The results on the mean-variance hedging problem in Gouriéroux, Laurent and Pham (1998), Rheinländer and Schweizer (1997) and Arai (2005) are extended to discontinuous semimartingale models. When the numéraire method is used, we only assume the Radon-Nikodym derivative of the variance-optimal signed martingale measure (VSMM) is non-zero almost surely (but may be strictly negative). When discussing the relation between the solutions and the Galtchouk-Kunita-Watanabe decompositions under the VSMM, we only assume the VSMM is equivalent to the reference probability.
We consider the mean-variance hedging problem under partial Information. The underlying asset price process follows a continuous semimartingale and strategies have to be constructed when only part of the information in the market is available. We show that the initial mean variance hedging problem is equivalent to a new mean variance hedging problem with an additional correction term, which is formulated in terms of observable processes. We prove that the value process of the reduced problem is a square trinomial with coefficients satisfying a triangle system of backward stochastic differentia…
We give an explicit solution of robust mean-variance hedging problem in the single period model for some type of contingent claims. The alternative approach is also considered.
For a large class of vanilla contingent claims, we establish an explicit Föllmer-Schweizer decomposition when the underlying is a process with independent increments (PII) and an exponential of a PII process. This allows to provide an efficient algorithm for solving the mean variance hedging problem. Applications to models derived from the electricity market are performed.
In a financial market model, we consider the variance-optimal semi-static hedging of a given contingent claim, a generalization of the classic variance-optimal hedging. To obtain a tractable formula for the expected squared hedging error and the optimal hedging strategy, we use a Fourier approach in a general multidimensional semimartingale factor model. As a special case, we recover existing results for variance-optimal hedging in affine stochastic volatility models. We apply the theory to set up a variance-optimal semi-static hedging strategy for a variance swap in both the Heston and the 3/…
We study the pricing and the hedging of claim ψ which depends on the default times of two firms A and B. In fact, we assume that, in the market, we can not buy or sell any defaultable bond of the firm B but we can only trade defaultable bond of the firm A. Our aim is then to find the best price and hedging of ψ using only bond of the firm A. Hence, we solve this problem in two cases: firstly in a Markov framework using indifference price and solving a system of Hamilton-Jacobi-Bellman equations, secondly, in a more general framework, using the mean variance hedging approach and solving backwar…
We derive analytic expressions for the variance-optimal hedging strategy and its mean-square hedging error in the lognormal SABR and in the rough Bergomi model. In the SABR model, we show that the variance-optimal hedging strategy coincides with the Delta adjustment of Bartlett [Wilmott magazine 4/6 (2006)]. We show both mathematically and in simulation that the efficiency of the variance-optimal strategy (in comparison to simple Delta hedging) depends strongly on the leverage parameter rho and - in a weaker sense - also on the roughness parameter H of the model, and give a precise quantificat…
We consider hedging of a contingent claim by a 'semi-static' strategy composed of a dynamic position in one asset and static (buy-and-hold) positions in other assets. We give general representations of the optimal strategy and the hedging error under the criterion of variance-optimality and provide tractable formulas using Fourier-integration in case of the Heston model. We also consider the problem of optimally selecting a sparse semi-static hedging strategy, i.e. a strategy which only uses a small subset of available hedging assets. The developed methods are illustrated in an extended numeri…
For a large class of vanilla contingent claims, we establish an explicit Föllmer-Schweizer decomposition when the underlying is an exponential of an additive process. This allows to provide an efficient algorithm for solving the mean variance hedging problem. Applications to models derived from the electricity market are performed.
This paper presents hedging strategies for European and exotic options in a Levy market. By applying Taylor's Theorem, dynamic hedging portfolios are con- structed under different market assumptions, such as the existence of power jump assets or moment swaps. In the case of European options or baskets of European options, static hedging is implemented. It is shown that perfect hedging can be achieved. Delta and gamma hedging strategies are extended to higher moment hedging by investing in other traded derivatives depending on the same underlying asset. This development is of practical importan…
We solve the problem of mean-variance hedging for general semimartingale models via stochastic control methods. After proving that the value process of the associated stochastic control problem has a quadratic structure, we characterize its three coefficient processes as solutions of semimartingale backward stochastic differential equations and show how they can be used to describe the optimal trading strategy for each conditional mean-variance hedging problem. For comparison with the existing literature, we provide alternative equivalent versions of the BSDEs and present a number of simple ex…
We determine the variance-optimal hedge when the logarithm of the underlying price follows a process with stationary independent increments in discrete or continuous time. Although the general solution to this problem is known as backward recursion or backward stochastic differential equation, we show that for this class of processes the optimal endowment and strategy can be expressed more explicitly. The corresponding formulas involve the moment, respectively, cumulant generating function of the underlying process and a Laplace- or Fourier-type representation of the contingent claim. An examp…
We consider the discretized version of a (continuous-time) two-factor model introduced by Benth and coauthors for the electricity markets. For this model, the underlying is the exponent of a sum of independent random variables. We provide and test an algorithm, which is based on the celebrated Foellmer-Schweizer decomposition for solving the mean-variance hedging problem. In particular, we establish that decomposition explicitely, for a large class of vanilla contingent claims. Interest is devoted in the choice of rebalancing dates and its impact on the hedging error, regarding the payoff regu…
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 $P$.
