnear algebra, th homogeneous equatons (7) and (9) can be consstent only f (10) Or f ( r) f r (11) For some (1 n), whch are dependent only on the state varables and tme. Ths proves the result n Equaton (1.13?). Substtutng from equatons (4) and (5) nto equaton (10), we obtan f f 1 f f m kss k k rf s t, k k Whch reduces to f f f 1 ( m s) kss k k rf, k k t (1) Droppng the subscrpts to f, we deduce that any securty whose prce, f, s contngent on the state varables (1 n) and tme,t, satsfes the second order dfferental equaton

10 f f 1 f ( m s) kss k k rf t, k k (13) Applcaton Two underlyng traded assets V and S. When we have ths case, the equaton (13) assumes the form 1 1 vv Fvv vsvsvsfvs ss Fss (1 Dv) VFv ( rd ) SF F rf s s Ths equaton has been used by Magrabe [4] to evaluate an opton to exchange S by V and Myers & Mad [5] for opton to abandon wth V beng a value of the proect and S ts uncertan salvage value. Opton to exchange S by V. In ths case F(V,S) s a homogeneous of degree 1 functon,.e. FcVcS (, ) cfvs (, ) and FV (, S, ) SFV ( / S,1, ) f F/ S, XV/ S, K1 r D s s, v s vsvs Therefore the equaton for F now reads 1 S X Fxx Ds Dv XFx DsF F whereby ( ) 0 Dv Ds CX (,, sd,, D) Xe Nd ( ) 1 e Nd ( s ) s and fnally we have obtaned that v

11 Dv Ds FV (, S, ) Ve Nd ( ) Se Nd ( s ) d V S D D s s Here {ln( / ) ( s v / ) }/ Se D s s a future prce for the uncertan varable wth yeld D s. One of possble applcaton of the model s consderng frm s operatons as a seres of European optons to exchange the uncertan varable producton cost (S) for the uncertan revenue (V). Real Captal Investment Opportuntes as Collectons of Optons on Real Assets (real optons) For ths approach let us assume we now have the conventonal NPV (net present value) technque,.e. the rsk-less world: For one perod we wll have : C1 NPV I wth 1 r C 1 beng a cash flow at the end of the perod and I an nvestment at the begnnng. For T perods we can calculate : NPV C O (1 ) (1 r) T T 1 r 0 where C O s a cash nflow at the end of perod s an analogous cash outflow In case that we are n front of a Rsk-adverse world, the standard procedure gves

12 T ˆ T C E{ C} NPV I I 1 (1 r) 1 (1 k) k r p p E{} r r cov(, r rm) Er { m} r = s the market prce of rsk var( rm ) r s the market rsk m s a dscount rsk premum The basc nadequacy of ths approach and other dscount cash flow (DCF) approaches to captal budgetng s that they gnore management s flexblty to adapt and revse later decsons, vz. to revew ts mplct operatng strategy. The tradtonal NPV approach makes mplct assumptons concernng an expected scenaro of cash flows and presumes management s commtment to a defnte operatng strategy. In dong ths, an expected pattern of cash flows over a specfed proect lfe s dscounted at a rsk-adusted rate to arrve at the proect s NPV (what s reflected n the above formulae). Ths rate s usually derved from the prces of a twn traded fnancal securty. Only proects wth postve NPV are to be accepted. In the real world of uncertanty and competton the realzaton of cash flows would dffer from what management orgnally expected. As new nformaton arrves and uncertanty about future cash flows s gradually resolved, management may fnd that exstng (or created) flexblty to depart from the orgnal proect desgn allows t to revse the ntal operatng strategy. For nstance, management may be able to abandon, defer, expand, contract, or some other way, alter a proect at varous stages of ts lfe. Ths flexblty ntroduces specfc elements smlar to those of fnancal optons, n partcular asymmetrc dstrbuton. Then the true expected value, or expanded expected NPV ncorporates manageral operatng flexblty and strategc adaptablty. It exceeds the statc or passve expected NPV by an opton premum reflectng that flexblty. Quantfyng Flexblty. Real Opton Calculus. Ablty to create a rsk-less replcatng portfolo f the underlyng asset s traded or to obtan a certanty-equvalent expected growth rate by subtractng an approprate rsk premum ( allows one more convenent valuaton n a rsk-neutral world, where r F e Eˆ( F T ) the rsk-neutral expectaton of a future opton payoff (at maturty T), F T, can be dscounted at the rate, r,.e. (European calls) where and F max( S E,0) T T ds S dt dw

