SENSITIVITY ESTIMATES FROM CHARACTERISTIC FUNCTIONS

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1 Proceedings of te 27 Winter Simulation Conference S. G. Henderson, B. Biller, M.-H. Hsie, J. Sortle, J. D. Tew, and R. R. Barton, eds. SENSITIVITY ESTIMATES FROM CHARACTERISTIC FUNCTIONS Paul Glasserman Columbia Business Scool Uris Hall, 322 Broadway, Room 11 New York, N.Y. 127, U.S.A. Zongjian Liu Department of IE & OR, Columbia University S.W. Mudd Building, 5 West 12t Street, Room 313A New York, N.Y. 127, U.S.A. ABSTRACT We investigate te application of te likeliood ratio metod LRM for sensitivity estimation wen te relevant density for te underlying model is known only troug its caracteristic function or Laplace transform. Tis problem arises in financial applications, were sensitivities are used for managing risk and were a substantial class of models ave transition densities known only troug teir transforms. We quantify various sources of errors arising wen numerical transform inversion is used to sample troug te caracteristic function and to evaluate te density and its derivative, as required in LRM. Tis analysis provides guidance for setting parameters in te metod to accelerate convergence. 1 INTRODUCTION Stocastic simulation is used widely in te financial industry for te pricing and edging of options and oter derivative securities. Under standard conditions, te price of a derivative security can be represented as te epectation of its discounted payoff. A typical pricing simulation involves simulating pats of te underlying asset or assets, evaluating te discounted payoff on eac pat, and averaging over te pats. Suc simulations are often used as muc for edging as for pricing, and edging requires calculation of sensitivities of prices wit respect to model parameters, including te initial values of te underlying assets. For sensitivity calculations, te likeliood ratio metod LRM or score function metd is attractive wen te payoff is discontinuous in te parameters. To fi ideas, let VX denote a discounted payoff, wic is a function of te random variable X, and suppose X as a density g θ depending on a parameter θ. Te key LRM identity is d dθ E θ[vx] = E θ [VXġθX ], 1 g θ X wit E θ denoting epectation wit respect to g θ, and ġ θ denoting te derivative of g θ wit respect to te parameter θ. Wen tis identity olds as it does under mild regularity conditions, te epression inside te epectation on te rigt provides an unbiased estimator of te sensitivity on te left. We will write tis estimator as VXS θ X wit S θ = ġ θ /g θ te score function. Te application of 1 requires evaluation of te density g θ and its derivative, and tis can limit te scope of te metod. Here we investigate te application of LRM wen te density is not eplicitly available but is known troug its caracteristic function or troug its Laplace transform. Tis problem arises for broad classes of models used in financial applications, including models driven by Lévy processes see, e.g., Cont and Tankov 24 and te affine class of jump-diffusion models studied in Duffie et al. 2. An eample of a Lévy-driven model is one tat models te price of te underlying asset troug a process S T = S epat X T, in wic X T is te time-t value of a Lévy process wit X =, and S and a are constants. A Lévy process as stationary independent increments, so its increments ave infinitely divisible distributions; suc distributions are often specified troug teir caracteristic functions, via te Lévy-Kincine formula as in, e.g., Sato 1999, p.37. An etensively studied case of a Lévy-driven model is te Variance Gamma model; in te notation of Madan, Carr and Cang 1998, wit parameters ρ, ν, and θ, te Laplace transform of X T is given by L vg t = E[e tx T 1 T/ν ] = 1θνt ρ 2 νt 2 2 /2 for t in a neigborood of te origin. Tere is no closed-form epression for te density of X T. We analyze a metod in wic numerical transform inversion te Fourier series metod of Abate and Witt /7/$ IEEE 932

