European Option Pricing of electricity under exponential functional of Lévy processes with Price-Cap principle
Martin Kegnenlezom, Patrice Takam Soh, Antoine-Marie Bogso, Yves Emvudu Wono
aa r X i v : . [ q -f i n . P R ] J un European Option Pricing of electricity under exponentialfunctional of L ´evy processes with Price-Cap principle
Martin Kegnenlezom ∗ , Patrice Takam Soh † , Antoine-Marie Bogso ‡ , andYves Emvudu Wono § June 27, 2019
Abstract
We propose a new model for electricity pricing based on the price cap principle. The particularity of themodel is that the asset price is an exponential functional of a jump L´evy process. This model can capture bothmean reversion and jumps which are observed in electricity market. It is shown that the value of an Europeanoption of this asset is the unique viscosity solution of a partial integro-differential equation (PIDE). A numericalapproximation of this solution by the finite differences method is provided. The consistency, stability and con-vergence results of the scheme are given. Numerical simulations are performed under a smooth initial condition.
Keywords:
Mean reverting jump-diffusion, option pricing, price-cap, integro-differential equation, Viscositysolution subclass MSC:
In finance, options are tools that help to guard against risks. But, it is difficult to know the value of an option beforethe maturity date since, for this end, one has to estimate the value of the underlying in future. In the early of 1970’s,Black and Scholes [4] brought a major contribution in the evaluation of options. In the case where the underlying isa share that does not pay dividends, they construct a risk-neutral portofolio that replicates the winning profile of anoption, which allows to perform the theoretical value of a European option under a closed formula. In the case ofthe exponential L´evy Black and Scholes model, this formula is derived from some classical results in discounting,statistics, stochastic and differential calculus. On the contrary, the valuation of options remains an open topic inthe case of jump-diffusion processes due to the additional jump term that complicates calculation of option prices.This has been investigated by several authors as [5], [24] and [30]. One may distinguish two main approachesused by these authors. The first one, which relies on Fourier transform-based methods, has been introduced byCarr and Madan [9] to price and analyse European option prices. Many other authors follow the same idea toevaluate option prices. Among them one find [10], [14], [22] and [31]. The advantage of this approach relies onits high computational efficiency when the characteristic function is available. The idea here is to apply the directdiscounted expectation method to evaluate the integral of the discounted payoff and risk neutral density functionof the underlying process. Continuous and dicrete Fourier transforms were sucessfully applied to pricing optionsof three exponential L´evy models: the classical Black-Scoles model (which is a continuous exponential L´evymodel), the Merton jump diffusion model (which is an exponential L´evy model with finite arrival rate of jumps),and the variance Gamma model (which is an exponential L´evy model with infinite arrival rate of jumps). However, ∗ University of Yaound´e I, Department of Mathematics, P.O. Box 812 Yaound´e, Cameroon. Phone: (+237)699 299 181, Email: [email protected] † University of Yaound´e I, Department of Mathematics, P.O. Box 812 Yaound´e, Cameroon. Phone: (+237)699 299 181, Email: [email protected] ‡ University of Yaound´e I, Department of Mathematics, P.O. Box 812 Yaound´e, Cameroon. Phone: (+237)652 620 452, Email: [email protected] § University of Yaound´e I, Department of Mathematics, P.O. Box 812 Yaound´e, Cameroon. Phone: (+237)699 299 181, Email:[email protected]
We suppose that the underlying asset risk-neutral dynamics is in the formd S t = ( α ( t ) S t − β ( t )) d t + σ t S t d W t + ( J − ) S t d q t , (1)with α ( t ) = I ( t ) − EF ( t ) and β ( t ) : = CP ( t ) + SQ ( t ) − FU ( t ) , where: S t represents the electricity spot prices process; I ( t ) represents the inflation rate; EF ( t ) represents the efficiency factor; σ ( t ) represents the volatility; W represents the standard Brownian motion; CP ( t ) represents the subvention; SQ ( t ) represents the service quality penalties, if any;2 U ( t ) represents the uncontrollable cost; J represents the proportional random jump size;d q a Poisson process such that; d q t = (cid:26) − ℓ d t ℓ d t .with the following assumptions: Assumption 1:
The proportional random jump size J is log-Normally distributed, with E [ J ] =
1. Hence,ln J ∼ N (cid:16) − σ J , σ J (cid:17) . Assumption 2:
The random jump size J , the Poisson process d q t and the diffusion d W t are independent.This model has been partly inspired from the economic principle price cap proposed by Littlechild in [23] forregulated telecommunication market in UK. This principle has been recently applied to electricity market [19].We will further assume that the electricity parameters α ( . ) , β ( . ) and σ ( . ) are bounded. It follows from the Itˆoformula for jump diffusion process that the exact solution of (1) is given by S t = S e X t − Z t β ( s ) e X t − X s ds , (2)where X t = Z t α ( s ) − σ ( s ) ds + Z t σ ( s ) dW s + Z t ln Jdq s . This part aims to evaluate the price of an option (Put or Call) in the regulated electricity market under risk-neutralprobability, Q , with the terminal payoff, H T , which is given by: C t = E [ e − r ( T − t ) H T | F t ] , (3)where r represents a free risk discounting rate, T denotes the maturity, and K represents the strike price. Let S T bethe solution of (1) at T, which is the equation of the underlying. H T = H ( S T ) , with H ( S ) = ( S − K ) + for EuropeanCall or H ( S ) = ( K − S ) + for European Put. From the Markov property, C t becomes C ( t , S ) = E [ e − r ( T − t ) H T | S t = S ] . (4) Proposition 3.1.