13 for S traded r and s the dvdend yeld. The trangle S T N 1 S generates a sample of traectores wth the error ~ s / n. The maor drawback t s lmted to European - type optons wth no early exercse or ntermedate decsons. Fnte-Dfference Methods For ths method to apply we wll use the followng Kolmogorov BS equaton 1 SFSS ( r) SFS Ft rf0 where, r are constant. And the boundary condtons FST (, ) max( ES,0) (termnal for put) F(0, T) E (lower boundary) FST (, ) / S 0 (upper boundary) Let us consder K-BS equaton and derve a fnte-dff. approxmaton. Let F(s,t) = F(h,q) then we have F F F h O h S ( 1, 1, )/ ( ) SS 1,, 1, F ( F F F )/ h O( h ) F ( F F )/ q O( q) t, 1, cf cf cf F 0 1,, 1,, 1 and the mplct scheme gves 1 ( ), 0 1 ( c r q c r ) q

14 1 1 1 Ths s a system of (3xN) lnear equatons such that and so on all down tll = 0. The explct scheme. F and F s ss 1. After some calculatons we have got the next equaton Further, when we use the and the coeffcents are determned by the next set of relatons representatons let us make the substtuton for each tme step backward (lattce approach). The coeffcents p and p 0 are the rsk-neutral probabltes that the state varable, S, beng n state at tme wll ump up (to state +1), ump down (to state -1) or stay n the same state () by the next perod (tme +1) respectvely and all of them should be non-negatve or otherwse an nstablty may arse. So the explct scheme gves the equaton that says that the current opton prce s obtaned from the expected one perod future opton values (usng the probabltes n a trnomal tree), dscounted back at the rsk-less rate n a rsk-neutral world Some smplfcatons for obtanng solutons 1. the frst smplfcaton we can do s a log-transformaton of S, vz. X = lns that leads to the followng equaton 1 1 F ( ) 0 ( ) XX r F X Ft rf wth F (1 rq) p F p F p F ( ) 1 0, 1, 1, 1 1, 1 1 p rqc 0 p 1 q1 ( p p ) (0,..., M), (0,..., N)

15 1 p h r hq ( / ) ( / )/ 0 0 p h q 1 ( / ) 0 So the p s varables are ndependent of the state and can be chosen always to be nonnegatve: q h h r ( / ), / 1 and X follows a trnomal ump process X p p p 0 +h 0 and -h E( X) p h p ( h) ( r / ) q Var( X) E( X ) [ E( X)] p h p ( h) ( ( / r / ) q q r ) q Unfortunately we have obtaned Var(DX) < process. q for the varance of the contnuous. The second smplfcaton conssts n the followng scheme. Removng the term rf by f ( X, ) e r F( S, ) and usng the mplct dff. scheme (at ) we come to (*) wth (**). 3. The thrd one, s done by playng wth the probabltes, consderng the fntedfference equaton as a phenomenologcal one, then we have and 1 1 p ( / h) ( r / )/ hq ( r /) q/ h 0 p 1 ( / h) q ( r /) q/ h

16 E( X) ( r /) q and Var( X) q 4. The next smplfcaton wll be done by a more general transformaton 1 r X S r f X t e F S t (log., detrended and normalzed transformaton) gves 1 or ln /, (, ) (, ) f XX f t 0 (heat eq.) f (1 q) p f p f p f 1 0, 1, 1, 1 1, 1 wth p q h p q h q h 0 /, 1 / / 1 now E( X) ( p p ) h0, Var( X) ( p p ) h q therefore n the case h = q we have p 1 0, p 0 the bnomal tree results (a partcular case of the bnomal tree). Specal bnomal approach At ths stage we can establsh the goal: to desgn method applcable both to the valuaton of complex fnancal optons and to the valung of captal budgetng proects wth multple real optons. In our case the underlyng asset, V the gross present value of the expected cash flows from mmedately undertakng the real proect (rather than S). Assume dv V dt dw Wth the followng parametrc defnton

