Remark_NN1 ._SENT Adding_VVG to_PRP (_PUL 5.9_CRD )_PUR the_AT0 f0_NN1 (_PUL x_ZZ0 )_PUR from_PRP (_PUL 5.2_CRD )_PUR ,_PUN we_PNP see_VVB that_CJT asymptotic_AJ0 expression_NN1 for_PRP density_NN1 fA_NN1 (_PUL x_ZZ0 )_PUR (_PUL for_PRP Δt≪1_CRD and_CJC x≫1_CRD )_PUR satisfies_VVZ bounds_NN2 obtained_VVN from_PRP (_PUL 5.10_CRD )_PUR so_AV0 that_DT0 fA_NN1 (_PUL x_ZZ0 )_PUR ∼O_NP0 (_PUL 1_CRD /_PUN x2_CRD )_PUR for_PRP x≫1_CRD ._SENT 
6._CRD The_AT0 density_NN1 with_PRP arbitrary_AJ0 correlation_NN1 between_PRP the_AT0 number_NN1 of_PRF jumps_NN2 in_PRP the_AT0 supply_NN1 and_CJC demand_NN1 
In_PRP this_DT0 section_NN1 we_PNP allow_VVB for_PRP the_AT0 possibility_NN1 that_CJT the_AT0 number_NN1 of_PRF jumps_NN2 ,_PUN ΔNt_NN2 in_PRP the_AT0 demand_NN1 (_PUL i.e._AV0 ,_PUN numerator_NN1 )_PUR has_VHZ an_AT0 arbitrary_AJ0 correlation_NN1 ,_PUN ρJ_NP0 ,_PUN with_PRP the_AT0 supply_NN1 (_PUL i.e._AV0 ,_PUN denominator_NN1 )_PUR ._SENT For_PRP positive_AJ0 reals_NN2 λ01_CRD ,_PUN λ02_CRD ,_PUN λ12_CRD and_CJC k_ZZ0 ,_PUN l∈ℕ_NN1 ,_PUN we_PNP write_VVB the_AT0 bivariate_AJ0 Poisson_NP0 probability_NN1 (_PUL see_VVB [_PUL 35_CRD ]_PUR )_PUR as_PRP equation_NN1 (_PUL 6.1_CRD )_PUR where_CJS 𝔼_NP0 [_PUL ΔNt_NP0 ]_PUR =_NP0 Var_NP0 [_PUL ΔNt_NP0 ]_PUR =_NP0 λ01+λ12_NP0 =_NP0 :_PUN λ1_CRD ,_PUN 𝔼_NP0 [_PUL ΔÑ_NP0 t_ZZ0 ]_PUR =_NP0 Var_NP0 [_PUL ΔÑ_NP0 t_ZZ0 ]_PUR =_NP0 λ02+λ12_NP0 =_NP0 :_PUN λ2_CRD ._SENT Summing_VVG over_PRP l_ZZ0 yields_VVZ the_AT0 individual_AJ0 probabilities_NN2 ,_PUN equation_NN1 ,_PUN and_CJC similarly_AV0 for_PRP ℙ_NP0 (_PUL ΔÑ_NP0 t_ZZ0 =_NN1 l_ZZ0 )_PUR ._SENT Also_AV0 ,_PUN the_AT0 covariance_NN1 is_VBZ given_VVN by_PRP Cov_NP0 [_PUL ΔNt_NP0 ,_PUN ΔÑ_NP0 t_ZZ0 ]_PUR =_NN1 λ12_CRD and_CJC ,_PUN hence_AV0 ,_PUN the_AT0 correlation_NN1 by_PRP equation_NN1 ._SENT 
Thus_AV0 the_AT0 correlation_NN1 in_PRP the_AT0 number_NN1 of_PRF jumps_NN2 can_VM0 be_VBI adjusted_VVN through_PRP the_AT0 parameters_NN2 λ01_CRD ,_PUN λ02_CRD ,_PUN and_CJC λ12_AJ0 ._SENT The_AT0 analysis_NN1 of_PRF Sections_NN2 2_CRD and_CJC 3_CRD can_VM0 now_AV0 be_VBI adapted_VVN for_PRP this_DT0 general_AJ0 case_NN1 by_PRP noting_VVG that_CJT the_AT0 joint_AJ0 density_NN1 for_PRP the_AT0 random_AJ0 variables_NN2 R̂_NP0 i_CRD (_PUL ki_NN2 )_PUR defined_VVN by_PRP (_PUL 2.4_CRD )_PUR is_VBZ still_AV0 given_VVN by_PRP (_PUL 2.7_CRD )_PUR and_CJC (_PUL 2.8_CRD )_PUR ._SENT At_PRP this_DT0 point_NN1 we_PNP do_VDB not_XX0 assume_VVI independence_NN1 of_PRF ΔNt_NN1 and_CJC ΔÑ_NN1 t_ZZ0 but_CJC use_VVB the_AT0 joint_AJ0 probability_NN1 density_NN1 above_AV0 ,_PUN and_CJC (_PUL 2.9_CRD )_PUR is_VBZ now_AV0 replaced_VVN by_PRP equation_NN1 ,_PUN and_CJC similarly_AV0 ,_PUN (_PUL 2.12_CRD )_PUR is_VBZ identical_AJ0 except_CJS for_PRP the_AT0 p_ZZ0 (_PUL k1_CRD ,_PUN k2_CRD )_PUR substitution_NN1 ._SENT The_AT0 result_NN1 (_PUL 2.14_CRD )_PUR is_VBZ then_AV0 equation_NN1 ._SENT (_PUL 6.3_CRD )_PUR One_CRD needs_NN2 to_TO0 show_VVI convergence_NN1 of_PRF the_AT0 sums_NN2 above_AV0 ._SENT Noting_VVG that_DT0 μR1_NN1 /_PUN σR1∼k1_CRD /_PUN 21_CRD and_CJC the_AT0 exponential_NN1 is_VBZ bounded_VVN by_PRP unity_NN1 ,_PUN one_PNI can_VM0 prove_VVI the_AT0 boundedness_NN1 of_PRF the_AT0 series_NN0 by_PRP noting_VVG that_DT0 equation_NN1 ._SENT 
