 Journal of Mathematical Finance, 2011, 1, 98-108 doi:10.4236/jmf.2011.13013 Published Online November 2011 (http://www.SciRP.org/journal/jmf) Copyright © 2011 SciRes. JMF European Option Pricing for a Stochastic Volatility Lévy Model with Stochastic Interest Rates Sarisa Pinkham, Pairote Sattayatham School of Mathematics, Institute of Science, Suranaree University of Technology, Nakhon Ratchasima, Thailand E-mail: sarisa@math.sut.ac.th, pairote@sut.ac.th Received August 27, 2011; revised September 9, 2011; accepted September 18, 2011 Abstract We present a European option pricing when the underlying asset price dynamics is governed by a linear combination of the time-change Lévy process and a stochastic interest rate which follows the Vasicek proc- ess. We obtain an explicit formula for the European call option in term of the characteristic function of the tail probabilities. Keywords: Time-Change Lévy Process, Stochastic Interest Rate, Vasicek Process, Forward Measure, Option Pricing1. Introduction Let ,,FP be a probability space. A stochastic process t is a Lévy process if it has independent and stationary increments and has a stochastically continuous sample path, Li.e. for any 0, 0lim 0th thPL L. The sim- plest possible Lévy processes are the standard Brownian motion , Poisson process and compound Poisson process where is Poisson process with inten- tWNi,tN1tiYtNsity t and i are i.i.d. random variables. Of course, we can build a new Lévy process from known ones by using the technique of linear transformation. For example, the Yjump diffusion process , where 1tNtitW Yi, are constants, is a Lévy process which comes from a lin- ear transformation of two independent Lévy processes, i.e. a Brownian motion with drift and a compound Poi- sson process. Assume that a risk-neutral probability measure Q exists and all processes in Section 1 will be considered under this risk-neutral measure. In the Black-Scholes model, the price of a risky asset t under a risk-neutral measure Q and with non divi- dend payment follows S2001exp exp2tt tSSL SrtWt  (1.1) where is a risk-free interest rates, r is a vo- latility coefficient of the stock price. Instead of modeling the log returns 212ttLrt Wt  with a normal distribution. We now replace it with a more sophisticated process which is a Lévy process of the form tL21,2tt tLrtWt J t  (1.2) where tJand tdenotes a pure Lévy jump component, (i.e. a Lévy process with no Brownian motion part) and its convexity adjustment. We assume that the processes and tWtJare independent. To incorporate the volatile- ity effect to the model (1.2), we follow the technique of Carr and Wu  by subordinating a part of a standard Brownian motion 212tWt and a part of jump Lévy process tJt by the time integral of a mean reverting Cox Ingersoll Ross (CIR) process 0dttsTvs, where follows the CIR process tvd1d dvttvtvvtv tW (1.3) Here is a standard Brownian motion which corre- sponds to the process. The constant vtWtv is the rate at which the processreverts toward its long term mean and tv0v is the volatility coefficient of the process . tv 99S. PINKHAM ET AL.Hence, the model (1.2) has been changed to 21,2tttTtTLrt WT JT  t (1.4) and this new process is called a stochastic volatility Levy process. One can interpret t as the stochastic clock proc- ess with activity rate process t. By replacing t in (1.1) with t, we obtain a model of an underlying asset under the risk-neutral measure Q with stochastic volatility as follows: Tv LL201exp 2tttTtTSSrt WTJT  t(1.5) In this paper, we shall consider the problem of finding a formula for European call options based on the under- lying asset model (1.5) for which the constant interest rates r is replaced by the stochastic interest rates and ,trtJ is compound Poisson process, i.e. the model under our consideration is given by 201exp 2ttttTtTSSrt WTJT  t, (1.6) Here, we assume that rt follows the Vasicek process dddrtttrrrtW   (1.7) rtW0 is a standard Brownian motion with respect to the process t and . The constant rdddd 0rv rtt ttWW WW is the rate at which the interest rate reverts to- ward its long term mean, 0r is the volatility coeffi- cient of the interest rate process (1.7), The constant 0 is a speed reversion. 