17 the nstantaneous expected return on the proect, the nstantaneous standard devaton, q=t then X=lnV follows an arthmetc Brownan moton. If we analyze the case under rsk neutralty, for whch r, we obtan As t can be easly seen, the ncrements, X, are ndependent, dentcally and normally dstrbuted wth X ln V / V ( r / ) t t t t W E( X) ( r / ) t and Var( X) t Denote q, then E( X) q and Var( X) q, Consder the dscrete process wth the same mean and Var and q / N the bnomal tree s p +h p p p -h r 1 Then ph h =q, p =(1+q/H)/or E( X) ph(1 p)( h) phh Var( X ) ph (1 p) h ( ph h) h ( ph h) h ( q) q, h q( q) mq or

18 q/ h 1, 0 p 1 and there are not external constrants for stablty. Implementaton For ths am we wll follow the next four man steps: 1. The standard parameters are specfed affectng opton values V r,,, T, (Any dvdend yeld ), set of costs outlays I,s and the number of subntervals N The cash flows, CF, and ther tmng (f dscrete) and the type, tmng and other characterstcs of the embedded real optons are specfed as well.. Calculatons of the algorthm parameters; tme step q = T / N, drft = (r - )/ 1/ value step h = q ( q) probablty 1 p (1 q / h ) 3. Determnaton of termnal values (at = N) for each state,: X ( ) X h and F( ) R( ) then 0 4. Backward teratve process: for each step ( = N,,1) and every second state, calculate opportunty values usng nformaton from step ( +1) as Adustments for cash flows (dvdends): At each cash nflow (ex-dvdend) tme, determne downward extenson of trangular path and shft () for each state : R() R( ) CF. At each cash outflow (exercse) tme: V e R V h () and () max (),0 R() R() I R e pr p R rq / '( ) ( 1) (1 ) ( 1) Adustments for multple real optons: The proect can be outlned as shown n the pcture:

19 Three nvestment outlays I 1, I, and I 3 durng the buldng stage, then possble cash nflows. The followng Optons are avalable : 1) to wat up to T 1 years; ) to abandon early by forgong a preplanned outlay I, 3) to contract the scale of operatons by c% thereby savng part I 3 of a planned outlay I 3, 4) to expand producton be e%, makng an extra outlay I 4, 5) to swtch the proect from current to ts best future alternatve use or to abandon for ts salvage value (S). Swtch use (abandon for salvage S): R = max(r, S) expand by e% by nvestng addtonal : R = R+max(eV - I 4, 0) contract by c%, savng : R = R+max(I 3 - cv, 0) abandon by defaultng on : R = max(r - I, 0) defer untl next perod: r R max( e E( R 1), R ) Adustments for exogenous compettve arrvals (umps) must be made at approprate tmes. Applcatons. Power Plant At ths part we wll follow the work [10].Valung a power plant usng real opton theory has two man purposes n compettve markets: 1. Accurately determne ts value.. To facltate the use of rsk management tools developed for fnancal markets n order to hedge both asset value and earnngs. (For nstance, a power plant can be hedged usng forward electrcty contracts.

20 Ignorng non-fuel costs, the net proft per hour for a power plant s E F NP q P HP where the nvolved varables are q s dspatch (output) level (MW) E P s electrcty spot prce ($/MWH) F P s the nput fuel spot prce ($/MMBtu) H s the plant heat rate (MMBtu fuel per MWh electrcty) The quantty ( PE HPE) s the spark spread (sp-sp) and If sp-sp > 0 then q must be maxmal If sp-sp < 0 then shut down.e. the nstantaneous plant pay-off per unt capacty s an opton to exchange one asset E P wth exercse prce max( P E HP F,0) HP F max( P E, HP F ) F HP. F HP for another There are three types of generatng plants (unts): 1. base-load (low nput costs) n-the money opton wth low prce enough to work. md-load 3. peakers (hgh nput costs) out-of-the-money opton wth hgh prce needed to work E P, or a call opton on the asset Ths s a good-but-not-enough approach (lnear opton to exchange) for t may msprce the plant value and mslead on the optmal operatng polcy. The followng mportant characterstcs (restrctons) are not nvolved: 1. Mnmum on (up) and off (down) tmes.. Mnmum ramp (start-up) tme (e.g. heatng the boler) 3. Mnmum generaton level 4. Response rate constrants (tme requred to effect a dscrete change n the dspatch level) 5. Non-constant heat rate (heat rate H vares wth the generaton level) 6. Varable start-up cost (cost to start-up depends on the tme spent off-lne)

