The Model Set-up

Table of Links

Abstract and 1. Introduction

2. The Model Set-up
3. Price Impact in a Market with No Transaction Costs
4. Price Impact in a Market with Transaction Costs

Appendix A. Proofs of Section 3

Appendix B. Proofs of Section 4

References

2. The Model Set-up

Remark 2.1. Considering only the diffusion part of the investors’ random endowment in (3) comes without loss of generality. Indeed, as stated in [HMKP21], since the solution to the investors’ objective functional (see (5) below) is invariant to any additional finite variation part or to endowment’s shocks that are orthogonal to the market. In other words, under suitable integrability assumptions, (3) could be generalized to:

and impose the following standing assumption:

The above investment strategy shall be called the competitive frictionless optimal portfolio of investor m. Note that the first term of (6) is the classic quadratic optimal portfolio (also called Merton’s portfolio), while the second reflects the investor’s hedging needs.

Under competitive market structure with no transaction costs, the local return process ν of (1) is determined simply by the market-clearing condition, where a zero-net supply is imposed:

Without transaction costs and when all investors are price takers, we readily get from (6) and (7), the explicit form for the equilibrium return process, henceforth called frictionless competitive equilibrium returns:

where the aggregate investors’ risk exposure and risk tolerance are defined as:

Due to Assumption 2.2, we directly verify that µ ∈ H2 . Equilibrium (8) is indicative for the discussion that follows. Say d = 1, for simplicity. Assuming that the aggregate investors’ exposure to market shocks is higher than the corresponding noise demand, the equilibrium returns increase when aggregate risk tolerance decreases and the asset’s volatility increases. This is intuitive: lower risk tolerance and higher risk means that investors decrease their demand for the asset and hence equilibrium returns increase. Note that higher equilibrium returns essentially imply that investors require higher expected returns in order to invest on the asset, a situation that is consistent to lower demand. On the other hand, higher exposure to market shocks increases investors’ demand and hence decreases the required expected returns at the equilibrium.