Bond market completeness under stochastic strings with distribution-valued strategies

Abstract

In stochastic string models, the bond market is complete if trading strategies are distribution-valued processes

Acknowledgments

The authors are very grateful to the editors Michael Dempster and Jim Gatheral and two anonymous referees for their truly helpful comments and suggestions over the previous versions of the paper. We also thank the valuable comments received from Alejandro Balbás, Natividad Blasco, and attendants at the XV Iberian-Italian Congress of Financial and Actuarial Mathematics and the XXII Spanish Finance Forum.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Aihara and Bagchi (2005) claim to have obtained bond market completeness in their model, but Barski et al. (2011, Remark 3.9) affirm the following: ‘It was pointed out by one of the reviewers that the market considered in Aihara and Bagchi (2005) is approximately complete but the limit passage, performed in the Proof of Theorem 4.1 to get completeness, is not correct.’

2 Jarrow and Madan (1999) prove a version of the Second Fundamental Theorem that states the equivalence between the uniqueness of the EMM and market completeness just if a certain fundamental operator is open. We will return to this result in section 6.

3 The stochastic string modeling was initiated in Santa-Clara and Sornette (2001). For technical reasons, we will focus here on the reformulation and extension of this framework that is made in Bueno-Guerrero et al. (2015, 2016).

4 Nevertheless, as Santa-Clara and Sornette (2001, p. 169) pointed out:

Economics has little to say with respect to the differentiability of forward rate curves, and unfortunately, this issue can not be resolved empirically. […] The decision to use string shocks that produce differentiable curves is thus fundamentally a matter of taste.

5 By admissibility we mean that for each t, $c(t,\cdot ,\cdot )$ is symmetric, positive semidefinite and satisfies $\left|c\right(t,x,y\left)\right|\le 1$ and $c(t,x,x)=1,\mathrm{\forall }x,y\ge 0$ Santa-Clara and Sornette 2001.

6 For example, in Bueno-Guerrero et al. (2018), this expression is applied, with Malliavin calculus techniques, to continuous arithmetic and geometric Asian call options, obtaining hedging portfolios that have a part with a continuum of bonds.

7 Bueno-Guerrero et al. (2016) actually work with $L_p^2\left({\mathbb{R}}^{+}\right)$ for consistency purposes. It is not difficult to check that the results in their paper can also be obtained working with $L^2\left({\mathbb{R}}^{+}\right)$.

8 Our model fits into the model of Jarrow and Madan just by exchanging their dual pair $(\mathbb{X},\mathbb{Y})$ by our pair $(\mathcal{D},{\mathcal{D}}^{\mathrm{\prime }})$. However, we believe that our approach is more sound from a financial point of view since it uses specific market prices of risk.

9 We adopt the notation $\overline{\sigma }(t,T)$ and $\sigma (t,x)\equiv \overline{\sigma }(t,t+x)$ for volatilities in the parameterizations of the maturity time T and time to maturity x, respectively.

10 Infinitely differentiable HJM volatilities are common in the literature. See, for example, Ritchken and Sankarasubramanian (1995) and Mercurio and Moraleda (2000).

Additional information

Funding

Moreno and Navas gratefully acknowledge financial support by Ministerio de Economia y Competitividad, Spain [grant number ECO2017-85927-P] and Consejería de Innovación, Ciencia y Empresa, Junta de Andalucía [grant number P12-SEJ-1733]. All the authors gratefully acknowledge financial support by Consejería de Educación, Cultura y Deportes, Junta de Comunidades de Castilla-La Mancha [grant number PPII-2014-030-P], [grant number SBPLY/19/180501/000172]. The usual caveat applies.