2 1992 is used bot to sample troug a Laplace transform or caracteristic function and to compute an LRM estimator. We quantify various sources of errors in order to provide guidance for setting parameters to accelerate convergence. Tere are general metods for sampling from transforms see Devroye 1981 and specific metods for specific distributions tat do not require numerical inversion, but tese do not address te problem of evaluating te score function. In separate work, we investigate alternatives to numerical transform inversion based on approimations to te score function; a side benefit of te metod we discuss ere is tat it can serve as a bencmark for approimations. Te rest of tis paper is organized as follows. In Section 2, we specify a sampling metod in wic we precompute a table of values of te cumulative distribution function CDF; tis involves discretization and truncation of te domain of te CDF. In Section 3, we review te metod of Abate and Witt 1992 and discuss its application to our problem. Section 4 summarizes te error in calculating prices, and Section 5 does te same for price sensitivities. We illustrate te results numerically in Section 6. We outline a proof of our error analysis in an appendi; complete proofs of all our results will be provided in a full-lengt article. 2 OUTLINE OF THE METHOD For simplicity, we limit our discussion to scalar X in 1. Let G θ denote te CDF associated wit g θ. Our first task is to sample X from G θ wen te distribution is known only troug a transform. We will accomplis tis by tabulating values of G θ calculated troug numerical transform inversion, and ten using te table to generate samples. We could restrict ourselves to working wit te caracteristic function; tere is little practical difference between sifting te integration contour to invert a caracteristic function and working directly wit te Laplace transform in te comple plane, so we present te inversion steps using te latter. Te two-sided Laplace transform of a function f is given by L f t = e t fd, were t = σ iω is a comple variable. Tis transform is two-sided because te lower limit of integration is rater tan zero. For background on two-sided Laplace transforms, see Widder 1941, Capter VI. For te transform L gθ of g θ, we suppose tat te region of convergence includes an interval σ l,σ u, were σ l < and σ u >. By Widder 1941, p.242, Teorem 5b, we ave L Gθ t = L gθ t/t for Ret,σ u, and we ave LḠθ t = L gθ /t for Ret σ l, and Ḡ θ = 1 G θ. Under mild condition on g θ, Lġθ = = θ e t θ g θd e t g θ d = θ L g θ. 3 We assume tat te region of convergence of Lġθ also includes σ l,σ u. Using numerical transform inversion, we can approimate te value of G θ at any. We will build an approimation Ĝ θ to te function G θ by inverting te transform at a fied set of values and interpolating between tese values. In more detail, we calculate Ĝ θ as follows: 1. Pick a grid on te -ais: j, j J} were J is a finite inde set, j j 1 = δ for j J. Let j min = min j J} and = jmin. Define j ma and ma accordingly. 2. Let G j denote te approimation to G θ j calculated troug numerical transform inversion. Set G min G jmin and G ma G jma For any [ j 1, j ], use piecewise linear interpolation to get Ĝ θ : Ĝ θ = j 1 δ G j j G j 1. 4 δ 4. For <, let Ĝ θ = ; for > ma, let Ĝ θ = 1. We defer te selection of δ, and ma for later discussion. To ensure tat Ĝ θ is monotone increasing, we require G j G j 1, for all j J. Wile tis is not automatically guaranteed because of numerical error in transform inversion, we will enforce tis property in te metod of te net section. We denote by ˆX a random variable wit distribution Ĝ θ. Te density of ˆX is denoted by ĝ θ and equals dĝ θ /d, wic is a piecewise constant function: G ĝ θ = j G j 1 /δ, if [ j 1, j, j J;, if < or > ma. 5 We sample from Ĝ θ as follows: 1. Generate U U[, 1]. 2. Find te inde j suc tat G j 1 U < G j. 3. Set ˆX = Uδ j 1G j j G j 1 G j G j 1. 6 By sampling from Ĝ θ, we can estimate E θ [V ˆX], wit E θ indicating tat ˆX Ĝ θ. In order to estimate te sensitivity E θ [V ˆXŜ θ ˆX], were Ŝ θ = ĝ θ /ĝ θ and 933

