Variance-optimal Martingale Measure Research Articles

Mean-variance portfolio selection with non-linear wealth dynamics and random coefficients

This paper studies the continuous time mean-variance portfolio selection problem with one kind of non-linear wealth dynamics. To deal the expectation constraint, an auxiliary stochastic control problem is firstly solved by two new generalized stochastic Riccati equations from which a candidate portfolio in feedback form is constructed, and the corresponding wealth process will never cross the vertex of the parabola. In order to verify the optimality of the candidate portfolio, the convex duality (requires the monotonicity of the cost function) is established to give another more direct expression of the terminal wealth level. The variance-optimal martingale measure and the link between the non-linear financial market and the classical linear market are also provided. Finally, we obtain the efficient frontier in closed form. From our results, people are more likely to invest their money in riskless asset compared with the classical linear market.

Quadratic Hedging and Optimization of Option Exercise Policies

Quadratic hedging of option payoffs generates the variance optimal martingale measure. When an option features an exercise policy and its cash flows are hedged according to this approach, it may be tempting to optimize such a policy under this measure. Because the variance optimal martingale measure may not be an equivalent probability measure, focusing on American options we show that the resulting exercise policy may be unappealing. This drawback can sometimes be remedied by imposing time consistency on exercise policies, but in general persists even in this case, which compounds the familiar issue that valuing an option using this measure may not result in an arbitrage free value. An alternative and known approach that bypasses both of these pitfalls consists of optimizing option exercise policies under an equivalent martingale measure and anchoring quadratic hedging to the resulting value of this policy. Additional research may assess on realistic applications the magnitude of the limitations associated with optimizing option exercise policies based on the variance optimal martingale measure.

Discrete-Time Quadratic-Optimal Hedging for European Contingent Claims

We revisit the problem of optimally hedging a European contingent claim (ECC) using a hedging portfolio consisting of a risky asset that can be traded at pre-specified discrete times. The objective function to be minimized is either the second-moment or the variance of the hedging error calculated in the market probability measure. The main outcome of our work is to show that unique solutions exist in a larger class of admissible strategies under integrability and non-degeneracy conditions on the hedging asset price process that are weaker than previously thought possible. Specifically, we do not require the hedging asset price process to be square-integrable, and do not use the bounded mean-variance trade off assumption. Our criterion for admissible strategies only requires the cumulative trading gain, and not the incremental trading gains, to be square integrable. We derive explicit expressions for the second-moment and the variance of the hedging error to arrive at the respective optimal hedging strategies. We use the expressions mentioned above to also give explicit solutions to two constrained mean-variance frontier problems, namely, minimizing the variance subject to a lower bound on the mean profit, and maximizing the mean profit subject to an upper bound on the variance. Further, we explain the connections between our solution and that of the previous formulations. Finally, we identify the associated variance-optimal martingale measure and provide an expression for the L2-approximation price of the hedged ECC in that measure.

Mean-Variance Hedging Based on an Incomplete Market with External Risk Factors of Non-Gaussian OU Processes

We prove the global risk optimality of the hedging strategy of contingent claim, which is explicitly (or called semiexplicitly) constructed for an incomplete financial market with external risk factors of non-Gaussian Ornstein-Uhlenbeck (NGOU) processes. Analytical and numerical examples are both presented to illustrate the effectiveness of our optimal strategy. Our study establishes the connection between our financial system and existing general semimartingale based discussions by justifying required conditions. More precisely, there are three steps involved. First, we firmly prove the no-arbitrage condition to be true for our financial market, which is used as an assumption in existing discussions. In doing so, we explicitly construct the square-integrable density process of the variance-optimal martingale measure (VOMM). Second, we derive a backward stochastic differential equation (BSDE) with jumps for the mean-value process of a given contingent claim. The unique existence of adapted strong solution to the BSDE is proved under suitable terminal conditions including both European call and put options as special cases. Third, by combining the solution of the BSDE and the VOMM, we reach the justification of the global risk optimality for our hedging strategy.

The use of BSDEs to characterize the mean–variance hedging problem and the variance optimal martingale measure for defaultable claims

In this paper, we consider the problem of mean–variance hedging of a defaultable claim. We assume the underlying assets are jump processes driven by Brownian motion and default processes. Using the dynamic programming principle, we link the existence of the solution of the mean–variance hedging problem to the existence of solution of a system of coupled backward stochastic differential equations (BSDEs). First we prove the existence of a solution to this system of coupled BSDEs. Then we give the corresponding solution to the mean–variance hedging problem. Finally, we give some existence conditions and characterize the well known variance optimal martingale measure (VOMM) using the solution to the first quadratic BSDE with jumps that we derived from the previous stochastic control problem. We conclude with an explicit example of our credit risk model giving a numerical application in a two defaults case.

MEAN VARIANCE HEDGING IN A GENERAL JUMP MARKET

We consider a financial market in which the discounted price process S is an ℝd-valued semimartingale with bounded jumps, and the variance-optimal martingale measure (VOMM) Q opt is only known to be a signed measure. We give a backward semimartingale equation (BSE) and show that the density process Z opt of Q opt with respect to P is a possibly non-positive stochastic exponential if and only if this BSE has a solution. For a general contingent claim H, we consider the following generalized version of the classical mean-variance hedging problem [Formula: see text] where [Formula: see text]. We represent the optimal strategy and the optimal cost of the mean-variance hedging by means of another backward martingale equation (BME) and an appropriate predictable process δ both with a straightforward intuitive interpretation.

Mean Variance Hedging in a General Jump Model

We consider the mean-variance hedging of a contingent claim H when the discounted price process S is an -valued quasi-left continuous semimartingale with bounded jumps. We relate the variance-optimal martingale measure (VOMM) to a backward semimartingale equation (BSE) and show that the VOMM is equivalent to the original measure P if and only if the BSE has a solution. For a general contingent claim, we derive an explicit solution of the optimal strategy and the optimal cost of the mean-variance hedging by means of another BSE and an appropriate predictable process δ

Change of filtrations and mean–variance hedging

We consider the mean–variance hedging (MVH) problem (under measure P) of two kinds of investors for two different levels of information, described by two filtrations and such that . Under the assumption that there exists a measure such that all -martingales are -martingales, we give the variance-optimal martingale measure (VOMM) with respect to and through a couple of stochastic Riccati equation (SRE)s, which can be viewed as the same SRE with differential terminal value under . Then we derive an explicit form of the optimal mean–variance strategy and the optimal costs with respect to and . We describe the concept of -no-value-to-investment in the means of mean–variance, and for a given contingent claim , we compare their optimal costs with respect to and .

On the structure of general mean-variance hedging strategies

We provide a new characterization of mean-variance hedging strategies in a general semimartingale market. The key point is the introduction of a new probability measure $P^{\star}$ which turns the dynamic asset allocation problem into a myopic one. The minimal martingale measure relative to $P^{\star}$ coincides with the variance-optimal martingale measure relative to the original probability measure $P$.

Investment with Sequence Losses in an Uncertain Environment and Mean-Variance Hedging

In a market with a discontinuous filtration, whose price is influenced by a random factor, we study an optimization problem of an investor who is facing a sequence of losses driven by a Cox process. We give a form of variance-optimal martingale measure by changing the filtration. By using the solutions of the stochastic Riccati equation and another associated backward stochastic equation, we obtain a solution of the optimization problem of the investor.

A Counterexample Concerning the Variance-Optimal Martingale Measure

The present note addresses an open question concerning a sufficient characterization of the variance-optimal martingale measure. Denote by S the discounted price process of an asset and suppose that Q* is an equivalent martingale measure whose density is a multiple of 1 − \varphi • S_T for some S-integrable process \varphi. We show that Q* does not necessarily coincide with the variance-optimal martingale measure, not even if \varphi • S is a uniformly integrable Q*-martingale.

Mean-Variance Hedging and Optimal Investment in Heston's Model with Correlation

This paper solves the mean-variance hedging problem in Heston's model with a stochastic opportunity set moving systematically with the volatility of stock returns. We allow for correlation between stock returns and their volatility (so-called leverage effect). Our contribution is threefold: using a new concept of opportunity-neutral measure we present a simplified strategy for computing a candidate solution in the correlated case. We then go on to show that this candidate generates the true variance-optimal martingale measure; this step seems to be partially missing in the literature. Finally, we derive formulas for the hedging strategy and the hedging error.

SOME REMARKS ON MEAN-VARIANCE HEDGING FOR DISCONTINUOUS ASSET PRICE PROCESSES

Mean-variance hedging for the discontinuous semimartingale case is obtained under some assumptions related to the variance-optimal martingale measure. In the present paper, two remarks on it are discussed. One is an extension of Hou–Karatzas' duality approach from the continuous case to discontinuous. Another is to prove that there is the consistency with the case where the mean-variance trade-off process is continuous and deterministic. In particular, one-dimensional jump diffusion models are discussed as simple examples.

Some properties of the variance-optimal martingale measure for discontinuous semimartingales

We focus on properties of the variance-optimal martingale measure for discontinuous semimartingales. In particular, we give sufficient conditions for the variance-optimal martingale measure to be a probability measure, and for the density process of the variance-optimal martingale measure to satisfy the reverse Hölder inequality, respectively. Moreover, we study relationship with mean–variance hedging.

An extension of mean-variance hedging to the discontinuous case

Our goal in this paper is to give a representation of the mean-variance hedging strategy for models whose asset price process is discontinuous as an extension of Gourieroux, Laurent and Pham (1998) and Rheinlander and Schweizer (1997). However, we have to impose some additional assumptions related to the variance-optimal martingale measure.

On the Structure of General Mean-Variance Hedging Strategies

We provide a new characterisation of mean-variance hedging strategies in a general semimartingale market. The key point is the introduction of a new probability measure P* which turns the dynamic asset allocation problem into a myopic one. The minimal martingale measure relative to P* coincides with the variance-optimal martingale measure relative to the original probability measure P.

A reduced form model for ESO valuation

In this paper we extend a reduced form model for the valuation of employee share options (ESOs) to incorporate employee departure, and company takeover. We also allow for performance linked vesting and other exotic features specific to ESOs. We clarify the assumptions underlying the reduced form model, and discuss their implications. We analyze the probabilistic structure of the model which includes an explicit characterization of the set of equivalent martingale measures, as well as the computation of the variance optimal martingale measure and the minimal martingale measure. Moreover, we deduce an additive decomposition of the relative entropy. Particular ESO specifications are studied emphasizing different aspects of the proposed framework. In this context, we also provide strict no-arbitrage bounds for ESO prices by applying optimal stopping. Furthermore, possible limitations of the proposed model are explored by examining departures from the crucial assumptions of no-arbitrage, i.e. by considering the effects of the employee having inside information.

A Semimartingale Backward Equation and the Variance-Optimal Martingale Measure under General Information Flow

We consider a financial market model, where the dynamics of asset prices are given by an Rd-valued continuous semimartingale and the information flow is right-continuous. Using the dynamic programming approach we express the variance-optimal martingale measure in terms of the value process of a suitable optimization problem and show that this value process uniquely solves the corresponding semimartingale backward equation. We consider two extreme cases when this equation admits an explicit solution. In particular, we give necessary and sufficient conditions in order that the variance-optimal martingale measure coincides with the minimal martingale measure as well as with the martingale measure appearing in the second extreme case.

Mean-Variance Hedging for Discontinuous Semimartingales

Mean-variance hedging is well-known as one of hedging methods for incomplete markets. Our end is leading to mean-variance hedging strategy for incomplete market models whose asset price process is given by a discontinuous semimartingale and whose mean-variance trade-off process is not deterministic. In this paper, on account, we focus on this problem under the following assumptions: (1) the local martingale part of the stock price process is a process with independent increments; (2) a certain condition restricting the number and the size of jumps of the asset price process is satisfied; (3) the mean-variance trade-off process is uniformly bounded; (4) the minimal martingale measure coincides with the variance-optimal martingale measure.

A Semimartingale Bellman Equation and the Variance-Optimal Martingale Measure. Correction

Abstract Corrections are given for [Mania and Tevzadze, Georgian Math. J. 7: 765–792, 2000], most of which consist in replacing “ess sup” by “ess inf” in all the exposed formulas.