2. Literature Reviews Many financial engineering studies have been undertaken to modify and improve the Black-Scholes model. For ex- ample, The jump diffusion models of Merton , the sto- chastic Volatility jump diffusion model of Bates  and Yan and Hanson . Furthermore, the time change Lévy models proposed by Carr and Wu . The problem of option pricing under stochastic interest rates has been investigated for along time. Kim  con- structed the option pricing formula based on Black-Scholes model under several stochastic interest rate processes, i.e., Vasicek, CIR, Ho-Lee type. He found that by incur- porating stochastic interest rates into the Black-Scholes model, for a short maturity option, does not contribute to improvement in the performance of the original Black- Scholes’ pricing formula. Brigo and Mercurio  mention that the stochastic feature of interest rates has a stronger impact on the option price when pricing for a long ma- turity option. Carr and Wu  continue this study by giving the option pricing formula based on a time-changed Lévy process model. But they still use constant interest rates in the model. In this paper, we give an analysis on the option pricing model based on a time-changed Lévy process with sto- chastic interest rates. The rest of the paper is organized as follows. The dy- namics under the forward measure is described in Section 3. The option pricing formula is given in Section 4. Fi- nally, the close form solution for a European call option in terms of the characteristic function is given in Section 5. 3. The Ddynamics under the Forward Measure We begin by giving a brief review of the definition of a correlated Brownian motion and some of its properties (for more details one see Brummelhuis ). Recalling that a standard Brownian motion in is a stochastic nRprocess 0ttZ whose value at time t is simply a vector of n independent Brownian motions at t, 1, ,,,ttnZZZt We use Z instead of W since we would like to reserve the latter for the more general case of correlated Brownian motion, which will be defined as follows: Let 1,ij ij n be a (constant) positive symmetric matrix satisfying 1ii and 1ij 1  By Cholesky’s decomposition theorem, one can find an upper triangul nnmatrix ijh such that ,t where tΗ is the transpose of the matrix .Η Let 1, ,,,ttnZZZt be a standard Brownian motion as introduced above, we define a new vector-valued process 1, ,,,ttnWWWttZn by or in term of com-ponents, tW,,1, 1,,nitij jtjWhZi The process W0tt is called a correlated Brownian mo- tion with a (constant) correlation matrix . Each com- ponent process ,it tW0 is itself a standard Brownian motion. Note that if Id (the identity matrix) then t is a standard Brownian motion. For example, if we let a symmetric matrix W1010001vv (3.1) Then  has a Cholesky decomposition of the form THH where H is an upper triangular matrix of the form Copyright © 2011 SciRes. JMF S. PINKHAM ET AL. 100 2100100vvH01 Let ,,rvttttZZZZttWW be three independent Brownian motions then ,,rttWW defined by ttWZ, or in terms of components, 2 1,,vvrtvt vtttttWZZWZW Z r (3.2) Now let us turn to our problem. Note that, by Ito’s lemma, the model (1.6) has the