Hence_AV0 ,_PUN we_PNP can_VM0 state_VVI the_AT0 conclusion_NN1 analogous_AJ0 to_PRP Theorem_NN1 2.3_CRD ._SENT In_PRP addition_NN1 ,_PUN the_AT0 analysis_NN1 for_PRP R1_CRD and_CJC R2_CRD conditioned_VVN on_PRP positivity_NN1 can_VM0 be_VBI similarly_AV0 carried_VVN out_AVP as_AV0 in_PRP Section_NN1 3_CRD ,_PUN with_PRP the_AT0 following_AJ0 result_NN1 ._SENT 
Theorem_NN1 6.1_CRD ._SENT Let_VVB Ra_NP0 :_PUN =_NP0 R1_CRD /_PUN R2_CRD where_AVQ R1_NP0 =_NP0 D_NP0 ,_PUN R2_NP0 =_NP0 S_NP0 are_VBB described_VVN by_PRP jump-diffusion_NN1 processes_NN2 (_PUL i.e._AV0 ,_PUN Brownian_AJ0 motion_NN1 plus_PRP a_AT0 compound_NN1 Poisson_NP0 process_NN1 )_PUR through_PRP (_PUL 2.3_CRD )_PUR ._SENT Assume_VVB that_CJT the_AT0 two_CRD Poisson_NP0 processes_NN2 ,_PUN ΔNt_NN1 and_CJC ΔÑ_NN1 t_ZZ0 are_VBB bivariate_AJ0 Poisson_NP0 with_PRP means_NN0 λ1_CRD =_NP0 λ01+λ12_CRD and_CJC λ2_CRD =_NP0 λ02+λ12_CRD respectively_AV0 ,_PUN and_CJC covariance_NN1 λ12_NN1 ._SENT Let_VVB ρ∈_NP0 (_PUL −1_CRD ,_PUN 1_CRD )_PUR be_VBI the_AT0 correlation_NN1 between_PRP R1_CRD and_CJC R2_CRD conditioned_VVN on_PRP a_AT0 fixed_AJ0 pair_NN0 of_PRF jumps_NN2 (_PUL k1_CRD ,_PUN k2_CRD )_PUR ._SENT With_PRP Gε_NP0 be_VBI defined_VVN via_PRP (_PUL 2.20_CRD )_PUR ,_PUN the_AT0 density_NN1 fc_NN0 of_PRF Rc_NP0 :_PUN =_NP0 Gε_NP0 (_PUL Ra_NP0 )_PUR satisfies_VVZ the_AT0 asymptotic_AJ0 relation_NN1 equation_NN1 ._SENT The_AT0 density_NN1 fc_NN0 (_PUL y_ZZ0 |_NP0 R1_CRD ,_PUN R2_CRD >_NP0 0_CRD )_PUR of_PRF Rc_NP0 :_PUN =_NP0 Gε_NP0 (_PUL Ra_NP0 )_PUR conditioned_VVN on_PRP positive_AJ0 R1_CRD and_CJC R2_CRD also_AV0 satisfies_VVZ the_AT0 asymptotic_AJ0 relation_NN1 equation_NN1 ._SENT 
The_AT0 total_AJ0 correlation_NN1 ,_PUN ρT_NN1 ,_PUN as_CJS a_AT0 function_NN1 of_PRF the_AT0 parameters_NN2 is_VBZ calculated_VVN in_PRP the_AT0 Appendix_NN1 C_ZZ0 ._SENT 
Summarizing_VVG the_AT0 results_NN2 ,_PUN we_PNP see_VVB that_CJT the_AT0 issue_NN1 of_PRF correlation_NN1 between_PRP supply_NN1 and_CJC demand_NN1 is_VBZ complex_AJ0 ._SENT Given_VVN k1_CRD jumps_NN2 or_CJC shocks_NN2 for_PRP demand_NN1 and_CJC k2_NN1 for_PRP supply_NN1 ,_PUN we_PNP can_VM0 impose_VVI a_AT0 correlation_NN1 ,_PUN ρ∈_NP0 (_PUL −1_CRD ,_PUN 1_CRD )_PUR ,_PUN between_PRP supply_NN1 and_CJC demand_NN1 conditioned_VVN on_PRP (_PUL k1_CRD ,_PUN k2_CRD )_PUR ._SENT However_AV0 ,_PUN the_AT0 overall_AJ0 correlation_NN1 (_PUL not_XX0 conditioned_VVN on_PRP k1_CRD and_CJC k2_CRD )_PUR for_PRP demand_NN1 and_CJC supply_NN1 will_VM0 have_VHI a_AT0 smaller_AJC magnitude_NN1 if_CJS the_AT0 number_NN1 of_PRF jumps_NN2 are_VBB independent_AJ0 ,_PUN and_CJC is_VBZ calculated_VVN in_PRP Appendix_NN1 C_ZZ0 ._SENT 
