Modelling spikes and pricing swing options in electricity markets

Transcription

1 Modelling spikes and pricing swing opions in elecriciy markes Ben Hambly Sam Howison Tino Kluge April 24, 27 Absrac Mos elecriciy markes exhibi high volailiies and occasional disincive price spikes, which resul in demand for derivaive producs which proec he holder agains high prices. In his paper we examine a simple spo price model ha is he exponenial of he sum of an Ornsein-Uhlenbeck and an independen mean revering pure jump process. We derive he momen generaing funcion as well as various approximaions o he probabiliy densiy funcion of he logarihm of he spo price process a mauriy T. Hence we are able o calibrae he model o he observed forward curve and presen semi-analyic formulae for premia of pah-independen opions as well as approximaions o call and pu opions on forward conracs wih and wihou a delivery period. In order o price pah-dependen opions wih muliple exercise righs like swing conracs a grid mehod is uilised which in urn uses approximaions o he condiional densiy of he spo process. 1 Inroducion A disincive feaure of elecriciy markes is he formaion of price spikes which are caused when he maximum supply and curren demand are close, ofen when a generaor or par of he disribuion nework fails unexpecedly. The occurrence of spikes has far reaching consequences for risk managemen and pricing purposes. In his conex a parsimonious model wih some degree of analyic racabiliy has clear advanages, and in his paper we propose and examine in deail a simple mean-revering spo price process exhibiing spikes. In our model he spo price process S is defined o be he exponenial of he sum of hree componens: a deerminisic periodic funcion f characerising seasonaliy, an Ornsein-Uhlenbeck (OU) process X and a mean-revering process wih a jump componen o incorporae spikes Y : S = exp(f() + X + Y ), dx = αx d + σ dw, dy = βy d + J dn, (1) where N is a Poisson-process wih inensiy λ and J is an independen idenically disribued (iid) process represening he jump size. We assume W, N and J o be muually independen processes. 1

2 Our model generalises a number of earlier models. For example, he commonly used model of [Lucia and Schwarz, 22] is obained by seing β = and aking J =. In his model S is log-normally disribued giving analyic opion price formulae very similar o hose in he Black-Scholes model. To allow for a sochasic seasonaliy a furher componen can be insered ino he model and, as long as his process has a normal disribuion, analyic racabiliy is mainained. The main disadvanage of he Lucia and Schwarz models is heir inabiliy o mimic price spikes. To overcome his, jumps can be added o he model; for example he case β = in our model, dx = αx d + σ dw + J dn, S = exp(f() + X ). (2) This model is briefly menioned in [Clewlow and Srickland, 2, Secion 2.8]. Analyic resuls are given in [Deng, 2] based on ransform analysis described in [Duffie e al., 2]. Calibraion o hisorical daa and he observed forward curve is discussed in [Carea and Figueroa, 25] where pracical resuls for he UK elecriciy marke are given. For hese models o exhibi ypical spikes he mean-reversion rae α mus be exremely high, oherwise he jumps do no rever quickly enough. In [Benh e al., 25] a se of independen pure mean-revering jump processes of he form S = i w i Y (i), dy (i) = α i Y (i) d + σ i dl (i), are inroduced, where w i are some posiive weighs and he L (i) are independen increasing càdlàg pure jump processes. The spo price process is a linear combinaion of he pure jump processes and as here is no exponenial funcion involved, posiiviy of he spo is achieved by allowing posiive jumps only. An advanage of his formulaion is ha semi-analyic formulae for opion prices on forwards wih a delivery period can be derived. However, a full analysis of his class of models sill seems o be in is early sages. The model (1) we consider in his paper is an exension of (2), in which we allow for wo differen mean-reversion raes, one for he diffusive par and one for he jump par. The inroducion of a mean revering spike process Y allows us o choose a higher mean-reversion rae β in order for he jump o rever much more quickly and so mimic a price spike. This is crucial for modelling he NordPool marke bu migh no be needed in markes where he speed of mean-reversion α is generally very high, like in he UKPX or EEX. Reurning o our model (1) and solving for X and Y, we have X = X e α +σ N e α( s) dw s, Y = Y e β + e β( τi) J τi, (3) where τ i ( indicaes he random) ime of he occurrence of he i-h jump. Thus, given X = x, X N x e α, σ2 2α (1 e2α ). Properies of he spike process Y are no as obvious and will be examined in he following secion. A his poin we make no assumpion on he jump size J bu will laer give resuls for exponenially and normally disribued jump sizes. Noe also ha, alhough X and Y are boh Markov processes, he price process S is no. We will herefore assume ha all hree componens of he price process, i.e. he ime values f(), X and Y are individually observable, implying we expec jumps no o be small. Figure 2 i=1

3 X Y S exp(f()) Figure 1: Simulaed sample pahs of X, Y and S of he spo model (1). We use he following parameers which are of he same order of magniude as calibraed values of he Nord Pool spo marke, excep he seasonaliy funcion f( ) which has been chosen arbirarily: f() = ln(1) +.5 cos(2π), α = 7, σ = 1.4, β = 2, J exp(1/µ J ), µ J =.4, λ = 4. 1 shows a simulaed sample pah of he processes X, Y and he composed process S. In Secion 2 we derive imporan sochasic properies of he process, including he momen generaing funcion and various approximaions o is probabiliy densiy funcion. Pricing of a variey of derivaive conracs will be discussed in Secions 3 and 4, using he resuls obained in Secion 2. 2 Properies of he model for spo prices 2.1 The spike process The following resul is known and given in [Duffie e al., 2] in a more general framework. Lemma 2.1 (momen generaing funcion of he spike process, Y ) Le {J 1, J 2,...} be a series of iid random variables wih he momen generaing funcion Φ J (θ) := E[e θj ]. Le {τ 1, τ 2,...} be he random jump imes of a Poisson process N wih inensiy λ. Then he process Y wih iniial condiion Y = has he momen generaing funcion ( Φ Y (θ, ) := E[e θy ] = exp λ Φ J (θ e βs) ) 1 ds. (4) Furhermore, he firs wo momens of Y are given by E[Y ] = Φ Y (, ) = λ β E[J](1 e β ), E[Y 2 ] = Φ Y (, ) = E[Y ] 2 + λ 2β E[J 2 ](1 e 2β ), 3