3 ĝ θ = ĝ θ / θ, we compute ĝ θ as follows: ĝ θ = Ġ j Ġ j 1 /δ, if [ j 1, j, j J;, if < or > ma, 7 were Ġ j Ġ θ j is calculated troug numerical inversion of te transform of Ġ θ. So, as we compute eac G j to construct te approimation Ĝ θ, we also compute Ġ j in order to be able to evaluate ĝ θ. Once tese values are computed and stored, sampling is easy and fast, so te key question is te quality of te approimation; i.e., te difference between E θ [V ˆX] and E θ [VX], and te difference between E θ [V ˆXŜ θ ˆX] and E θ [VXS θ X]. Tese differences ave several sources, including numerical errors in transform inversion and discretization errors in te approimation Ĝ θ. In te net section, we discuss transform inversion and te associated error analysis. 3 THE FOURIER-SERIES METHOD FOR LAPLACE INVERSION Abate and Witt 1992 defined and analyzed a Fourierseries inversion formula for te one-sided Laplace transform, and we follow teir approac. Etending it to te two-sided case see Cai, Kou and Liu 27 yields, for a function f and its two-sided Laplace transform L f, f = eσ [ Re[L f σ iω]cosω π Im[L f σ iω]sinω ] dω. 8 We abbreviate tis formula as f = I L f. Employing te trapezoidal rule to numerically evaluate te infinite integral in 8 wit a step size gives Iσ,L f = eσ 2π L fσ eσ π [ Re[L f σ ik]cosk Im[L f σ ik]sink ], 9 were σ can be any point in σ l,σ u and can be cosen to depend on. As in Abate and Witt 1992, we truncate te infinite sum in 9; let Iσ, L f denote te truncation of te series in 9 to te first N terms. We call T p = N te truncation point. Applying te Fourier-series metod to L gθ, we obtain I σ,l gθ and σ, L gθ. Te discretization error at resulting from step size is e d σ = I σ,l gθ g θ ; we can sow tat e d σ see Appendi A. Te truncation error is e t σ = σ, L gθ I σ,l gθ. Tus, Iσ, L gθ = g θ e d σ e t σ. Likewise, we define ė d σ = I σ,lġθ ġ θ and ė t σ = Iσ, Lġθ I σ,lġθ. We will apply te Fourier-series metod in a way tat ensures monotonicity of G j, j J, and ensures tat G jmin approaces and G jma approaces 1 as jmin and jma approac and, respectively. First, we make te following observation about te beavior of te inversion metod at etreme values of : Proposition 1 For any σ,σ u, for any σ σ l,, σ, L Gθ as, σ, L Gθ as ; σ, LḠθ as, σ, LḠθ as. Proof: By looking at te formula of Iσ, L Gθ and Iσ, LḠθ, we ave Iσ, L Gθ = Oe σ and Iσ, LḠθ = Oe σ, wic yields te conclusion. From tis result we see tat, in order for te G j to approac and 1 at etreme values of jmin and jma, we can pick σ,σ u and σ σ l,, and let G j = σ, j L Gθ, if j ; 1 σ, j LḠθ, if j >. 1 For te monotonicity of te G j, we will use te following property of te Fourier-series metod, wic can be verified by direct differentiation: Proposition 2 Let f be a density wit CDF F. Suppose te interval σ 1,σ 2 is witin te region of convergence of L F and L f, were σ 1 < and σ 2 >. Ten for any σ,σ 2, d d I σ, L F = Iσ, L f. 11 Similarly, if F is te complementary CDF, ten for any σ σ 1,, d d I σ, L F = Iσ, L f

