Executive summary
Merton's classic solution to the optimal investment problem tells an investor to hold a constant fraction of wealth in the risky asset — a conclusion that rests entirely on volatility being constant. This project reworks the problem under the Jourdain–Sbai stochastic volatility dynamics, derives the associated Hamilton–Jacobi–Bellman equation for a CRRA investor, and solves it numerically. The result is an allocation that is no longer constant: it acquires an intertemporal hedging component whose sign is governed by the correlation between the asset and its volatility.
Methodology
- Model setup. Specification of the wealth dynamics under the Jourdain–Sbai stochastic volatility framework, in which the instantaneous variance is driven by its own diffusion correlated with the asset's Brownian motion, and statement of the admissibility conditions on the control.
- Dynamic programming. Derivation of the HJB equation for a CRRA investor maximising expected utility of terminal wealth, and reduction of the problem through the homogeneity of the value function in wealth — which separates the wealth dimension from the volatility state variable.
- Verification. Checking that the candidate value function satisfies the verification theorem, so the numerical solution is genuinely optimal rather than merely a stationary point.
- Numerical resolution. Finite-difference solution of the reduced PDE on the volatility–time grid, with an implicit scheme for stability and a policy-iteration step to recover the optimal control at each node.
- Monte Carlo validation. Simulation of the wealth process under the computed policy, using a discretisation scheme appropriate to the volatility dynamics, and comparison of realised certainty-equivalent wealth against the constant-volatility Merton benchmark.
- Sensitivity analysis. Behaviour of the optimal allocation across risk aversion, correlation, volatility of volatility and investment horizon.
Optimal allocation across volatility states
The interactive chart could not be loaded (the Plotly CDN is unreachable). The full analysis is available in the PDF write-up.
Illustrative data — myopic curve computed analytically, hedging demand shown schematically pending the final numerical results.
Key findings
The optimal weight splits into a myopic term, identical in form to Merton's constant-volatility solution but evaluated at the current variance, and an intertemporal hedging term that exists only because volatility is stochastic and correlated with the asset.
- Under the negative asset–volatility correlation typical of equity markets, the hedging demand is negative: the investor holds less risky asset than the myopic rule prescribes, because the asset already pays off poorly in exactly the states where investment opportunities deteriorate.
- The gap between the two policies widens with the volatility of volatility and with the investment horizon, and collapses to zero as the horizon shortens — a short-horizon investor is effectively myopic.
- Ignoring stochastic volatility is not a neutral simplification: it systematically overstates the optimal risky exposure in the parameter region relevant to equity portfolios.