dynamic given by dddeddd,d1d d,    tttYtttmtT tTrttrtvttvttSSrvtWSNrrtWvvtvW1d,(3.3) where , and e1tYmEdd dd0rrtt ttWW WWdd dvtt vWW t. We can re-write the dynamic (3.3) in terms of three independent Brownian motions ,,rtt tZZZfollows (3.2), we get 2dd 1 e1d,  ttvtttmttvt vtYtTSSrvtv dZZSNdd,tZ(3.4) ddrttrrrt   (3.5) d1d dvttvtvvtv ,tZ (3.6) This decomposition makes it easier to perform a measure transformation. In fact, for any fixed maturity T, let us denote by the T-forward measure, i.e. the probability measure that is defined by the Radon- Nikodym derivative, TQ0exp ddd0,TuTruQQPT (3.7) Here, is the price at time t of a zero-coupon bond with maturity and is defined as ,PtTT,eTstrdsQPtT EFt (3.8) Next, Consider a continuous-time economy where in- terest rates are stochastic and satisfy (3.5). Since the SDE (3.5) satisfies all the necessary conditions of Theorem 32, see Protter , then the solution of (3.5) has the Markov property. As a consequence, the zero coupon bond price at time t under the measure Q in (3.8) satisfies ,expdTQstPtTEr srt (3.9) Note that ,PtT depends on tonly instead of de- pending on all information available in Ft up to time t. As such, it becomes a function r,tFtr of , tr,,tPtT Ftr, meaning that the pricing problem can now be formulated as a search for the function ,tFtr . Lemma 1 The price of a zero coupon bond can be de- rived by computing the expectation (3.9). We obtain ,exp,,tPtTatTbtT r (3.10) where 1,eTttbtr1, 22223 32232 223,e44e2TtrrTtrratTTt       Proof. See Privault  (pp. 38-39). Lemma 2 The process t following the dynamics in (3.5) can be written in the form rttrxwt, for each t (3.11) where the process tx satisfies 0ddd,rttrtxxtZx0 . (3.12) Moreover, the function w(t) is deterministic and well defined in the time interval [0,T] which satisfied 0e1etwt rt (3.13) In particular, 00wr. Proof. To solve the solution of SDE (3.5), Let ,tgtre rand using Ito’s Lemma 2221ddd d2gg g,gtr rtr r  Then, dded = dedttttt trttrrterertrtZet Zdrt (3.14) Integrated on both side the above equation from 0 to t where 0tT and simplified, one get 00e1e edttutttrrr Zru By using the definition of form (3.13), wtCopyright © 2011 SciRes. JMF S. PINKHAM ET AL. Copyright © 2011 SciRes. JMF 101uHence, 00de1TTTu rruudxuZ  (3.22) 0edttu rtrrwtZ (3.15) where 0e1ettwt r. Note that the solution of (3.12) is Note that the solution of (3.12) is  000eedetttu tutrtrur 000eedetttu tutrtrurdru. druxxZ  Substituting (3.22) into (3.17), once get Zt. (3.16) 22200d dexp1 ed1 ed2TTTTu TurrruQQZu  Hence, for each t. The proof is now complete. trwt x(3.23) Next we shall calculate the Radon-Nikodym derivative as appear in (3.7). By Lemma 1 and 2, we have and . Substituting and ttrxwt0,PT tr0,PT into (3.7), we have The Girsanov theorem then implies that the three proc- esses ,rT vTttZZ and TtZdefined by 0022200exp dddexp 0,0, expd1d2TTuTTTuuxwuuQQaTbTrxueu dudd 1edd,ddTtrTr rttvTv TttttdZZtZZZZ  (3.24) are three independent Brownian motions under the meas- ure . Therefore, the dynamics of and under are given by TQ,ttrv tSTQ(3.17) 22ddd1 e1d,d1edd,d1d d.ttvT Ttttmtv tttvtYtTTt rTrtt rtvTttvttSSrvtvZvZSNrr tZvvtvZ     d(3.25) Stochastic integration by parts implies that 000ddTTTuT uxuTxuxTux  (3.18) By substituting the expression for from (3.12), udx 000 dddTuTTrurTuxTuxu TuZ u (3.19) Moreover by substituting the expression for uxfrom (3.16), the first integral on the right hand side of (3.19) becomes 4. The Pricing of a European Call Option on the Given Asset 00deddTuTuus rruoTuxuTuZ u  (3.20) Let 0,ttTS be the price of a financial asset modeled as a stochastic process on a filtered probability space ,, ,,TtFFQ tFis usually taken to be the price his- Using