4 Because σ, L gθ = g θ e d σe t σ and e d σ, we may conclude tat σ, L gθ is nonnegative for all sufficiently large N, at any point at wic g θ is strictly positive. From Proposition 2, we see tat nonnegativity of σ, L gθ implies monotonicity of σ, L Gθ and σ, LḠθ. In practice, we do not know ow large N needs to be, so we apply te following rule: if it appens tat G j < G j 1 for some j, we simply let G j = G j 1 to make G j, j J a monotonically increasing sequence. Te steps we use to construct te sequence G j are as follows: 1. Let = E θ [X] = L g θ. We start from in constructing our grid. Compute G by 1. For tis value we use a very large truncation point to get an accurate value for G. 2. Let j = jδ and and j = jδ. Compute G ± j by 1. After getting G j and G j, we adjust teir values by te following rule: If G j < G j 1 ten set G j = G j 1 ; if G j > G j 1 ten set G j = G j We continue for j = 1,2,... until we find j ma > and j min < suc tat G jma 1 or ma jma is large enoug, and G jmin or jmin is large enoug in te negative direction. We will eplain ow to determine te magnitude of ma and in te net section. We ten set J = j min, j min 1,..., j ma 1, j ma } and use j, j J} as our grid. In te net two sections, we discuss te errors in estimating prices and sensitivities using te Fourier-series metod. It will be important to keep in mind tat we use σ σ l, in computing values at >, and we use σ,σ u for all <. 4 ERROR ANALYSIS FOR PRICES In tis section, we analyze te error in estimating a price, i.e., te difference between E θ [V ˆX] and E θ [VX]. For simplicity, we let I σ L gθ =,L gθ, if ; σ,l gθ, if >, 13 and let e d = 1 > }e d σ 1 }e d σ and e t = 1 > }e t σ 1 }e t σ, were 1 } is te indicator function. We can decompose te error using Eθ [V ˆX] E θ [VX] = Vĝ θ d Vg θ d Vĝ θ d V VI L gθ d 14 L gθ g θ d. 15 We will analyze 15 first, and ten turn to 14. Note tat = I L gθ g θ d Ve d e t d Ve d d Ve t d 16 V In order to bound te error, we need to impose some conditions. Our condition on L gθ is te following: Assumption 1 For any σ in σ l,σ u, as ω, and Re[L gθ σ iλω] = Oλ α R Re[L gθ σ iω] Im[L gθ σ iλω] = Oλ α I Im[L gθ σ iω] uniformly in λ 1, for some α R > 1 and α I > 1. Tis assumption is not very restrictive. For eample, it olds if Re[L gθ σ iω] and Im[L gθ σ iω] are regularly varying functions of ω wit negative indices, or if logre[l gθ σ iω] and logim[l gθ σ iω] are regularly varying functions wit positive indices. See, e.g., Bingam, Goldie and Teugels 1987 for background on regular variation. We impose te following condition on te payoff function V : Assumption 2 For >, V C v e v, and for <, V C v e v, for some constants C v >, v, σ l, and v σ u,. Tis assumption is more tan sufficient to ensure tat E θ [VX] eists, and it is satisfied by many standard option payoffs. For fied σ σ l, and σ,σ u, let M ± T p = L gθ σ ± it p. We now ave te following: Teorem 1 Under Assumptions 1 and 2, we can find σ σ l, and σ,σ u suc tat Ve d d = Oe C/, 935

5 for some constant C >, and Ve t d = OmaM T p,m T p }. Proof: See Appendi A. Troug 16, tis result determines te order of 14. We turn net to 15 and decompose tis error term as min Vĝ θ d VI VI L gθ d L gθ d V L gθ d ma Vĝ θ I L gθ d For te last term, we ave te following result: Lemma 1 If V is continuous on te interval [ j 1, j ], ten j Vĝ θ I j 1 If furtermore V is differentiable, ten j Vĝ θ I j 1 L gθ d = Oδ 2. L gθ d = Oδ 3. Troug tis lemma, we arrive at te following result: Teorem 2 If V is differentiable almost everywere, ten Vĝ θ I L gθ d = Oδ 2, and tere are positive constant C min and C ma for wic and min V VI ma L gθ d = Oe C min, L gθ d = Oe C ma ma. Proof: Given Lemma 1, we only need to establis te two tail errors. Since I L gθ = Oe σ wen and L gθ = Oe σ wen, te result follows. Tis result indicates tat we can set and ma large enoug in absolute value to make min V L gθ d and negligible compared to VI ma Vĝ L gθ d L gθ d. Wit tis specification, we can combine Teorems 1 and 2 to quantify te pricing error: Corollary 3 Under te foregoing conditions, Eθ [V ˆX] E θ [VX] = Oδ 2 Oe C/ OmaM T p,m T p }. 5 ERROR ANALYSIS FOR SENSITIVITIES In tis section, we analyze te error in estimating te sensitivity, i.e., E θ [V ˆXŜ θ ˆX] E θ [VXS θ X]. Muc as in te previous section, we define I σ Lġθ =,Lġθ if σ,lġθ if >, and we let ė d = 1 > }ė d σ 1 }ė d σ and ė t = 1 > }ė t σ 1 }ė t σ. We bound te error in te sensitivity estimate as Eθ [V ˆXŜ θ ˆX] E θ [VXS θ X] = V ĝ θ d Vġ θ d V ĝ θ d V Lġθ d VI Lġθ ġ θ d. Te form of tis bound is very similar to tat used for te error in te price estimate, but now wit derivatives of g θ. We require tat ġ θ d <, and muc as in Assumption 1, we impose Assumption 3 For any σ in σ l,σ u, as ω, Re[Lġθ σ iλω] = Oλ α R Re[Lġθ σ iω] 936