4 and in paricular we have var[y ] = λ 2β E[J 2 ](1 e 2β ). Remark 2.2 (asympoics for β ) As remarked above, in pracice he imescale 1/β for mean-reversion of spikes is much shorer han any of: he conrac lifeime T ; he diffusive mean-reversion ime 1/α; he volailiy imescale 1/σ 2 ; he mean arrival ime of spikes 1/λ. We herefore calculae approximaions for he momen generaing funcion of he spike process, and for is disribuion, as β. To analyse he behaviour of he momen generaing funcion for large β we make he subsiuion u = θ e βs in he inegrand o obain ( λ θ ) Φ J (u) 1 Φ Y (θ) = exp du. β θ e β u For fixed θ,, as β we have θ e β Φ J (u) 1 u du = θ e β E[J] + O(e 2β ), because Φ J (u) = 1 + E[J]u + O(u 2 ), u, and so ( ( λ θ )) Φ J (u) 1 Φ Y (θ) = exp du θ e β E[J] + O(e 2β ). (5) β u Example 2.3 (exponenially disribued jump size) If J Exp(1/µ J ) wih mean jump size µ J, hen Φ J (θ) = 1/(1 θµ J ), θµ J < 1. We obain ( 1 θµj e β ) λ/β Φ Y (θ, ) =, θµ J < 1. 1 θµ J As, we have Φ Y (θ, ) (1 θµ J ) λ/β so he saionary disribuion for Y is Γ(λ/β, 1/µ J ). As β, we also have Φ Y (θ, ) = 1 + θµ J λ/β + O(β 2 ) (1 θµ J ) λ/β. Thus Y is approximaely Γ(λ/β, 1/µ J ) disribued for large β. The mean and variance of he spike process Y wih Y = are E[Y ] = λµ J β (1 e β ), var[y ] = λµ2 J β (1 e 2β ). 2.2 The combined process Having examined he properies of he spike process Y we conclude properies of he sum X + Y and consequenly of he price S = exp(f() + X + Y ). Theorem 2.4 Le he spo process S be defined by (1) and le Z := ln S = f() + X + Y wih X and Y given. The momen generaing funcion of Z is hen E e θz = exp (θf() + θx e α +θ 2 σ2 4α (1 e 2α ) + θy e β +λ Φ J (θ e βs) ) 1 ds. 4 (6)

5 Proof. The processes X and Y are independen so he expecaion of he produc is he produc of he expecaions. The momen generaing funcion of Y is given in Lemma 2.1 which yields he resul. The expecaion value of he spo process a ime, S immediaely follows by seing θ = Approximaions We now derive approximaions o he densiy funcions of he spike process a mauriy T for large mean-reversion values β of he spike process. Alhough we can always compue he densiy by Laplace inversion of he momen generaing funcion, an explici expression for he densiy allows for more efficien algorihms and more explici opion pricing formulae. Here we only provide an expression for he densiy of a runcaed spike process ỸT as defined below. However, knowledge of he densiy of ỸT alone will help us o efficienly consruc a grid o price swing opions. We sar by defining he runcaed spike process Ỹ, showing ha Ỹ provides a good approximaion o he value Y for large values of β, deriving a general formula for he densiy funcion of ỸT and finally making i more explici by considering an exponenial jump size disribuion. For very high mean reversion raes β and small jump inensiies λ, he dominan conribuion o he densiy of he spike process comes from he las jump. We herefore inroduce he runcaed spike process { J N e β( τ N ) N >, Ỹ := (7) N =. Noe ha we only consider Y saring from ; any oher saring poin can be incorporaed by adding he iniial value. Lemma 2.5 Ỹ is idenically disribued as { J 1 e βτ 1 τ 1, Z := τ 1 >. Proof. We use he reversibiliy propery ha if N = {N ; R + } is a Poisson process hen ˆN = { N ; R + } is also a Poisson process. As τ N is he jump ime of he las jump before, his ranslaes o he firs jump of he reversed process and hence τ N and τ 1 are idenically disribued, given N >. If N = hen here has been no jump in [, ] and he same applies for he reversed process and so his is equivalen o τ 1 >. Lemma 2.6 (momen generaing funcion of he runcaed spike process) The random variable Ỹ of he runcaed spike process a ime wih iniial condiion Ỹ = has he momen generaing funcion ( ) ΦỸ (θ) = 1 + λ Φ J (θ e βs ) 1 e λs ds, and he firs wo momens are given by E[Ỹ] = λ ( β + λ E[J] 1 e (β+λ)), E[Ỹ 2 ] = λ ( 2β + λ E[J 2 ] 1 e (2β+λ)). 5