21 Stochastc dynamc programmng s a tool to solve the problem, vz. to calculate plant values and optmal operatng polces. Two tasks Developng a lattce for the underlyng stochastc varables Backward dynamc programmng to compute the value & the optmal operatng polces. For a method, see n Hull & Whte 1993 for path dependent opton evaluaton. Also ths approach s good for energy ppelnes and storage facltes. Model: evaluaton of thermal power unts over a short-term horzon (a week). Tme spacng (decson makng nterval) s 1 hour: 0,1,,,T Table 14 Parameter descrpton unts t on mnmum up tme hours t off mnmum down tme hours t cold addtonal tme over t off hours wth varable cost t ramp tme requred to brng a unt hours on lne q mn mnmum dspatch level MW q max maxmum dspatch level MW H(q) heat rate MMBtus/ MWh Operatng state constrants s s the operatng state of the plant consstng of plant condton and ts duraton N s the total number of the plant states: 1 s N wth

22 N t off t cold t ramp t on Table 15 Plant condton States Off-lne 1 s t t o ff cold Ramp (unable to sell power) toff tcold s toff tcold tramp On-lne toff tcold tramp stoff tcold tramp ton State transton dagram Example: Condton t, t, t 1, t on off ramp cold Duraton (hours) Off-lne Ramp 5 On-lne 6 7 Mnmum 8 Maxmum State 1: a plant has ust gone off-lne

23 State : a plant has been off-lne for an hour State 3: a plant remans off-lne hours State 4: off-lne for 3 or more hours State 5: start up State 6: on-lne at the mnmum dspatch level State 7: on-lne for one or more hours ( hours t on ) at the mnmal dspatch level State 8: on-lne for one or more hours at the maxmum dspatch level Possble transtons s s shown at the dagram mean that s A(s). Prce Processes P P wth ndex specfyng the pont n E F Let us assume the prces are dscrete: t and t the prce space. Assume also that the spot prce of electrcty follows a mean-revertng geometrc Brownan moton process: E E dln P p a ln P dt dw where p E s a drft parameter, a E s the mean reverson rate, s the volatlty and E dw E s the Wener generator. In the Hull book t s shown that a trnomal tree may be used to represent ths process. Ths tree s determned by the prce space and the transton probabltes we have dscussed. The drft term s assumed tme-dependent to calbrate to an observed forward prce curve. In the smplest model the fuel prces are put constant, however ths restrcton can be removed by consderng two-factor model. Some more smplfcaton of the model s to assume the plant heat rate to be constant and playng wth only one stochastc process, the spark spread drectly. Costs and Revenues Each operatng state has an assocated cost or revenue. We assume the followng form f Here K fx s the fxed cost n all states. Transton costs (from state s to state s`) may be accounted for. Dspatch and response rate constrants mn E E E E K 1 s t t t ( q, s) qpt H( q) Pt K fx otherwse t E F max fx off cold ramp q0 f s 1,..., t off t cold t startup q qq otherwse

