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

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.
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  • 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.

    Mathematics Subject Classification: 60H10; 60K35.


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

    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).

    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]} $
    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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