In_PRP this_DT0 section_NN1 ,_PUN we_PNP have_VHB considered_VVN the_AT0 problem_NN1 when_AVQ the_AT0 number_NN1 of_PRF jumps_NN2 in_PRP the_AT0 supply_NN1 and_CJC demand_NN1 ,_PUN ΔNt_NN1 and_CJC ΔÑ_NN1 t_ZZ0 has_VHZ an_AT0 arbitrary_AJ0 correlation_NN1 ,_PUN ρ_NP0 (_PUL P_ZZ0 )_PUR ,_PUN given_VVN by_PRP (_PUL 6.2_CRD )_PUR in_PRP which_DTQ case_NN1 one_PNI can_VM0 obtain_VVI the_AT0 full_AJ0 range_NN1 (_PUL −1_CRD ,_PUN 1_CRD )_PUR of_PRF correlation_NN1 for_PRP the_AT0 supply_NN1 and_CJC demand_NN1 ,_PUN as_CJS stated_VVN in_PRP Theorem_NN1 6.1_CRD ._SENT 
The_AT0 limiting_AJ0 case_NN1 in_PRP which_DTQ the_AT0 correlation_NN1 between_PRP supply_NN1 and_CJC demand_NN1 is_VBZ given_VVN by_PRP −1_CRD ,_PUN i.e._AV0 ,_PUN anti-correlation_NN1 ,_PUN has_VHZ been_VBN considered_VVN in_PRP Theorem_NN1 5.1_CRD ._SENT 
7._CRD Conclusions_NN2 
We_PNP have_VHB shown_VVN that_CJT fat_AJ0 tails_NN2 can_VM0 be_VBI obtained_VVN solely_AV0 by_PRP modeling_VVG price_NN1 adjustment_NN1 through_PRP supply_NN1 and_CJC demand_NN1 ,_PUN each_DT0 of_PRF which_DTQ are_VBB random_AJ0 variables_NN2 consisting_VVG of_PRF Brownian_AJ0 motion_NN1 and_CJC shocks_NN2 from_PRP a_AT0 Poisson_NP0 distribution_NN1 ._SENT This_DT0 consistent_AJ0 with_PRP the_AT0 empirical_AJ0 evidence_NN1 for_PRP many_DT0 asset_NN1 markets_NN2 ._SENT The_AT0 classical_AJ0 theory_NN1 embodied_VVN in_PRP (_PUL 1.1_CRD )_PUR is_VBZ based_VVN partly_AV0 on_PRP empirical_AJ0 observations_NN2 near_PRP the_AT0 mean_AJ0 (_PUL and_CJC not_XX0 the_AT0 tails_NN2 )_PUR of_PRF price_NN1 changes_NN2 ._SENT The_AT0 fact_NN1 that_CJT the_AT0 normal_AJ0 distribution_NN1 is_VBZ a_AT0 reasonable_AJ0 model_NN1 near_PRP the_AT0 mean_NN1 –_PUN and_CJC is_VBZ generally_AV0 convenient_AJ0 mathematically_AV0 –_PUN does_VDZ not_XX0 provide_VVI any_DT0 evidence_NN1 that_CJT the_AT0 tail_NN1 of_PRF density_NN1 should_VM0 behave_VVI as_PRP the_AT0 normal_AJ0 does_VDZ ,_PUN i.e._AV0 ,_PUN f_ZZ0 (_PUL x_ZZ0 )_PUR ∼e−x2_CRD /_PUN 2._CRD Prices_NN2 evolve_VVB through_PRP re-balancing_VVG supply_NN1 and_CJC demand_NN1 ._SENT In_PRP this_DT0 paper_NN1 we_PNP have_VHB assumed_VVN a_AT0 model_NN1 that_CJT embodies_VVZ the_AT0 perspective_NN1 of_PRF supply_NN1 /_PUN demand_NN1 without_PRP invoking_VVG the_AT0 fine_AJ0 microstructure_NN1 ,_PUN and_CJC show_VVB that_CJT the_AT0 quotient_NN1 of_PRF two_CRD Levy_NP0 processes_NN2 of_PRF jump-diffusion_NN1 type_NN1 has_VHZ a_AT0 tail_NN1 distribution_NN1 that_DT0 is_VBZ a_AT0 power_NN1 law_NN1 ._SENT 