6 Proof. By Lemma 2.5 we only need o deermine he momen generaing funcion of Z := J e βτ I τ, τ Exp(λ), where I A denoes he indicaor funcion for he even A. Given he jump ime τ we have E[e θz τ = s] = Φ J (θ e βs I s ), and so E[e θz ] = E[E[e θz τ]] = = Φ J (θ e βs I s ) e λs ds Φ J (θ e βs ) e λs ds + e λ. The firs wo momens are given by E[Ỹ] = Φ Ỹ () and E[Ỹ 2 ] = Φ Ỹ (). Remark 2.7 (poinwise convergence of he momen generaing funcions) The momen generaing funcion of he runcaed spike process converges poinwise o he momen generaing funcion of he spike process for eiher λ or β wih and θ fixed. Firs consider λ. Fix all oher parameers and se g(s; β, θ) := Φ J (θ e βs ) 1, hen ( ) Φ Y (θ) = exp λ g(s; β, θ) ds = 1 + λ g(s; β, θ) ds + O(λ 2 ), ΦỸ (θ) = 1 + λ g(s; β, θ) e λs ds = 1 + λ g(s; β, θ) ds + O(λ 2 ). To see he convergence for β wih λ, and θ fixed, firs noe ha from (5) we have Also from Lemma 2.6, Φ Y (θ) = 1 + λ β ΦỸ (θ) = 1 + λ = 1 + λ β θ θ Φ J (u) 1 u du + O(1/β 2 ). ( ) Φ J (θ e βs ) 1 e λs ds Φ J (u) 1 θ e β u ( u θ ) λ/β du by seing θ e βs = u. Now as β, (u/θ) λ/β 1 excep in a small region u = O(θ e β/λ ), which makes a negligible (exponenially small) conribuion o he inegral. Likewise we may replace he lower limi of inegraion by and incur a similarly small error. Hence θ ΦỸ (θ) = 1 + λ Φ J (u) 1 du + o( 1 β u β ) ( ) 1 = Φ Y (θ) + o. β Two examples of he approximaed and exac momen generaing funcion using our sandard parameers can be seen in Figure 2. 6

7 Phi Momen generaing funcion of he spike process Y hea mgf approximaion Phi Momen generaing funcion of he spike process Y hea mgf approximaion Figure 2: Momen generaing funcion of Y and Ỹ, denoed by mgf and approximaion, respecively. In he lef we use J Exp(1/µ J ) and in he righ J N (µ J, µ 2 J ). The same parameers as in Figure 1 are used and we se = 1. Lemma 2.8 (disribuion of he runcaed spike process) Le he jump size disribuion have densiy funcion f J. Then he runcaed spike process Ỹ as defined above has he cdf wih FỸ (x) = e λ I x + fỹ (x) = λ β 1 x 1 λ/β x e β x x fỹ (y) dy,, f J (y) y λ/β dy, x. (8) Proof. Based on Lemma 2.5 i suffices o deermine he disribuion of Ỹ = JZI τ, Z := e βτ, τ Exp(λ). I follows ha Z is he β λh power of an uniformly disribued random variable on [, 1] and is densiy is given by f Z (x) = λ β x (1 λ β ) I x [,1]. As P(τ > ) = e λ we obain he cdf of ZI τ as F ZIτ (x) = e λ I x + f ZIτ (x) = λ β x (1 λ β ) I x [e β,1], f ZIτ (y) dy, and he disribuion of he produc of wo independen random variables J and ZI τ is hen given by c F JZIτ (c) = e λ I c + f JZIτ (x) dx, f JZIτ (c) = f ZIτ (c/x) f J(x) x dx. 7

8 densiy e-4 1e-5 1e-6 disribuion 1e Y mone carlo, T=1 approximaion, T=1 mone carlo, T=1/365 approximaion, T=1/365 Figure 3: Disribuion of he spike process (Y ) a T wih a jump size of J Exp(1/µ J ). We use approximaion (9) and compare i wih he exac densiy as produced by a Mone-Carlo simulaion. We use he same parameers as in Figure 1. Wih f ZIτ (c/x) = λ 1 I β c 1 λ x [c,c e β ]x 1 λ β, β c >, f ZIτ (c/x) = λ 1 I β c 1 λ x [c e β,c] x 1 λ β, β c <, he desired resul follows. Example 2.9 (exponenial jump size disribuion) Le J Exp(1/µ J ) be exponenially disribued. The disribuion of he runcaed spike process Ỹ is fỹ (x) = λ βµ λ/β J Γ(1 λ/β, x/µ J ) Γ(1 λ/β, x e β /µ J ) x 1 λ/β, x >, (9) where Γ(a, x) is he incomplee Gamma funcion. The approximaion is a good fi o he exac densiy for ypical marke parameers as can be seen in Figure 3. The only discrepancy occurs a Y = where he densiy has a singulariy. We use his approximaion in Secion 4.1 o efficienly generae a grid o price swing opions. 3 Opion pricing The elecriciy marke wih he model presened is obviously incomplee. No only are we faced wih a non-hedgeable jump risk bu also we canno use he underlying process (S ) o hedge derivaives due o inefficiencies in soring elecriciy. Hence, he discouned spo price 8

