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Time-Consistent Mean-Variance Portfolio Selection in Discrete and Continuous Time

Christoph Czichowsky

arXiv:1205.4748v1q-fin.PMmath.OC

TL;DR

Mean-variance portfolio selection is time-inconsistent because it does not satisfy Bellman’s optimality principle. The paper formulates local mean-variance efficiency in a general semimartingale setting, then shows that discrete-time solutions converge to the continuous-time solution under stated conditions.

  • Problem

    Mean-variance portfolio selection is time-inconsistent, so its conditional criterion does not satisfy Bellman’s optimality principle.

    Source: §1 · p. 2, §2 · p. 7

  • Method

    The paper defines time-consistent local mean-variance efficiency and characterizes locally optimal strategies using the structure condition and Föllmer–Schweizer decomposition.

    Source: §1 · p. 3, §3 · p. 8

  • Results

    As the partition mesh tends to zero, the discrete-time locally mean-variance efficient strategies converge in L2(M) to the continuous-time strategy under the stated assumptions.

    Source: §5 · p. 39

  • Takeaways & Limitations

    The continuous-time formulation is supported as the natural extension of the discrete-time recursive formulation.

    Source: §5 · p. 29

    The formulation assumes a finite horizon and a filtration satisfying the usual completeness and right-continuity conditions.

    Source: §2 · p. 4

Abstract

from arXiv · show

It is well known that mean-variance portfolio selection is a time-inconsistent optimal control problem in the sense that it does not satisfy Bellman's optimality principle and therefore the usual dynamic programming approach fails. We develop a time- consistent formulation of this problem, which is based on a local notion of optimality called local mean-variance efficiency, in a general semimartingale setting. We start in discrete time, where the formulation is straightforward, and then find the natural extension to continuous time. This complements and generalises the formulation by Basak and Chabakauri (2010) and the corresponding example in Bj\"ork and Murgoci (2010), where the treatment and the notion of optimality rely on an underlying Markovian framework. We justify the continuous-time formulation by showing that it coincides with the continuous-time limit of the discrete-time formulation. The proof of this convergence is based on a global description of the locally optimal strategy in terms of the structure condition and the F\"ollmer-Schweizer decomposition of the mean-variance tradeoff. As a byproduct, this also gives new convergence results for the F\"ollmer-Schweizer decomposition, i.e. for locally risk minimising strategies.

1 Introduction

Mean-variance portfolio selection is explicit in one period but becomes delicate across time because its criterion violates Bellman’s optimality principle. The paper develops a locally optimal, time-consistent formulation beyond the Markovian setting and shows its continuous-time version is the limit of discrete-time solutions.

1 Introduction

  • The mean-variance criterion does not satisfy Bellman’s optimality principle, making multiperiod and continuous-time portfolio selection more delicate than the one-period problem.The one-period criterion measures return by expectation and risk by variance, yielding an explicit mean-variance efficient strategy.Source: §1 · p. 2
  • The paper seeks a strategy that is optimal for conditional mean-variance preferences and remains optimal on every remaining interval, without relying on a Markovian framework.The authors note that a general time-consistent formulation was an open problem, while Basak and Chabakauri’s formulation used a Markovian framework.Source: §1 · pp. 2–3
  • The authors adapt Strotz’s recursive approach by optimizing only over future strategies the investor will actually follow, rather than all available strategies.In discrete time this yields backward recursion; continuous time requires combining the recursion with a limit argument.Source: §1 · pp. 2–3
  • The formulation introduces local mean-variance efficiency for a general semimartingale framework and connects the optimal strategy to the structure condition and Föllmer–Schweizer decomposition.The authors state that this characterization gives necessary and sufficient conditions for existence.Source: §1 · p. 3
  • The continuous-time formulation coincides with the continuous-time limit of the discrete-time formulation, supported by a global characterization of the locally optimal strategy.The authors also report new convergence results for the Föllmer–Schweizer decomposition and locally risk-minimizing strategies.Source: §1 · pp. 3–4
  • Cui et al.’s alternative allows withdrawals and a free cash-flow stream, but the authors distinguish it from their notion of time consistency.The related strategy achieves the same terminal mean-variance pair as the Markowitz solution, but is not time consistent in the authors’ sense.Source: §1 · p. 4

2 Formulation of the problem and preliminaries

The section sets up mean-variance portfolio selection in a semimartingale market and explains why the usual static Markowitz objective does not yield time-consistent conditional decisions. It motivates replacing global optimization with local optimization over the next trading interval.

