Asset-liability management with Epstein-Zin utility under stochastic interest rate and unknown market price of risk

Technical Summary: Asset-Liability Management with Epstein-Zin Utility Under Stochastic Interest Rate and Unknown Market Price of Risk

Problem Formulation
This paper addresses a continuous-time consumption–investment problem within an Asset-Liability Management (ALM) framework. The investor operates under partial information, where the market price of risk is unobservable, and faces a terminal liability $K_T$ at the horizon $T$. The investor's preferences are modeled using Epstein-Zin recursive utility, which disentangles the coefficient of relative risk aversion ($\gamma$) from the elasticity of intertemporal substitution ($\psi$).

The financial market extends the classical Kim–Omberg model to include:

1. A money market account with a stochastic short-term interest rate following a Vasicek (Ornstein-Uhlenbeck) process.
2. A stock and a zero-coupon bond, both with stochastic volatility and interest rate dependence.
3. An unobservable stochastic market price of risk process $R_t$, which drives the stock's excess return.

The investor observes only the interest rate and the stock price, necessitating the use of stochastic filtering to estimate the latent market price of risk. The objective is to maximize the Epstein-Zin utility of the consumption stream and the terminal net wealth ($X_T - K_T$), subject to the constraint that strategies must be adapted to the observation filtration rather than the full filtration.

Methodology
The solution approach involves three primary technical stages:

1. Reduction to Observable Filtration:
   Using stochastic filtering theory, the unobservable market price of risk $R_t$ is replaced by its conditional mean $m_t$ and variance $v_t$ given the observation filtration $\mathcal{F}^{r,S}_t$. The dynamics of the wealth process are reformulated in terms of the "innovation process" (a Brownian motion under the observation filtration), rendering all coefficients observable.

2. Ansatz and FBSDE Formulation:
   The presence of the terminal liability prevents the direct application of standard homothetic ansatzes used in recursive utility literature. The author proposes a new ansatz for the value process:
   $$\hat{V}_t = e^{-\delta\theta t} \frac{(X_t + e^{\int_0^t r_s ds} Y_t)^{1-\gamma}}{1-\gamma}$$
   where $Y_t$ is an auxiliary process representing the discounted future liability exposure. Applying the martingale optimality principle leads to a coupled linear Forward-Backward Stochastic Differential Equation (FBSDE) with unbounded random coefficients. This equation involves the wealth process $X_t$ and the auxiliary process $Y_t$ (and its diffusion term $Z_t$).

3. Decoupling and Solvability:
   Standard well-posedness theories for coupled FBSDEs (relying on monotonicity or Lipschitz conditions) are inapplicable due to the unbounded nature of the coefficients. The author develops a decoupling reduction method that exploits the specific structure of the equation. Unlike previous approaches that reduce coupled linear FBSDEs to Backward SDEs (BSDEs), this method reduces the problem to a system of Forward SDEs.

   • Under specific exponential integrability assumptions on the market price of risk, the author proves the existence and uniqueness of the solution.
   • The solution yields explicit expressions for the optimal consumption and portfolio strategies.

4. Malliavin Calculus and Representation:
   To facilitate numerical implementation, the paper establishes the Malliavin differentiability of the solution. Using the Clark-Ocone formula under a change of measure, the optimal investment strategy is represented as conditional expectations involving Malliavin derivatives. This representation allows for efficient approximation via Monte Carlo simulation.

Key Contributions
The paper claims the following specific contributions:

• Novel Framework: It formulates the first continuous-time consumption–investment problem combining Epstein-Zin recursive utility, terminal liabilities, and partial information. The liability is allowed to depend on the entire path of risky assets.
• Structural Innovation: The introduction of terminal liabilities necessitates a new ansatz for the value process, leading to a coupled linear FBSDE with unbounded coefficients—a class not previously encountered in recursive utility optimization.
• Analytical Breakthrough: The author develops a decoupling reduction method that solves this specific class of FBSDEs by reducing them to Forward SDEs, bypassing the limitations of existing well-posedness theories.
• Explicit Solutions: The methodology yields explicit expressions for the optimal consumption strategy, portfolio allocation, and value function in a stochastic volatility market with stochastic interest rates and unobservable risk premiums.
• Computational Feasibility: By establishing Malliavin differentiability, the optimal strategy is expressed in a form amenable to Monte Carlo simulation.
• Quantification of Learning Value: The paper quantifies the welfare loss incurred when investors ignore the opportunity to learn about the market price of risk (i.e., acting as if the risk premium is constant).

Results

• Optimal Strategies: The optimal consumption rate $c^*_t$ and portfolio weights $\pi^*_t$ are derived explicitly in terms of the filtered state variables and the solution to the reduced Forward SDE system.
• Value Function: An explicit closed-form expression for the value function is provided, dependent on initial wealth, the expected discounted liability, and model parameters.
• Welfare Loss Analysis: Numerical analysis reveals that:
  • In the presence of liabilities, the welfare loss (the percentage of initial wealth an investor would sacrifice to access learning) increases with initial wealth. This contrasts with the liability-free case, where welfare loss is independent of initial wealth.
  • Welfare loss decreases with the coefficient of relative risk aversion ($\gamma$) and increases with the elasticity of intertemporal substitution ($\psi$).
  • The welfare loss is generally smaller under CRRA utility (where $\gamma\psi=1$) than under general Epstein-Zin preferences, as higher risk aversion in the CRRA case is tied to lower substitution elasticity, reducing the value of learning.

Significance
The paper positions itself as providing the first analysis of an Epstein-Zin ALM problem under partial information. Its significance lies in bridging the gap between theoretical recursive utility optimization and practical ALM constraints (liabilities) under realistic market conditions (unobservable risk premiums). By overcoming the mathematical challenges posed by unbounded coefficients in coupled FBSDEs, the work extends the applicability of the Kim–Omberg framework and offers a computationally tractable method for evaluating the economic value of information in asset management.