EXPLICIT OPTION PRICING FORMULA FOR A MEAN-REVERTING ASSET IN ENERGY MARKET

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1 Journal of Numerical and Applied Mahemaics Vol.1 (96), 8, pp ANATOLY SWISHCHUK EXPLICIT OPTION PRICING FORMULA FOR A MEAN-REVERTING ASSET IN ENERGY MARKET Some commodiy prices, like oil and gas, exhibi he mean reversion, unlike sock price. I means ha hey end over ime o reurn o some long-erm mean. In his paper we consider a risky asse S following he mean-revering sochasic process. The aim of his paper is o obain an explici expression for a European opion price based on S, using a change of ime mehod. A numerical example for he AECO Naural Gas Index (1 May April 1999) is presened. Mahemaics Subjec Classificaions. 6H1, 91B8 Key words and phrases. Mean-revering asse, European call opion, opion pricing formula, risk-neural measure, Black-Scholes formula, AECO Naural Gas Index 1. Inroducion Some commodiy prices, like oil and gas, exhibi he mean reversion, unlike sock price. I means ha hey end over ime o reurn o some longerm mean. In his paper we consider a risky asse S following he meanrevering sochasic process given by he following sochasic differenial equaion ds = a(l S )d + σs dw, where W is a sandard Wiener process, σ > is he volailiy, he consan L is called he long-erm mean of he process, o which i revers over ime, and a > measures he srengh of mean reversion. This mean-revering model is a one-facor version of he wo-facor model made popular in he conex of energy modelling by Pilipovic (1997). Black s model (1976) and Schwarz s model (1997) have become a sandard approach o he problem of pricing opions on commodiies. These models have he advanage of mahemaical convenience, 1

2 ANATOLIY SWISHCHUK in ha hey give rise o closed-form soluions for some ypes of opions (See Wilmo ()). Bos, Ware and Pavlov () presened a mehod for evaluaion of he price of a European opion based on S, using a semi-specral mehod. They did no have he convenience of a closed-form soluion, however, hey showed ha values for cerain ypes of opions may neverheless be found exremely efficienly. They used he following parial differenial equaion (see, for example, Wilmo, Howison and Dewynne (1995)) C + R(S, )C S + σ S C SS/ = rc for opion prices C(S, ), where R(S, ) depends only on S and, and corresponds o he drif induced by he risk-neural measure, and r is he risk-free ineres rae. Simplifying his equaion o he singular diffusion equaion hey were able o calculae numerically he soluion. The aim of his paper is o obain an explici expression for a European opion price, C(S, ), based on S, using a change of ime mehod (see Swishchuk (7)). This mehod was once applied by he auhor o price variance, volailiy, covariance and correlaion swaps for he Heson model (see Swishchuk (4)).. Mean-Revering Asse Model (MRAM) Le (Ω, F, F, P ) be a probabiliy space wih a sample space Ω, σ-algebra of Borel ses F and probabiliy P. The filraion F, [, T ], is he naural filraion of a sandard Brownian moion W, [, T ], such ha F T = F. Some commodiy prices, like oil and gas, exhibi he mean reversion, unlike sock price. I means ha hey end over ime o reurn o some longerm mean. In his paper we consider a risky asse S following he meanrevering sochasic process given by he following sochasic differenial equaion ds = a(l S )d + σs dw, (1) where W is an F -measurable one-dimensional sandard Wiener process, σ > is he volailiy, consan L is called he long-erm mean of he process, o which i revers over ime, and a > measures he srengh of mean reversion. 3. Explici Opion Pricing Formula for European Call Opion for MRAM under Physical Measure In his secion, we are going o obain an explici expression for a European opion price, C(S, ), based on S, using a change of ime mehod and physical measure.