9 process in he risk-neural measure is no necessarily a maringale. From now on we assume he model is specified in he risk-neural measure Q as S = exp(f() + X + Y ), dx = αx d + σ dw, dy = βy d + J dn, (1) where W is a Brownian moion under Q and N a Poisson process wih inensiy λ under Q. For simpliciy of noaion we use he same parameers as in (1) bu noe ha hey migh differ from he parameers under he real world measure. As is reasonable, he risk-neural model has a jump srucure ha is similar o ha observed under P. Lemma 3.1 (seasonal funcion consisen wih he forward curve) Le = and F [T ] be he forward a ime mauring a ime T ; hen he risk-neural seasonaliy funcion is given by T f(t ) = ln F [T ] X e αt Y e βt σ2 4α (1 e 2αT ) λ Φ J (e βs) 1 ds. (11) Proof. The forward price is F [T ] = E Q [S T ] and so he resul follows from (6). This resul forms an imporan par of he calibraion of he model. As he model is incomplee he calibraion procedure depends on he se of liquid derivaives used. Here we assume a coninuous forward curve is observable in he marke, i.e. values of F [T ] are given. This is no a realisic assumpion bu here are ways o generae a coninuous curve consisen wih discreely observed prices, see Figure 4 and [Kluge, 26, Secion ] for more deails. For he sake of simpliciy we adop a policy of choosing parameers from he real world measure P if hey are no uniquely deermined by he se of observed derivaive prices. This is equivalen o saying we choose a risk-neural measure Q which changes as few parameers of he model as possible. The volailiy parameer σ remains unchanged by any equivalen measure change so we can always deermine i from hisorical daa. If we only see forward prices in he marke we can also calibrae all oher parameers o hisorical daa excep he seasonal funcion. In brief, his can be done by de-seasonalising he daa using he realworld seasonaliy and hen making a firs esimaion of α from which we can sar filering suspeced spikes. From he reduced daase, α can be re-esimaed and suspeced spikes filered recursively. Having deermined all parameers from hisorical daa we finally calculae he risk-neural seasonaliy funcion f from he observed forward curve based on (11). 3.1 Pricing pah-independen opions If he payoff of an opion on he spo a mauriy T is given by g(s T ) hen is arbirage free price a ime is given by V (x, y, ) = e r(t ) E Q [g(s T ) X = x, Y = y]. Alhough we do no have an expression for he densiy of S T we know is momen generaing funcion and so can apply Laplace ransform mehods o calculae he expecaion value. 9

10 3 25 NordPool forward inerpolaed (spline) inerpolaion forwards on 1/6/21 spo seasonaliy 2 NOK/MWh Jan 1999 Jan 2 Jan 21 Jan 22 Jan 23 Jan 24 Jan 25 Figure 4: Inerpolaion of he forward curve by a seasonal funcion and spline correcion. Three years worh of spo hisory daa has been used o calibrae a seasonaliy funcion which is hen used as a firs approximaion of he forward curve. The difference beween he seasonaliy funcion and he observed forward prices is hen correced by a piecewise quadraic polynomial. 1

11 For an overview see [Con and Tankov, 24, Secion ] 1 or [Carr and Madan, 1998] and [Lewis, 21]. Consider, for example, pu or call opions. Le Z = ln S and le Φ Z (θ) be is momen generaing funcion, as given in (6). Now define is runcaed momen generaing funcion by G ν (x, ) := E [ ] x e νz I {Z x} = e νy df Z (y), which can be compued using a generalisaion of Lévy s inversion heorem: G ν (x) = Φ Z (ν) 2 The price of a pu opion is hen 1 π I ( Φ Z (ν + iθ) e iθx) dθ. θ E[(K S T ) + ] = K E[I ST K] E[S T I ST K] = KG (ln K) G 1 (ln K), and by pu-call pariy we obain he price of a call opion. 3.2 Pricing opions on Forwards For a forward conrac a ime, undersood o be oday, mauring a T he srike of a zero-cos forward is given by F [T ] = E Q [S T F ]. The mos common opions on forwards are pus or calls mauring a he same ime as he underlying forward, i.e. he payoff is given by (F [T ] T K) + which is equivalen o (S T K) +. We can price hese conracs based on he dynamics of he spo and using mehods developed above. However, by analysing he dynamics of he forward curve implied by he spo price model we will gain furher insighs and be able o relae he price of an opion o he Black-76 formula [Black, 1976], which is sill widely used in pracice. Recall ha he expecaion value of S T is equal o he momen generaing funcion given in (6) a θ = 1. For F [T ] = E Q [S T X, Y ] we obain T ) F [T ] = exp (f(t ) + X e α(t ) +Y e β(t ) + σ2 4α (1 e 2α(T ) ) + λ Φ J (e βs ) 1 ds. For fixed T, he dynamics of he forward mauring a T is hen df [T ] F [T ] ( ) ( ) = λ Φ J (e β(t ) ) 1 d + σ e α(t ) dw + exp(j e β(t ) ) 1 dn. (13) The forward is a maringale under Q by definiion, and so he drif erm compensaes he jump process. For large ime o mauriies T, a jump in he underlying process has only very limied effec on he forward. More precisely, if he relaive change in he underlying is exp(j ) 1 he forward changes relaively by exp(j e β(t ) ) 1. In addiion o he jump componen he dynamics follows a deerminisic volailiy process saring wih a low 1 They describe he mehod in erms of a complex valued characerisic funcion and Fourier inversion, bu by allowing complex values he mehod can also be wrien in erms of Laplace ransforms. (12) 11

