The term structure of sharpe ratios and arbitrage-free asset pricing in continuous time

Patrick Beißner; Emanuela Rosazza Gianin

doi:10.3934/puqr.2021002

Previous Article
Improved Hoeffding inequality for dependent bounded or sub-Gaussian random variables
PUQR Home
This Issue
Next Article
G-Lévy processes under sublinear expectations

March 2021, 6(1): 23-52. doi: 10.3934/puqr.2021002

The term structure of sharpe ratios and arbitrage-free asset pricing in continuous time

Patrick Beißner ^1, and Emanuela Rosazza Gianin ^2,,

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

Email: patrick.beissner@anu.edu.au, emanuela.rosazza1@unimib.it

Received June 15, 2019 Revised November 09, 2020 Accepted December 17, 2020 Published March 2021

Fund Project: The authors wish to thank two Referees for their careful reading and for the useful remarks that contributed to improving the paper.　This research started with the research group “Robust Finance: Strategic Power, Knightian Uncertainty, and the Foundations of Economic Policy Advice” at ZIF in Bielefeld, Germany. The financial support, as well as the stimulating discussions, are gratefully acknowledged. The authors also thank Fabio Bellini, Freddy Delbaen, Giulia Di Nunno, Frank Riedel, and Carlo Sgarra for comments.　Emanuela Rosazza Gianin is also grateful to Bernt Øksendal for stimulating and helpful discussions on this subject and on BSVIEs.

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.

Keywords: Volterra equations, BSDEs, Asset pricing, Time inconsistency, Arbitrage-free, Incomplete markets, Term structures, Sharpe ratio.

Mathematics Subject Classification: 60H10; 60K35.

Citation: Patrick Beißner, Emanuela Rosazza Gianin. The term structure of sharpe ratios and arbitrage-free asset pricing in continuous time. Probability, Uncertainty and Quantitative Risk, 2021, 6 (1) : 23-52. doi: 10.3934/puqr.2021002