2 Formulation of the problem and preliminaries

  • The market has one riskless asset, d risky semimartingale assets, and self-financing strategies whose gains are square-integrable.The setup assumes a finite horizon and a filtration satisfying the usual completeness and right-continuity conditions.Source: §2 · p. 4
  • The paper assumes a square-integrable representative for the price process without loss of generality, preserving the same wealth processes as the original market.The authors note that this parameterization can extend local risk minimization and quadratic hedging to general semimartingales in future work.Source: §2 · p. 5
  • The static Markowitz formulation optimizes terminal wealth over the whole interval using the one-period mean-variance criterion evaluated at time zero.The section relates its solution to an auxiliary stochastic optimal control problem that can be treated dynamically.Source: §2 · pp. 5–6
  • The static optimizer is not conditionally optimal later, so plugging it into the conditional problems violates time consistency and Bellman’s optimality principle.The investor’s preference for a precommitment strategy or a dynamic time-consistent strategy depends on their preferences.Source: §2 · p. 7
  • The conditional objective contains an adjustment term depending on future strategy choices, which gives the investor an incentive to deviate after the next period.At the later time, the prior gains are known and no longer pose risk to be minimized, so the investor’s choice can differ.Source: §2 · p. 7
  • The section proposes optimizing only over the next infinitesimal interval while using the optimal future strategy, following the recursive approach associated with Strotz.This local criterion is presented as intermediate between fixed-horizon dynamic optimization and step-by-step one-period optimization.Source: §2 · p. 8

3 Discrete Time

In discrete time, local mean-variance efficiency is obtained by backward recursion, optimizing each step against the conditional criterion while holding the future strategy fixed. The section characterizes existence through the structure condition and the mean-variance tradeoff, and interprets the strategy as a myopic position plus an intertemporal hedge.

3 Discrete Time

  • The discrete-time development restricts to one risky asset to motivate the continuous-time case.Trading occurs at fixed times, with each position held over the next period.Source: §3 · p. 8
  • A locally mean-variance efficient strategy optimizes the conditional criterion one step at a time, backward from the terminal date, and is time-consistent in the recursive sense.The criterion varies only the next-period position while retaining the strategy planned for the remaining horizon.Source: §3 · p. 8
  • The recursive optimality conditions do not depend on the strategy used before each step, yielding a recursive formula and uniqueness.The paper notes that this formula had appeared in a Markovian framework.Source: §3 · p. 9
  • The strategy combines the myopically mean-variance efficient position with an intertemporal hedging demand that locally minimizes risk from fluctuations in expected future gains.The additional demand vanishes when the investment opportunity set, or the terminal mean-variance tradeoff, is deterministic.Source: §3 · p. 12
  • A locally mean-variance efficient strategy exists only when the price process satisfies the structure condition and the terminal mean-variance tradeoff admits the stated decomposition.The decomposition uses a square-integrable initial term, a stochastic integral, and a square-integrable martingale strongly orthogonal to the martingale component of prices.Source: §3 · pp. 12–13
  • When the decomposition integrand is admissible, the decomposition of the terminal mean-variance tradeoff is a Föllmer–Schweizer decomposition.The paper cautions that the integrand is not generally admissible, so the strategy characterization need not always coincide with that decomposition.Source: §3 · pp. 12–13

4 Continuous Time

The section develops a continuous-time notion of local mean-variance efficiency based on local perturbations along refining partitions, and characterizes its existence using the structure condition and a global decomposition of the mean-variance tradeoff. It also characterizes and computes the efficient strategy, which combines myopic efficiency with locally risk-minimising intertemporal hedging, through Föllmer–Schweizer decompositions and iteration.

4 Continuous Time

  • LMVE requires local perturbations along any increasing sequence of partitions tending to the identity to asymptotically decrease the mean–variance tradeoff, measured on operational time B.The continuous-time formulation coincides with the discrete-time formulation under the usual embedding.Source: §4 · p. 14
  • The paper decomposes the local perturbation criterion into three terms and controls their asymptotics separately along refining partitions to characterize LMVE.The analysis establishes the asymptotic behavior of the first term and shows the other terms vanish.Source: §4 · pp. 15, 17, 18
  • The approach generalizes local risk minimisation to general semimartingales, removing the continuity of A and structure-condition assumptions in cited results.The paper identifies this extension as a byproduct of its convergence analysis.Source: §4 · p. 17
  • A continuous-time LMVE exists exactly when S satisfies the structure condition and K_T has the stated L1 decomposition with the required integrability and orthogonality properties.When K_T is in L2(P), the decomposition’s integrand belongs to Θ and the decomposition coincides with the Föllmer–Schweizer decomposition.Source: §4 · pp. 20–21
  • If S satisfies the structure condition and E(-λ • M) is regular and satisfies R2(P), an LMVE exists uniquely, using the unique Föllmer–Schweizer decomposition integrand of K_T.The corollary assumes K_T is in L2(P).Source: §4 · p. 22

Computing the LMVE strategy

  • The strategy can be obtained explicitly from the price process’s canonical decomposition and a Föllmer–Schweizer decomposition of the terminal tradeoff, making the approach more explicit than solving the static problem via the variance-optimal martingale measure.If the candidate density process is not regular or fails R2(P), the procedure may still produce a candidate that can be checked directly for square-integrability.Source: §4 · p. 23
  • If explicit derivation is unavailable, the Föllmer–Schweizer decomposition can be solved as a linear BSDE using numerical schemes that reduce to conditional expectations and martingale-representation integrands.Monte Carlo simulation and Malliavin derivatives have been suggested for this computation.Source: §4 · p. 23