6 and Im[Lġθ σ iλω] = Oλ α I Im[Lġθ σ iω] uniformly in λ 1, for some α R > 1 and α I > 1. For fied σ and σ, let Ṁ ± T p = Lġθ σ ± it p. Wit tese assumptions and definitions, te analysis in te previous section goes troug wit appropriate modification, leading to te following result: Teorem 4 Under Assumptions 2 and 3, using te same σ and σ as in Teorem 1, V ė d d = Oe Ċ/, for some positive constant Ċ, and Vė t d = OmaṀ T p,ṁ T p }. 6 A NUMERICAL EXAMPLE In te previous sections, we ave focused on te bias in estimating prices and sensitivities. As a measure of overall simulation error, we use mean square error MSE, wic is te sum of te squared bias and te estimator variance. If we use N s simulation trials, ten te MSE for te price estimate is MSE price = OmaM T p,m T p } Oe C/ Oδ 2 Var price N s, and for te sensitivity, te MSE is MSE sen = OmaṀ T p,ṁ T p } Oe Ċ/ Oδ 2 2 Var sen N s, were Var price and Var sen denote te variance per replication of te price estimate and sensitivity estimate, respectively. Several factors affect te two MSEs, including te truncation parameter T p, te step size, te grid parameter δ, and te number of pats N s. To make eac MSE converge to, we need to cange all of tese factors simultaneously, and, for efficiency, we sould do so at rates consistent wit 2 teir impact on te MSE. In tis section, we use te Variance Gamma VG model as in, e.g., Madan, Carr and Cang 1998 to illustrate ow to cange te values of te factors appropriately based on te error analysis. Te function we use is te discounted payoff for a European call option, VX = e rt mas T K,, were T is te maturity of te option and S T follows formula 22 in Madan, Carr and Cang 1998, in wic S T = S epat X T, X is a VG process, and a = r 1 ν log1 θν ρ2 ν/2, 17 wit r a constant interest rate and ρ, ν, and θ parameters of te model. Te Laplace transform of X T appears in 2. Te region of convergence of te Laplace transform is te vertical strip in te comple plane tat intersects te real ais on te interval θν θ 2 ν 2 2ρ 2 ν ρ 2, θν θ 2 ν 2 2ρ 2 ν ν ρ 2 ν For any σ in tis interval, L vg σ iω as a power decay as ω wit rate 2T/ν. Terefore, te MSE for te price in VG model is MSE price,vg = OTp 2T/ν 2 Oe C/ Oδ 2 Var price. 18 N s To reduce te MSE, we need to increase T p, decrease, decrease δ, and increase N s. Te purpose of our error analysis is to guide te allocation of computational effort. We increase or decrease tese parameters to equate te magnitude of te error reduction in eac source of error. From 18, we see tat if T p increases by a factor of 1, ten sould decrease by a factor of Cν/2T log1, δ sould decrease by a factor of 1 T/ν, and te number of replications sould increase by a factor of 1 4T/ν. Our coice of C is specified in te proof of Teorem 1 in te Appendi. Wit tese canges, te RMSE te square root of te MSE for te price estimate sould decrease by a factor of 1 2T/ν. Te rate of decrease of te RMSE is constrained by te slowest rate in 18; if we were to cange te parameters T p,, δ, and N s witout equating te overall rates of decrease in te corresponding error terms, we would be allocating too muc computational effort to some parts of te algoritm, insufficient effort to oters. All of tese statements sould be understood in te big-o sense provided by our results.. 937

