# American Institute of Mathematical Sciences

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

 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.

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:

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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.

##### References:
Illustration of an EMM-string in the set ${\cal Q}$ of equivalent martingale measures.
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).
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 $\{x_\tau\}_{\tau\in[t,T]}$ 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
 Approach Claim $X$ Payoff Stream $\{x_\tau\}_{\tau\in[t,T]}$ 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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