3 PRICING FORMULA FOR A MEAN-REVERTING ASSET Explici Soluion of MRAM. Le Then, from () and (1) we obain V := e a (S L). () dv = ae a (S L)d + e a ds = σ(v + e a L)dW. (3) Using change of ime approach o he equaion (3) (see Ikeda and Waanabe (1981) or Ellio (198)) we obain he following soluion of he equaion (3) V = S L + W (φ 1 ), or (see ()), S = e a [S L + W (φ 1 )] + L, (4) where W () is an F -measurable sandard one-dimensional Wiener process, φ 1 is an inverse funcion o φ : We noe ha φ = σ (S L + W (s) + e aφs L) ds. (5) φ 1 = σ (S L + W (φ 1 ) + e as L) ds, (6) which follows from (5) and he following ransformaions: dφ = σ (S L+ W ()+e aφ L) d σ (S L+ W ()+e aφ L) dφ = d = σ (S L + W (s) + e aφs L) dφ s = σ φ 1 (S L + W (s) + e aφs L) dφ s = σ (S L + W (φ 1 s ) + e as L) ds. φ Some Properies of he Process W (φ 1 -measurable and F -maringale. process W (φ 1 Then ) is F := F φ 1 ). We noe ha E W (φ 1 ) =. (7) Le s calculae he second momen of W (φ 1 ) (see (6)): E W (φ 1 ) = E < W (φ 1 ) >= Eφ 1 = σ E(S L + W (φ 1 s ) + e as L) ds = σ [(S L) + L(S L)(e a 1) + L (e a 1) a a + E W (φ 1 s )ds]. (8)

4 4 ANATOLIY SWISHCHUK From (8), solving his linear ordinary nonhomogeneous differenial equaion wih respec o E W (φ 1 ), de W (φ 1 ) d we obain = σ [(S L) + L(S L)e a + L e a + E W (φ 1 )], E W (φ 1 ) = σ [(S L) eσ 1 + L(S L)(e a e σ ) + L (e a e σ ) ]. σ a σ a σ (9) 3.3. Explici Expression for he Process W (φ 1 ). I is urns ou ha we can find he explici expression for he process W (φ 1 ). From he expression (see Secion 3.1) V = S L + W (φ 1 ), we have he following relaionship beween W () and W (φ 1 ) : d W (φ 1 ) = σ [S() L + Le a + W (φ 1 s )]dw (). I is a linear SDE wih respec o W (φ 1 ) and we can solve i explicily. The soluion has he following look: W (φ 1 ) = S()(e σ σw () 1)+L(1 e a )+ale σw () σ e as e σw (s)+ σ s ds. (1) I is easy o see from (1) ha W (φ 1 ) can be presened in he form of a linear combinaion of wo zero-mean maringales m 1 () and m () : where and W (φ 1 ) = m 1 () + Lm (), m 1 () := S()(e σw () σ 1) m () = (1 e a ) + ae σw () σ e as e σw (s)+ σ s ds. Indeed, process W (φ 1 ) is a maringale (see Secion 3.), also i is wellknown ha process e σw () σ and, hence, process m 1 () is a maringale. Then he process m (), as he difference beween wo maringales, is also maringale. In his way, we have Em 1 () =,

5 since As for m () we have PRICING FORMULA FOR A MEAN-REVERTING ASSET 5 Ee σw () σ = 1. Em () =, since from Iô s formula we have d(ae σw () σ eas σ σw (s)+ s e ds) = aσe σw () σ and, hence, Eae σw () σ eas e σ σw (s)+ s + ae σw () σ e a e σw ()+ σ = aσe σw () σ + ae a d, eas e e as e σw (s)+ σ s ds = e a 1. d σ σw (s)+ s I is ineresing o see ha he las expression, he firs momen for η() := ae σw () σ e as e σw (s)+ σ s ds, dsdw () dsdw () does no depend on σ. I is rue no only for he firs momen bu for all he momens of he process η() = ae σw () σ eas σ σw (s)+ s e ds. Indeed, using Iô s formula for η n () we obain dη n () = na n σe nσw () nσ ( eas e and σ σw (s)+ s deη n () = nae a Eη n 1 ()d, n 1. ds) n dw () + an(η ()) n 1 e a d, This is a recursive equaion wih iniial funcion (n = 1) Eη() = e a 1. Afer calculaions we obain he following formula for Eη n () : Eη n () = (e a 1) n Some Properies of he Mean-Revering Asse S. From (4) we obain he mean value of he firs momen for mean-revering asse S : ES = e a [S L] + L. I means ha ES L when +. Using formulae (4) and (9) we can calculae he second momen of S : ES = (e a (S L) + L) + σ e a [(S L) eσ 1 σ + L(S L)(e a e σ ) a σ + L (e a e σ ) a σ ].