In the paper, a mean-square minimization problem under terminal wealth constraint with partial observations is studied. The problem is naturally connected to the mean-variance hedging problem under incomplete information. A new approach to solving this problem is proposed. The paper provides a solution when the underlying pricing process is a square-integrable semimartingale. The proposed method for the study is based on the martingale representation. In special cases, the Clark-Ocone representation can be used to obtain explicit solutions. The results and the method are illustrated and suppor…
We consider a time-consistent mean-variance portfolio selection problem of an insurer and allow for the incorporation of basis (mortality) risk. The optimal solution is identified with a Nash subgame perfect equilibrium. We characterize an optimal strategy as solution of a system of partial integro-differential equations (PIDEs), a so called extended Hamilton-Jacobi-Bellman (HJB) system. We prove that the equilibrium is necessarily a solution of the extended HJB system. Under certain conditions we obtain an explicit solution to the extended HJB system and provide the optimal trading strategies…
We consider a Poisson process $η$ on a measurable space $(\BY,\mathcal{Y})$ equipped with a partial ordering, assumed to be strict almost everwhwere with respect to the intensity measure $λ$ of $η$. We give a Clark-Ocone type formula providing an explicit representation of square integrable martingales (defined with respect to the natural filtration associated with $η$), which was previously known only in the special case, when $λ$ is the product of Lebesgue measure on $\R_+$ and a $σ$-finite measure on another space $\BX$. Our proof is new and based on only a few basic properties of Poisson p…
Kramkov and Sirbu (2006, 2007) have shown that first-order approximations of power utility-based prices and hedging strategies can be computed by solving a mean-variance hedging problem under a specific equivalent martingale measure and relative to a suitable numeraire. In order to avoid the introduction of an additional state variable necessitated by the change of numeraire, we propose an alternative representation in terms of the original numeraire. More specifically, we characterize the relevant quantities using semimartingale characteristics similarly as in Cerny and Kallsen (2007) for mea…
We consider the problem of ESO valuation in continuous time. In particular, we consider models that assume that an appropriate random time serves as a proxy for anything that causes the ESO's holder to exercise the option early, namely, reflects the ESO holder's job termination risk as well as early exercise behaviour. In this context, we study the problem of ESO valuation by means of mean-variance hedging. Our analysis is based on dynamic programming and uses PDE techniques. We also express the ESO's value that we derive as the expected discounted payoff that the ESO yields with respect to an…
Optimal B-robust estimate is constructed for multidimensional parameter in drift coefficient of diffusion type process with small noise. Optimal mean-variance robust (optimal V -robust) trading strategy is find to hedge in mean-variance sense the contingent claim in incomplete financial market with arbitrary information structure and misspecified volatility of asset price, which is modelled by multidimensional continuous semimartingale. Obtained results are applied to stochastic volatility model, where the model of latent volatility process contains unknown multidimensional parameter in drift …
In this paper, we prove the global risk optimality of the hedging strategy of contingent claim, which is explicitly (or called semi-explicitly) 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…
Variance-Optimal Hedging in the Rough Hawkes--Heston Model
School of Mathematical Sciences, Fudan University（复旦大学数学科学学院）；FinTech Thrust, Hong Kong University of Science and Technology (Guangzhou)（香港科技大学（广州）金融科技学域）
We consider the mean-variance hedging problem under partial information in the case where the flow of observable events does not contain the full information on the underlying asset price process. We introduce a martingale equation of a new type and characterize the optimal strategy in terms of the solution of this equation. We give relations between this equation and backward stochastic differential equations for the value process of the problem.
Machine learning methods for solving nonlinear partial differential equations (PDEs) are hot topical issues, and different algorithms proposed in the literature show efficient numerical approximation in high dimension. In this paper, we introduce a class of PDEs that are invariant to permutations, and called symmetric PDEs. Such problems are widespread, ranging from cosmology to quantum mechanics, and option pricing/hedging in multi-asset market with exchangeable payoff. Our main application comes actually from the particles approximation of mean-field control problems. We design deep learning…