7 In our eamples, we use te following values for te VG process and te call option payoff: S = 1 K = 1 r =.5 T = 1 ρ =.2 θ =.15 We compare results at ν = 1 and ν =.5. Using te formula in Madan, Carr and Cang 1998 for te prices of European call options, we get te values in Table 1, against wic we compare te simulation estimates. Table 1: European call prices for VG model ν Call Price To test our sensitivity estimates, we calculate sensitivities wit respect to te model parameter ρ and te initial price S of te underlying asset. By applying finite difference approimations to te formula for option prices, we get te derivative values in Table 2. Table 2: Derivatives for VG model Parameter Derivatives ν = 1 ν =.5 S ρ To apply LRM, we need to move te dependence on S and ρ into te density; recall from 17 tat a is a function of ρ. We terefore work wit te random variable logs at X T, wose Laplace transform is S t ep attl vgt. For te parameter S, te Laplace transform of te partial derivative is tl vg t/s ; for te parameter ρ, te Laplace transform of te derivative is S t ep attl vgt/ ρ. In bot cases, te sensitivity MSE is MSE sen,vg = OTp 2T/ν1 2 Oe C/ Oδ 2 Var sen.19 N s Te impact of te truncation point T p in te sensitivity MSE 19 differs from tat in te price MSE 18 and results in a slower overall rate of convergence. For eample, wit ν = 1, we get 2T/ν = 2, so te optimal RMSE for te price is OTp 2 wereas for te sensitivity it is OTp 1. Tus, to decrease te price RMSE by a factor of 1, we increase te truncation point by a factor of 1, but to decrease te sensitivity RMSE by a factor of 1 we increase te truncation point by a factor of 1. A similar comparison applies wit ν =.5. In eac case, we also cange, δ and N s consistent wit 19 and 18. Table 3 sows numerical results for price estimates wit ν = 1. From eac row to te net, we multiply T p by 1 and cange te oter parameters at te corresponding rates. Te initial values are set somewat arbitrarily by equating = δ 2 = e C/. In te Error column, we report te difference between te simulation mean and te formula price. In general agreement wit our analysis, te error decreases by rougly a factor of 1 from eac row to te net. In order to get reliable estimates for our comparison, we use a larger number of replications tan would be optimal under our analysis. In practice, we would try to set te value of N s to make te standard error approimately equal to te bias. Tp 2 Table 3: Results for prices, wit ν = E E E Tables 4 and 5 sow numerical results for te sensitivities wit ν = 1. Te error decreases by approimately 1 from one row to te net, in line wit our analysis. Table 4: Results for sensitivities to S, wit ν = E E E Table 5: Results for sensitivities to ρ, wit ν = E E E Tables 6, 7 and 8 sow numerical results for ν =.5. In tis case, te modulus of te Laplace transform decays more quickly, so we start wit a smaller value of T p and increase it by a factor of 4 1 from one row to te net. Tis sould decrease te price error by a factor of 1 and te sensitivity error by a factor of 1 3/4 5.6 in eac case. Te results in te tables are rougly in line wit tese predictions, toug te convergence in Table 6 is a bit slower tan epected. 7 SUMMARY We ave proposed and tested a metod for estimating price sensitivities by simulation using te likeliood ratio metod wen te underlying density is known only troug its caracteristic function or Laplace transform. Te metod 938