12 volailiy σ e αt a = and increasing o σ a mauriy. Wihou he jump componen here are clear similariies wih he Black-76 model. For pricing purposes we need o find he disribuion of F [T ] T F [T ]. We have ln F [T ] T = f(t ) + X T + Y T, ln F [T ] = f(t ) + X e α(t ) +Y e β(t ) + σ2 4α (1 e 2α(T ) ) + λ Eliminaing he seasonaliy componen f(t ), and using he relaions T X T X e α(t ) = σ e α(t s) dw s, Y T Y e β(t ) = we finally ge in erms of is iniial condiion T N T Φ J (e βs ) 1 ds. i=n J τi e β(t τi), T ln F [T ] T = ln F [T ] + σ + σ2 4α (1 e 2α(T ) ) + λ e α(t s) dw s + T N T i=n J τi e β(t τi) Φ J (e βs ) 1 ds. (14) Wihou he jump componen, F [T ] T would be log-normally disribued. In order o relae he pricing of opions o he Black-76 formula even in he presence of spike risks, we assume ha F [T ] T is log-normally disribued in a firs approximaion. We basically ignore he heavy ails caused by he spike risk and so expec o underesimae prices of far ou-of-he-money calls bu should do well wih a-he-money calls. We define he approximaion by maching he firs wo momens bu ake ino accoun ha by definiion F [T ] is a maringale for a fixed mauriy T and in order o keep he same propery we se ( ln F [T ] T ln F [T ] + ξ, ξ N 1 ) 2 ˆσ2 (T ), ˆσ 2 (T ), and se ˆσ 2 (T ) := var[ln F [T ] T F ], i.e. [ ˆσ 2 (T ) = var σ T e α(t s) dw s + N T i=n J i e β(t i) = σ2 2α (1 e 2α(T ) ) + λ 2β E[J 2 ](1 e 2β(T ) ). Remark 3.2 (erm srucure of implied volailiy) Comparing his resul wih he seing of Black-76 [Black, 1976] where df = F σ dw and so F T = F exp(ξ) wih ξ N ( 1 2 σ2 (T ), σ 2 (T ) ), we conclude ha ˆσ is he implied Black-76 volailiy and in a firs approximaion given by σ 2 ˆσ 2 2α (1 e 2α(T ) ) + λ 2β E[J 2 ](1 e 2β(T ) ), (15) T 12 ]

13 2.5 2 Term srucure of implied Black-76 vol wihou jumps small jumps big jumps Price of a call opion implied vol price mauriy T 4 wihou jumps 2 wih jumps wih big jumps mauriy T Figure 5: Implied volailiies and prices. The lef graph shows implied volailiies wih respec o ime o mauriy where approximaion (15) is used. The hree lines correspond o no jumps (µ J = ), small jumps (µ J =.4) and big jumps (µ J =.8). In he righ graph he corresponding prices of an a he money call are ploed. Parameers are r = ln(1.5), α = 7, β = 2, σ = 1.4, λ = 4., F [T ] = 1, K = 1. Noe, for an exponenial disribued jump size J we have E[J 2 ] = 2µ 2 J. which is shown in Figure 5. I can be seen ha he spike process has a much more significan impac on he implied volailiy for shor mauriies raher han for long erm mauriies. As far as he price of an a he money call is concerned, he addiional jump risk adds an almos consan premium o he price o be paid wihou any jump risk. Remark 3.3 (implied volailiy across srikes) The approximaion does no predic a change of implied volailiy across srikes. However, he jump risk inroduces a skew as can be seen in Figure 5 where he exac soluion based on Secion 3.1 has been used o calculae implied volailiies. The bigger he mean jump size and hence he bigger E[J 2 ], he more profound is he skew. 3.3 Pricing opions on Forwards wih a delivery period As elecriciy is a flow variable, forwards always specify a delivery period. The resuls of he previous secion can herefore only be seen as an approximaion o opion prices on forwards wih shor delivery periods, like one day. Here we only consider opions on forwards mauring a he beginning of he delivery period, i.e. he payoff is given by some funcion of F [,T 2 ] a ime. An opion on such a forward is concepually similar o an Asian opion in he Black- Scholes world. One mehod of pricing Asian opions is o approximae he disribuion of he inegral by a log-normal disribuion and can be done by maching he firs wo momens, see [Turnbull and Wakeman, 1991] for example. Once he parameers of he approximae log-normal disribuion have been deermined, pricing opions comes down o pricing in he Black-Scholes or Black-76 seing. The srike price of a zero cos forward wih a delivery period is generally given by a weighed average of insananeous forwards of he form F [T T2 1,T 2 ] = w(t ;, T 2 )F [T ] dt, 13

14 small jumps big jumps Implied Black-76 volailiy, T=.2 vol K 35 sample pah 9 sample pah S S Figure 6: Implied volailiies across srikes and sample pahs. The upper graph shows he implied volailiy for one mauriy T =.2 based on he exac soluion. The approximae soluion (15) yields.82 and.85 for he small and big jumps, respecively. Sample pahs of he model wih he same parameers are drawn in he lower wo graphs, where he lef pah is generaed wih a low mean jump size (µ J =.4) and he righ wih a high mean jump size (µ J =.8). All he oher parameers are he same as in Figure 5. 14