Assume that the European option C given byC : ( , T ) × ( , ∞ ) → R ( t , S ) C ( t , S ) (5) is C , , with ∂ C / ∂ S and ∂ C / ∂ S bounded, then C satisfies the partial integro-differential equation: ∂ C ∂ t ( t , S ) + ( α ( t ) S − β ( t )) ∂ C ∂ S ( t , S ) + σ ( t ) S ∂ C ∂ S ( t , S ) − rC ( t , S )+ ℓ Z R ν ( dx )[ C ( t , xS ) − C ( t , S )] = on ( , T ) × ( , ∞ ) with the terminal condition C ( T , S ) = H ( S ) , ∀ S > , where ℓ represents the intensity of thePoisson process under risk-neutral measure, and the measure ν ( dx ) is the jump size distribution.Proof. The proof consists of applying Itˆo formula with jump as in [15] to the martingale ˜ C ( t , S t ) = e − rt C ( t , S t ) .Then the result follows from the fact that the drift term is equal to zero.By construction ˜ C is a martingale. Applying the Itˆo formula to ˜ C we obtain: d ˜ C t = e − rt (cid:20) − rC ( t , S t ) + ∂ C ∂ t ( t , S t ) + σ ( t ) S t ∂ C ∂ S ( t , S t ) (cid:21) dt + e − rt ∂ C ∂ S ( t , S t ) dS t + e − rt (cid:20) C ( t , JS t − ) − C ( t , S t − ) + ( J − ) S t − ∂ C ∂ S ( t , S t − ) (cid:21) dq t . d ˜ C t = e − rt (cid:20) − rC ( t , S t ) + ∂ C ∂ t ( t , S t ) + σ ( t ) S t ∂ C ∂ S t ( t , S t ) (cid:21) dt + e − rt (cid:20) ( α ( t ) S t − β ( t )) ∂ C ∂ S ( t , S t ) dt + ∂ C ∂ S ( t , S ) S t σ ( t ) dW t (cid:21) + e − rt [ C ( t , JS t − ) − C ( t , S t − )] dq t . Adding and subtracting ℓ Z R ν ( dx )( C ( t , S t x ) − C ( t , S t )) dt , one has d ˜ C t = a ( t ) dt + dM t , where a ( t ) = e − rt (cid:20) ∂ C ∂ t ( t , S t ) + ( α ( t ) S t − β ( t )) ∂ C ∂ S ( t , S t ) + σ ( t ) S t ∂ C ∂ S ( t , S t ) − rC ( t , S t ) + ℓ Z R ν ( dx )( C ( t , S t x ) − C ( t , S t )) (cid:21) and dM t = e − rt (cid:20) ∂ C ∂ S ( t , S ) S t σ ( t ) dW t + ( C ( t , JS t − ) − C ( t , S t − )) d ˜ q t (cid:21) , with ˜ q t = q t − ℓ t . We now show that M t is a martingale. Since the payoff function H is Lipschitz. Then, C is alsoLipschitz with respect to the second variable S . Indeed: | C ( t , x ) − C ( t , y ) | = e − ( T − t ) | E [ H ( S t e X T − X t − Z Tt β ( s ) e X T − X s ds ) | S t = x ] − E [ H ( S t e X T − X t − Z Tt β ( s ) e X T − X s ds ) | S t = y ] |≤ c e − r ( T − t ) E [ e R Tt α ( s ) − ( / ) σ ( s ) ds + R Tt σ ( s ) dW s + R Tt ln Jdq s ] | x − y | , for every fixed t . Since e − R Tt ( / ) σ ( s ) ds + R Tt σ ( s ) dW s is a martingale and we also have from assumption 1 that E [ e R Tt ln Jdq s ] =
1, then weget | C ( t , x ) − C ( t , y ) | ≤ c | x − y | e R Tt α ( s ) ds ≤ c | x − y | , with c = ce R Tt α ( s ) ds .Therefore the predictable random function ϕ ( t , x ) = C ( t , xS t − ) − C ( t , S t − ) satisfies: E (cid:2) Z T Z R ν ( dx ) | ϕ ( t , x ) | dt (cid:3) ≤ E [ Z T dt Z R ν ( dx ) c ( x + ) S t ] ≤ Z R c ( x + ) ν ( dx ) E (cid:2) Z T S t dt (cid:3) < ∞ , where the last inequality holds because the distribution ν ( dx ) of the jump sizes is assumed log-normal. Indeed, wehave R R x ν ( dx ) < ∞ , hence E [ R T S t dt ] < ∞ . Therefore, the compensated Poisson integral Z T Z R e − rt [ C ( t , xS t − ) − C ( t , S t − )] d ˜ q t is a square integrable martingale. Since C is Lipschitz, ∂ C ∂ S ( t , . ) ∈ L ∞ and (cid:13)(cid:13) ∂ C ∂ S ( t , . ) (cid:13)(cid:13) L ∞ ≤ c . Thus, E (cid:2) Z T S t | ∂ C ∂ S ( t , S t ) | dt (cid:3) ≤ c E [ Z T S t dt ] < ∞ . R T ∂ C ∂ S ( t , S t ) S t σ ( t ) dW t is a squareintegrable martingale. Therefore, M t is also a square integrable martingale, implying ˜ C t − M t is a square integrablemartingale. But ˜ C t − M t = R t a ( t ) dt is also a continuous process with finite variation, so, from Theorem 4.13-450in [18], one must have a ( t ) = Q -almost surely, leading to the PIDE (6).Note that the smoothness (particularly the uniform boundedness of derivatives) assumption made on the Eu-ropean call option is not generally verified as discussed in [11]. In this case, option prices should be consideredas a viscosity solution of the PIDE obtained in Proposition 3.1. The following proposition gives the link betweenoption prices and the viscosity solution of the PIDE. Proposition 3.2. ( Option prices as viscosity solutions )The forward value of the European option defined by (4) is the (unique) viscosity solution of the Cauchy problem (6) .Proof.
Existence and uniqueness of viscosity solutions for such parabolic integro-differential equations are dis-cussed in [2] in the case (the one considered here) where ν is the finite measure. In what follows, we propose anumerical solution to the PIDE which converges to the viscosity solution as proven in [12]. In this section we present a numerical procedure for solving the PIDE (6) obtained in Proposition 3.1. Introducingthe change of variable x = ln SS and τ = T − t and defining: u ( τ , x ) = e r τ C ( T − τ , S e x ) , we obtain: u ( τ , x ) = E (cid:20) H ( S t e X T − X T − τ − Z TT − τ β ( s ) e X T − X s ds ) | S t = S e x (cid:21) = E [ H ( Y x τ )] , (7)where Y x τ = S e x + X T − X T − τ − R TT − τ β ( s ) e X T − X s ds . We then obtain a PIDE in terms of u , given by: ∂ u ∂τ = L u , on ( , T ] × Ou ( , x ) = H ( S e x ) , x ∈ O , u ( τ , x ) = , x ∈ O c , (8)where O ⊂ R is an open interval which is not necessarily bounded, L u ( τ , x ) = (cid:18) α ( T − τ ) − σ ( T − τ ) − β ( T − τ ) S (cid:19) ∂ u ∂ x ( τ , x ) + σ ( T − τ ) ∂ u ∂ x ( τ , x )+ ℓ Z R [ u ( τ , x + y ) − u ( τ , x )] g ln J ( y ) dy , with g ln J denoting the density function of ln J .The main idea in this method is to split the operator L into two parts as in [11]. We replace the differentialpart with a finite difference approximation, and the integral part with a trapezoidal quadrature approximation. Wetreat the integral part with an explicit time stepping in order to avoid the inversion problem of the dense matrix L J associated to the discretization of the integral term. We then rewrite the PIDE (8) as follow: ∂ u ∂τ = ( L D + L J ) u , on ( , T ] × Ou ( , x ) = H ( S e x ) , x ∈ O , u ( τ , x ) = , x ∈ O c , (9)where L D u ( τ , x ) = (cid:18) α ( T − τ ) − σ ( T − τ ) − β ( T − τ ) S (cid:19) ∂ u ∂ x ( τ , x )+ σ ( T − τ ) ∂ u ∂ x ( τ , x ) , (10) L J u ( τ , x ) = ℓ Z R [ u ( τ , x + y ) − u ( τ , x )] g ln J ( y ) dy . (11)5ence, we obtain the approximate problem using the following explicit-implicit time stepping scheme: u n + − u n ∆ t = L D u n + + L J u n . (12)Before showing the stability of this scheme and applying discretization, the equation must be localized to a boundeddomain. To numerically solve the Cauchy problem (9), we first truncate the space domain to a bounded interval ( − A l , A r ) .Usually this leads to defining some boundary conditions at x = − A l and x = A r . But here, we are in an ellipticlocal PIDE due to the presence of an integral term. Thus, we need to extend the function u ( τ , . ) to a subset { x + y : x ∈ ( − A l , A r ) , y ∈ supp g ln J } , where supp g ln J = R + , is the support of g ln J . Let u A ( τ , x ) be the solution ofthe following localization problem: ∂ u l , r ∂τ = ( L D + L J ) u l , r , on ( , T ] × ( − A l , A r ) u l , r ( , x ) = H ( S e x ) , x ∈ ( − Al , Ar ) u l , r ( τ , x ) = , x / ∈ ( − A l , A r ) . (13)We will show in the following proposition that the localization error decays exponentially with the domain size A . Proposition 4.1.