24 Ths s denoted as q B(s). These constrants mpose no restrcton on how fast a plant can change ts dspatch level: f t s on-lne, t can be dspatched at any output level. The thrd dmenson to the state descrptor (n addton to plant condton and duraton) s two dscrete levels of the plant dspatch, mn and max. There s a restrcton for that transton, say one hour. Soluton method Here we face the optmzaton problem that may be formulated n a set of tme perods. An optmal polcy wth n perods remanng may be determned by selectng the polcy that maxmzes the sum of net revenue n perod n plus the expected net revenue n the subsequent n 1 remanng perods. The optmal polcy for ths problem s to solve, t1 t s A( s ) F () s max f (,) q s p max F ( s ) g ( s, s ) t t t qb( s) Here F t (s) denotes the value of the power plant over the perod t to T condtonal on beng n energy Prce State at tme t and operatng state s ; p t represents the probablty of movng from prce state at tme t to prce state at tme t + 1. Ths equaton states that the value of the plant over the remanng perods (from tme t to T) s the sum of the net revenue n perod t and the expected value of the power plant from tme t + 1 to T whch s condtonal on the plant operatng state at tme t + 1. We select the operatng state s that results n the maxmum plant value (net the state transton cost), condtonal that t s feasble transton from state s. Ths maxmzaton determnes the optmal operatng state transton polcy for the plant. The plant value at tme 0, F0,0( s ) s obtaned by solvng the equaton recursvely, backward from tme T for all possble Prce States and operatng States s, to tme 0 whch has only a sngle known prce state. In addton to plant value, a key result of the soluton s the optmal operatng polcy that conssts of the optmal plant output n each on-lne state as a functon f prce state and tme, and the optmal state transton strategy as a functon of the current operatng state, prce state, and tme. The optmal operatng polcy should be used by the plant operators to maxmze the plant value. Usually the optmal state transton strategy may be expressed n terms of a set of exercse boundares. For example, f the current state s on-lne, the optmal transton n the next perod wll be to reman on-lne for all values of the spark spread greater than a certan crtcal value and go off-lne for all spark spreads that are less. In general those boundares vary through tme. Smple example: t, t 1, t t 0 on off cold startup H = const., K fx, q mn = 0.5 MW, q max = 1. See the dagram (???????)

25 Feasble operatng state transtons condton duraton 0 1 Off-lne 1 On-lne 3 Plant value and optmal operatng polcy: NV + Opton s q s F s q s F (,) (3,) (3,3) (,) (3,3) (3,3) s q s F (-,-) (-,-) (-,-) s q s F (-,-) (-,-) (-,-) s q s F (,) (3,) (3,1) 1-4 s q s F (,) (3,) (3,3) - State transton decsons should take nto account not ust mmedate net revenue but also the opportunty cost n terms of future decson-makng flexblty; the smple exchange

26 opton approach does not consder ths. Ths phenomenon explans why electrcty prces have gone to zero or even negatve for short tme perods n some markets. Conclusons We saw how real optons theory may be appled to value power generaton assets. In partcular, the model we develop s capable of handlng constrants related to mnmum on-and off-tmes, ramp tmes, mnmum dspatch levels and response rates. The optmal operatng polcy also may be very much affected. Real optons theory supples a methodology for quantfyng the value of the operatng flexblty of real assets and for determnng optmal operatng polces. It s possble to mprove greatly the effectvty of operatng optons and to reveal "hdden" asset value. Understandng the sources of asset value and ts senstvty to fuel and electrcty prces s also crtcal for companes seekng to determne a sutable hedgng polcy through ether forward sales or other dervatves contracts. Effectve applcatons of real optons theory demands that managers become famlar wth ts underlyng assumptons n order to understand ts strengths and weaknesses as well. The pay-off for companes s the ablty to effectvely leverage a company's assets to acheve an optmal trade-off between rsk and payoff. Lterature 1. F. Black, and M. Scholcs.. The prcng of optons and corporate labltes, Journal of Poltcal Economy, 8(1973) ,.. K. B. Connolly. Buyng and Sellng Volatlty. Wley, Chchester, J.C. Hull. Optons, Futures, and other Dervatve Securtes. Prentce Hall, NJ, J.C. Cox, S.A. Ross, and M. Rubnsten. Opton prcng: A smplfed Approach. J.Fn.Econ. 7(1979) W. Margrabe. The value of an opton to exchange one asset for another. J.Fnance, 33(1978) S.C. Myers and S. Mad. Abandonment value and proect lfe. Adv.Future & Opton Research, 4(1990) L. Trgeorgs. Real Optons. MIT Press, Cambrdge, Mass V. Makhankov. Fnance Calculus. Markov Processes and Some Mathematcal Models of Investment. Lecture Notes, Moscow, 1998, Santa Fe, T. Copeland and V. Antkarov. Real Optons. A Practtoner s Gude. TEXERE, NY, J. Cowan. An Introducton to the Mathematcal Theory of Opton Prcng and Related Topcs. Lecture Notes, Santa Fe, Bos, D. Gardner and Y. Xhuang. Valuaton of power generaton assets: A real opton approach. ALGO Research Quarterly. 3(000)9-0.