6 6 ANATOLIY SWISHCHUK Combining he firs and he second momens we have he variance of S : V ar(s ) = ES (ES ) = σ e a [(S L) eσ 1 σ + L(S L)(e a e σ ) a σ + L (e a e σ ) ]. a σ From he expression for W (φ 1 ) (see (1)) and for S() in (4) we can find he explici expression for S() hrough W () : S() = e a [S L + W (φ 1 )] + L = e a [S L + m 1 () + Lm ()] + L = S()e a e σw () σ + ale a e σw () σ eas e σ σw (s)+ s ds, (11) where m 1 () and m () are defined as in Secion Explici Opion Pricing Formula for European Call Opion for MRAM under Physical Measure. The payoff funcion f T for European call opion equals f T = (S T K) + := max(s T K, ), where S T is an asse price defined in (4), T is an expiraion ime (mauriy) and K is a srike price. In his way (see (11)), f T = [e at (S L + W (φ 1 T )) + L K]+ = [S()e at e σw (T ) σ T + ale at e σw (T ) σ T T eas e To find he opion pricing formula we need o calculae σ σw (s)+ s ds K] +. C T = e rt Ef T = e rt E[e at (S L + W (φ 1 T )) + L K]+ = 1 π e rt + max[s()e at e σy T σ T + ale at e σy T σ T T eas e σy s+ σ s Le y be a soluion of he following equaion: ds K, ]e y dy. (1) or S() e at e σy T σ T + ale at e σy T σ T T eas e σy s+ σ s ds = K (13) y = ln( K S() ) + ( σ + a)t σ T

7 PRICING FORMULA FOR A MEAN-REVERTING ASSET 7 From (1)-(13) we have: al T ln(1 + S() eas e σy s+ σ s ds) σ T (14) C T = 1 π e rt + max[s()e at e σy T σ T where + ale at e σy T σ T = 1 T eas e σy s+ σ s π e rt + [S()e at e σy T σ y + ale at e σy T σ T = 1 T T eas e σy s+ σ s ds K, ]e y dy ds K]e y dy π e rt + [S()e at e σy T σ e y dy e rt K[1 Φ(y )] y + Le (r+a)t 1 = BS(T ) + A(T ), + π y y (ae σy T σ T T T eas e σy s+ σ s ds)e y dy BS(T ) := 1 + e rt [S()e at e σy T σ T e y dy e rt K[1 Φ(y )], π and where A(T ) := Le (r+a)t 1 + π y (ae σy T σ T Φ(x) = 1 π x Afer calculaion of BS(T ) we obain T eas e σy s+ σ s ds)e y dy, (15) (16) (17) e y dy. (18) BS(T ) = e (r+a)t S()Φ(y + ) e rt KΦ(y ), (19) y + := σ T y and y := y () and y is defined in (14). Consider A(T ) in (17). Le F T (dz) be a disribuion funcion for he process η(t ) = ae σw (T ) σ T T e as e σw (s)+ σ s ds, which is a par of he inegrand in (17). As M. Yor [15, 16] menioned here is sill no closed form probabiliy densiy funcion for ime inegral of an exponenial Brownian moion, while he bes resul is a funcion wih a double inegral.

8 8 ANATOLIY SWISHCHUK We can use Yor s resul [15] o ge F T (dz) above. Using he scaling propery of Wiener process and change of variables, we can rewrie our expression for S() in (11) in he following way S(T ) = S()e Bv T + 4 σ ale Bv T A v T, where T = σ T, v = a + 1, B 4 σ = σw ( 4 ), B v σ T = vt + B T, A v T = T e Bv s ds. Also, he process η(t ) may be presened in he following way using hese ransformaions η(t ) = 4ae at e B σt 4 A σ T. 4 σ We sae here he resul obained by Yor [37] for he join probabiliy densiy funcion of A v T and B v T. Theorem (M. Yor [37]). The join probabiliy densiy funcion of A v T and B v T saisfies P (A v T du, BT v dx) = e vx v/ exp 1 + ex u θ(ex u, )dxdu u, where >, u >, x R and θ(r, ) = r (π 3 ) e π 1/ + e s r cosh s sinh(s) sin( πs )ds. Using his resul we can wrie he disribuion funcion for η(t ) in he following way P (η(t ) u) = P ( 4ae at σ e B σ T 4 A σ T 4 = P (e B σ T 4 A σ T 4 = F T (u). u) σ e at 4a u) (1) In his way, A(T ) in (17) may be presened in he following way: + A(T ) = Le (r+a)t zf T (dz). Afer calculaion of A(T ) we obain he following expression for A(T ) : y A(T ) = Le (r+a)t [(e at 1) y zf T (dz)], since Eη(T ) = e at 1. Finally, summarizing (1)-(1), we have obained he following Theorem.