Assume k H k ∞ < ∞ and C τ = E " e sup η ∈ [ , τ ] | X T − X T − η | < ∞ . Let u l , r ( τ , x ) and u ( τ , x ) be respectivelythe solutions of the Cauchy problems (9) and (13) . Then | u ( τ , x ) − u l , r ( τ , x ) | ≤ C τ k H k ∞ e − max ( A l , A r )+ | x | , ∀ x ∈ ( − A l , A r ) (14) where the constant C τ does not depend on A r and A l .Proof. Let M x τ = sup η ∈ [ , τ ] | x + X T − X T − η | . Then u l , r ( τ , x ) = E (cid:2) { M x τ < max ( Al , Ar ) } H ( Y x τ ) (cid:3) and u ( τ , x ) = E [ H ( Y x τ )] . (15)Hence | u − u l , r | = (cid:12)(cid:12)(cid:12) E (cid:2) H ( Y x τ ) { M x τ ≥ max ( A l , A r ) } (cid:3) (cid:12)(cid:12)(cid:12) ≤ k H k ∞ (cid:12)(cid:12) E (cid:2) { M x τ ≥ max ( A l , A r ) } (cid:3)(cid:12)(cid:12) ≤ k H k ∞ Q ( M x τ > max ( Al , Ar ) . (16)Theorem 25.18 in [29] and the fact that R R e | x | ν ( dx ) < ∞ imply C τ = E " e sup η ∈ [ , τ ] | X T − X T − η | < ∞ . (17)But Q ( M x τ > max ( Al , Ar )) = Q ( e M x τ > e max ( A l , A r ) ) (18)(19)and, since sup η ∈ [ , τ ] | x + X T − X T − η | ≤ sup η ∈ [ , τ ] | X T − X T − η | + | x | , then { M x τ > max ( Al , Ar ) } ⊂ ( sup η ∈ [ , τ ] | X T − X T − η | ≥ max ( Al , Ar ) − | x | ) . Hence Q (cid:16) e M x τ > e max ( Al , Ar ) (cid:17) ≤ Q e sup η ∈ [ , τ ] | X T − X T − η | > e max ( Al , Ar ) −| x | ! . (20)6ow, using Markov’s inequality we obtain Q e sup η ∈ [ , τ ] | X T − X T − η | > e max ( Al , Ar ) ! ≤ E " e sup η ∈ [ , τ ] | X T − X T − η | e max ( Al , Ar ) −| x | . (21)Comparing these last inequalities with ((16)) gives the desired result. To compute numerically the integral term of the PIDE (9), we need to reduce the region of integration to a boundedinterval which leads to the truncation of large jumps. We then estimate the error resulting from this approximation.Precisely, suppose a new process, ˜ S t , is characterized by the fact that logarithm of the jump size, ln ˜ J , is boundedin [ B l , B r ] , with the associated measure { y ∈ [ Bl , Br ] } ν , where B l and B r are real. We further suppose, without lossgenerality, that B l < B r >
0. In this case the corresponding solution to the associated PIDE is denoted by˜ u ( τ , x ) . We analyse, in the following proposition, the difference | u − ˜ u | . Proposition 4.2.
One has: | u ( τ , x ) − ˜ u ( τ , x ) | ≤ C τ (cid:16) C e −| B l | + C e − B r (cid:17) , (22) where C τ = C h τ ℓ S + ℓ R TT − τ β ( s )( T − s ) ds i .Proof. Firstly, let ˜ X t be a new L´evy process defined by:˜ X t = Z t α ( s ) − σ ( s ) ds + Z t σ ( s ) dW s + Z t ln ˜ Jdq s , (23)and let: ˜ u ( τ , x ) = E [ H ( ˜ Y x τ )] , (24)where ˜ Y x τ = S e x + ˜ X T − ˜ X T − τ − R TT − τ β ( s ) e ˜ X T − ˜ X s ds . Setting R τ = X T − ˜ X T − ( X τ − ˜ X τ ) and using the Lipschitz propertyon H , we obtain: | u ( τ , x ) − ˜ u ( τ , x ) | = | E [ H ( Y x τ )] − E [ H ( ˜ Y x τ )] |≤ c E h | S ( e x + ˜ X T − ˜ X T − τ + R T − τ − e x + ˜ X T − ˜ X T − τ ) − Z TT − τ β ( s )( e ( ˜ X T − ˜ X s + R s − e ˜ X T − ˜ X s ds | (cid:21) ≤ c (cid:16) S E [ e ˜ X T − ˜ X T − τ | e R T − τ − | ]+ Z TT − τ β ( s ) E [ e ˜ X T − ˜ X s | e R s − | ] ds (cid:19) . (25)Since R τ and ˜ X T − ˜ X τ are independent, we have | u ( τ , x ) − ˜ u ( τ , x ) | ≤ c e x ( S E [ e ˜ X T − ˜ X T − τ ] E [ | e R T − τ − | ] + Z TT − τ β ( s ) E [ e ˜ X T − ˜ X s ] E [ | e R s − | ] ds ) . (26)Moreover, (cid:16) e ˜ X T − ˜ X T − u , u ∈ [ , T ] (cid:17) being a martingale, E [ e ˜ X T − ˜ X T − τ ] = E [ e ˜ X T − ˜ X s ] =
1. As a consequence, | u ( τ , x ) − ˜ u ( τ , x ) | ≤ c e x ( S E [ | e R T − τ − | ] + Z TT − τ β ( s ) E [ | e R s − | ] ds ) . (27)Since, for every a ∈ R , | e a − | = ( e a − ) + ( e a − ) + and ( e a − ) + ≤ | a | , then | u ( τ , x ) − ˜ u ( τ , x ) | ≤ c e x ( S E [ | R T − τ | ] + Z TT − τ β ( s ) E [ | R s | ] ds ) . (28)7ut: E [ | R T − τ | ] ≤ ℓ Z TT − τ (cid:20) − Z Bl − ∞ yg ln J ( y ) dy + Z + ∞ Br yg ln J ( y ) dy (cid:21) ds ≤ τ ℓ (cid:18) − e −| Bl | Z Bl − ∞ ye | y | g ln J ( y ) dy + e −| Br | Z + ∞ Br ye | y | g ln J ( y ) dy (cid:19) ≤ τ ℓ S (cid:16) C e −| Bl | + C e −| Br | (cid:17) , (29)where C = − R Bl − ∞ ye | y | g ln J ( y ) dy and C = R + ∞ Br ye | y | g ln J ( y ) dy . Replacing (29) into (25), we get: | u ( τ , x ) − ˜ u ( τ , x ) | ≤ C (cid:20) τ ℓ S + ℓ Z TT − τ β ( s )( T − s ) ds (cid:21) (cid:16) C e −| Bl | + C e −| Br | (cid:17) , (30)where C = c e x From Proposition4.1 and 4.2, ˜ u converges to u when | B l | and | B r | grow to infinity. Define a uniform grid on ( , T ] × ( − A l , A r ) by τ n = n ∆ t , n = M , x i = i ∆ x − A l , i = , .., N , with ∆ t = T / M and ∆ x = A r + A l N . Let ( u ni ) be the solution of the numerical