Example 4.14

  • An example shows that an LMVE strategy can exist despite nonunique Föllmer–Schweizer decompositions and LMVE strategies, while the static mean-variance problem has no solution.The price process satisfies the structure condition, but admits no equivalent martingale measure.Source: §4 · pp. 23–24

Continuous-time MMVE

  • A continuous-time MMVE strategy exists exactly when the structure condition holds and the terminal mean-variance tradeoff is square-integrable; it is then unique and equals 1/γ times λ.Source: §4 · p. 26

Continuous-time LMVE and MMVE

  • The LMVE strategy consists of the MMVE strategy plus locally risk-minimising intertemporal hedging against risk from the stochastic investment opportunity set.The added hedging demand is driven by the mean-variance tradeoff process; LMVE and MMVE coincide when its terminal value is deterministic.Source: §4 · p. 27

Deterministic mean-variance tradeoff

  • When the terminal mean-variance tradeoff is deterministic, its Föllmer–Schweizer decomposition is constant and LMVE coincides with MMVE; under additional continuity or determinism conditions, the static optimum has a stated representation.The static-optimum representation additionally requires that S be continuous or the entire mean-variance tradeoff process be deterministic.Source: §4 · p. 27

Fixed-point characterization

  • The LMVE strategy is a fixed point of an operator, and bounded continuous mean-variance tradeoff makes that operator a contraction whose iteration converges from any starting strategy.The iteration is a continuous-time analogue of the discrete-time backward recursion and gives an equilibrium interpretation as an intrapersonal game.Source: §4 · p. 28

5 Convergence of solutions

This section establishes convergence of appropriately discretized models to the continuous-time formulation. Under the stated conditions, discrete-time LMVE strategies converge in L2(M) to the continuous-time LMVE strategy as the partition mesh tends to zero.

5 Convergence of solutions

  • The proof directly controls the solution structure because the discrete Doob decomposition is not obtained by simply discretizing the continuous-time martingale and finite-variation components.The discrete decomposition also contains a martingale term representing discretization error.Source: §5 · pp. 29–31
  • Under Theorem 5.5’s assumptions, discrete Föllmer–Schweizer integrands converge in L2(PM) to the continuous-time integrand when claims converge in L2(P) and the partition mesh vanishes.The result applies to the stated sequence of discretized markets and claims Hn converging to H.Source: §5 · p. 34
  • The cited result establishes that, for each fixed p ∈ N0, the sequence converges to ξ∞,p in L2(M) as n → ∞.Source: §5 · p. 34
  • A contraction with a common modulus independent of the discretization makes the discrete fixed-point iterations converge uniformly in n as the iteration count grows.These iterations construct the discrete Föllmer–Schweizer integrands.Source: §5 · pp. 35–37
  • The convergence holds for increasing partitions when the preceding convergence and boundedness conditions on K_n are satisfied.The stated bound is sup_{n≥n_0} ||K_n^T||_{L∞(P)} ≤ b for some n_0 ∈ N and b ∈ (0, 1).Source: §5 · p. 39
  • As |τ_n| → 0, the discrete-time LMVE strategy bϑ_n converges to the continuous-time strategy bϑ in L2(M).The result follows by combining the preceding convergence results for the LMVE strategies.Source: §5 · p. 39
  • The proof uses the Föllmer–Schweizer decomposition of the mean-variance tradeoff and combines Lemma 5.2, Corollary 5.4, and Theorem 5.5.Boundedness of K implies λ • M is in bmo2, yielding the stated L2(M) convergence.Source: §5 · p. 39

A Representative square-integrable portfolios

The appendix constructs representative square-integrable portfolios that generate the same wealth processes as the original market. It uses a pointwise characterization of σ-square-integrability to establish their existence.

A Representative square-integrable portfolios

  • σ-square-integrability can be characterized pointwise, making it sufficient for finding representative square-integrable portfolios.The proof relates global square-integrability of a strategy to σ-square-integrability of its stochastic integral, then uses a one-to-one correspondence with square-integrable integrands.Source: A Representative square-integrable… · p. 40
  • The proof uses the canonical characteristics of S to establish the pointwise conditions needed for representative portfolios.These characteristics describe the drift, continuous covariation, and jump compensator relative to B.Source: A Representative square-integrable… · p. 40
  • The constructed representative portfolios preserve all wealth processes: ΘS • S = ΘeS • eS.Each representative strategy belongs to ΘS, and the transformed market is defined by eSi = ϕi • S.Source: A Representative square-integrable… · pp. 39–40
  • A strictly positive bounded predictable scaling converts a σ-square-integrable semimartingale into a square-integrable one, and conversely.This lets the proof construct representative portfolios by switching between the two integrability classes.Source: A Representative square-integrable… · pp. 40–41
  • In multiple dimensions, the construction accounts for linear dependence among components by using a subspace of admissible integrands.The one-dimensional case is simpler; the multidimensional proof constructs bounded elements of V and scales them to obtain square-integrable portfolios.Source: A Representative square-integrable… · p. 41
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