15 where for a selemen a mauriy T 2 he weighing facor is given by w(t ;, T 2 ) = 1/(T 2 ) and for insananeous selemen he discouning alers he weighing o w(t ;, T 2 ) = r e rt /(e r e rt 2 ). The second momen of F [,T 2 ] is given by [ ( T2 ) 2 ] E Q w(t )F [T ] dt F = T2 T2 [ ] w(t )w(t ) E Q F [T ] F [T ] F dt dt, [ ] and he expecaion of he produc of wo individual forwards E Q F [T ] F [T ] F derived using he soluion of he forward (12) as follows: and so ln F [T ] = ln F [T ] T1 + e α(t T1) σ σ2 + λ 4α (e 2α(T ) e 2α(T ) ) T T1 = ln F [T ] Φ J (e βs ) 1 ds λ T1 + e α(t T1) σ σ2 4α (e 2α(T ) e 2α(T ) ) λ e α( s) dw s + e β(t ) T Φ J (e βs ) 1 ds e α( s) dw s + e β(t ) T1 ln F [T ] + ln F [T ] = ln F [T ] + ln F [T ] ( ) + e α(t T1) + e α(t ) σ N T1 i=n J τi e β(t1 τi) N T1 i=n J τi e β(t1 τi) Φ J (e β(t ) e βs ) 1 ds, T1 e α( s) dw s + (e ) N T 1 β(t T1) + e β(t ) J τi e β(t1 τi) i=n σ2 4α (1 + e 2α(T T ) )(e 2α(T ) e 2α(T ) ) ln Φ Y (e β(t ) ) ln Φ Y (e β(t ) ), which gives [ ] [ ( ) ] E Q F [T ] F [T ] F = E Q exp ln F [T ] + ln F [T ] F = F [T ] F [T ] Φ Y (e β(t T1) + e β(t ) ) Φ Y (e β(t T1) )Φ Y (e β(t ) ) ) exp ( σ2 4α (1 + e 2α(T T ) )(e 2α(T T1) e 2α(T ) ) ( ) σ 2 exp 4α (1 + e α(t T ) ) 2 (e 2α(T T1) e 2α(T ) ). 15 can be

16 .25.2 disribuion mone carlo approximaion densiy Figure 7: Disribuion of F [,T 2 ]. Parameers are as before, see Figure 5, and µ J =.4, = 1, T 2 = The red curve is based on a Mone-Carlo simulaion wih 1 6 sample pahs; he green curve is he log-normal approximaion maching he firs wo momens. F How well he momen maching procedure works is shown in Figure 7 where he densiy of a forward F [,T 2 ] is compared o he densiy obain by he approximaion. The shapes of boh densiies are similar bu differences in values are clearly visible. As a resul one migh no expec call opion prices based on he approximae disribuion o be very close o he exac prices for all srikes K bu sill close enough o be useful. For an a-he-money srike call opion, prices for varying mauriies are shown in Figures 8 and 9. As i urns ou, he approximaion gives very good resuls for shor delivery periods and is sill wihin a 5% range for delivery periods of one year. 4 Pricing swing opions Swing conracs are a broad class of pah dependen opions allowing he holder o exercise a cerain righ muliple imes over a specified period bu only one righ a a ime 2 or per imeinerval like a day. Such a righ could be he righ o receive he payoff of a call opion. Oher possibiliies include he mixure of differen payoff funcions like calls and pus or calls wih differen srikes. Anoher very common feaure is o allow he holder o exercise a muliple of a call or pu opion a once, where he muliple is called volume. This generally involves furher resricion on he volume, like upper and lower bounds for each righ and for he sum of all rades. Swing conracs can be seen as an insurance for he holder agains excessive rises in elecriciy prices. Assuming he prices generally rever o a long erm mean, even a small number N of exercise opporuniies suffices o cover he main risks and hence make he premium of he conrac cheaper. Someimes, swing conracs are bundled wih forward conracs. The 2 This will also involve a refracion period in which no furher righ can be exercised. 16

17 16 14 Price of a call opion, T1=1. approximaion monecarlo 12 1 price duraion T2-T1 Figure 8: Value of an a-he-money call opion on a forward F [,T 2 ] depending on he delivery period T 2. Parameers are as before, see Figure 5, and µ J =.4, = 1, K = 1. The Mone-Carlo resul is based on 1 sample pahs for each duraion. Noe, any forward F [,T 2 ] delivers exacly 1 MWh over he delivery period [, T 2 ] approximaion monecarlo Price of a call opion, T1=1. price duraion T2-T1 Figure 9: Same as in Figure 8 bu where he volume of he forward is proporional o he delivery period, i.e. we assume a consan consumpion of 1MWh per year, which is abou 114 W. Here, he price is simply T 2 imes he price of a sandard call opion on F [,T 2 ]. 17

18 forward conrac hen supplies he holder wih a consan sream of energy o a fixed predeermined price. If he srike price of he call opions of he swing conrac is se o he forward price, he swing conrac will allow for flexibiliy in he volume he cusomer receives for he fixed price. They can eiher swing up or swing down he volume of energy and hence he name swing conrac. One canno assume ha he holder always exercises he conrac in an opimal way o maximise expeced profi bu hey migh also exercise according o heir own inernal energy demands. I is only very recenly ha aricles on numerical pricing mehods for swing opions have appeared in he lieraure. We can idenify a few main approaches all based on dynamic programming principles. A Mone-Carlo mehod and ideas of dualiy heory are uilised in [Meinshausen and Hambly, 24] o derive lower and upper bounds for swing opion prices. The main advanages of he mehod being heir flexibiliy as i can be easily adaped o any sochasic model of he underlying and is abiliy o produce confidence inervals of he price. Mone-Carlo echniques are also used in [Ibanez, 24] and [Carmona and Touzi, 24], where he laer uses he heory of he Snell envelope o deermine he opimal exercise boundaries and also uilises he Malliavin calculus for he compuaion of greeks. A consrucive soluion o he perpeual swing case for exponenial Brownian moion is also given in [Carmona and Touzi, 24]. Unforunaely, hese mehods only work for he mos basic versions of swing conracs where a each ime only one uni of an opion can be exercised. More general swing conracs wih a variable volume per exercise and an overall consrain can be priced wih a ree based mehod inroduced in [Jaille e al., 24]. In he above papers a discree-ime model for he underlying is used where one ime sep corresponds o he ime frame in which no more han one righ can be exercised, i.e. one day in mos of he raded conracs. A special case where he number of exercise opporuniies is equal o he number of exercise daes is considered in [Howison and Rasmussen, 22] and a coninuous opimal exercise sraegy derived which yields a parial inegro-differenial equaion for he opion price. Our mehod is based on he ree approach of [Jaille e al., 24] wih some sligh modificaions o adap i o he peculiariies of our model for he underlying elecriciy price process. 4.1 The grid approach The ree mehod of [Jaille e al., 24] requires a discree ime model of he underlying. This is due o he fac ha heir swing conracs allow he holder o exercise a mos one opion wihin a specified ime inerval, say a day, and his is bes modelled if he underlying process has he same ime discreisaion. Assuming (S ) is some coninuous sochasic process for he spo price we obain a discree model by observing i on discree poins in ime only, i.e. S, S 1, S 2,..., S m, wih =, i+1 = i +, m = T and = 1 365, indicaing we can exercise on a daily basis. Le he mauriy dae T be fixed and he payoff a ime for simpliciy 3 be given by (S K) + for some srike price K and we assume only one uni of he underlying can be exercised any ime period. Le V (n, s, ) denoe he price of such a swing opion a ime and spo price s 3 We could assume any general payoff funcion. 18