9 PRICING FORMULA FOR A MEAN-REVERTING ASSET 9 Theorem 3.1. Opion pricing formula for European call opion for meanrevering asse under physical measure has he following look: C T = e (r+a)t S()Φ(y + ) e rt KΦ(y ) + Le (r+a)t [(e at 1) y zf T (dz)], () where y is defined in (14), y + and y in (), Φ(y) in (18), and F T (dz) is a disribuion funcion in (1). Remark. From (1)-() we find ha European Call Opion Price C T for mean-revering asse lies beween he following boundaries: or (see (19)), BS(T ) C T BS(T ) + Le (r+a)t [e at 1], e (r+a)t S()Φ(y + ) e rt KΦ(y ) C T e (r+a)t S()Φ(y + ) e rt KΦ(y ) + Le (r+a)t [e at 1]. 4. Mean-Revering Risk-Neural Asse Model (MRRNAM) Consider our model (1) ds = a(l S )d + σs dw. (3) We wan o find a probabiliy P equivalen o P, under which he process e r S is a maringale, where r > is a consan ineres rae. The hypohesis we made on he filraion (F ) [,T ] allows us o express he densiy of he probabiliy P wih respec o P. We denoe his densiy by L T. I is well-known (see Lamperon and Lapeyre (1996), Proposiion 6.1.1, p. 13), ha here is an adoped process (q()) [,T ] such ha, for all [, T ], In his case, L = exp[ q(s)dw s 1 q (s)ds] a.s. dp T dp = exp[ q(s)dw s 1 T q (s)ds] = L T. In our case, wih model (17), he process q() is equal o q() = λs, (4)

10 1 ANATOLIY SWISHCHUK where λ is he marke price of risk and λ R. Hence, for our model L T = exp[ λ T S(u)dW u 1 T λ S (u)du]. Under probabiliy P, he process (W ) defined by W := W + λ S(u)du (5) is a sandard Brownian moion (Girsanov heorem) (see Ellio and Kopp (1999)). Therefore, in a risk-neural world our model (3) akes he following look: ds = (al (a + λσ)s )d + σs dw, or, equivalenly, ds = a (L S )d + σs dw, (6) where a := a + λσ, L := al a + λσ, (7) and W is defined in (5). Now, we have he same model in (6) as in (1), and we are going o apply our mehod of changing of ime o his model (6) o obain he explici opion pricing formula. 5. Explici Opion Pricing Formula for European Call Opion for MRRNAM In his secion, we are going o obain explici opion pricing formula for European call opion under risk-neural measure P, using he same argumens as in secions 3-7, where in place of a and L we are going o ake a and L a a := a + λσ, L L := al a + λσ, where λ is a marke price of risk (See secion 3) Explici Soluion for he Mean-Revering Risk-Neural Asse Model. Applying ()-(6) o our model (6) we obain he following explici soluion for our risk-neural model (6): S = e a [S L + W ((φ ) 1 )] + L, (8) where W () is an F -measurable sandard one-dimensional Wiener process under measure P and (φ ) 1 is an inverse funcion o φ : φ = σ (S L + W (s) + e a φ s L ) ds. (9)