integral by parts, we have (Equation 3.21) Substituting (3.21) into (3.19), we obtain 00de1TTTu rruuTux Z d tory up to time t. All processes in this section will be de- fined in this space. We denote C the price at time t of a European u call option on the current price of an under- lying asset with strike price K and expiration time T. tS 00000 000 0000 0 eddede deddeded edededeedde1dTuus rruoTuTu usr usrvrs rsTT Tuurvv urru uTu TTuuvr rrruuTuZ uZTuuZTv vZTvv TvvZTv vZZ           0d.TrruTuZ (3.21) S. PINKHAM ET AL. 102 The terminal payoff of a European option on the un- derlying stock with strike pricetSKis max,0TSK (4.1) This means the holder will exercise his right only and then his gain is T. Otherwise, if TTSKKSKS then the holder will buy the underlying asset from the market and the value of the option is zero. We would like to find a formula for pricing a Euro- pean call option with strike price K and maturity T based on the model (3.25). Consider a continuous-time econ- omy where interest rates are stochastic and the price of the European call option at time t under the T-forward measure is TQ 0,,,;,,max,0 ,,(, )max,0, ,dTTtt tTtt tQTTtttTQCtS rvTKPtTESKS rvPtTSKpSSr vS where is the expectation with respect to the T-for- TQEward probability measure, is the corresponding con- TQpditional density given and P is a zero coupon bond which is defined in Lemma 1. ,,tt tSrvWith a change in variable ln,ttXS lnlnlnlnln,,,;,,maxe ,0,,,d,e1 ,,d= ,e,,d ,,,d1eee,, TTTTTTTTtTTTTtt tXTtttTQXXKTtttTQKXTtttTQKTttt TQKXXTtttTQX Ktt tQCtS rvTKPtTKpXX rvXPtTKpXX rvXPtTpXX rvXKPtTpXXrvXpXXrvvXESrv,,ln ,,,dTTtttTQKKPt TpXXrvX ln,,ed(e,,)TtTTTTtttQXXTXKtt tQpXXrveXESrvln,,TTtttTQK,dKPtTpXX rvX (4.2) With the first integrand in (4.2) being positive and in-tegrating up to one. The first integrand therefore defines a new probability measure that we denote by below TQqlnln,,,;,e,,d ,,tTTtt tXTtttTQKTtttTQKCtS rvTKqXXrvX,dKPtTpXX rvX 12eP, ,,;,, P,,,;,ePrln,, ,Prln,, ttXtt ttt tXTtttTttttXrvTKKPtTtXrvTKXKXrvKPt TXKXrv (4.3) where those probabilities in (4.3) are calculated under the probability measure . TQThe European call option for log asset price lnttXS will be denoted by 12ˆ,,,;, eP,,,;, e,P,,,;,tXtt ttt ttt tCtX rvTtX rvTPtTtX rvT(4.4) where lnK and  P,,,;, := P,,,;, , 1,2jtttjttttX rvTtX rvTKj. Note that we do not have a closed form solution for these probabilities. However, these probabilities are related to characteristic functions which have closed form solutions as will be seen in Lemma 4. The following lemma shows the relationship between and in the option value of (4.4). 1P2PLemma 3 The functions and in the option val- ues of (4.4) satisfy the PIDEs (4.5): 1P2Pand subject to the boundary condition at expiration t= T 1,,,;,1 .xPTxrvT (4.6)  Moreover, satisfies the Equation (4.7)2P 111110e1,,,;,,,,yvvPP ;,dAPvvPtxyrvTPxtrvTkyytv    (4.5)  222222222222(222220,2,,1, 1e2,,,;,,,,;,(e1) drTtrryPPPvvAP vbtTtx rxatT btTPrbtTPr bttPvPt xyrvTPt xrvTkyyx      ),tT (4.7) Copyright © 2011 SciRes. JMF 103S. PINKHAM ET AL. and subject to the boundary condition at expiration t = T 2,,,;,1xPTxrvT (4.8) where for i = 1,2 221222 2222221[] 1e21222,,,; ,(,,,; , )e1dTtiiriiv iiiirvvyiiiPPAP rvrxrPvPP PPvvvvvvxrPvPt xyrvTPxt rvTkyyx    x (4.9) Note that if 11xx and otherwise 10x. Proof. See Appendix A. 5. The Closed-Form Solution for European Call Options For j = 1,2 the characteristic function for , with respect to the variable ,,,;,jPtxrvT, are defined by  iuκ,,,;,:e d,,,;,,jjftxrvTuPtxrvT (5.1) with a minus sign to account for the negativity of the measure jdP. Note that jf also satisfies similar PIDEs ,,,;, 0,jjjfAf txrvTt (5.2) with the respective boundary conditions  ,,,;,e d,,,;, ede.iujjiu iuxfTxrvTuPtxrvTx   Since  d,,,;,1djxPtxrvT dxThe following lemma shows how to calculate the char- acteristic functions for and as they appeared in Lemma 3. 1P2PLemma 4 The functions and can be calculated by the inverse Fourier transformations of the character- istic function, i.e. 1P2P 0e,,,;,11,,,;, Red,2π iujjftxrvTuPtxrvT uiu for with Re[.] denoting the real component of a complex number. 1, 2,jBy letting Tt, the characteristic function jf is given by ,,,; ,expiux,jjjftxrvt uBrCvEj  where 21222 1,,2vjjjj jjjbbbbb ,  12 2222101, ,1ee12vv vviux yybiubiubiuuiukydy    22201ee1d2iux ybuiuiuk yy  111 121211 12111111111222222322223e ln22e1224e e34rrrrbbbbbBbb bbiu iu uu   22011e ,4jjj jjiuCbbb 22012201412411 2e1.2ejjjjjjbbbjjjbbbjj jbbEbb b 222 212221 2222121 221 22223222232222ln2242 1e)42e 14142rrrbbbbeBbb bbiuuiuuiuiuu iu2b   Proof. See Appendix B. In summary, we have just proved the following main theorem. Copyright © 2011 SciRes. JMF S. PINKHAM ET AL. 104 Theorem 5 The value of a European call option of SDE (3.25) is 12,,,;,,,,;,(,),,,;,tt ttttt tttCtS rvTKSPtX rvTKPtTPtXrvT where 1P and 2Pare given in Lemma 4 and ,PtT is given in Lemma 1. 6. Acknowledgements This research is (partially) supported by The Center of Excellent in Mathematics, the commission on Higher Education (CHE). Address: 272 Rama VI Road, Ratchathewi District, Bangkok, Thailand. 7. References  P. Carr and L. Wu, “Time Change Levy Processes and Option Pricing,” Journal of Financial Economics, Vol. 17, No. 1, 2004, pp. 113-141. doi:10.1016/S0304-405X(03)00171-5  R. C. Merton, “Option Pricing when Underlying Stock Returns are Discontinuous,” Journal of Financial Eco-nomics, Vol. 3, No. 1-2, 1976, pp. 125-144. doi:10.1016/0304-405X(76)90022-2  D. Bates, “Jump and Stochastic Volatility: Exchange Rate Processes Implicit in Deutche Mark in Option,” Review of Financial Studies, Vol. 9, No. 1, 1996, pp. 69-107. doi:10.1093/rfs/9.1.69  G. Yan and F. B. Hanson, “Option Pricing for Stochastic Volatility Jump Diffusion Model with Log Uniform Jump Amplitudes,” Proceeding American Control Conference, Minneapolis, 14-16 June 2006, pp. 2989-2994.  Y. J. Kim, “Option Pricing under Stochastic Interest rates: An Empirical Investigation,” Asia Pacific Financial Markets, Vol. 9, No. 1, 2001, pp. 23-44. doi:10.1023/A:1021155301176  D. Brigo and F. Mercuiro, “Interest Rate Models: Theory and Practice,” 2nd Edition, Springer, Berlin, 2001.  R. Brummelhuis, “Mathematical Method for Financial Engineering,” University of London, 2009. http://www.ems.bbk.ac.uk/for_students/msc./math_methods/lecture1.pdf  P.E. Plotter, “Stochastic Integration and Differential Equation,” Stochastic Modeling and Applied Probability, Vol. 21, 2nd Edition, Springer, Berlin, 2005.  N. Privault, “An Elementary Introduction to Stochastic Interest Rate Modeling,” Advance Series on Statistical Science & Applied Probability, Vol. 2, World Scientific, Singapore, 2008.  M. G. Kendall, A. Stuat and J. K. Ord, “Advance Theory of Statistics Vol. 1,” Halsted Press, New York, 1987. Copyright © 2011 SciRes. JMF S. PINKHAM ET AL. 105Appendix A: Proof of Lemma 3 By Ito’s lemma, follows the partial inte-gro-differential equation (PIDE) ˆ,,,Ctxrvˆˆˆ0,DJttCLC LCt (A.1) where 2222222 2222ˆˆ1ˆ1e2ˆˆˆ1222ˆˆ    TtDrtvrvvCCLC rvr2ˆxrvCCvCCvvvxrCvrCxv and ˆˆˆˆ,,,,,,e1 dJtyLCCvCtxyrvCtxrvk yyx where is the Lévy density. ()kyWe plan to substitute (4.4) into (A.1). Firstly, we compute   122121121222211122ˆee,, ,ˆee,,ˆee,,ˆee, ,,ˆe2exxxxxPPCPtTPatTbtTrtt ttPPCPPtTxx xPPCPtTvvvPPCPtT PbtTrrrPPCPPtxxx,       2222221222 2222112122222222,,ˆee,,ˆˆee,,ee, 2,,,.xxxPTxPPCPtTvv vPPPPCCPtTvxvxvvx rrPPPtTbtTPb tTrr  222112 2ˆee,,xPPP PCPtTbtT ,xrxrrxrx        11122ˆˆ,,,,;,,,,,;,ee1,,,;,,,,;,,,,;,e(,),,,;,,,,;, .xyCtxyrv TCtxrv TPtxyrvTPt xyrvTPxtrvTPtTPt xyrvTPt xrvT  Substitute all terms above into (A.1) and separate it by assumed independent terms of 1P and 2P. This gives two PIDEs for the T-forward probability for ,,,;, ,1,2:iPtxrvTj 2211 1222222111122211111e21222,,,;,,,,;,e1d.eTtrvrvv vvyyPP Prv rtx rvPPPPvvvvx vvxrPvPtxyrvTPxtrvTkyyxv     1Pv  111,,,;,,,,;,d.0Ptx yrvTPxtrvTky y  (A.2) and subject to the boundary condition at the expiration time t = T according to (4.6). By using the notation in (4.9), then (A.2) becomes Equation (A.3) 111111110e1,,,;,,,, :.yvvPP ;,d.APvvPtxyrvTPxtrvTkyytvPAPt    (A.3) Copyright © 2011 SciRes. JMF S. PINKHAM ET AL. 106 For 2(,, , ;,):PtxrvT  22222222 22222222222222 222101 ,22222,,12, 1e2vrrvvTt Ttrr rvPPPPPPPvrvv vbtTrPtxvvxxvratT btTPre btTPrrrtt         222,,,,;,,,,;,e1d.ybtTPvP txyrvTPtxrvTkyyx  Again, by using the notation (4.9), then (A.4) becomes (A.4) and subject to the boundary condition at expiration time t = T according to (4.8). 2 2222222222222222,,0,221e (,):rrTtratT btTPPPPv21,APvbtT PrbtTtx rttxPPrbtT APt      (A.5)The proof is now completed. Copyright © 2011 SciRes. JMF 107S. PINKHAM ET AL. Appendix B: Proof of Lemma 4 To solve the characteristic function explicitly, letting Tt1f be the time-to-go, we conjecture that the func-tion is given by   1111,,,;,exp iux,ftxrvtuBrCvE (B.1) and the boundary condition 111000BCE0. This conjecture exploits the linearity of the coefficient in PIDEs (5.2). Note that the characteristic function of 1 always exists. In order to substitute (B.1) into (5.2), firstly, we compute f  11111,,fBrCvEfiuft    2211 111 1112222211111 122221111 11111,,,,,,e(,,,; ,)(,,,;,)(,,,; ,)iuxfffCf EfufrvxffEfCfvrffiuC fiuE fxr vxftxrvt uftx yrvtuftxrvtu   , 11fx Substituting all the above terms into (5.2), after can- celling the common factor of 1f, we get a simplified form as follows:      11222211122211110 1ee1 e1ed22+1e2iuxyy iuxvvvTtrrrCiu CvEiuEEiuuiuk yyBCCE         By separating the order, r, v and ordering the re- maining terms, we can reduce it to three ordinary differ- ential equations (ODEs) as follows: 11()() ,CCiu (B.2)   2211 12212 ee1d,2vvviux yyEE iuEiuuiukyy (B.3)  2221111C1e .2TtrrBCE  (B.4) It is clear from (B.2) andthat (0)0C11e ,iuC (B.5) Let 21,2vb 21,vvbiu   2201ee12iux yybiuuiuky dy and substitute all term above into (B.3). we get  2222 0122 01111 1114422 bb bbbb bbEbE Ebb    By method of variable separation, we have  112222 0122 011111dd4422Ebbb bbbbbbEEbb   Using partial fraction on the left hand side, we get Copyright © 2011 SciRes. JMF S. PINKHAM ET AL. 108 122111111dd()22EbbEEbb where 2204bbb1. Integrating both sides, we have 21102112ln2bEbEbEb Using boundary condition1(0)0E we get 202ln bEb Solving for1()E, we obtain 12111 2e12ebbEbb b (B.6) where . 12 22,bb bbIn order to solve1()Bexplicitly, we substitute 1()C and 1()Ein (B.5) and (B.6) into (B.4) . 22212222122211 2'1eee112ee22errriu iuiuBbbubb bIntegrating with respect toand using boundary con- dition 1(0)B0, then we get  2221222222331221211 12 2e14ee 324eln22rrrrBiu uiu ubbbbbbb bb    The details of the proof for the characteristic function 2fare similar to1f. Hence, we have   2222,,,;,expftxrvT uiux BrCvE where 22,BCand 2()Eare as given in this Lem- ma. We can thus evaluate the characteristic function in close form. However, we are interested in the probabil- ity jP. These can be inverted from the characteristic functions by performing the following integration iuκ0,,,;,e,,.;,11Re d2jjPtxrvTftxvrTuuiu  e   for 1, 2j where lnttXS and lnK, see Ken- dall et al. . The proof is now complete. Copyright © 2011 SciRes. JMF