19 which has n ou of N exercise righs lef. The dynamic programming principle allows us o wrie { } V (n, s, ) = max e r E Q [V (n, S +, + ) S = s], e r E Q [V (n 1, S +, + ) S = s] + (s K) +, n < N, (16) and V (n, s, T ) = (S K) +, < n N and V (, s, ) =. The condiional expecaions can be wrien E Q [V (n, S +, + ) S = s] = V (n, x, + )f S (x; s) dx, where f S (x; s) is he densiy of S + given S = s. Discreising he spo variable we approximae E Q [V (n, S +, + ) S = s i ] V (n, s j, + )f S (s j ; s i ) s j. j This is only one possible approximaion; ohers migh be o use higher order inegraion rules or using only a few grid poins in he sum based on he fac ha f S (x; s) for s x large. For a rinomial ree one only uses hree grid poins, i.e. E Q [V (n, S +, + ) S = s i ] 1 j= 1 V (n, s i+j, + )p i,i+j, p i,j being he probabiliy of going from node i o node j. However, such a ree approach is no well suied o our case for wo reasons. Firs, he ime sep size is deermined by he shores ime beween wo possible exercise daes, which is mainly one day for swing conracs. This limis he accuracy of he algorihm as a refinemen of he grid in spo direcion will no improve he resul. Second, in he presence of jumps, a hree-poin approximaion for he condiional densiy is insufficien due o he heavy ails in he disribuion. As a resul, we keep our mehod general and say E Q [V (n, S +, + ) S = s i ] j V (n, s j, + )p i,j, where p i,j is an approximaion o he densiy f S (s j ; s i ) s j (i can accommodae higher-order inegraion rules and boundary approximaions). Wih he noaion Vi,k n := V (n, s i, k ) we can hen wrie he mehod as Vi,k n = max e r Vj,k+1 n p i,j, e r V n 1 j,k+1 p i,j + (s i K) +, j j (17) Vi,k =, Vi,m n =. 4.2 Numerical resuls We now urn o he model of ineres, (1), which exhibis spikes. Assume ha he meanreversion process (X ) and he spike process (Y ) are individually observable and so he value 19

20 funcion V of a swing opion depends on boh variables and he general pricing principle (16) becomes { e r E Q [V (n, X +, Y +, + ) X = x, Y = y], V (n, x, y, ) = max e r E Q [V (n 1, X +, Y +, + ) X = x, Y = y] + (e f()+x+y K) +.. In order o calculae condiional expecaions we need o define ransiion probabiliies. Given one sars a node (X, Y ) = (x i, y j ) he probabiliy o arrive a node (X +, Y + ) = (x k, y l ) is approximaely given by p i,j,k,l f X+ X =x i (x k )f Y+ Y =y i (y l ) x y, because X and Y are independen. The condiional densiy of he mean-revering process (X ) is known as X + given X = x is normally disribued wih N (x e α, σ2 2α (1 e 2α )). As we do no have a closed form expression for he densiy of he spike process we use approximaions developed in Secion 2.1. For an exponenial jump size disribuion J Exp(1/µ J ) for example we use approximaion (9) for he spike process a ime given zero iniial condiions. The inroducion of a second space dimension increases he complexiy of he algorihm considerably, essenially by a facor proporional o he square of he number of grid poins in he y-direcion. To price he swing conrac shown in Figure 1 which has 365 exercise daes and up o 1 exercise opporuniies, our C++ implemenaion requires abou 1 minues o complee he calculaion on an Inel P4, 3.4GHz, and for a grid of 12 6 poins in x and y direcion, respecively. The same compuaion bu wih no spikes and a grid of 12 1 poins only akes abou one second. Based on Figure 1 we make wo observaions. Firs, he price per exercise righ decreases wih he number of exercise righs. This is he correc qualiaive behaviour one would expec because n swing opions each wih one exercise righ 4 only, offer more flexibiliy han one swing opion wih n exercise righs. 5 Second, he premium added due o he spike risk is much more significan for opions wih small numbers of exercise righs han for a large number. This is also inuiively clear, as an opion wih say 1 exercise righs will mainly be used agains high prices caused by he diffusive par and only occasionally agains spiky price explosions. In Figure 11 we show how sensiive swing opion prices are o changes in marke parameers. There we consider a swing opion wih a duraion of 6 days and up o 2 exercise opporuniies. In each graph we change one parameer by 2% up and down. The mos significan change is caused by a change in he volailiy parameer σ. Noe, he longerm variance of he mean-revering process (X ) is σ2 2α and we expec some direc relaionship beween he long-erm variance and he opion price. Hence, a change in he mean-reversion parameer α is inversely proporional o he price and quaniaively changes he price less han he volailiy σ. The mean-reversion parameer β of he spike process has a similar effec on he opion price as α has, bu where he influence slighly decreases wih he number of opions. This is consisen wih previous observaions of he impac of jumps on opion prices as seen in Figure 1. This effec is much more clearly visible for he oher jump parameers λ and µ J 4 This is acually an American opion. 5 The righs of a swing opion can only be exercised one a a ime. 2