11 We noe ha PRICING FORMULA FOR A MEAN-REVERTING ASSET 11 (φ ) 1 = σ (S L + W ((φ ) 1 ) + e a s L ) ds, (3) where a and L are defined in (7). 5.. Some Properies of he Process W ((φ ) 1 ). Using he same argumen as in Secion 4, we obain he following properies of he process W ((φ ) 1 ) in (5). This is a zero-mean P -maringale and E W ((φ ) 1 ) =, E [ W ((φ ) 1 )] = σ [(S L ) eσ 1 σ + (L ) (e a e σ ) a σ ], + L (S L )(e a e σ ) a σ (31) where E is he expecaion wih respec o he probabiliy P and a, L and (φ ) 1 are defined in (7) and (3), respecively Explici Expression for he Process W (φ 1 ). I is urns ou ha we can find he explici expression for he process W (φ 1 ). From he expression V = S L + W (φ 1 ), we have he following relaionship beween W () and W (φ 1 ) : d W (φ 1 ) = σ [S() L + Le a + W (φ 1 s )]dw (). I is linear SDE wih respec o W (φ 1 ) and we can solve i explicily. The soluion has he following look: W (φ 1 ) = S()(e σw () σ 1) + L(1 e a ) + ale σw () σ s ds. eas e σw (s)+ σ (3) I is easy o see from (3) ha W (φ 1 ) can be presened in he form of a linear combinaion of wo zero-mean P -maringales m 1() and m () : W (φ 1 ) = m 1() + L m (), where and m 1() := S()(e σw () σ 1) m () = (1 e a ) + a e σw () σ e a s e σw (s)+ σ s ds.

12 1 ANATOLIY SWISHCHUK Indeed, process W (φ 1 ) is a maringale (see Secion 5.), also i is wellknown ha process e σw () σ and, hence, process m 1() is a maringale. Then he process m (), as he difference beween wo maringales, is also maringale. In his way, we have since As for m () we have since from Iô s formula we have and, hence, d (a e σw () σ = a σe σw () σ E P m 1() =, E P e σw () σ = 1. E P m () =, ea s e σw (s)+ σ s s ea s e σw (s)+ σ + a e σw () σ e a e σw ()+ σ d = a σe σw () σ + a e a d, E P a e σw () σ ea s e σw (s)+ σ s ds) dsdw () dsdw () e a s e σw (s)+ σ s ds = e a 1. I is ineresing o see ha in he las expression, he firs momen for η () := a e σw () σ e a s e σw (s)+ σ s ds, does no depend on σ. This is rue no only for he firs momen bu for all he momens of he process η () = a e σw () σ ea s e σw (s)+ σ s ds. Indeed, using he Iô s formula for (η ()) n we obain and d(η ()) n = n(a ) n σe nσw () nσ ( + a n(η ()) n 1 e a d, ea s e σw (s)+ σ s de(η ()) n = na e a E(η ()) n 1 d, n 1. ds) n dw () This is a recursive equaion wih iniial funcion (n = 1) Eη () = e a 1. Afer calculaions we obain he following formula for E(η ()) n : E(η ()) n = (e a 1) n.

13 PRICING FORMULA FOR A MEAN-REVERTING ASSET Some Properies of he Mean-Revering Risk-Neural Asse S. Using he same argumen as in Secion 5, we obain he following properies of he mean-revering risk-neural asse S in (18): E S = e a [S L ] + L V ar (S ) := E S (E S ) = σ e a [(S L ) eσ 1 σ + (L ) (e a e σ ) ], a σ + L (S L )(e a e σ ) a σ (33) where E is he expecaion wih respec o he probabiliy P and a, L and (φ ) 1 are defined in (7) and (3), respecively. From he expression for W (φ 1 ) (see (3)) and for S() in (8) (see also (9)-(3)) we can find he explici expression for S() hrough W () : S() = e a [S L + W (φ 1 )] + L = e a [S L + m 1() + L m ()] + L eas e σw (s)+ σ s = S()e a e σw () σ + ale a e σw () σ ds, (34) where m 1() and m () are defined as in secion Explici Opion Pricing Formula for European Call Opion for MRAM under Risk-Neural Measure. Proceeding wih he same calculaions (15)-() as in Secion 3, where in place of a and L we ake a and L in (7), we obain he following Theorem. Theorem 5.1. Explici opion pricing formula for European call opion under risk-neural measure has he following look: C T = e (r+a )T S()Φ(y + ) e rt KΦ(y ) + L e (r+a )T [(e a T 1) y zf T (dz)], (35) where y is he soluion of he following equaion y = ln( K S() ) + ( σ + a )T σ T a L T ln(1 + S() ea s e σy s+ σ s ds) σ, (36) T y + := σ T y and y := y, (37) a := a + λσ, L := al a + λσ, and FT (dz) is he probabiliy disribuion as in (1), where insead of a we have o ake a = a + λσ.