In_PRP essence_NN1 ,_PUN exponential_NN1 decay_NN1 in_PRP the_AT0 classical_AJ0 finance_NN1 models_NN2 is_VBZ a_AT0 feature_NN1 of_PRF the_AT0 assumptions_NN2 made_VVN without_PRP compelling_AJ0 theoretical_AJ0 arguments_NN2 ._SENT Prices_NN2 are_VBB not_XX0 distributed_VVN by_PRP a_AT0 random_AJ0 process_NN1 ;_PUN rather_AV0 they_PNP are_VBB the_AT0 consequence_NN1 of_PRF changes_NN2 in_PRP supply_NN1 and_CJC demand_NN1 which_DTQ have_VHB randomness_NN1 ._SENT In_PRP fact_NN1 ,_PUN it_PNP is_VBZ almost_AV0 a_AT0 tautology_NN1 to_TO0 say_VVI that_CJT all_DT0 price_NN1 change_NN1 is_VBZ a_AT0 direct_AJ0 consequence_NN1 of_PRF changes_NN2 in_PRP supply_NN1 and_CJC demand_NN1 ,_PUN which_DTQ have_VHB randomness_NN1 ._SENT If_CJS the_AT0 supply_NN1 and_CJC demand_NN1 were_VBD deterministic_AJ0 ,_PUN there_EX0 would_VM0 be_VBI no_AT0 randomness_NN1 in_PRP price_NN1 change_NN1 ._SENT Agammaegate_NP0 supply_NN1 and_CJC demand_NN1 involve_VVB the_AT0 sum_NN1 of_PRF a_AT0 large_AJ0 number_NN1 of_PRF decision_NN1 makers_NN2 so_AV0 that_CJT the_AT0 Central_AJ0 Limit_NN1 Theorem_NN1 can_VM0 be_VBI applied_VVN ._SENT The_AT0 addition_NN1 of_PRF a_AT0 compound_NN1 Poisson_NP0 process_NN1 to_PRP the_AT0 randomness_NN1 incorporates_VVZ the_AT0 influx_NN1 of_PRF a_AT0 smaller_AJC number_NN1 of_PRF large_AJ0 orders_NN2 that_CJT are_VBB manifested_VVN as_PRP jumps_NN2 in_PRP supply_NN1 or_CJC demand_NN1 ._SENT 
A_AT0 natural_AJ0 way_NN1 to_PRP model_NN1 price_NN1 change_NN1 is_VBZ through_PRP an_AT0 excess_AJ0 demand_NN1 model_NN1 which_DTQ involves_VVZ a_AT0 quotient_NN1 of_PRF supply_NN1 and_CJC demand_NN1 ._SENT The_AT0 important_AJ0 mathematical_AJ0 idea_NN1 is_VBZ that_CJT a_AT0 quotient_NN1 of_PRF two_CRD normals_NN2 is_VBZ not_XX0 normal_AJ0 ,_PUN though_CJS it_PNP resembles_VVZ a_AT0 normal_AJ0 near_PRP the_AT0 mean_NN1 for_PRP a_AT0 range_NN1 of_PRF parameters_NN2 ._SENT The_AT0 tail_NN1 of_PRF the_AT0 density_NN1 of_PRF a_AT0 quotient_NN1 of_PRF normals_NN2 has_VHZ previously_AV0 been_VBN shown_VVN to_TO0 have_VHI a_AT0 power_NN1 law_NN1 decay_NN1 ._SENT In_PRP this_DT0 paper_NN1 ,_PUN we_PNP showed_VVD that_CJT this_DT0 persists_VVZ even_AV0 with_PRP the_AT0 introduction_NN1 of_PRF shocks_NN2 in_PRP the_AT0 form_NN1 of_PRF a_AT0 Poisson_NP0 distribution_NN1 when_CJS there_EX0 is_VBZ an_AT0 arbitrary_AJ0 correlation_NN1 in_PRP the_AT0 number_NN1 of_PRF shocks_NN2 and_CJC an_AT0 arbitrary_AJ0 total_AJ0 correlation_NN1 between_PRP supply_NN1 and_CJC demand_NN1 ._SENT 
Another_DT0 important_AJ0 issue_NN1 is_VBZ that_CJT the_AT0 price_NN1 adjustment_NN1 equation_NN1 (_PUL 1.5_CRD )_PUR is_VBZ not_XX0 necessarily_AV0 linear_AJ0 ._SENT Hence_AV0 ,_PUN we_PNP consider_VVB appropriate_AJ0 nonlinear_AJ0 functions_NN2 ,_PUN Gε_NP0 ,_PUN (_PUL 2.20_CRD )_PUR \_CJC that_DT0 are_VBB characterized_VVN by_PRP an_AT0 exponent_NN1 q−1_NN1 for_PRP q_ZZ0 >_NP0 0._CRD The_AT0 salient_AJ0 feature_NN1 of_PRF this_DT0 function_NN1 is_VBZ that_CJT supply_NN1 and_CJC demand_NN1 enter_VVB on_PRP equal_AJ0 footing_NN1 ._SENT 