8 Table 6: Results for prices, wit ν = E E E Table 7: Results for sensitivities to S, wit ν = E E E Table 8: Results for sensitivities to ρ, wit ν = E E E uses numerical transform inversion and incurs several types of error; we ave presented results on te convergence rates of tese errors and illustrated tese results in te Variance Gamma model. In tis eample, te main determinant of te overall convergence rate is te truncation point used in te transform inversion, and tis parameter results in slower convergence of sensitivity estimates tan of price estimates. A THEOREM 1: SKETCH OF PROOF In proving te first statement in te teorem, we use te Poisson summation formula in Abate and Witt 1992, Section 5, and get e d σ = k=,k 2πσk ep g θ 2πk. 2 Because g θ is nonnegative, e d σ, and e d σ = if and only if g θ 2πk 1 = for all nonzero k. To simplify notation, we now write g and L instead of g θ and L gθ. Because L is finite on σ l,σ u, for any σ in tis interval we ave g < e σ for all sufficiently large. In particular, we can coose ε > sufficiently small to ave σ l,ε σ l ε < v and σ u,ε σ u ε > v, and ten ave g < e σ l,ε, for all sufficiently large, and g < e σ u,ε for all sufficiently large. Now suppose > and take σ = σ, a negative number. For sufficiently small >, 2 gives e d σ 2πσ k ep 1 k= 2πσ k ep Te first term is less tan or equal to σ l,ε 2πk g 2πk 2ep σ l,ε 2πσ σ l,ε, if is sufficiently small. It follows tat Ve d σ d [ C v 2e σ l,εv e 2πσ σ l,ε 1 e 2πσ k e 2πv k k=. 21 ] e v 2πk/ g 2πk d Te first term on te rigt can be integrated to get 2C v ep 2πσ σ l,ε. σ l,ε v We bound te second term by 1 C v e 2πσ vk k= C v L v e v 2πk/ g 2πk d 1 e 2πσ vk k= 2C v L v e 2πσ v. So, if we let σ = σ l v /2, ten Ve d σ d = Oe C 1/, wit C 1 = πσ l v. By a similar argument, wen wen <, we can let σ = σ u v /2 and C 2 = πσ u v to get Ve d σ d = Oe C 2/. Setting C = minc 1,C 2 } concludes te proof of te first statement in te teorem. 939

9 To illustrate te argument for te second part of te teorem, we simplify to V 1. Note tat, using Assumption 1, e t σ d π Re[Lσ it p ik] e σ coskt p d Im[Lσ it p ik] e σ sinkt p d Re[Lσ π it p 1 k N ] σ σ 2 kt p 2 Im[Lσ it p 1 k N ] kt p σ 2 kt p 2 Re[Lσ π it p ] k σ 1 N α R σ 2 kt p 2 Im[Lσ it p ] 1 k N α I π Lσ it p 1 k N α I = O Lσ it p kt p σ 2 kt p 2 1 k σ N α R σ 2 kt p 2 kt p σ 2 kt p 2 Similarly, te integral from to is O Lσ it p. Tis conclusion continues to old for V satisfying Assumption 2. Duffie, D., J. Pan, and K. Singleton. 2. Transform analysis and option pricing for affine jump-diffusions. Econometrica 68: Madan, D. and P. Carr and E. Cang Te variance gamma process and option pricing. European Finance Review 2: Sato, K.-I Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge, UK. Widder, D.V Te Laplace Transform. Princeton University Press. AUTHOR BIOGRAPHIES PAUL GLASSERMAN is te Jack R. Anderson Professor in te Decision, Risk, and Operations Division of Columbia Business Scool. His researc interests include modeling and computational problems in risk management and te pricing of derivative securities. He is autor of te book Monte Carlo Metods in Financial Engineering Springer, 24. His address is pg2@columbia.edu and is web page is ZONGJIAN LIU is a PD candidate in te Department of Industrial Engineering and Operations Researc at Columbia University. His researc interests include security pricing and portfolio management. His address is zl2115@ columbia.edu and is web page is ttp://www. columbia.edu/ zl2115/. ACKNOWLEDGMENTS Tis researc is supported in part by NSF grants DMS7463 and DMI344. REFERENCES Abate, J. and W. Witt Te Fourier-series metod for inverting transforms of probability distributions. Queueing Systems: Teory and Applications 1:5 88. Bingam, N.H., C.M. Goldie and J.L. Teugels Regular Variation. Cambridge University Press, Cambridge, UK. Cai, N., S.G. Kou and Z. Liu. 27. Manuscript in preparation. Cont, R., and P. Tankov. 24. Financial Modelling wit Jump Processes, Capman & Hall/CRC, Boca Raton, Florida. Devroye, L On te computer generation of random variables wit a given caracteristic function. Computers and Matematics wit Applications 7: