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G-Lévy processes under sublinear expectations
The term structure of sharpe ratios and arbitrage-free asset pricing in continuous time
1. | Research School of Economics, The Australian National University, Canberra, Australia |
2. | Department of Statistics and Quantitative Methods, University of Milano-Bicocca, 20126 Milano, Italy |
Motivated by financial and empirical arguments and in order to introduce a more flexible methodology of pricing, we provide a new approach to asset pricing based on Backward Volterra equations. The approach relies on an arbitrage-free and incomplete market setting in continuous time by choosing non-unique pricing measures depending either on the time of evaluation or on the maturity of payoffs. We show that in the latter case the dynamics can be captured by a time-delayed backward stochastic Volterra integral equation here introduced which, to the best of our knowledge, has not yet been studied. We then prove an existence and uniqueness result for time-delayed backward stochastic Volterra integral equations. Finally, we present a Lucas-type consumption-based asset pricing model that justifies the emergence of stochastic discount factors matching the term structure of Sharpe ratios.
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show all references
1See Elliott and Madan [37] for an alternative choice of an EMM by means of an extended Girsanov principle.
2From a technical stance, our notion of an EMM-string is comparable with a shifting of martingale measures in Biagini et al. [12], see Example 1.
3From the reverse perspective, the range of the SR forward curve already contains a lower bound for the degree of incompleteness of the financial market.
4See Lettau and Wachter [54], Hansen et al. [43] for research on maturity-dependent risk pricing.
5Dividend strips are discounted sums of dividends over a small time interval with a constant length and a varying position on the time axis.
6For an application of the HJM approach to volatility surface modeling see Schweizer and Wissel [66] and Carmona and Nadtochiy [20].
7For the pricing under a given EMM, time consistency directly follows from the law of iterated expectations, see also Pelsser and Stadje [61] for more general results in this direction.
8For a definitive version see Delbaen and Schachermayer [27].
9Roughly speaking, a set of probability measures is m-stable whether any pasting of different probability measures within
10Note that, although
11The concept of forward measures, introduced by Jarrow [47], is conceptually different from the idea of changing equivalent martingale measures; see also Musiela and Rutkowski [58]. Such measures rest on a numéraire change via different maturities of zero-coupon bond valuations. In contrast, for an EMM-string it is essential to have an incomplete market setting. Conversely, the forward measure has the same structure under complete markets as under incomplete markets, as the SDF remains fixed.
12One may ask if the existence result holds under the assumption that the maturity time
13Such fragility can be caused by learning. We follow the perspective of Kurz [52] and consider the belief as time dependent. An alternative viewpoint refers to the concept of optimal beliefs, as considered in the study by Brunnermeier and Parker [18], in which a forward–looking agent maximizes average felicity over beliefs.
14Alternatively, consider
15Note that the scalar product on
16For the variational argument in the proof of Theorem 2, it suffices to consider simple processes.
17See also Lepeltier and San Martin [53] and Kobylanski [49] for more general existence results.
18See also Luo and Tangpi [56] for a general existence result.
References:
[1] |
Agram, N., Dynamic risk measure for BSVIE with jumps and semimartingale issues, Stochastic Analysis and Applications, 2019, 37(3): 1-16. Google Scholar |
[2] |
Aliprantis, C. D., Separable utility functions, Journal of Mathematical Economics, 1997, 28(4): 415-444.
doi: 10.1016/S0304-4068(97)00805-7. |
[3] |
Andries, M., Eisenbach, T.M. and Schmalz, M.C., Horizon-dependent risk aversion and the timing and pricing of uncertainty. FRB of New York Staff Report 703, 2018. Google Scholar |
[4] |
Bansal, R. and Yaron, A., Risks for the long run: a potential resolution of asset pricing puzzles, The Journal of Finance, 2004, 59(4): 1481-1509.
doi: 10.1111/j.1540-6261.2004.00670.x. |
[5] |
Barrieu, P. and El Karoui, N., Pricing, hedging and optimally designing derivatives via minimization of risk measures. In: Indifference Pricing: Theory and Applications, Carmona, R. (ed.), Princeton University Press, Princeton, 2005: 77−144. Google Scholar |
[6] |
Beissner, P., Lin, Q. and Riedel, F., Dynamic consistent α-maxmin expected utility. Center for Mathematical Economics. Working Paper 535, 2016. Google Scholar |
[7] |
Berg, T., The term structure of risk premia: new evidence from the financial crisis. ECB Working Paper Series No 1165, 2010. Google Scholar |
[8] |
Berger, M. A., A Malliavin-type anticipative stochastic calculus, The Annals of Probability, 1988, 16(1): 231-245.
doi: 10.1214/aop/1176991897. |
[9] |
Berger, M. A. and Mizel, V. J., A Fubini theorem for iterated stochastic integrals, Bulletin of the American Mathematical Society, 1978, 84(1): 159-160.
doi: 10.1090/S0002-9904-1978-14452-8. |
[10] |
Berger, M. A. and Mizel, V. J., Theorems of Fubini type for iterated stochastic integrals, Transactions of the American Mathematical Society, 1979, 252: 249-274.
doi: 10.1090/S0002-9947-1979-0534121-3. |
[11] |
Berger, M. A. and Mizel, V. J., An extension of the stochastic integral, The Annals of Probability, 1982, 10(2): 435-450.
doi: 10.1214/aop/1176993868. |
[12] |
Biagini, F., Föllmer, H. and Nedelcu, S., Shifting martingale measures and the birth of a bubble as a submartingale, Finance and Stochastics, 2014, 18(2): 297-326.
doi: 10.1007/s00780-013-0221-8. |
[13] |
Binsbergen, v. J., Brandt, M. and Koijen, R., On the timing and pricing of dividends, The American Economic Review, 2012, 102(4): 1596-1618.
doi: 10.1257/aer.102.4.1596. |
[14] |
Binsbergen, v. J., Hueskes, W., Koijen, R. and Vrugt, E., Equity yields, Journal of Financial Economics, 2013, 110(3): 503-519.
doi: 10.1016/j.jfineco.2013.08.017. |
[15] |
Bismut, J.-M., Conjugate convex functions in optimal stochastic control, Journal of Mathematical Analysis and Applications, 1973, 44(2): 384-404.
doi: 10.1016/0022-247X(73)90066-8. |
[16] |
Björk, T., Arbitrage theory in continuous time, Oxford University Press, 2009. Google Scholar |
[17] |
Brunnermeier, M. K., Papakonstantinou, F. and Parker, J. A., Optimal time-inconsistent beliefs: misplaning, procrastination, and commitment, Working Paper, 2013. Google Scholar |
[18] |
Brunnermeier, M. K. and Parker, J. A., Optimal expectations, American Economic Review, 2005, 95(4): 1092-1118.
doi: 10.1257/0002828054825493. |
[19] |
Campbell, J. Y. and Cochrane, J. H., By force of habit: a consumption-based explanation of aggregate Stock Market behavior, Journal of Political Economy, 1999, 107(2): 205-251.
doi: 10.1086/250059. |
[20] |
Carmona, R. and Nadtochiy, S., Local volatility dynamic models, Finance and Stochastics, 2009, 13(1): 1-48.
doi: 10.1007/s00780-008-0078-4. |
[21] |
Chen, Z. and Epstein, L., Ambiguity, risk, and asset returns in continuous time, Econometrica, 2002, 70(4): 1403-1443.
doi: 10.1111/1468-0262.00337. |
[22] |
Coquet, F., Hu, Y., Mémin, J. and Peng, S., Filtration-consistent nonlinear expectations and related g-expectations, Probability Theory and Related Fields, 2002, 123(1): 1-27.
doi: 10.1007/s004400100172. |
[23] |
Csiszar, I., I-Divergence geometry of probability distributions and minimization problems, Annals of Probability, 1975, 3(1): 146-158.
doi: 10.1214/aop/1176996454. |
[24] |
Delbaen, F., Representing martingale measures when asset prices are continuous and bounded, Mathematical Finance, 1992, 2(2): 107-130.
doi: 10.1111/j.1467-9965.1992.tb00041.x. |
[25] |
Delbaen, F., The structure of m-stable sets and in particular of the set of risk neutral measures. In Memoriam Paul-André Meyer, Springer, 2006: 215−258. Google Scholar |
[26] |
Delbaen, F., Peng, S. and Rosazza Gianin, E., Representation of the penalty term of dynamic concave utilities, Finance and Stochastics, 2010, 14(3): 449-472.
doi: 10.1007/s00780-009-0119-7. |
[27] |
Delbaen, F. and Schachermayer, W., A general version of the fundamental theorem of asset pricing, Mathematische Annalen, 1994, 300(1): 463-520.