scheme which we define below: Firstly, to approximatethe integral terms, we use the trapezoidal quadrature rule with the same resolution ∆ x . Let K l and K r be such that [ B l , B r ] ⊂ [( K l − / ) ∆ x , ( K r + / ) ∆ x ] , then: Z B r B l ( u ( τ , x i + y ) − u ( τ , x i )) g ln J ( y ) dy ≃ K r ∑ j = K l ν j ( u i + j − u i ) , (31)where ν j = R ( j + / ) ∆ x ( j − / ) ∆ x g ln J ( y ) dy . Notice that to compute the integral term, we need to extend the solution to [ − A l + B l , Ar + B r ] . Hence, we assume that this solution is zero except in [ − A l , A r ] . The derivatives are discretizedusing the finite difference method thus: (cid:18) ∂ u ∂ x (cid:19) i ≃ u i + − u i + u i − ( ∆ x ) (cid:18) ∂ u ∂ x (cid:19) i ≃ u i + − u i ∆ x if f ( τ , x ) ≥ u i − u i − ∆ x if f ( τ , x ) ≤ , (32)where f ( τ , x ) = α ( T − τ ) − σ ( T − τ ) − β ( T − τ ) S e x .Using (31) and (32), and supposing f ( τ , x ) <
0, we obtain the following relation: u n + i − u ni ∆ t = ( L D u ) n + i + ( L J u ) ni , (33)where ( L D u ) ni = f ( τ n , x i ) u ni + − u ni ∆ x + σ ( T − τ n ) u ni + − u ni + u ni − ( ∆ x ) ( L J u ) ni = K r ∑ j = K l ν j ( u ni + j − u ni ) . (34)Finally, we replace the problem (9) with the following time-stepping numerical scheme: Initialisation u i = H ( S e x i ) if i ∈ { , ... N − } For n=0,...,M-1 u n + i − u ni ( ∆ t ) = ( L D u ) n + i + ( L J u ) ni if i ∈ { , .., N − } u n + i = / ∈ { , .., N − } . (35)After defining the numerical scheme, we study some of its important properties, particularly, consistency, mono-tonicity, stability and convergence. 8 .4 Consistency The follow proposition shows that (35) is consistent with (9).
Proposition 4.3. ( Consistency )The finite difference scheme (35) is locally consistent with equation (9) : That is, ∀ v ∈ C ∞ ([ , T ] × ( A l , A r )) , and ∀ ( τ n , x i ) ∈ [ , T ] × R , one has: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) v n + i − v ni ( ∆ t ) − ( L D v ) n + i − ( L J v ) ni − ∂ v ∂τ ( τ n , x i ) − ( L D + L J ) v ( τ n , x i ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = r ni ( ∆ t , ∆ x ) → as ( ∆ t , ∆ x ) → ( , ) . In other words, ∃ c > such that: | r ni ( ∆ t , ∆ x ) | ≤ c ( ∆ t + ∆ x ) . Proof.
Let a = v n + i − v ni ∆ t − ∂ v ∂τ ( τ n , x i ) a = ( L D v ) n + i − L D v ( τ n , x i ) a = ( L J v ) ni − L J v ( τ n , x i ) . (37)Using the second order Taylor expansion with respect to τ , we obtain v n + i ≈ v ni + ∆ t ∂ v ∂τ ( τ n , x i ) + ( ∆ t ) ∂ v ∂τ ( τ n , x i ) . Plugging this relation in the first equation in ((37)) we get: | a | = ∆ t (cid:12)(cid:12)(cid:12)(cid:12) ∂ v ∂τ ( τ n , x i ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ∆ t (cid:13)(cid:13)(cid:13)(cid:13) ∂ v ∂τ ( τ n , x i ) (cid:13)(cid:13)(cid:13)(cid:13) ∞ = ∆ t M , (38)where M = (cid:13)(cid:13)(cid:13)(cid:13) ∂ v ∂τ (cid:13)(cid:13)(cid:13)(cid:13) ∞ . We now show that | a | is bounded. By the mean-value theorem, there exists θ ∈ ] τ n , τ n + [ such that: L D v ( τ n + , x i ) − L D v ( τ n , x i ) ≈ ∆ t ∂ τ L D v ( τ n + ∆ t θ , x i ) . If one replaces this relation in the second equation in ((37)), then: | a | = (cid:12)(cid:12)(cid:12) ( L D v ) n + i − L D v ( τ n + , x i ) + ∆ t ∂ τ L v ( τ n + ∆ t θ , x i ) (cid:12)(cid:12)(cid:12) . (39)Next, taking Taylor expansion of v of order 4 gives: v n + i + ≈ v ni + ∆ x ∂ v ∂ x ( τ n + , x i ) + ( ∆ x ) ∂ v ∂ x ( τ n + , x i ) ; + ( ∆ x ) ∂ v ∂ x ( τ n + , x i ) + ( ∆ x ) ∂ v ∂ x ( τ n + , x i ) v n + i − ≈ v n + i − ∆ x ∂ v ∂ x ( τ n + , x i ) + ( ∆ x ) ∂ v ∂ x ( τ n + , x i ) − ( ∆ x ) ∂ v ∂ x ( τ n + , x i ) + ( ∆ x ) ∂ v ∂ x ( τ n + , x i ) , hence v n + i + − v n + i + v n + i − ( ∆ x ) ≈ (cid:18) ( ∆ x ) ∂ v ∂ x + ( ∆ x ) ∂ v ∂ x (cid:19) . Putting this last result in ((39)) gives: | a | ≤ ( ∆ x ) | f ( τ n + , x i ) | (cid:12)(cid:12)(cid:12)(cid:12) ∂ v ∂ x + ( ∆ x ) ∂ v ∂ x (cid:12)(cid:12)(cid:12)(cid:12) + ( ∆ x ) σ ( T − τ n + ) (cid:12)(cid:12)(cid:12)(cid:12) ∂ v ∂ x (cid:12)(cid:12)(cid:12)(cid:12) + ∆ t | ∂ τ L v ( τ n + ∆ t θ , x i ) | , (40)9ince α , σ , β and f are bounded functions. Also, since the derivatives ∂ m + n v (cid:14) ∂τ n ∂ x m are bounded, it implies: | a | ≤ ( ∆ x ) M + ∆ tM . (41)One also has: | a | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K r ∑ j = K l ν j ( v ni + j − v ni ) − Z B r B l ( v ( τ , x i + y ) − v ( τ , x i )) g ln J ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K r ∑ j = K l Z ( j + / ) ∆ x ( j − / ) ∆ x ( v ni + j − v ni ) g J ( y ) dy − Z B r B l ( v ( τ , x i + y ) − v ( τ , x i )) g J ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Since [ B l , B r ] ⊂ [( K l − / ) ∆ x , ( K r + / ) ∆ x ] , then we have: | a | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K r ∑ j = K l Z ( j + / ) ∆ x ( j − / ) ∆ x ( v ( τ n , x i + y j ) − v ( τ n , x i + y ) g ln J ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , and using Taylor’s expansion of order one, we get: | a | ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K r ∑ j = K l Z ( j + / ) ∆ x ( j − / ) ∆ x ( y j − y ) ∂ v ∂ x ( τ n , x i + ψ ) g ln J ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , ψ ∈ ] x i + y , x i + y j [ . From the scheme, we have ∆ x ( j − / ) ≤ y ≤ ∆ x ( j + / ) , which leads to − ∆ x ≤ y j − y ≤ ∆ x | a | ≤ ∆ x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K r ∑ j = K l Z ( j + / ) ∆ x ( j − / ) ∆ x ∂ v ∂ x ( τ n , x i + ψ ) g ln J ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∆ x (cid:13)(cid:13)(cid:13)(cid:13) ∂ v ∂ x (cid:13)(cid:13)(cid:13)(cid:13) ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K r ∑ j = K l Z ( j + / ) ∆ x ( j − / ) ∆ x g ln J ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ∆ x M , (42)where M = (cid:13)(cid:13)(cid:13)(cid:13) ∂ v ∂ x (cid:13)(cid:13)(cid:13)(cid:13) ∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) K r ∑ j = K l R ( j + / ) ∆ x ( j − / ) ∆ x g ln J ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Finally, (38), (41) and (42) imply | r ni ( ∆ t , ∆ x ) | ≤ ∆ t (cid:18) M + M (cid:19) + ∆ x (cid:18) ∆ xM + M (cid:19) → ( ∆ t , ∆ x ) → ( , ) . Two properties are important to show convergence to viscosity solutions: stability and monotonicity of scheme.
Definition 4.4.
Stability
The scheme (32) is stable if, and only if, for some bounded initial conditions, its solution exists and is boundedindependently of ∆ t and ∆ x, and uniformly bounded on [ , T ] × R . That is to say, ∃ C > , ∀ ∆ t > , ∀ ∆ x > , i ∈ Z , n ∈ { , ..., M } , | u ni | ≤ C . We will say that a given vector v is positive if all its elements are positive. We write u ≥ v if u − v ≥
0. Inthis part we show the stability property of the scheme, which in turn implies the discrete comparison principle, aproperty which has an important interpretation in finance. This property makes possible the fact that the optionsvalues computed using our numerical scheme will check arbitrage inequalities: Inequality between payoffs leadingto inequality between options values.
Proposition 4.5. ( Stability and the discrete comparison principle )If ∆ t ≤ (cid:14) ∑ K r j = K l ν j , the scheme (32) is stable, and hence verifies the discrete comparison principle:u ≥ v = ⇒ ∀ n ∈ N ∗ , u n ≥ v n . roof. Firstly, consider (32) in the form: − cu n + i − + ( + a ∆ t ) u n + i − b ∆ tu n + i + = − ∆ t K r ∑ j = K l ν j ! u ni + ∆ t K r ∑ j = K l u ni + j ν j , (43)where c =
12 1 ( ∆ x ) σ ( T − τ n + ) ≥ a = ∆ x f ( τ n + , x i ) + ( ∆ x ) σ ( T − τ n + ) ≥ b = ∆ x f ( τ n + , x i ) +
12 1 ( ∆ x ) σ ( T − τ n + ) ≥ . (44)The positivity of a and b arises from g being positive. If g is not positive, we change the approximation of thefirst-order derivatives in the scheme used. In either case, one has: a = b + c ⇒ a ∆ t = b ∆ t + c ∆ t ⇒ + a ∆ t > ( b + c ) ∆ t . It follows that | + a ∆ t | > | − c ∆ t | + | − b ∆ t | , implying the matrix of linear system on ( u n + , ..., u n + N ) has a strictdominant diagonal, hence invertible. Therefore, the solution of the linear system exists and is unique. We nowshow by mathematical induction that this solution is bounded. That is, if k H k ∞ ≤ ∞ is the bounded initial condition,then, ∀ n ∈ N , k u n k ∞ ≤ k H k ∞ . (45)By definition of u , we have k u k ∞ ≤ k H k ∞ . Assume (45) holds for n . To show that it holds for n +
1, we supposeon the contrary that k u n + k ∞ > k H k ∞ . By the definition of k . k ∞ , ∃ i ∈ { , ..., n } such that | u n + i | = k u n + k ∞ , and ∀ i ∈ Z , | u n + i | < | u n + i | .Since a = b + c , we can write, k u n + k ∞ = | u n + i | = − c ∆ t | u n + i | + ( + a ∆ t ) | u n + i | − b ∆ t | u n + i | . (46)Moreover, as | u n + i − | < | u n + i | and | u n + i + | < | u n + i | we have k u n + k ∞ − c ∆ t | u n + i − | + ( + a ∆ t ) | u n + i | − b ∆ t | u n + i + | . (47)Using (43) and (47), and the fact that ∆ t ≤ (cid:14) ∑ K r j = K l ν j , we obtain: k u n + k ∞ ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ∆ t K r ∑ j = K l ν j ! u ni + ∆ t K r ∑ j = K l u ni + j ν j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ − ∆ t K r ∑ j = K l ν j ! | u ni | + ∆ t K r ∑ j = K l | u ni + j ν j |≤ − ∆ t K r ∑ j = K l ν j ! k u n k ∞ + ∆ t K r ∑ j = K l ν j k u n k ∞ = k u n k ∞ ≤ k H k ∞ , which contradicts our assumption. Hence k u n k ∞ ≤ k H k ∞ . Proposition 4.6. ( Monotonicity )Let u n and v n be two solutions to (32) corresponding to some initial conditions f and h respectively, satisfyingf ( x ) ≥ h ( x ) ∀ x ∈ R . If ∆ t ≤ (cid:14) ∑ K r j = K l ν j , then u n ≥ v n , ∀ n ∈ N .Proof. Define w n = u n − v n . We show that w n ≥ ∀ n ∈ N . As in Proposition 4.5, we proceed by induction. Byconstruction, we have w i = f ( x i ) − b ( x i ) ≥ ∀ i ∈ Z . Let w n ≥
