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Optimal portfolios with anticipating information on the stochastic interest rate

Bernardo D'Auria, José Antonio Salmerón

TL;DR

This work studies optimal portfolio selection when an informed trader has anticipatory information about the future path of a stochastic short-rate, using enlargement of filtrations under Jacod's hypothesis. It develops two modeling frameworks—a general affine diffusion for the short rate (encompassing Vasicek) and a Markov-modulated regime-switching variant—and derives explicit expressions for the additional expected logarithmic utility and the corresponding optimal strategy under the enlarged information flow. The main contributions are explicit formulas for the information drift and the optimal portfolio in the presence of anticipating information, plus a detailed treatment of non-Gaussian information via vector measures and a numerical demonstration of gains under different information structures and correlations. The results underscore the potential for substantial, even unbounded, gains from precise anticipating information about the short rate, while also providing tractable methodology for incorporating such information into continuous-time portfolio optimization with logarithmic utility.

Abstract

By employing the technique of enlargement of filtrations, we demonstrate how to incorporate information about the future trend of the stochastic interest rate process into a financial model. By modeling the interest rate as an affine diffusion process, we obtain explicit formulas for the additional expected logarithmic utility in solving the optimal portfolio problem. We begin by solving the problem when the additional information directly refers to the interest rate process, and then extend the analysis to the case where the information relates to the values of an underlying Markov chain. The dynamics of this chain may depend on anticipated market information, jump at predefined epochs, and modulate the parameters of the stochastic interest rate process. The theoretical study is then complemented by an illustrative numerical analysis.

Optimal portfolios with anticipating information on the stochastic interest rate

TL;DR

This work studies optimal portfolio selection when an informed trader has anticipatory information about the future path of a stochastic short-rate, using enlargement of filtrations under Jacod's hypothesis. It develops two modeling frameworks—a general affine diffusion for the short rate (encompassing Vasicek) and a Markov-modulated regime-switching variant—and derives explicit expressions for the additional expected logarithmic utility and the corresponding optimal strategy under the enlarged information flow. The main contributions are explicit formulas for the information drift and the optimal portfolio in the presence of anticipating information, plus a detailed treatment of non-Gaussian information via vector measures and a numerical demonstration of gains under different information structures and correlations. The results underscore the potential for substantial, even unbounded, gains from precise anticipating information about the short rate, while also providing tractable methodology for incorporating such information into continuous-time portfolio optimization with logarithmic utility.

Abstract

By employing the technique of enlargement of filtrations, we demonstrate how to incorporate information about the future trend of the stochastic interest rate process into a financial model. By modeling the interest rate as an affine diffusion process, we obtain explicit formulas for the additional expected logarithmic utility in solving the optimal portfolio problem. We begin by solving the problem when the additional information directly refers to the interest rate process, and then extend the analysis to the case where the information relates to the values of an underlying Markov chain. The dynamics of this chain may depend on anticipated market information, jump at predefined epochs, and modulate the parameters of the stochastic interest rate process. The theoretical study is then complemented by an illustrative numerical analysis.

Paper Structure

This paper contains 12 sections, 13 theorems, 95 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Affine model and notation
  3. Optimization problem
  4. Optimal portfolio
  5. Non-linear type information
  6. Half-bounded interval
  7. Bounded interval
  8. Non-atomic information
  9. Markovian modulation
  10. Numerical example
  11. Conclusions
  12. Appendix

Key Result

Proposition 2.4

Let $M=(M_t, \, 0 \leq{t} \leq{T})$ be an $\mathbb{F}$-local martingale and let $G$ be an $\mathcal{F}_T$-measurable random variable satisfying Jacod's hypothesis. Then, the process is a $\mathbb{G}$-local martingale.

Figures (4)

  • Figure 1: Plots for $\max_{k}\{\pi_t^{\mathbb{H}}(k)\}$ and $\min_{k}\{\pi_t^{\mathbb{H}}(k)\}$. The black is for the Natural trader, middle-gray for the Interval trader and light-gray for the Precise one.
  • Figure 2: Comparison of the average of the logarithmic gains for $\rho = 0.75$ (left) and $\rho = 0.25$ (right). The black is for the Natural trader, middle-gray for the Interval trader and light-gray for the Precise one.
  • Figure 3: Plot for $\max_{k}\{\pi_t^{\mathbb{H}}(k)\}$ and $\min_{k}\{\pi_t^{\mathbb{H}}(k)\}$. The black is for the Natural trader, middle-gray for the Markov-modulated one and light-gray for the Precise one.
  • Figure 4: Comparison of the average of the logarithmic gains for $c = 0.25$ (left) and $c = 1.25$ (right). The black is for the Natural trader, middle-gray for the Markov-modulated one and light-gray for the Precise one.

Theorems & Definitions (35)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.4
  • Lemma 3.1
  • proof
  • Proposition 3.2
  • proof
  • Remark 3.3
  • Example 3.4
  • Remark 3.5
  • ...and 25 more