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January  2017, 2: 8 doi: 10.1186/s41546-017-0021-8

## Measure distorted arrival rate risks and their rewards

 Robert H. Smith School of Business, University of Maryland, College Park, MD 20742, USA

Received  January 16, 2017 Revised  June 02, 2017 Published  December 2017

Risks embedded in asset price dynamics are taken to be accumulations of surprise jumps. A Markov pure jump model is formulated on making variance gamma parameters deterministic functions of the price level. Estimation is done by matrix exponentiation of the transition rate matrix for a continuous time finite state Markov chain approximation. The motion is decomposed into a space dependent drift and a space dependent martingale component. Though there is some local mean reversion in the drift, space dependence of the martingale component renders the dynamics to be of the momentum type. Local risk is measured using market calibrated measure distortions that introduce risk charges into the lower and upper prices of two price economies. These risks are compensated by the exponential variation of space dependent arrival rates. Estimations are conducted for the S&P 500 index (SPX), the exchange traded fund for the financial sector (XLF), J. P. Morgan stock prices (JPM), the ratio of JPM to XLF, and the ratio of XLF to SPX.
Citation: Dilip B. Madan. Measure distorted arrival rate risks and their rewards. Probability, Uncertainty and Quantitative Risk, 2017, 2 (0) : 8-. doi: 10.1186/s41546-017-0021-8
##### References:
 [1] Angelos, B:The Hunt Variance Gamma Process with Applications to Option Pricing. University of Maryland, PhD. Dissertation (2013) [2] Artzner, P, Delbaen, F, Eber, M, Heath, D:Coherent Measures of Risk. Math. Finance 9, 203-228 (1999) [3] Barndorff-Nielsen, O, Shephard, N:Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics (with discussion). J. R. Stat. Soc. Ser. B 63, 167-241 (2001) [4] Bass, RF:Uniqueness in law for pure jump Markov processes. Probab. Theory Relat. Fields 79, 271-287(1988) [5] Carr, P, Geman, H, Madan, D, Yor, M:The fine structure of asset returns:An empirical investigation. J.Bus 75(2), 305-332 (2002) [6] Cherny, A, Madan, DB:Markets as a counterparty:An Introduction to Conic Finance. Int. J. Theor. Appl.Finance 13, 1149-1177 (2010) [7] Cont, R, Minca, A:Recovering Portfolio Default Intensities Implied by CDO Quotes. Math. Finance 23, 94-121 (2013) [8] Cousin, A, Crépey, S, Kan, YH:Delta-hedging correlation risk. Rev. Deriv. Res 15, 25-56 (2012) [9] Delbaen, F, Schachermayer, W:A general version of the fundamental theorem of asset pricing. Math. Ann 300, 463-520 (1994) [10] Hunt, G:Martingales et Processus de Markov. Dunod, Paris (1966) [11] Khintchine, AY:Limit laws of sums of independent random variables. ONTI, Moscow, Russian (1938) [12] Kusuoka, S:On Law Invariant Coherent Risk Measures. Adv. Math. Econ 3, 83-95 (2001) [13] Lévy, P:Théorie de l'Addition des Variables Aléatoires. Gauthier-Villars, Paris (1937) [14] Madan, DB:Asset pricing theory for two price economies. Ann. Finance 11, 1-35 (2015a) [15] Madan, DB:Estimating parametric models of probability distributions. Methodol. Comput. Appl. Probab 17, 823-831 (2015b) [16] Madan, DB, Schoutens, W:Applied Conic Finance. Cambridge University Press, Cambridge (2016a) [17] Madan, DB, Schoutens, W:Conic Asset Pricing and The Costs of Price Fluctuations (2016b). Working Paper, Robert H. Smith School of Business Madan, DB:Instantaneous Portfolio Theory (2016a). Available at https://ssrn.com/abstract=2804718 [18] Madan, DB:Momentum and reversion in risk neutral martingale probabilities. Quant. Finan 14, 777-787(2016b). Available at https://ssrn.com/abstract=2251300 [19] Madan, DB:Risk Premia in Option Markets. Ann. Finance 12, 71-94 (2016c) [20] Madan, D, Carr, P, Chang, E:The variance gamma process and option pricing. Eur Financ Rev 2, 79-105(1998) [21] Madan, DB, Pistorius, M, Stadje, M:On Dynamic Spectral Risk Measures and a Limit Theorem (2016). [22] forthcoming Finance and Stochastics Madan, DB, Seneta, E:The variance gamma (VG) model for share market returns. J. Bus 63, 511-524(1990) [23] Pistorius, M, Mijatović, A:Continuously Monitored Barrier Options under Markov Processes. Math. Finance 23, 1-38 (2011). Also available at https://ssrn.com/abstract=1462822 [24] Sato, K:Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999) [25] Skiadis, C:Asset Pricing Theory. Princeton University Press, Princeton (2009) [26] Stroock, DW:Diffusion processes associated with Lévy generators. Probab. Theory Relat. Fields 32, 209-244 (1975) [27] Stroock, DW, Varadhan, SRS:Multidimensional diffusion processes, a series of comprehensive studies in mathematics. vol. 233. Springer-Verlag, Berlin (1979)

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##### References:
 [1] Angelos, B:The Hunt Variance Gamma Process with Applications to Option Pricing. University of Maryland, PhD. Dissertation (2013) [2] Artzner, P, Delbaen, F, Eber, M, Heath, D:Coherent Measures of Risk. Math. Finance 9, 203-228 (1999) [3] Barndorff-Nielsen, O, Shephard, N:Non-Gaussian Ornstein-Uhlenbeck-based models and some of their uses in financial economics (with discussion). J. R. Stat. Soc. Ser. B 63, 167-241 (2001) [4] Bass, RF:Uniqueness in law for pure jump Markov processes. Probab. Theory Relat. Fields 79, 271-287(1988) [5] Carr, P, Geman, H, Madan, D, Yor, M:The fine structure of asset returns:An empirical investigation. J.Bus 75(2), 305-332 (2002) [6] Cherny, A, Madan, DB:Markets as a counterparty:An Introduction to Conic Finance. Int. J. Theor. Appl.Finance 13, 1149-1177 (2010) [7] Cont, R, Minca, A:Recovering Portfolio Default Intensities Implied by CDO Quotes. Math. Finance 23, 94-121 (2013) [8] Cousin, A, Crépey, S, Kan, YH:Delta-hedging correlation risk. Rev. Deriv. Res 15, 25-56 (2012) [9] Delbaen, F, Schachermayer, W:A general version of the fundamental theorem of asset pricing. Math. Ann 300, 463-520 (1994) [10] Hunt, G:Martingales et Processus de Markov. Dunod, Paris (1966) [11] Khintchine, AY:Limit laws of sums of independent random variables. ONTI, Moscow, Russian (1938) [12] Kusuoka, S:On Law Invariant Coherent Risk Measures. Adv. Math. Econ 3, 83-95 (2001) [13] Lévy, P:Théorie de l'Addition des Variables Aléatoires. Gauthier-Villars, Paris (1937) [14] Madan, DB:Asset pricing theory for two price economies. Ann. Finance 11, 1-35 (2015a) [15] Madan, DB:Estimating parametric models of probability distributions. Methodol. Comput. Appl. Probab 17, 823-831 (2015b) [16] Madan, DB, Schoutens, W:Applied Conic Finance. Cambridge University Press, Cambridge (2016a) [17] Madan, DB, Schoutens, W:Conic Asset Pricing and The Costs of Price Fluctuations (2016b). Working Paper, Robert H. Smith School of Business Madan, DB:Instantaneous Portfolio Theory (2016a). Available at https://ssrn.com/abstract=2804718 [18] Madan, DB:Momentum and reversion in risk neutral martingale probabilities. Quant. Finan 14, 777-787(2016b). Available at https://ssrn.com/abstract=2251300 [19] Madan, DB:Risk Premia in Option Markets. Ann. Finance 12, 71-94 (2016c) [20] Madan, D, Carr, P, Chang, E:The variance gamma process and option pricing. Eur Financ Rev 2, 79-105(1998) [21] Madan, DB, Pistorius, M, Stadje, M:On Dynamic Spectral Risk Measures and a Limit Theorem (2016). [22] forthcoming Finance and Stochastics Madan, DB, Seneta, E:The variance gamma (VG) model for share market returns. J. Bus 63, 511-524(1990) [23] Pistorius, M, Mijatović, A:Continuously Monitored Barrier Options under Markov Processes. Math. Finance 23, 1-38 (2011). Also available at https://ssrn.com/abstract=1462822 [24] Sato, K:Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999) [25] Skiadis, C:Asset Pricing Theory. Princeton University Press, Princeton (2009) [26] Stroock, DW:Diffusion processes associated with Lévy generators. Probab. Theory Relat. Fields 32, 209-244 (1975) [27] Stroock, DW, Varadhan, SRS:Multidimensional diffusion processes, a series of comprehensive studies in mathematics. vol. 233. Springer-Verlag, Berlin (1979)
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