0, and suppose that: inf i ∈ Z w n + i <
0. Since ∀ i ∈ \{ , ..., N } , w n + i =
0, this implies that ∃ i ∈ { , ..., N } such that w n + i = inf i ∈ Z w n + i . Using (43) and ∆ t ≤ ∑ K r j = K l ν j ,we have that inf i ∈ Z w n + i = w n + i = − c ∆ tw n + i + ( + a ∆ t ) w n + i − b ∆ tw n + i ≥ − c ∆ tw n + i − + ( + a ∆ t ) w n + i − b ∆ tw n + i + = − ∆ t K r ∑ j = K l ν j ! w ni + ∆ t K r ∑ j = K l w ni + j ν j ≥ , which is a contradiction. Therefore, inf i ∈ Z w n + i ≥
0, and hence w n + ≥ As proved above, our scheme (35) is locally consistent, stable, monotone and verifies the discrete comparison prin-ciple. In the usual approach to the convergence of finite difference schemes for PDE’s, consistency and stabilityensure convergence under regularity assumptions on the solution. These conditions are not sufficient here becausethe solution may not be smooth, and higher order derivatives may not exist. This is where the notion of viscositysolutions are introduced. In the second order parabolic/elliptic PDEs Barles and Souganidis [7] showed that forelliptic (or parabolic) PDEs, any locally consistent, stable and monotone finite difference scheme converge uni-formly, on each compact subset [ , T ] × R , to the unique continuous viscosity solution. Rama Cont and EkaterinaVoltchkova in [11] showed that the solution of a numerical scheme converges uniformly on each compact subsetof [ , T ] × R to the unique viscosity solution even when the subsolution and the supersolution constructed using anumerical scheme may not have uniform continuity properties. The PIDE studied in this paper relies on the sameassumptions as in [11], except that here, we are in the case of a finite activity measure since the sizes of the jumpsis log-normal. Then we used the same technics to showed the convergence of the explicit-implicit scheme (35) tothe viscosity solution of problem (9). Proposition 4.7. ( Convergence of the explicit-implicit scheme )Let H be a bounded piecewise continuous initial condition, then solution u ( ∆ t , ∆ x ) of the numerical scheme convergesuniformly on each compact subset of [ , T ] × R to the solution u of continuous problem (9) .Proof. Define u ( τ , x ) = lim inf ( ∆ t , ∆ x ) → ( , )( t , y ) → ( τ , x ) u ( ∆ t , ∆ x ) ( t , y ) andu ( τ , x ) = lim sup ( ∆ t , ∆ x ) → ( , )( t , y ) → ( τ , x ) u ( ∆ t , ∆ x ) ( t , y ) . (48)The aim of this proof consists to show the following equalities u ( τ , x ) = u ( τ , x ) = u ( τ , x ) . Before showing thisequalities some preparatory results are needed.We start by giving an equivalent expression for (35). u ( τ n , x i ) = F [ u ( τ n − ∆ t , . )]( x i ) , n = , ..., M , i ∈ , ..., N , u ( , x i ) = H ( S e x i ) , i ∈ , ..., N , (49) u ( τ n , x i )) = , n = , ..., M , / ∈ , ..., N . One can define super and subsolution of 49 by the following definition
Definition 4.8.
A function w is a supersolution of 49 ifw ( τ n , x i ) ≥ F [ w ( τ n − ∆ t , . )]( x i ) , n = , ..., M , i ∈ , ..., N , w ( , x i ) ≥ H ( S e x i ) , i ∈ , ..., N , (50) w ( τ n , x i )) ≥ , n = , ..., M , / ∈ , ..., N . A function z is a subsolution of 49 ifz ( τ n , x i ) ≤ F [ z ( τ n − ∆ t , . )]( x i ) , n = , ..., M , i ∈ , ..., N , z ( , x i ) ≤ H ( S e x i ) , i ∈ , ..., N , (51) z ( τ n , x i )) ≤ , n = , ..., M , i / ∈ , ..., N .
12o Avoid the problem of uniform continuity and smoothness which may not hold for u and u define in (48), itis convenient to consider smooth super and subsolutions of 9 and super and subsolutions of 49, and to derive thelink with u and u . The following results extends the comparison principle to the super and subsolutions. Lemma 4.9.
For any supersolution w and subsolution z of 49 we have z ≤ u ≤ w.Proof. For ( i / ∈ , ..., N ) or ( n = i ∈ , ..., N ) the above inequalities are satisfied by definition. For n = , ..., M , i ∈ , ..., N from monotonicity of the scheme we have z ( τ n , x i ) ≤ F [ z ( τ n − ∆ t , . )]( x i ) ≤ F [ u ( τ n − ∆ t , . )]( x i ) = u ( τ n , x i )= F [ u ( τ n − ∆ t , . )]( x i ) ≤ F [ w ( τ n − ∆ t , . )]( x i ) ≤ w ( τ n , x i ) . Lemma 4.10.
Let w and z be a smooth supersolution and subsolution of 9 respectively. Then for all ε , there exists ∆ > such that ∀ ∆ t , ∆ x , ≤ ∆ , ∀ n ≥ , ∀ i ∈ Z , z ( τ n , x i ) − ε < u ( τ n , x i ) < w ( τ n , x i ) + ε Proof.