21 1.2 1 Value of swing conracs, 1 year delivery no spikes wih spikes value/righs exercise righs Figure 1: Value of a one year swing opion per exercise righ. Marke parameers of he underlying are as before, see Figure 1, i.e. α = 7, β = 2, σ = 1.4, λ = 4, J Exp(1/µ J ) wih µ J =.4, f() =, r =, and iniial condiions X = and Y =. The swing conrac delivers over a ime period of one year T [, 1] wih up o 1 call righs and a srike price of K = 1, where a righ can be exercised on any day. As a comparison he price of he same swing opion is ploed bu where he underlying does no exhibi spikes, i.e. λ =. which have he greaes impac on opions wih only a few exercise righs. For one exercise righ, a 2% change in he jump size µ J has an even greaer effec on he price han a 2% change in volailiy σ. A possible explanaion is ha we deal wih he exponenial of an exponenially disribued jump size which is very heavy ailed. References [Benh e al., 25] Benh, F., Kallsen, J., and Meyer-Brandis, T. (25). A non-gaussian Ornsein-Uhlenbeck process for elecriciy spo price modeling and derivaive pricing. preprin. [Black, 1976] Black, F. (1976). The pricing of commodiy conracs. Journal of Financial Economics, 3: [Carmona and Touzi, 24] Carmona, R. and Touzi, N. (24). Opimal muliple sopping and valuaion of swing opions. o appear in Mahemaical Finance. [Carr and Madan, 1998] Carr, P. and Madan, D. (1998). Fourier ransform. J. Compu. Fin., 2: Opion valuaion using he fas [Carea and Figueroa, 25] Carea, A. and Figueroa, M. (25). Pricing in elecriciy markes: a mean revering jump diffusion model wih seasonaliy. Applied Mahemaical Finance, 12(4):

22 .45.4 Value of swing conracs alpha=8.4 alpha= Value of swing conracs sigma=1.68 sigma=1.12 value/righs value/righs exercise righs exercise righs.45.4 Value of swing conracs bea=24 bea= Value of swing conracs lambda=4.8 lambda=3.2 value/righs value/righs exercise righs exercise righs.45.4 Value of swing conracs muj=.48 muj= Value of swing conracs no spikes value/righs value/righs exercise righs exercise righs Figure 11: Sensiiviy of swing opion prices wih respec o model parameers. A swing opion wih 6 exercise daes and up o 2 righs is considered, where he red curve is based on he parameers of Figure 1. In each graph one marke parameer is shifed up by 2% (green line) and down by 2% (blue line). We always plo he opion price divided by he number of exercise righs. 22

23 [Clewlow and Srickland, 2] Clewlow, L. and Srickland, C. (2). Energy Derivaives: Pricing and Risk Managemen. Lacima Publicaions, London, UK. [Con and Tankov, 24] Con, R. and Tankov, P. (24). Financial Modelling wih Jump Processes. Chapman & Hall/CRC. [Deng, 2] Deng, S. (2). Sochasic models of energy commodiy prices and heir applicaions: Mean-reversion wih jumps and spikes. Working paper PWP-73. [Duffie e al., 2] Duffie, D., Pan, J., and Singleon, K. (2). Transform analysis and asse pricing for affine jump-diffusions. Economerica, 68(6): [Howison and Rasmussen, 22] Howison, S. and Rasmussen, H. (22). Coninuous swing opions. working paper. [Ibanez, 24] Ibanez, A. (24). Valuaion by simulaion of coningen claims wih muliple early exercise opporuniies. Mahemaical Finance, 14(2): [Jaille e al., 24] Jaille, P., Ronn, E., and Tompadis, S. (24). Valuaion of commodiybased swing opions. Managemen Science, 5: [Kluge, 26] Kluge, T. (26). Pricing Swing Opions and oher Elecriciy Derivaives. PhD hesis, Universiy of Oxford. [Lewis, 21] Lewis, A. (21). A simple opion formula for general jump-diffusion and oher exponenial Lévy processes. working paper. [Lucia and Schwarz, 22] Lucia, J. and Schwarz, E. (22). Elecriciy prices and power derivaives: Evidence from he nordic power exchange. Review of Derivaives Research, 5(1):5 5. [Meinshausen and Hambly, 24] Meinshausen, N. and Hambly, B. (24). Mone carlo mehods for he valuaion of opions wih muliple exercise opporuniies. Mahemaical Finance, 14(4): [Turnbull and Wakeman, 1991] Turnbull, S. M. and Wakeman, L. M. (1991). A quick algorihm for pricing european average opions. Journal of Financial and Quaniaive Analysis, 26(3):