Mind the cap!—constrained portfolio optimisation in Heston's stochastic volatility model

AbstractWe consider a portfolio optimisation problem for a utility-maximising investor who faces convex constraints on his portfolio allocation in Heston's stochastic volatility model. We apply existing duality methods to obtain a closed-form expression for the optimal portfolio allocation. In doing so, we observe that allocation constraints impact the optimal constrained portfolio allocation in a fundamentally different way in Heston's stochastic volatility model than in the Black Scholes model. In particular, the optimal constrained portfolio may be different from the naive ‘capped’ portfolio, which caps off the optimal unconstrained portfolio at the boundaries of the constraints. Despite this difference, we illustrate by way of a numerical analysis that in most realistic scenarios the capped portfolio leads to slim annual wealth equivalent losses compared to the optimal constrained portfolio. During a financial crisis, however, a capped solution might lead to compelling annual wealth equivalent losses.Keywords: Portfolio optimisationAllocation constraintsDynamic programmingHeston's stochastic volatility modelIncomplete marketsJEL Classifications: G11C61 Disclosure statementNo potential conflict of interest was reported by the author(s).Supplemental dataSupplemental data for this article can be accessed online at http://dx.doi.org/10.1080/14697688.2023.2271223.Notes1 Note that obtaining and formally verifying the optimality of a candidate portfolio process requires more than just a solution to the associated HJB PDE, as pointed out by Korn and Kraft (Citation2004).2 As any π∈Λ can only take finite values L[0,T]⊗Q-a.s., we do not need to distinguish between (−∞,β] and [−∞,β] or [α,∞) and [α,∞] for any −∞≤α,β≤∞.3 Technically, one can formulate this assumption less restrictively by expressing ‘No Blow-Up’ in terms of the time spent in each of the zones Z−, Z0 and Z+. However, as this would significantly complicate the presentation without adding major additional insights, it is omitted here.4 If ρ=0 all of these transition times will be infinite.5 Using a similar separation with respect to the zones Z−, Z0 and Z+ and equation (B6), it is also possible to determine a closed-form expression for A from lemma 2.1.6 Equation (Equation18(18) b1−bη(κρσ+η2)<κ22σ2,(18) ) corresponds to part (i) of Assumption 2.4. In the setting of Kraft (Citation2005), part (ii) of Assumption 2.4 is also implied by (Equation18(18) b1−bη(κρσ+η2)<κ22σ2,(18) ) and so does not have to be mentioned explicitly.7 Note that this is different from classic mean-variance optimisation, where the variance of the terminal portfolio wealth Vv0,π(T) is constrained.8 Q.ai (Citation2022) reported that the average length of an S&P500 bear market (defined as a period with drawdown in excess of 20%) was 289 days.9 Since we exclusively work with power utility functions in this paper, we may without loss of generality assume that the WEL is independent of wealth.10 If π is deterministic and Jπ is the unique solution to the Feynman–Kac PDE, one can use an exponentially affine ansatz to characterise Jπ in terms of the solutions to a system of ODEs. If the ODE solutions are given, then the WEL Lπ(0,z0) is known in closed form. We provide a description of this approach in lemma B.6 and corollary B.7 in the supplementary material. In our studies, we approximated the corresponding ODE solutions by an Euler method.11 For an overview, consider e.g. table 4 in Escobar-Anel and Gschnaidtner (Citation2016). Here, the authors consider values of κ=3.5, σ=0.3, ρ=−0.4 as an ‘Average Case’ for their reviewed literature.