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[37]	Elliott, R., A discrete time equivalent martingale measure, Mathematical Finance, 1998, 8(2): 127-152. doi: 10.1111/1467-9965.00048. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[44]	Harrison, J. M. and Kreps, D. M., Martingales and arbitrage in multiperiod securities markets, Journal of Economic Theory, 1979, 20(3): 381-408. doi: 10.1016/0022-0531(79)90043-7. Google Scholar
[45]	Heath, D., Jarrow, R. and Morton, A., Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation, Econometrica, 1992, 60(1): 77-105. doi: 10.2307/2951677. Google Scholar
[46]	Hu, Y. and Øksendal, B., Linear Volterra backward stochastic integral equations, Stochastic Processes and their Applications, 2019, 129(2): 626-633. doi: 10.1016/j.spa.2018.03.016. Google Scholar
[47]	Jarrow, R., The pricing of commodity options with stochastic interest rates, Advances in Futures and Options Research, 1987, 2: 19-45. Google Scholar
[48]	Karatzas, I. and Kou, S. G., On the pricing of contingent claims under constraints, The Annals of Applied Probability, 1996, 6(2): 321-369. doi: 10.1214/aoap/1034968135. Google Scholar
[49]	Kobylanski, M., Backward stochastic differential equations and partial differential equations with quadratic growth, Annals of Probability, 2000: 558-602. Google Scholar
[50]	Kromer, E. and Overbeck, L., Differentiability of BSVIEs and dynamic capital allocations, International Journal of Theoretical and Applied Finance, 2017, 20(07): 1750047. doi: 10.1142/S0219024917500479. Google Scholar
[51]	Krusell, P. and Smith, A. A., Consumption−savings decisions with quasi−geometric discounting, Econometrica, 2003, 71(1): 365-375. doi: 10.1111/1468-0262.00400. Google Scholar
[52]	Kurz, M., On the structure and diversity of rational beliefs, Economic Theory, 1994, 4(6): 877-900. doi: 10.1007/BF01213817. Google Scholar
[53]	Lepeltier, J. P. and San Martin, J., Existence for BSDE with superlinear quadratic coefficient, Stochastics: An International Journal of Probability and Stochastic Processes, 1998, 63(3-4): 227-240. Google Scholar
[54]	Lettau, M. and Wachter, J. A., Why is long-horizon equity less risky? a duration-based explanation of the value premium, The Journal of Finance, 2007, 62(1): 55-92. doi: 10.1111/j.1540-6261.2007.01201.x. Google Scholar
[55]	Lucas, R.E., Asset prices in an exchange economy, Econometrica, 1978, 46(6): 1429-1445. doi: 10.2307/1913837. Google Scholar
[56]	Luo, P. and Tangpi, L., BSDEs on finite and infinite horizon with time-delayed generators, Working paper, http://arxiv.org/abs/1509.01991v1. Google Scholar
[57]	Mania, M. and Schweizer, M., Dynamic exponential utility indifference valuation, The Annals of Applied Probability, 2005, 15(3): 2113-2143. doi: 10.1214/105051605000000395. Google Scholar
[58]	Musiela, M. and Rutkowski, M., Continuous-time term structure models: forward measure approach, Finance and Stochastics, 1997, 1(4): 261-291. doi: 10.1007/s007800050025. Google Scholar
[59]	Palhares, D., Cash-flow maturity and risk premia in CDS markets, Working Paper, The University of Chicago Booth School of Business and Division of the Social Sciences, Department of Economics, 2013. Google Scholar
[60]	Pardoux, E. and Peng, S., Adapted solution of a backward stochastic differential equation, Systems and Control Letters, 1990, 14: 55-61. doi: 10.1016/0167-6911(90)90082-6. Google Scholar
[61]	Pelsser, A. and Stadje, M., Time-consistent and market-consistent evaluations, Mathematical Finance, 2014, 24(1): 25-65. doi: 10.1111/mafi.12026. Google Scholar
[62]	Peng, S., Backward SDE and related g-expectations. In: Backward Stochastic Differential Qquations, Pitman Research Notes in Mathematics Series (El Karoui N. and Mazliak L. eds.), Longman, Harlow, 1997, 364: 141−159. Google Scholar
[63]	Peng, S., Nonlinear expectations, nonlinear evaluations and risk measures. Stochastic Methods in Finance, Springer, 2004, 1856: 165−253. Google Scholar
[64]	Rosazza Gianin, E., Risk measures via g-expectations, Insurance: Mathematics and Economics, 2006, 39(1): 19-34. doi: 10.1016/j.insmatheco.2006.01.002. Google Scholar
[65]	Rouge, R. and El Karoui, N., Pricing via utility maximization and entropy, Mathematical Finance, 2000, 10(2): 259-276. doi: 10.1111/1467-9965.00093. Google Scholar
[66]	Schweizer, M. and Wissel, J., Term structures of implied volatilities: absence of arbitrage and existence results, Mathematical Finance, 2008, 18(1): 77-114. Google Scholar
[67]	Wang, T. and Yong, J., Comparison theorems for some backward stochastic Volterra integral equations, Stochastic Processes and Their Applications, 2015, 125(5): 1756-1798. doi: 10.1016/j.spa.2014.11.013. Google Scholar
[68]	Yong, J., Backward stochastic Volterra integral equations and some related problems, Stochastic Processes and Their Applications, 2006, 116(5): 779-795. doi: 10.1016/j.spa.2006.01.005. Google Scholar
[69]	Yong, J., Backward stochastic Volterra integral equations - a brief survey, Applied Mathematics-A Journal of Chinese Universities, 2013, 28(4): 383-394. doi: 10.1007/s11766-013-3189-4. Google Scholar
[70]	Yong, J., Continuous-time dynamic risk measures by backward stochastic Volterra integral equations, Applicable Analysis, 2007, 86(11): 1429-1442. doi: 10.1080/00036810701697328. Google Scholar
[71]	Zhang, J., Backward Stochastic Differential Equations: From Linear to Fully Nonlinear Theory. Springer, 2017. Google Scholar

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 $ {\cal Q} $ corresponds to another probability belonging again to that set.

10Note that, although $ \theta(t,\tau) $ is defined for any $ t \in [0,T] $ , it is only relevant for any $ t \leq \tau $ because of the interpretation and definition of an EMM-string and in similarity to Volterra equations.

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 $ T $ is sufficiently small. Concerning this issue, we thank an anonymous Referee of a different journal for her/his comment inducing a discussion of this issue from both a financial and a mathematical point of view. On one hand, from a mathematical point of view there is no guarantee that an existence and uniqueness result holds true for a TD–BSVIE with an arbitrary time horizon $ T $ (similarly to the case of a TD–BSDE). On the other hand, however, once the EMM-string is fixed, the maturity-based pricing (20) satisfies (21) by construction under no restriction on the length of maturity $ T. $ This allows the application of the maturity-based approach for pricing under no restriction on the length of maturity.