Choose q such that 0 < q ( T + ) < ε and let ˜ w ( τ , x ) = w ( τ , x ) + q ( + τ ) , notice that a constant function isalways a solution. In fact one can see from the definition that the scheme is linear.If i / ∈ , ..., N , we have ˜ w ( τ n , x i ) = w ( τ n , x i ) + q ( τ + ) ≥ q ≥ . (52)If n = i ∈ , ..., N , ˜ w ( , x i ) = w ( , x i ) + q ≥ H ( S e x i ) . (53)If n ≥ i ∈ , ..., N from the consistency of the scheme we obtain˜ w ( τ n , x i ) − ˜ w ( τ n − ∆ t , x i )( ∆ t ) − L D ˜ w ( τ n − ∆ t , x i ) − L J ˜ w ( τ n − ∆ t , x i ) = w ( τ n , x i ) − w ( τ n − ∆ t , x i )( ∆ t ) − L D w ( τ n − ∆ t , x i ) − L J w ( τ n − ∆ t , x i ) + q > −→ ∂ w ∂τ ( τ , x ) − ( L D + L J ) w ( τ , x ) + q as ∆ t , ∆ x −→ ( , ) , ( τ n , x i ) −→ ( τ , x ) , uniformly on ( , T [ × O . Therefore for any sufficiently small ∆ >
0, for all ∆ t , ∆ x ≤ ∆ , we have ˜ w ( τ n , x i ) ≥ F [ ˜ w ( τ n − ∆ t , . )]( x i ) , ∀ n ≤ , ∀ i ∈ , ..., N ) . (55)Combining 52, 54 and 55, show that function ˜ w is supersolution of 49. Indeed, Lemma 4.9 implies that u ( τ n , x i ) ≤ ˜ w ( τ n , x i ) + q ( + T ) < w ( τ n , x i ) + ε , ∀ n ≥ , ∀ i ∈ Z , which is the desired property. The lower bound z ( τ n , x i ) − ε can be proved in the same manner and then completesthe proof.Following Lemma 52 and Lemma 54, we have the following main Lemma Lemma 4.11.
Let u and u be the function define by 48. For any smooth supersolution w ( τ , x ) and any subsolutionz ( τ , x ) of the problem 9, we have for ( τ , x ) ∈ [ , T ] × O,z ( τ , x ) ≤ u ( τ , x ) ≤ u ( τ , x ) ≤ w ( τ , x ) . (56) Proof.
By the definition of upper and lower limits, Lemma 4.10 implies desired property.13fter giving some properties needed we can start the proof of convergence (i.e. showed that u = u = u ). If H , H are smoothness functions on R such that H ≤ H ≤ H , then w ( τ , x ) = E [ H ( Y x τ )] and z ( τ , x ) = E [ H ( Y x τ )] are respec-tively a supersolution and a subsolution of the Cauchy problem 9. From Lemma 4.11 we obtain 56. Notice that If w ( τ , x ) − u ( τ , x ) , u ( τ , x ) − z ( τ , x ) could be made small this would imply that lim ( ∆ t , ∆ x ) → ( , )( τ n , x i ) → ( τ , x ) u ( ∆ t , ∆ x ) ( τ n , x i ) = u ( τ , x ) . Indeed, it remains to construct appropriate smooth approximations H and H .Let ζ , ...., ζ I be the discontinuity points of H . We suppose that the jumps of H are bounded by δ . Given ε > H , H smooth functions that satisfied the following relations H ( x ) ≤ H ≤ H ( x ) ∀ x ∈ R , | H ( x ) − H ( x ) | ≤ δ ∀ x ∈ I [ j = ( ζ j − ε , ζ j + ε ) , | H ( x ) − H ( x ) | ≤ ε ∀ x / ∈ I [ j = ( ζ j − ε , ζ j + ε ) . We have w ( τ , x ) − z ( τ , x ) = E [ H ( Y x τ ) − H ( Y x τ )] ≤ δ Q ( Y x τ ∈ I [ j = ( ζ j − ε , ζ j + ε )) + ε Q ( Y x τ / ∈ I [ j = ( ζ j − ε , ζ j + ε )) (57) ≤ δ Q ( Y x τ ∈ I [ j = ( ζ j − ε , ζ j + ε )) + ε . (58)Noting that T ε > { Y x τ ∈ I S j = ( ζ j − ε , ζ j + ε ) } = { Y x τ ∈ { ζ , ...., ζ I }} . Since Y x τ has an absolutely continuous distribution,so we have Q ( { Y x τ ∈ { ζ , ...., ζ I }} ) =
0. Consequently Q ( Y x τ ∈ I S j = ( ζ j − ε , ζ j + ε )) −→ ε −→ w ( τ , x ) − z ( τ , x ) −→ ε −→ z ( τ , x ) ≤ u ( τ , x ) ≤ w ( τ , x ) together with Lemma4.11 implies desired result which completes the proof. Remark 4.12.
For τ = the scheme does not converge to the initial condition at the discontinuous points of H.This is due to the fact that Q ( Y x τ ∈ I S j = ( ζ j − ε , ζ j + ε )) −→ Q ( S e x ∈ { ζ , ...., ζ I } ) = { x ∈{ ln ( ζ S ) ,..., ln ( ζ IS ) }} . However,this has no practical interest since it is not important to compute the solution numerically at τ = . In this section we discuss to the details of the implementation of our schemas and present numerical results andsome interpretation.Before started simulation parameters scheme are take as follow the parameters used to implement the followingTable 1: scheme parameters
T M N A l A r − .
096 0 . r Strike Product1, 2, 3, 4 α = . β = . σ J = . ℓ = . σ = . S =
50 0 . K =
45 Call τ (remaining time) C( τ ,S T ) European Call S T =45.4232 τ (remaining time) C( τ ,S T ) European Call S T =46.5731 τ (remaining time) C( τ ,S T ) European Call S T =48.9609 τ (remaining time) C( τ ,S T ) European Call S T =49.7009 Figure 1: Call price for four diffrents values of spot price at maturity versus remaining time to maturity τ (remaining time) C( τ ,S T ) European Call S T =45.5141S T =46.5731S T =48.9609S T =49.7009 Figure 2: Comparison of call values for four diffrents values of spot price at maturity versus remaining time tomaturity
45 50 550510
European CallS
C(0,S)
45 50 550246
European CallS
C(0.01,S)
45 50 5500.511.52
European CallS
C(0.05,S)
45 50 5500.10.20.30.4
European CallS
C(0.1,S)
Figure 3: Call price for four different remaining time to maturity versus spot price
45 46 47 48 49 50 51 52 53 54 55012345678910
European CallS C( τ ,S) τ =0 τ =0.01 τ =0.05 τ =0.1 Figure 4: Comparison of call price for four different versus spot priceFigure 3 and 4 illustrate a reality enough close to those in the classical financials markets. In the sens that infinancial market the values of call option before the maturity evolved in the form of a curve which towards to the15ayoff line progressively and as we approach maturity.
45 50 5502468101214 S C(0.01,S)
European Call
K=35K=40K=45
Figure 5: Call price for three diffrents values of strike versus spot price S, the other parameters is unchanged as intable 2
36 38 40 42 44 46 48 50 52 5400.0050.010.0150.020.0250.030.0350.040.045
European Callstrike
C(0.1,50.1001)
Figure 6: Call price versus strike price K, the other parameters in unchange as in table 2
34 36 38 40 42 44 46 48 50 52 5400.10.20.30.40.50.60.7
European Callstrike
C(0.1,S)
S=50.1001S=51.2658S=50.7557
Figure 7: Call price versus strike price K, the other parameters in unchange as in table 2From figures 5, 6 and 7 one can observed that the values of call decrease when strike price increase. This behaviourof the call values is from a risk management point of view what it is wished. Analysis plots of figures 8 and 9 whichillustrate call option price as a function of remaining time and spot price one can said that for a large remainingtime jumps effects are not perceptible and then can effect call. Whereas a small remaining time to maturity callprices increase suddenly which express the effect of jump. We must therefore say that the jump term which allowsto take into account certain reality of the electricity market is not inconsiderable since that it impact on the callprices are quite noticeable. 16 τ European Callspot price
Call price Value
Figure 8: Call price versus remaining time and spot price σ J = . K =
45, the other parameters is unchanged asin table 2
European Callspot price τ Call price Value
Figure 9: Call price versus remaining time and spot price σ J = . K =
40, the other parameters is unchangedas in table 2
The characterization of european options prices in terms of the classical solution, or, in general, in the terms ofthe viscosity solution of a PIDE allows the use of numerical methods to obtain efficient approximations of optionvalues. This has been a centre of research in recent times in the case of exponential L´evy models with finite arrivalor infinite arrival rate of jumps. Some authors use the finite difference method to approximate the PIDE solution(see [2]), while others like [12] approximate viscosity solutions in the case of nonsmoothness of option prices. Inboth cases, success (in terms of efficient approximation) has been obtained. In this paper we used their approachto evaluate European call option when the underlying is electricity. The motivation behind our approach arosefrom the fact that the electricity prices model presented here, by hypothesis, possesses most of the properties (as intheir case) of an exponential L´evy model, and the Markov process property. We focus on the pricing of call optionbecause put option can be deduced from the put-call using parity formula. In the mathematical point of view,numerical results have confirmed the established theoretical results. In the finance point of view, numerical resultspresent an interpretation which was coherent with some realities in the electricity market, when it is regulatedunder price cap.A limitation of this work is that it can be adapted only to option pricing with short maturity in a regulatedmarket. In future work, we plan to study option pricing on future option.
Acknowledgements
We thank the African Center of Excellence in Technologies, Information and Communication ( CETIC ) whichplaced at our disposal its infrastructures. This helped us to improve conditions of work.17 eferences [1] J. Acton and I. Vogelsang,
Symposuim on Price Cap Regulation: Introduction , Rand Journal of Economics,Vol.20, page 369, 1989.[2] O. Alvarez and A. Tourin,
Viscosity solutions of nonlinear integro-differential equation , Annales de l’InstitutHenri Poincar,13(3), pp. 293-317, 1996.[3] H. Amann,
Linear and Quasilinear Parabolic Problems: Abstract Linear Theory , Birkh¨auser Vol. 1., 1995.[4] F. Black and M. Scholes,
The Pricing of options and Corporate Liabilities , The journal of Political Economy,Vol. 81, No. 3, pp. 634-654, 1973.[5] D. Bates,
Jumps and stochastic volatility: Exchange rate processes implicit in Deutchemark option , Reviewof Financial Studies, 9, 69-108, 1996.[6] G. Bales, R. Buckdahn, and E. Pardoux,
Backward Stochastic Differential Equations and Integral-PartialDifferential Equations , Stochast. Stochast. Reports 60, 57-83, 1997.[7] G. Barles and P. E. Souganidis,
Convergence of Approximation Schemes for Fully Nonlinear Second OrderEquations , Asymptotic Anal., pp. 271-283, 1991.[8] A. Cartea, M. Figueroa,
Pricing in electricity markets: A mean reverting jump diffusion model with season-ality. , Applied Mathematical Finance 12(4), 313-335, 2005.[9] P. Carr and D. B. Madan,
Option Valuation using the Fast Fourier Transform , Journal of ComputationalFinance, 2, N .4, Summer 61-73, 1999.[10] Chiarella, carl, Ziogas and Andrew, American call options under jump-diffusion processes- A Fourier trans-form approach , Appl. Math. Finance, Vol. 16, no. 1, pp37-79, 2009.[11] R. Cont, Ekaterina voltchkova,
Integro-differential equation for option prices in exponential Lvy models ,Finance and Stochastics,Springer, 2005.[12] R. Cont, Ekaterina voltchkova,
A Finite difference scheme fo option pricing in jump diffusion and exponentialL´evy models , Siam J. Numer. Anal. Vol. 42, No. pp.1596-1626, 2005.[13] M. G. Crandall, H. Ishi, and P. L. Lions,
User’s Guide to Viscosity Solutions of Second Oder Partial Differ-ential Equations , New Series of the American Mathematical Society, Vol. 27, Number 1, 1992.[14] S. Deng,
Pricing electricity derivatives under alternative stochastic spot price models , Proceedings of the33rd Hawaii Internatioal Conference on system Sciences, 2000.[15] A. Etheridge,
A Course in Financial Calculus , Cambridge University Press,first edition, 2002.[16] ENMAX,
Power Corporation 2007-2016 formula Based Ratemaking , Alberto Utilities Commission, Deci-sion N0. 2009-035, 2009.[17] A. Hirsa and D.B. Madan,
Pricing American option under Variance Gamma , Journal of Computational Fi-nance, vol. 7, pp. 63-80, 2004.[18] J. Jacod and A. N. Shiryaev,
Limit Theorems for Stochastic Processes , Springer, Berlin, 2nd edition, 2002.[19] P.L. Joskow,
Incentive Regulation and its Application to Electricity Networks , Review of Network Economics,vol 7,Issue 4, 2008.[20] P.L. Joskow,
Incentive Regulation in Theory and Pratice: Electricity Distribution And Transmission Network ,Prepared for the National bureau of Economic Reseach on Economic Conference Regulation, Working Paper05-18, Brooking Joint Center for Regulatory Studies, Washington DC, 2005.[21] J. J. Laffont, J. Tirole,
A Theory of Incentives in Procurement and Regulation , MIT Press, Cambridge, MA,1993. 1822] A. Lewis,
A Sample option Formula for General Jump-Diffusion and other exponential L´evy processes
Regulation of British Telecommunication’Profitability. Tech. Rep. , Department of Industry,London, 1983.[24] D. B. Madan, P. Carr and E. C. Chang,
The Variance gamma process and option pricing , European FinanceReview, 2, 79-105.[25] C. Merton,
Continuous-Time Finance , Harvard University, First revised edition, 2001.[26] D. Nualart and W. Schoutens,
Backward stochastic differential equations and Feynman-Kac formula for L´evyprocesses, with applications in finance , Bernoulli 7(5), 761-776, 2001.[27] Polipovic Dragana,
Energy Risk: Valuing and Managing Energy Derivatives
The Mac-Graw Hill Companies,2007.[28] C.Rama, P. Tankov,
Financial modelling with jump processes , Chapman and Hall/CRC press, 2003.[29] K. Sato,
Lvy Processes and Infinitely Divisible Distributions , Cambridge University Press, 1999.[30] L. Scott,
Pricing stock in a jump-diffusion model with stochastic volatility and interest rate: Application ofFourier inversion methods , Mathematical Finance, 7, 413-426, 1997.[31] H. Wong and P. Guan,