For_PRP this_DT0 family_NN1 of_PRF functions_NN2 we_PNP have_VHB calculated_VVN the_AT0 power_NN1 of_PRF the_AT0 decay_NN1 exponent_NN1 for_PRP the_AT0 density_NN1 of_PRF the_AT0 relative_AJ0 price_NN1 change_NN1 P−1ΔP_NN1 ,_PUN as_CJS f_ZZ0 (_PUL x_ZZ0 )_PUR ∼x−q−1_CRD ._SENT The_AT0 exponent_NN1 is_VBZ a_AT0 function_NN1 of_PRF the_AT0 shape_NN1 of_PRF the_AT0 price_NN1 function_NN1 ,_PUN Gε_NP0 ,_PUN so_AV0 that_CJT there_EX0 is_VBZ a_AT0 direct_AJ0 link_NN1 between_PRP the_AT0 form_NN1 of_PRF the_AT0 price_NN1 function_NN1 and_CJC the_AT0 decay_NN1 exponent_NN1 ._SENT Other_AJ0 possibilities_NN2 for_PRP the_AT0 function_NN1 Gε_NN1 for_PRP supply_NN1 and_CJC demand_NN1 yields_VVZ similar_AJ0 results_NN2 ._SENT Roughly_AV0 speaking_VVG if_CJS the_AT0 dependence_NN1 of_PRF G_ZZ0 on_PRP the_AT0 quotient_NN1 of_PRF demand_NN1 to_PRP supply_NN1 is_VBZ characterized_VVN by_PRP the_AT0 power_NN1 q−1_NN1 ,_PUN then_AV0 the_AT0 decay_NN1 will_VM0 be_VBI f_ZZ0 (_PUL x_ZZ0 )_PUR ∼_NP0 |_NP0 x_ZZ0 |_NP0 −α_NP0 with_PRP α_NP0 =_NP0 1+q_NP0 ._SENT 
Empirical_AJ0 studies_NN2 of_PRF the_AT0 tail_NN1 of_PRF distributions_NN2 for_PRP relative_AJ0 stock_NN1 price_NN1 changes_NN2 (_PUL denoted_VVN by_PRP r_ZZ0 below_AV0 )_PUR have_VHB mainly_AV0 shown_VVN (_PUL see_VVB [_PUL 4_CRD ,_PUN 23_CRD ,_PUN 24_CRD ,_PUN 47_CRD ,_PUN 51_CRD ,_PUN 53_CRD ]_PUR and_CJC references_NN2 therein_AV0 )_PUR that_DT0 large_AJ0 capitalization_NN1 stocks_NN2 have_VHB a_AT0 decay_NN1 rate_NN1 given_VVN by_PRP ζ_NP0 =_NP0 ̃_NP0 3_CRD or_CJC at_PRP least_AV0 within_PRP a_AT0 range_NN1 ζ∈_NN1 [_PUL 3_CRD ,_PUN 5_CRD ]_PUR where_AVQ P_ZZ0 (_PUL {_PUL |_NP0 r_ZZ0 |_NP0 >_NP0 x_ZZ0 }_PUR )_PUR ∼x−ζ_NP0 ,_PUN so_AV0 that_CJT f_ZZ0 (_PUL x_ZZ0 )_PUR ∼x−ζ−1_CRD ._SENT An_AT0 early_AJ0 paper_NN1 studied_VVD 40_CRD million_CRD data_NN0 points_NN2 and_CJC hypothesized_VVD a_AT0 ζ_NP0 =_NN1 3_CRD "_PUQ universal_AJ0 cubic_AJ0 law_NN1 ._SENT "_PUQ Thus_AV0 ,_PUN we_PNP have_VHB α_NP0 =_NP0 q+1_NP0 ,_PUN i.e._AV0 ,_PUN ζ_NP0 =_NP0 q_ZZ0 ._SENT This_DT0 would_VM0 predict_VVI that_CJT the_AT0 Gε_NN1 defined_VVN above_AV0 should_VM0 have_VHI q−1_CRD =_NP0 1_CRD /_PUN 3_CRD corresponding_VVG to_PRP α_NP0 =_NP0 4ζ_NP0 =_NP0 3._CRD This_DT0 means_VVZ that_CJT if_CJS the_AT0 nonlinear_AJ0 price_NN1 function_NN1 Gε_NP0 has_VHZ a_AT0 behavior_NN1 that_DT0 is_VBZ somewhat_AV0 less_AV0 responsive_AJ0 than_CJS linear_AJ0 ,_PUN the_AT0 implied_AJ0 decay_NN1 will_VM0 be_VBI within_PRP the_AT0 empirical_AJ0 range_NN1 ._SENT The_AT0 analysis_NN1 also_AV0 suggests_VVZ that_CJT it_PNP would_VM0 be_VBI useful_AJ0 to_TO0 do_VDI empirical_AJ0 research_NN1 on_PRP determining_VVG the_AT0 form_NN1 of_PRF the_AT0 function_NN1 Gε_NN1 ,_PUN as_CJS well_AV0 as_CJS examining_VVG the_AT0 correlation_NN1 between_PRP the_AT0 changes_NN2 in_PRP supply_NN1 and_CJC demand_NN1 to_TO0 determine_VVI how_AVQ close_AJ0 it_PNP is_VBZ to_PRP anti-correlation_NN1 ._SENT Some_DT0 