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 $ d({\mathbb P}_s,{\mathbb P}) \geq d({\mathbb P}_t,{\mathbb P}) \to 0 $ for some generalized distance $ d $ , as in Csiszar [23] and Frittelli [41].

15Note that the scalar product on $ {\mathbb L} $ yields $ \langle \psi(0,\cdot), c-e \rangle_0 = {E}^{\mathbb P}[\int_0^T \psi(0,\tau)(c_\tau-e_\tau ){\rm{d}}\tau ] $ .

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[37]	Elliott, R., A discrete time equivalent martingale measure, Mathematical Finance, 1998, 8(2): 127-152. doi: 10.1111/1467-9965.00048. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[44]	Harrison, J. M. and Kreps, D. M., Martingales and arbitrage in multiperiod securities markets, Journal of Economic Theory, 1979, 20(3): 381-408. doi: 10.1016/0022-0531(79)90043-7. Google Scholar
[45]	Heath, D., Jarrow, R. and Morton, A., Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation, Econometrica, 1992, 60(1): 77-105. doi: 10.2307/2951677. Google Scholar
[46]	Hu, Y. and Øksendal, B., Linear Volterra backward stochastic integral equations, Stochastic Processes and their Applications, 2019, 129(2): 626-633. doi: 10.1016/j.spa.2018.03.016. Google Scholar
[47]	Jarrow, R., The pricing of commodity options with stochastic interest rates, Advances in Futures and Options Research, 1987, 2: 19-45. Google Scholar
[48]	Karatzas, I. and Kou, S. G., On the pricing of contingent claims under constraints, The Annals of Applied Probability, 1996, 6(2): 321-369. doi: 10.1214/aoap/1034968135. Google Scholar
[49]	Kobylanski, M., Backward stochastic differential equations and partial differential equations with quadratic growth, Annals of Probability, 2000: 558-602. Google Scholar
[50]	Kromer, E. and Overbeck, L., Differentiability of BSVIEs and dynamic capital allocations, International Journal of Theoretical and Applied Finance, 2017, 20(07): 1750047. doi: 10.1142/S0219024917500479. Google Scholar
[51]	Krusell, P. and Smith, A. A., Consumption−savings decisions with quasi−geometric discounting, Econometrica, 2003, 71(1): 365-375. doi: 10.1111/1468-0262.00400. Google Scholar
[52]	Kurz, M., On the structure and diversity of rational beliefs, Economic Theory, 1994, 4(6): 877-900. doi: 10.1007/BF01213817. Google Scholar
[53]	Lepeltier, J. P. and San Martin, J., Existence for BSDE with superlinear quadratic coefficient, Stochastics: An International Journal of Probability and Stochastic Processes, 1998, 63(3-4): 227-240. Google Scholar
[54]	Lettau, M. and Wachter, J. A., Why is long-horizon equity less risky? a duration-based explanation of the value premium, The Journal of Finance, 2007, 62(1): 55-92. doi: 10.1111/j.1540-6261.2007.01201.x. Google Scholar
[55]	Lucas, R.E., Asset prices in an exchange economy, Econometrica, 1978, 46(6): 1429-1445. doi: 10.2307/1913837. Google Scholar
[56]	Luo, P. and Tangpi, L., BSDEs on finite and infinite horizon with time-delayed generators, Working paper, http://arxiv.org/abs/1509.01991v1. Google Scholar
[57]	Mania, M. and Schweizer, M., Dynamic exponential utility indifference valuation, The Annals of Applied Probability, 2005, 15(3): 2113-2143. doi: 10.1214/105051605000000395. Google Scholar
[58]	Musiela, M. and Rutkowski, M., Continuous-time term structure models: forward measure approach, Finance and Stochastics, 1997, 1(4): 261-291. doi: 10.1007/s007800050025. Google Scholar
[59]	Palhares, D., Cash-flow maturity and risk premia in CDS markets, Working Paper, The University of Chicago Booth School of Business and Division of the Social Sciences, Department of Economics, 2013. Google Scholar
[60]	Pardoux, E. and Peng, S., Adapted solution of a backward stochastic differential equation, Systems and Control Letters, 1990, 14: 55-61. doi: 10.1016/0167-6911(90)90082-6. Google Scholar
[61]	Pelsser, A. and Stadje, M., Time-consistent and market-consistent evaluations, Mathematical Finance, 2014, 24(1): 25-65. doi: 10.1111/mafi.12026. Google Scholar
[62]	Peng, S., Backward SDE and related g-expectations. In: Backward Stochastic Differential Qquations, Pitman Research Notes in Mathematics Series (El Karoui N. and Mazliak L. eds.), Longman, Harlow, 1997, 364: 141−159. Google Scholar
[63]	Peng, S., Nonlinear expectations, nonlinear evaluations and risk measures. Stochastic Methods in Finance, Springer, 2004, 1856: 165−253. Google Scholar
[64]	Rosazza Gianin, E., Risk measures via g-expectations, Insurance: Mathematics and Economics, 2006, 39(1): 19-34. doi: 10.1016/j.insmatheco.2006.01.002. Google Scholar
[65]	Rouge, R. and El Karoui, N., Pricing via utility maximization and entropy, Mathematical Finance, 2000, 10(2): 259-276. doi: 10.1111/1467-9965.00093. Google Scholar
[66]	Schweizer, M. and Wissel, J., Term structures of implied volatilities: absence of arbitrage and existence results, Mathematical Finance, 2008, 18(1): 77-114. Google Scholar
[67]	Wang, T. and Yong, J., Comparison theorems for some backward stochastic Volterra integral equations, Stochastic Processes and Their Applications, 2015, 125(5): 1756-1798. doi: 10.1016/j.spa.2014.11.013. Google Scholar
[68]	Yong, J., Backward stochastic Volterra integral equations and some related problems, Stochastic Processes and Their Applications, 2006, 116(5): 779-795. doi: 10.1016/j.spa.2006.01.005. Google Scholar
[69]	Yong, J., Backward stochastic Volterra integral equations - a brief survey, Applied Mathematics-A Journal of Chinese Universities, 2013, 28(4): 383-394. doi: 10.1007/s11766-013-3189-4. Google Scholar
[70]	Yong, J., Continuous-time dynamic risk measures by backward stochastic Volterra integral equations, Applicable Analysis, 2007, 86(11): 1429-1442. doi: 10.1080/00036810701697328. Google Scholar
[71]	Zhang, J., Backward Stochastic Differential Equations: From Linear to Fully Nonlinear Theory. Springer, 2017. Google Scholar

