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

Patrick Beißner; Emanuela Rosazza Gianin

doi:10.3934/puqr.2021002

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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

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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 $ {\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:

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[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.
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[8]	Berger, M. A., A Malliavin-type anticipative stochastic calculus, The Annals of Probability, 1988, 16(1): 231-245. doi: 10.1214/aop/1176991897.
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[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.
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[17]	Brunnermeier, M. K., Papakonstantinou, F. and Parker, J. A., Optimal time-inconsistent beliefs: misplaning, procrastination, and commitment, Working Paper, 2013.
[18]	Brunnermeier, M. K. and Parker, J. A., Optimal expectations, American Economic Review, 2005, 95(4): 1092-1118. doi: 10.1257/0002828054825493.
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[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.
[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.
[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.
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Figure 1. Illustration of an EMM-string in the set $ {\cal Q} $ of equivalent martingale measures.

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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).

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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

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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

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

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