studies_NN2 (_PUL e.g._AV0 ,_PUN [_PUL 40_CRD ]_PUR )_PUR have_VHB indicated_VVN that_CJT the_AT0 tail_NN1 exponent_NN1 varies_VVZ to_PRP some_DT0 extent_NN1 as_CJS one_PNI examines_VVZ different_AJ0 countries_NN2 and_CJC markets_NN2 such_DT0 as_PRP emerging_AJ0 markets_NN2 ._SENT While_CJS stock_NN1 price_NN1 movements_NN2 have_VHB been_VBN studied_VVN extensively_AV0 ,_PUN there_EX0 is_VBZ much_DT0 less_AV0 empirical_AJ0 work_NN1 on_PRP the_AT0 nature_NN1 of_PRF the_AT0 supply_NN1 and_CJC demand_NN1 curves_NN2 ,_PUN particularly_AV0 under_PRP extreme_AJ0 conditions_NN2 ._SENT This_DT0 would_VM0 be_VBI a_AT0 useful_AJ0 empirical_AJ0 work_NN1 that_CJT could_VM0 open_VVI up_AVP a_AT0 new_AJ0 avenue_NN1 to_TO0 understand_VVI rare_AJ0 events_NN2 as_CJS discussed_VVN below_AV0 ._SENT This_DT0 approach_NN1 can_VM0 be_VBI utilized_VVN not_XX0 only_AV0 for_PRP the_AT0 tail_NN1 but_CJC for_PRP the_AT0 entire_AJ0 distribution_NN1 so_CJS volatility_NN1 can_VM0 be_VBI modeled_VVN in_PRP its_DPS entirety_NN1 ._SENT 
Our_DPS model_NN1 is_VBZ applicable_AJ0 to_PRP any_DT0 price_NN1 formation_NN1 ,_PUN and_CJC is_VBZ not_XX0 restricted_VVN to_PRP financial_AJ0 instruments_NN2 ._SENT In_PRP some_DT0 markets_NN2 for_PRP goods_NN2 ,_PUN the_AT0 exponential_NN1 tail_NN1 may_VM0 hold_VVI [_PUL 1_CRD ,_PUN 37_CRD ]_PUR ._SENT Our_DPS analysis_NN1 may_VM0 provide_VVI a_AT0 link_NN1 between_PRP the_AT0 tails_NN2 of_PRF financial_AJ0 assets_NN2 and_CJC goods_NN2 /_PUN services_NN2 ._SENT In_PRP our_DPS model_NN1 ,_PUN this_DT0 would_VM0 be_VBI attained_VVN by_PRP a_AT0 function_NN1 G_ZZ0 that_DT0 is_VBZ very_AV0 slowly_AV0 responding_VVG for_PRP large_AJ0 imbalance_NN1 e.g._AV0 ,_PUN involving_VVG a_AT0 logarithmic_AJ0 function_NN1 ._SENT This_DT0 would_VM0 mean_VVI that_CJT as_CJS the_AT0 supply_NN1 /_PUN demand_NN1 balance_NN1 becomes_VVZ extreme_AJ0 ,_PUN the_AT0 relative_AJ0 price_NN1 does_VDZ not_XX0 continue_VVI to_TO0 change_VVI nearly_AV0 as_PRP much_DT0 ._SENT It_PNP seems_VVZ intuitively_AV0 reasonable_AJ0 that_CJT the_AT0 supply_NN1 and_CJC demand_NN1 curves_NN2 for_PRP goods_NN2 /_PUN services_NN2 have_VHB this_DT0 feature_NN1 ,_PUN since_CJS there_EX0 is_VBZ usually_AV0 no_AT0 possibility_NN1 of_PRF resale_NN1 ._SENT However_AV0 ,_PUN further_AJC research_NN1 is_VBZ needed_VVN on_PRP this_DT0 issue_NN1 ._SENT 
In_PRP general_AJ0 ,_PUN G_ZZ0 is_VBZ a_AT0 function_NN1 to_TO0 be_VBI calibrated_VVN ._SENT The_AT0 essential_AJ0 point_NN1 ,_PUN however_AV0 ,_PUN is_VBZ that_CJT G_ZZ0 is_VBZ a_AT0 deterministic_AJ0 function_NN1 ,_PUN while_CJS D_ZZ0 and_CJC S_ZZ0 are_VBB Levy_NP0 processes_NN2 of_PRF jump-diffusion_NN1 type_NN1 ._SENT To_TO0 simplify_VVI the_AT0 discussion_NN1 ,_PUN let_VVB us_PNP suppose_VVB that_CJT they_PNP are_VBB simply_AV0 Gaussian_AJ0 ._SENT The_AT0 central_AJ0 thesis_NN1 is_VBZ that_CJT price_NN1 change_NN1 is_VBZ essentially_AV0 deterministic_AJ0 given_VVN a_AT0 specific_AJ0 supply_NN1 and_CJC demand_NN1 ._SENT Suppose_VVB that_CJT one_PNI is_VBZ interested_AJ0 in_PRP estimating_VVG the_AT0 probability_NN1 of_PRF a_AT0 5_CRD %_NN0 drop_NN1 in_PRP an_AT0 asset_NN1 ._SENT In_PRP order_NN1 to_TO0 establish_VVI this_DT0 probability_NN1 directly_AV0 from_PRP empirical_AJ0 data_NN0 ,_PUN one_PNI would_VM0 need_VVI a_AT0 very_AV0 large_AJ0 amount_NN1 of_PRF data_NN0 since_CJS it_PNP occurs_VVZ infrequently_AV0 ._SENT However_AV0 ,_PUN using_VVG the_AT0 formalism_NN1 above_AV0 ,_PUN one_PNI would_VM0 need_VVI many_DT0 fewer_DT0 observations_NN2 that_CJT would_VM0 determine_VVI the_AT0 value_NN1 of_PRF the_AT0 price_NN1 drop_NN1 as_PRP a_AT0 function_NN1 of_PRF the_AT0 imbalance_NN1 in_PRP supply_NN1 and_CJC demand_NN1 ._SENT Once_AV0 this_DT0 connection_NN1 is_VBZ established_VVN ,_PUN it_PNP is_VBZ then_AV0 a_AT0 matter_NN1 of_PRF understanding_VVG the_AT0 probability_NN1 of_PRF that_DT0 imbalance_NN1 ._SENT The_AT0 supply_NN1 ,_PUN S_ZZ0 ,_PUN and_CJC demand_NN1 ,_PUN D_ZZ0 ,_PUN are_VBB each_DT0 Gaussian_AJ0 ,_PUN with_PRP means_NN0 and_CJC variances_NN2 that_CJT are_VBB easily_AV0 estimated_VVN with_PRP a_AT0 moderate_AJ0 amount_NN1 of_PRF data_NN0 (_PUL i.e._AV0 ,_PUN without_PRP necessarily_AV0 observing_VVG rare_AJ0 events_NN2 )_PUR so_CJS we_PNP know_VVB the_AT0 density_NN1 of_PRF the_AT0 normalized_AJ0 excess_NN1 demand_NN1 ,_PUN namely_AV0 ,_PUN (_PUL D−S_NP0 )_PUR /_PUN S_ZZ0 ._SENT 
In_PRP summary_NN1 ,_PUN we_PNP can_VM0 establish_VVI the_AT0 deterministic_AJ0 connection_NN1 between_PRP the_AT0 imbalance_NN1 of_PRF supply_NN1 /_PUN demand_NN1 and_CJC the_AT0 change_NN1 in_PRP price_NN1 through_PRP the_AT0 function_NN1 G_ZZ0 ._SENT Once_AV0 this_DT0 is_VBZ done_VDN ,_PUN we_PNP can_VM0 estimate_VVI the_AT0 means_NN0 and_CJC variances_NN2 of_PRF the_AT0 supply_NN1 and_CJC demand_NN1 individually_AV0 ._SENT The_AT0 mathematical_AJ0 results_NN2 then_AV0 yield_VVB the_AT0 density_NN1 of_PRF (_PUL D−S_NP0 )_PUR /_PUN S_NP0 ._SENT With_PRP the_AT0 approximation_NN1 of_PRF G_ZZ0 we_PNP then_AV0 obtain_VVB the_AT0 probability_NN1 of_PRF a_AT0 particular_AJ0 size_NN1 price_NN1 drop_NN1 ._SENT In_PRP particlar_NN1 ,_PUN we_PNP can_VM0 estimate_VVI the_AT0 distribution_NN1 tail_NN1 exponent_NN1 ,_PUN ζ_NP0 =_NP0 q_ZZ0 ._SENT 
Note_VVB that_CJT the_AT0 use_NN1 of_PRF a_AT0 smaller_AJC data_NN0 set_VVB is_VBZ not_XX0 just_AV0 a_AT0 matter_NN1 of_PRF convenience_NN1 ._SENT The_AT0 issue_NN1 is_VBZ that_CJT as_CJS we_PNP use_VVB much_DT0 larger_AJC sets_NN2 ,_PUN we_PNP are_VBB utilizing_VVG data_NN0 that_CJT may_VM0 be_VBI irrelevant_AJ0 ,_PUN for_PRP example_NN1 ,_PUN because_CJS it_PNP is_VBZ an_AT0 artifact_NN1 of_PRF a_AT0 different_AJ0 era_NN1 ._SENT Data_NN0 sets_NN2 of_PRF 252_CRD trading_NN1 days_NN2 and_CJC 2520_CRD correspond_VVB to_PRP one_CRD year_NN1 and_CJC ten_CRD years_NN2 ,_PUN respectively_AV0 ._SENT In_PRP the_AT0 year_NN1 2011_CRD ,_PUN for_PRP example_NN1 ,_PUN the_AT0 ten_CRD year_NN1 period_NN1 would_VM0 include_VVI the_AT0 post-Internet_AJ0 bubble_NN1 of_PRF 2001-2003_CRD ,_PUN as_CJS well_AV0 as_CJS the_AT0 2007-2008_CRD housing_NN1 debacle_NN1 and_CJC subsequent_AJ0 recovery_NN1 ._SENT So_AV0 data_NN0 from_PRP only_AV0 the_AT0 period_NN1 2010-2011_CRD would_VM0 be_VBI more_AV0 pertinent_AJ0 in_PRP terms_NN2 of_PRF estimating_VVG the_AT0 probabilities_NN2 of_PRF possible_AJ0 rare_AJ0 events_NN2 in_PRP 2012._CRD Then_AV0 using_VVG the_AT0 data_NN0 from_PRP 252_CRD or_CJC 504_CRD relevent_NN1 trading_NN1 days_NN2 ,_PUN one_PNI can_VM0 find_VVI the_AT0 most_AV0 extreme_AJ0 price_NN1 changes_NN2 and_CJC match_VVB them_PNP up_AVP with_PRP the_AT0 (_PUL D−S_NP0 )_PUR /_PUN S_ZZ0 values_NN2 ,_PUN so_AV0 that_CJT the_AT0 values_NN2 of_PRF G_ZZ0 for_PRP those_DT0 arguments_NN2 are_VBB set_VVN ._SENT This_DT0 is_VBZ a_AT0 consequence_NN1 of_PRF G_NP0 being_VBG deterministic_AJ0 ._SENT From_PRP this_DT0 data_NN0 set_VVB of_PRF 252_CRD or_CJC 504_CRD we_PNP can_VM0 reliably_AV0 determine_VVI the_AT0 mean_NN1 and_CJC variance_NN1 of_PRF each_DT0 of_PRF supply_NN1 and_CJC demand_NN1 ,_PUN noting_VVG that_CJT these_DT0 are_VBB Gaussian_AJ0 by_PRP virtue_NN1 of_PRF the_AT0 Central_AJ0 Limit_NN1 Theorem_NN1 for_PRP independent_AJ0 agents_NN2 ._SENT Using_VVG these_DT0 parameters_NN2 ,_PUN and_CJC the_AT0 analysis_NN1 of_PRF the_AT0 quotient_NN1 of_PRF Levy_NP0 jump-diffusion_NN1 processes_NN2 ,_PUN we_PNP can_VM0 calculate_VVI a_AT0 more_AV0 accurate_AJ0 probability_NN1 for_PRP the_AT0 extreme_AJ0 events_NN2 that_CJT were_VBD observed_VVN than_CJS would_VM0 be_VBI possible_AJ0 by_PRP using_VVG just_AV0 the_AT0 raw_AJ0 frequency_NN1 ._SENT 
The_AT0 problem_NN1 of_PRF fat_AJ0 tails_NN2 is_VBZ part_NN1 of_PRF the_AT0 general_AJ0 issue_NN1 of_PRF volatility_NN1 in_PRP markets_NN2 that_CJT has_VHZ been_VBN of_PRF great_AJ0 interest_NN1 in_PRP recent_AJ0 years_NN2 (_PUL see_VVB ,_PUN e.g._AV0 ,_PUN [_PUL 22_CRD ]_PUR )_PUR ._SENT The_AT0 methods_NN2 discussed_VVN can_VM0 be_VBI useful_AJ0 within_PRP the_AT0 general_AJ0 setting_NN1 of_PRF volatility_NN1 problems_NN2 ._SENT 
The_AT0 analysis_NN1 of_PRF this_DT0 paper_NN1 has_VHZ involved_VVN normal_AJ0 distributions_NN2 through_PRP the_AT0 Brownian_AJ0 motion_NN1 and_CJC the_AT0 distribution_NN1 of_PRF each_DT0 jump_NN1 in_PRP the_AT0 Poisson_NP0 process_NN1 ._SENT An_AT0 open_AJ0 problem_NN1 is_VBZ to_TO0 extend_VVI these_DT0 results_NN2 to_PRP a_AT0 broad_AJ0 set_NN1 of_PRF distributions_NN2 ._SENT Using_VVG results_NN2 such_DT0 as_PRP [_PUL 56_CRD ]_PUR ,_PUN one_PNI can_VM0 consider_VVI the_AT0 quotient_NN1 other_AJ0 distributions_NN2 for_PRP the_AT0 supply_NN1 and_CJC demand_NN1 and_CJC analyze_VVB the_AT0 tail_NN1 behavior_NN1 ._SENT 