of equivalent martingale measures.">Figure 1. Illustration of an EMM-string in the set $ {\cal Q} $ of equivalent martingale measures.

Figure Options

Download full-size image

Download as PowerPoint slide

. Method 1 uses the EMM-string at the time of evaluation; see (14). Method 2 uses the EMM-string at the maturity of the claim and employs the whole gray triangle; see (19).">Figure 2. Two ways to employ the random field of SR $ \theta(t,\tau)_{t\leq \tau} $ . Method 1 uses the EMM-string at the time of evaluation; see (14). Method 2 uses the EMM-string at the maturity of the claim and employs the whole gray triangle; see (19).

Figure Options

Download full-size image

Download as PowerPoint slide

Table 1. Summary of methods for pricing at time $ t $ . The first row is discussed in section 2.1. The recursive columns state the type of related backward stochastic equation. The SDF columns present the involved SDF. $ \psi $ is indexed by time–maturity pairs.

Approach	Claim $ X $		Payoff Stream $ \{x_\tau\}_{\tau\in[t,T]} $
Approach	1 SDF	2 Recursive	3 SDF	4 Recursive
$ p $ - classical	$ \dfrac{\psi(T,T)}{\psi(t,t)} $	BSDE	$ \left\{\dfrac{\psi(\tau,\tau)}{\psi(t,t)}\right\}_{\tau\in[t,T]} $	BSDE
$ p^* $ - time	$ \dfrac{\psi(t,T)}{\psi(t,t)} $	BSVIE	$ \left\{\dfrac{\psi(t,\tau)}{\psi(t,t)}\right\}_{\tau\in[t,T]} $	BSVIE
$ \hat p $ - maturity	$ \dfrac{\psi(T,T)}{\psi(t,T)} $	BSDE	$ \left\{\dfrac{\psi(\tau,\tau)}{\psi(t,\tau)}\right\}_{(t,\tau): \, t\leq\tau} $	TD–BSVIE

