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Sentiment lost: the effect of projecting the pricing kernel onto a smaller filtration set. (English) Zbl 1445.91064

Summary: This paper provides a theoretical analysis on the impacts of using a suboptimal information set for the estimation of the pricing kernel and, more in general, for the validity of the fundamental theorems of asset pricing. While inferring the risk-neutral measure from options data provides a naturally forward-looking estimate, extracting the real world measure from historical returns is only partially informative, thus suboptimal with respect to investors’ future beliefs. As a consequence of this disalignment, the two measures no longer share the same nullset, thus distorting the investors’ risk premium and the validity of the pricing measure. From a probabilistic viewpoint, the missing beliefs are totally unaccessible stopping times on the coarser filtration set, so that an absolutely continuous strict local martingale, once projected on it, becomes continuous with jumps. Some empirical examples complete the paper.

MSC:

91G20 Derivative securities (option pricing, hedging, etc.)
60G44 Martingales with continuous parameter
62P05 Applications of statistics to actuarial sciences and financial mathematics
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[1] Lucas, E. J., Asset prices in an exchange economy, Econometrica, 46, 6, 1429-1445 (1978) · Zbl 0398.90016 · doi:10.2307/1913837
[2] Rubinstein, M., The valuation of uncertain income streams and the pricing of options, Bell J. Econ., 7, 2, 407-425 (1976) · doi:10.2307/3003264
[3] Äit-Sahalia, Y.; Lo, A. W., Non-parametric estimation of state-price densities implicit in financial asset prices, J. Financ., 53, 2, 499-547 (1998) · doi:10.1111/0022-1082.215228
[4] Engle, R.; Rosenberg, V., Empirical pricing kernels, Rev. Financ. Stud., 64, 341-372 (2002) · doi:10.1016/S-405X(02)00128-9
[5] Jackwerth, J., Recovering risk aversion from option prices and realized returns, Rev. Financ. Stud, 13, 2, 433-451 (2000) · doi:10.1093/rfs/13.2.433
[6] Protter, P., Strict local martingales with jumps, Stoch. Processes Appl., 125, 4, 1352-1367 (2015) · Zbl 1322.60048 · doi:10.1016/j.spa.2014.10.018
[7] Brown, D.; Jackwerth, J., The pricing kernel puzzle: Reconciling index option data and economic theory, Contemp. Stud. Econ. Financ. Anal., 94, 155-183 (2012)
[8] Barone-Adesi, G.; Engle, R.; Mancini, L., A garch option pricing model in incomplete markets, Rev. Financ. Stud, 21, 3, 1223-1258 (2008) · doi:10.1093/rfs/hhn031
[9] Barone-Adesi, G.; Fusari, N.; Mira, A.; Sala, C., Option market trading activity and the estimation of the pricing kernel: a Bayesian approach, J. Econom. (2019) · Zbl 1456.62241 · doi:10.1016/j.jeconom.2019.11.001
[10] Schneider, P.; Trojani, F., (Almost) model-free recovery, J. Financ., 74, 1, 323-370 (2019) · doi:10.1111/jofi.12737
[11] Ross, S., The recovery theorem, J. Financ., 70, 2, 615-648 (2015) · doi:10.1111/jofi.12092
[12] Linn, M.; Shive, S.; Shumway, T., Pricing kernel monotonicity and conditional information, Rev. Financ. Stud., 31, 2, 493-531 (2018) · doi:10.1093/rfs/hhx095
[13] Chernov, M.; Ghysels, E., A study towards a unified approach to the joint estimation of objective and risk neutral measures for the purpose of options valuation, J. Financ. Econ., 56, 3, 407-458 (2000) · doi:10.1016/S-405X(00)00046-5
[14] Goulbev, Y.; Härdle, W.; Timonfeev, R., Testing monotonicity of pricing kernels, Asta Adv. Stat. Anal., 98, 4, 305-326 (2014) · Zbl 1443.62350 · doi:10.1007/s10182-014-5
[15] Hens, T.; Reichlin, C., Three solutions to the pricing kernel puzzle, Rev. Financ. Stud., 17, 3, 1065-1098 (2012) · Zbl 1417.91566 · doi:10.1093/rof/rfs008
[16] Stricker, C., Quasimartingales, martingales locales, et filtrations naturalles, Z Wahrscheinlichkeitstheorie Verw. Gebiete, 39, 1, 55-63 (1977) · Zbl 0362.60069 · doi:10.1007/BF01844872
[17] Follmer, H.; Protter, P., Local martingales and filtration shrinkage, ESAIM Probab. Stat., 14, 825-838 (2011)
[18] Johnson, G.; Helms, L., Class “d” supermartingales, Bull. Amer. Math. Soc, 69, 1, 59-62 (1963) · Zbl 0133.40402 · doi:10.1090/S-9904-1963-10857-5
[19] Doob, J., Stochastic Processes (1953), New York: Wiley, New York · Zbl 0053.26802
[20] Meyer, P.-A., Decomposition of supermartingales: the uniqueness theorem, Illinois J. Math., 7, 1, 1-17 (1963) · Zbl 0133.40401 · doi:10.1215/ijm/1255637477
[21] Itô, K.; Watanabe, S., Transformation of Markov processes by multiplicative functionals, Annals de L’institut Fourier, 15, 15-30 (1965) · Zbl 0141.15103
[22] Hardle, W.; Hlavka, Z., Dynamics of state price densities, J. Econom., 150, 1-15 (2009) · Zbl 1429.62470
[23] Harrison, J. M.; Kreps, D., Martingales and arbitrage in multiperiod securities markets, J. Econ. Theory, 20, 3, 381-408 (1979) · Zbl 0431.90019 · doi:10.1016/0022-(79)90043-7
[24] Delbaen, F.; Schachermayer, W., A general version of the fundamental theorem of asset pricing, Math. Ann., 300, 1, 463-520 (1994) · Zbl 0865.90014 · doi:10.1007/BF01450498
[25] Delbaen, F.; Schachermayer, W., Fundamental theorem of asset pricing for unbounded stochastic processes, Mathemathische Annalen, 334, 215-250 (1998) · Zbl 0917.60048 · doi:10.1007/s002080050220
[26] Itô, K., Extensions of stochastic integrals, Proceedings of International Symposium on Stochastic Differential Equations, 95-109 (1978) · Zbl 0409.60060
[27] Jarrow, R.; Protter, P.; Sezer, A., Information reduction via level crossings in a credit risk model, Financ. Stoch., 11, 2, 195-212 (2007) · Zbl 1143.91031 · doi:10.1007/s00780-006-1
[28] Elworthy, K.; Li, X.-M.; Yor, M., The importance of strictly local martingales: Applications to radial ornstein-uhlenbeck processes, Probab. Theory Relat. Fields, 115, 3, 325-355 (1999) · Zbl 0960.60046 · doi:10.1007/s004400050240
[29] Billingsley, P., Probability and Measure. (1995), New York: Wiley, New York · Zbl 0822.60002
[30] Jarrow, R., The third fundamental theorem of asset pricing, Ann. Financ. Econ, 07, 2, 1250007 (2012) · doi:10.1142/S2010495212500078
[31] Dybvig, P., Inefficient dynamic portfolio strategies or how to throw away a million dollars in the stock market, Rev. Financ. Stud, 1, 1, 67-88 (1988) · doi:10.1093/rfs/1.1.67
[32] Barone-Adesi, G.; Sala, C., Testing market efficiency with the pricing kernel, The Eur. J. Financ., 25, 13, 1166-1193 (2019) · doi:10.1080/1351847X.2019.1581638
[33] Jarrow, R., Option pricing and market efficiency, JPM Manage., 40, 1, 88-94 (2013) · doi:10.3905/jpm.2013.40.1.088
[34] Jarrow, R.; Kchia, Y.; Protter, P., How to detect an asset bubble, SIAM J. Finan. Math., 2, 1, 839-865 (2011) · Zbl 1239.91184 · doi:10.1137/10079673X
[35] Figlewski, S., Estimating the Implied Risk Neutral Density for the U.S. Market Portfolio. Volatility and Time Series Econometrics: Essays in Honor of Robert Engle (2008), Oxford, UK: Oxford University Press, Oxford, UK
[36] Bliss, R.; Panigirtzoglou, N., Option-implied risk aversion estimates, J. Financ., 59, 1, 407-446 (2004) · doi:10.1111/j.1540-6261.2004.00637.x
[37] Leisen, D., The shape of small sample biases in pricing kernel estimations, Quant. Financ., 17, 6, 943-958 (2017) · Zbl 1406.62121 · doi:10.1080/14697688.2016.1258486
[38] Audrino, F.; Huitema, R.; Ludwig, M. (2015)
[39] Bakshi, G.; Chabi-Yo, F.; Gao, X., A recovery that we can trust? Deducing and testing the restrictions of the recovery theorem, Rev. Financ. Stud., 31, 2, 532-555 (2018) · doi:10.1093/rfs/hhx108
[40] Borovicka, J.; Hansen, L.; Scheinkman, J., Misspecified recovery, J. Financ., 71, 6, 2493-2544 (2016) · doi:10.1111/jofi.12404
[41] Carr, P.; Yu, J., Risk, return and the Ross recovery, JOD., 20, 1, 38-59 (2012) · doi:10.3905/jod.2012.20.1.038
[42] Dubynskiy, S.; Goldstein, R. S. (2013)
[43] Jensen, C.; Lando, D.; Pedersen, H. L., Generalized recovery, J. Financ. Econ., 133, 1, 154-174 (2019) · doi:10.1016/j.jfineco.2018.12.003
[44] Massacci, F.; Williams, J.; Zhang, Y. (2016)
[45] Qin, L.; Linetsky, V., Positive eigenfunctions of Markovian pricing operators: Hansen-Scheinkman factorization, Ross recovery and long-term pricing, Oper. Res., 64, 1, 99-117 (2016) · Zbl 1338.90447 · doi:10.1287/opre.2015.1449
[46] Qin, L.; Linetsky, V.; Nie, Y., Long forward probabilities, recovery and the term structure of bond risk premiums, Rev. Financ. Stud., 31, 12, 4863-4883 (2018)
[47] Platen, E.; Heath, D., A Benchmark Approach to Quantitative Finance. MR22653260 (2006), Berlin: Springer Finance, Berlin · Zbl 1104.91041
[48] Fernzhold, D.; Karatzas, I., On optimal arbitrage, Ann. Appl. Probab, 20, 1179-1204 (2010) · Zbl 1206.60055 · doi:10.1214/09-AAP642
[49] Heston, S. L.; Loewenstein, M.; Willard, G. A., Options and bubbles, Rev. Financ. Stud, 20, 2, 359-390 (2007) · doi:10.1093/rfs/hhl005
[50] Jarrow, R.; Protter, P.; Shimbo, K., Asset price bubbles in incomplete markets, Math. Financ., 20, 2, 145-185 (2010) · Zbl 1205.91069 · doi:10.1111/j.1467-9965.2010.00394.x
[51] Protter, P., Stochastic Integration and Differential Equations (2005), Heidelberg: Springer Verlag, Heidelberg
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