doi: 10.1007/BF01450498. |
[28] |
Delong, L. and Imkeller, P., Backward stochastic differential equations with time delayed generators - results and counterexamples, The Annals of Applied Probability, 2010, 20(4): 1512-1536.
doi: 10.1214/09-AAP663. |
[29] |
Detemple, J. and Rindisbacher, M., Dynamic asset allocation: Portfolio decomposition formula and applications, Review of Financial Studies, 2010, 23(1): 25-100.
doi: 10.1093/rfs/hhp040. |
[30] |
Dos Reis, G. and Dos Reis, R. J. N., A note on comonotonicity and positivity of the control components of decoupled quadratic FBSDE, Stochastics and Dynamics, 2013, 13(04): 1350005.
doi: 10.1142/S0219493713500056. |
[31] |
Duffie, D., Dynamic asset pricing theory. Princeton University Press, 1996. Google Scholar |
[32] |
Duffie, D. and Epstein, L. G., Stochastic differential utility, Econometrica, 1992, 60(2): 353-394.
doi: 10.2307/2951600. |
[33] |
Duffie, D. and Skiadas, C., Continuous-time security pricing: A utility gradient approach, Journal of Mathematical Economics, 1994, 23(2): 107-131.
doi: 10.1016/0304-4068(94)90001-9. |
[34] |
Eisenbach, T. M. and Schmalz, M. C., Anxiety in the face of risk, Journal of Financial Economics, 2016, 121(2): 414-426.
doi: 10.1016/j.jfineco.2015.10.002. |
[35] |
El Karoui, N. and Quenez, M.-C., Dynamic programming and pricing of contingent claims in an incomplete market, SIAM Journal on Control and Optimization, 1995, 33(1): 29-66.
doi: 10.1137/S0363012992232579. |
[36] |
El Karoui, N., Peng, S. and Quenez, M.-C., Backward stochastic differential equations in finance, Mathematical Finance, 1997, 7(1): 1-71.
doi: 10.1111/1467-9965.00022. |
[37] |
Elliott, R., A discrete time equivalent martingale measure, Mathematical Finance, 1998, 8(2): 127-152.
doi: 10.1111/1467-9965.00048. |
[38] |
Epstein, L.G., Farhi, E. and Strzalecki, T., How much would you pay to resolve long-run risk? American Economic Review, 2014, 104(9): 2680-97.
doi: 10.1257/aer.104.9.2680. |
[39] |
Epstein, L. G. and Zin, S. E., Substitution, risk aversion, and the temporal behavior of consumption and asset returns: a theoretical framework, Econometrica, 1989, 57(9): 937-969. Google Scholar |
[40] |
Föllmer, H. and Schweizer, M., Hedging of contingent claims under incomplete information. In: Applied Stochastic Analysis (Davis M. H. A. and Elliott R. J. eds.), Stochastics Monographs, Gordon and Breach, London/New York, 1991, 5: 389-414. Google Scholar |
[41] |
Frittelli, M., The minimal entropy martingale measure and the valuation problem in incomplete markets, Mathematical Finance, 2000, 10(1): 39-52.
doi: 10.1111/1467-9965.00079. |
[42] |
Gabaix, X., Variable rare disasters: an exactly solved framework for ten puzzles in Macro-Finance, The Quarterly Journal of Economics, 2012, 127(2): 645-700.
doi: 10.1093/qje/qjs001. |
[43] |
Hansen, L. P., Heaton, J. C. and Li, N., Consumption strikes back? Measuring long-run risk, Journal of Political Economy, 2008, 116(2): 260-302.
doi: 10.1086/588200. |
[44] |
Harrison, J. M. and Kreps, D. M., Martingales and arbitrage in multiperiod securities markets, Journal of Economic Theory, 1979, 20(3): 381-408.
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Approach | Claim
|
Payoff Stream
|
|||
1 SDF | 2 Recursive | 3 SDF | 4 Recursive | ||
|
|
BSDE |
|
BSDE | |
|
|
BSVIE |
|
BSVIE | |
|
|
BSDE |
|
TD–BSVIE |
Approach | Claim
|
Payoff Stream
|
|||
1 SDF | 2 Recursive | 3 SDF | 4 Recursive | ||
|
|
BSDE |
|
BSDE | |
|
|
BSVIE |
|
BSVIE | |
|
|
BSDE |
|
TD–BSVIE |
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