14 14 ANATOLIY SWISHCHUK Remark. From (35) we can find ha European Call Opion Price C T for mean-revering asse under risk-neural measure lies beween he following boundaries: e (r+a )T S()Φ(y + ) e rt KΦ(y ) C T e (r+a )T S()Φ(y + ) e rt KΦ(y ) + L e (r+a )T [e a T 1], (38) where y, y, y + are defined in (36)-(37) Black-Scholes Formula Follows: L = and a = r. If L = and a = r we obain from (35) where C T = S()Φ(y + ) e rt KΦ(y ), (39) y + := σ T y and y := y, (4) and y is he soluion of he following equaion (see (36)) or S()e rt e σy T σ T = K y = ln( K ) + ( σ r)t S() σ. (41) T Bu (39)-(41) is exacly he well-known Black-Scholes resul! 6. Numerical Example: AECO Naural GAS Index (1 May April 1999) We shall calculae he value of a European call opion on he price of a daily naural gas conrac. To apply our formula for calculaing his value we need o calibrae he parameers a, L, σ and λ. These parameers may be obained from fuures prices for he AECO Naural Gas Index for he period 1 May 1998 o 3 April 1999 (see Bos, Ware and Pavlov (), p.34). The parameers peraining o he opion are he following: Price and Opion Process Parameers T a σ L λ r K 6 monhs From his able we can calculae he values for a and L : a = a + λσ = ,

15 PRICING FORMULA FOR A MEAN-REVERTING ASSET 15 and L = al a + λσ =.569. For he value of S we can ake S [1, 6]. Figure 1 (see Appendix) depics he dependence of mean value ES on he mauriy T for AECO Naural Gas Index (1 May 1998 o 3 April 1999). Figure (see Appendix) depics he dependence of mean value ES on he iniial value of sock S and mauriy T for AECO Naural Gas Index (1 May 1998 o 3 April 1999). Figure 3 (see Appendix) depics he dependence of variance of S on he iniial value of sock S and mauriy T for AECO Naural Gas Index (1 May 1998 o 3 April 1999). Figure 4 (see Appendix) depics he dependence of volailiy of S on he iniial value of sock S and mauriy T for AECO Naural Gas Index (1 May 1998 o 3 April 1999). Figure 5 (see Appendix) depics he dependence of European Call Opion Price for MRRNAM on he mauriy (monhs) for AECO Naural Gas Index (1 May 1998 o 3 April 1999) wih S() = 1 and K = Fuure Work As we could see in Secion 3. 4 (see also Figures 1, 3 and 4) he main drawbacks of one-facor mean-revering models are: 1) he long-erm mean L remains fixed over ime: needs o be recalibraed on a coninuous basis in order o ensure ha he resuling curves are marked o marke; ) he bigges drawback is in opion pricing: resuls in a model-implied volailiy erm srucure ha has he volailiies going o zero as expiraion ime increases (spo volailiies have o be increased o non-inuiive levels so ha he long erm opions do no lose all he volailiy value-as in he markeplace hey cerainly do no). To eliminae hese drawbacks we are going o consider wo-facor meanrevering model or ds = α(l S )d + σs dw 1 dl = ξl d + ηl dw ds = α(l S )d + σs dw 1 dl = ξ(n L )d + ηl dw (processes W 1 and W may be correlaed) and he change of ime mehod o obain explici opion pricing formula for hese models. We noe, ha one of a possible way o eliminae hese drawbacks is o consider he meanrevering Markov swiching model, see Chen and Forsyh (6). Anoher way is o consider above-menioned SDEs wih Lévy processes as driven processes in place of Wiener processes, see Schouens (3) for applicaions of Lévy processes in finance.

16 16 ANATOLIY SWISHCHUK 8. Acknowledgemens This research is suppored by NSERC gran RPG I would like o hank very much o Rober Ellio, John van der Hoek, Gordon Sick, Tony Ware and Graham Weir for valuable suggesions and commens, and o all he paricipans of he Lunch a he Lab (weekly finance seminar a he Mahemaical and Compuaional Finance Laboraory, Deparmen of Mahemaics and Saisics, Universiy of Calgary, Calgary, Albera, Canada) for ineresing discussion and useful remarks during my firs presenaion of his paper on Thursday, March 1, 5 (PPT presenaion on line: hp:// aswish/). This paper was also presened a he CMS 6 Summer Meeing, Calgary, Canada and QMF 7 Conference, Sydney, Ausralia. I would also like o hank very much o Professor Dmirii Silvesrov and organizers of he Inernaional School Finance, Insurance, and Energy Markes-Susainable Developmen, May 5-9, 8, Väserås, Sweden, for heir kind inviaion o submi his paper for proceedings of he school. Also, I would like o hank Anaoliy Malyarenko for many useful and helpful suggesions and remarks ha improved he paper. All remaining errors are mine. Bibliography 1. Bos, L. P., Ware, A. F. and Pavlov, B. S. () On a semi-specral mehod for pricing an opion on a mean-revering asse, Quaniaive Finance,, Black, F. (176) The pricing of commodiy conracs, J. Financial Economics, 3, Chen, Z. and Forsyh, P. (6) Sochasic models of naural gas prices and applicaions o naural gas sorage valuaion, Technical Repor, Universiy of Waerloo, Waerloo, Canada, November 4, 3p. (hp:// paforsy/regimesorage.pdf). 4. Ellio, R. (198) Sochasic Calculus and Applicaions, Springer-Verlag, New York. 5. Ellio, R. and Kopp, E. (1999) Mahemaics of Financial Markes, Springer- Verlag, New York. 6. Ikeda, N. and Waanabe, S. (1981) Sochasic Differenial Equaions and Diffusion Processes, Kodansha Ld., Tokyo. 7. Lamperon, D. and Lapeyre, B. (1996) Inroducion o Sochasic Calculus Applied o Finance, Chapmann & Hall. 8. Pilipovic, D. (1997) Valuing and Managing Energy Derivaives, New York, McGraw-Hill. 9. Schouens, W. (3 Lévy Processes in Finance: Pricing Financial Derivaives, Wiley. 1. Schwarz, E. (1997) The sochasic behaviour of commodiy prices: implicaions for pricing and hedging, J. Finance, 5,

17 PRICING FORMULA FOR A MEAN-REVERTING ASSET Swishchuk, A. (4) Modeling and valuing of variance and volailiy swaps for financial markes wih sochasic volailiies, Wilmo Magazine, Technical Aricle No, Sepember Issue, Swishchuk, A. (7) Change of ime mehod in mahemaical finance, CAMQ, Vol. 15, No. 3, Wilmo, P., Howison, S. and Dewynne, J. (1995) The Mahemaics of Financial Derivaives, Cambridge, Cambridge Universiy Press. 14. Wilmo, P. () Paul Wilmo on Quaniaive Finance, New York, Wiley. 15. Yor, M. (199) On some exponenial funcions of Brownian moion, Advances in Applied Probabiliy, Vol. 4, No. 3, Yor, M. and Masumoo, H. (5) Exponenial Funcionals of Brownian moion, I: Probabiliy laws a fixed ime, Probabiliy Surveys, Vol., Deparmen of Mahemaics and Saisics, Universiy of Calgary, 5 Universiy Drive NW, Calgary, AB, TN 1N4, Canada. aswish@ucalgary.ca 9. Appendix: Figures

18 18 ANATOLIY SWISHCHUK Fig. 1. Dependence of ES on T (AECO Naural Gas Index (1 May April 1999)) Fig.. Dependence of ES on S and T (AECO Naural Gas Index (1 May April 1999)) Fig. 3. Dependence of variance of S on S and T (AECO Naural Gas Index (1 May April 1999)) Fig. 4. ependence of volailiy of S on S and T (AECO Naural Gas Index (1 May April 1999)) Fig. 5. Dependence of European Call Opion Price on Mauriy (monhs) (S() = 1 and K = 3) (AECO Naural Gas Index (1 May April 1999))