Table Options

Download as excel

Approach	Claim $ X $		Payoff Stream $ \{x_\tau\}_{\tau\in[t,T]} $
Approach	1 SDF	2 Recursive	3 SDF	4 Recursive
$ p $ - classical	$ \dfrac{\psi(T,T)}{\psi(t,t)} $	BSDE	$ \left\{\dfrac{\psi(\tau,\tau)}{\psi(t,t)}\right\}_{\tau\in[t,T]} $	BSDE
$ p^* $ - time	$ \dfrac{\psi(t,T)}{\psi(t,t)} $	BSVIE	$ \left\{\dfrac{\psi(t,\tau)}{\psi(t,t)}\right\}_{\tau\in[t,T]} $	BSVIE
$ \hat p $ - maturity	$ \dfrac{\psi(T,T)}{\psi(t,T)} $	BSDE	$ \left\{\dfrac{\psi(\tau,\tau)}{\psi(t,\tau)}\right\}_{(t,\tau): \, t\leq\tau} $	TD–BSVIE

[1]	Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355
[2]	Jinsen Guo, Yongwu Zhou, Baixun Li. The optimal pricing and service strategies of a dual-channel retailer under free riding. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021056
[3]	Shiwen Niu, Hongmei Cheng, Rong Yuan. A free boundary problem of some modified Leslie-Gower predator-prey model with nonlocal diffusion term. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021129
[4]	Weiyi Zhang, Zuhan Liu, Ling Zhou. Dynamics of a nonlocal diffusive logistic model with free boundaries in time periodic environment. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3767-3784. doi: 10.3934/dcdsb.2020256
[5]	Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2543-2557. doi: 10.3934/dcds.2020374
[6]	Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027
[7]	V. V. Zhikov, S. E. Pastukhova. Korn inequalities on thin periodic structures. Networks & Heterogeneous Media, 2009, 4 (1) : 153-175. doi: 10.3934/nhm.2009.4.153
[8]	René Aïd, Roxana Dumitrescu, Peter Tankov. The entry and exit game in the electricity markets: A mean-field game approach. Journal of Dynamics & Games, 2021 doi: 10.3934/jdg.2021012
[9]	Ying Sui, Huimin Yu. Singularity formation for compressible Euler equations with time-dependent damping. Discrete & Continuous Dynamical Systems, 2021 doi: 10.3934/dcds.2021062
[10]	Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199
[11]	Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, 2021, 20 (3) : 995-1023. doi: 10.3934/cpaa.2021003
[12]	Olivier Ley, Erwin Topp, Miguel Yangari. Some results for the large time behavior of Hamilton-Jacobi equations with Caputo time derivative. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3555-3577. doi: 10.3934/dcds.2021007
[13]	Jumpei Inoue, Kousuke Kuto. On the unboundedness of the ratio of species and resources for the diffusive logistic equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2441-2450. doi: 10.3934/dcdsb.2020186
[14]	Yueqiang Shang, Qihui Zhang. A subgrid stabilizing postprocessed mixed finite element method for the time-dependent Navier-Stokes equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3119-3142. doi: 10.3934/dcdsb.2020222
[15]	Yongqiang Fu, Xiaoju Zhang. Global existence and asymptotic behavior of weak solutions for time-space fractional Kirchhoff-type diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021091
[16]	Pengyu Chen, Xuping Zhang, Zhitao Zhang. Asymptotic behavior of time periodic solutions for extended Fisher-Kolmogorov equations with delays. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021103
[17]	Amru Hussein, Martin Saal, Marc Wrona. Primitive equations with horizontal viscosity: The initial value and The time-periodic problem for physical boundary conditions. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3063-3092. doi: 10.3934/dcds.2020398
[18]	Tobias Breiten, Sergey Dolgov, Martin Stoll. Solving differential Riccati equations: A nonlinear space-time method using tensor trains. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 407-429. doi: 10.3934/naco.2020034
[19]	Guo-Bao Zhang, Ruyun Ma, Xue-Shi Li. Traveling waves of a Lotka-Volterra strong competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 587-608. doi: 10.3934/dcdsb.2018035
[20]	Dan Wei, Shangjiang Guo. Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2599-2623. doi: 10.3934/dcdsb.2020197

Impact Factor:

Tools

Metrics

Tools

Article outline

Figures and Tables

Impact Factor:

Tools

Figures and Tables

Impact Factor:

American Institute of Mathematical Sciences

The term structure of sharpe ratios and arbitrage-free asset pricing in continuous time

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

The term structure of sharpe ratios and arbitrage-free asset pricing in continuous time

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors