Approximate Method for Mean-Variance Hedging Strategy with Model Risk

The authors aim to develop numerical schemes of the two representative quadratic hedging strategies: locally risk-minimizing and mean-variance hedging strategies, for models whose asset price process is given by the exponential of a normal inverse Gaussian process, using the results of Arai et al. (Int J Theor Appl Financ 19:1650008, 2016) and Arai and Imai (A closed-form representation of mean-variance hedging for additive processes via Malliavin calculus, preprint. Available at https://arxiv.org/abs/1702.07556). Here normal inverse Gaussian process is a framework of Levy processes that frequently appeared in financial literature. In addition, some numerical results are also introduced.

The price of a financial claim at a fixed time can represented by arandom variable H . In an incomplete market H can be approximatedby a trading strategy known as minimum variance hedging. Minimumvariance hedging can be extended to a mean-variance optimal strat-egy where a riskier trading strategy attempts to exploit the expecteddifference between the hedging strategy and the financial claim. Thispaper shows that any mean-variance optimal hedging strategy can bedecomposed into a sum of two conceptually and mathematically dif-ferent trades. One is the minimum variance hedging strategy and theother is the mean-variance optimal price ‘direction’ trading strategythat is independent of the hedging strategy and is only dependent onthe price of the asset used for hedging. General explicit optimal price‘direction’ strategies are also formulated.Keywords: Mean-Variance Optimal Hedging, Minimum VarianceHedging, Optimal Price Trading, Trade Separation, Variance Opti-mal Martingale Measure.JEL Codes: C61, G11.

Unlike the majority of other hedging literatures in which variance is taken as the risk indicator, this article uses the Value-at-Risk (VaR) as the risk management tool of the hedged portfolio. This article adopts a bivariate Markov regime Switching Autoregressive Conditional Heteroscedastic (SWARCH) model to formulate the optimal VaR hedging strategy and then compares it with the other dynamic futures hedging strategies mentioned in the literature in hedging performance. Using Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) futures data, the in-sample and out-of-sample results shows that when VaR is used as the criterion to measure the futures hedging effectiveness, the performance of the dynamic hedging strategy is superior to that of the static hedging strategy, and the performance of the optimal VaR hedging strategy is better than that of the minimum variance and mean-variance hedging strategies. Besides, from the standpoint that the volatility of hedge ratio and hedged portfolio variance decline, no matter what kind of hedging strategy is adopted, the regime switching model is better in in-sample and out-of-sample hedging effectiveness than the Generalized Autoregressive Conditional Heteroscedastic (GARCH) model.

In this paper we consider the mean-variance hedging problem of a continuous state space financial model with the rebalancing strategies for the hedging portfolio taken at discrete times. An expression is derived for the optimal self-financing mean-variance hedging strategy problem, considering any given payoff in an incomplete market environment. To some extent, the paper extends the work of Cerný [1] to the case in which prices may assume any value within a continuous state space, a situation that more closely reflects real market conditions. An expression for the “fair hedging price” for a derivative with any given payoff is derived. Closed-form solutions for both the “fair hedging price” and the optimal control for the case of a European call option are obtained. Numerical results indicate that the proposed method is consistently better than the Black and Scholes approach, often adopted by practitioners.

We consider the problem of explicitly pricing and hedging an option written on a non-exchangeable asset when trading in a correlated asset is possible. This is a typical case of incomplete market where it is well known that the super-replication concept provides generally too high prices. We study several prices and in particular the instantaneous no-good-deal price (see Cochrane and Saa-Requejo in J Polit Econ 108(1):79–119, 2001) and the global one. We show numerically that the global no-good-deal price can be significantly higher that the instantaneous one. We then propose several hedging strategies and show numerically that the mean-variance hedging strategy can be efficient.

Asymptotic analysis of utility-based hedging strategies for small number of contingent claims

In this paper I consider a hedging problem in an illiquid market where there is a risk that the hedger’s order to buy or sell the underlying asset may be executed only partially. In this setting, I find a mean-variance optimal hedging strategy by the dynamic programming method. The solution contains a new endogenous state variable representing the current position in the underlying. The exogenous coefficients in the solution are given by recursive formulas which can be calculated efficiently in Markov models. I illustrate effects of the partial execution risk in several examples.

In this work, we study the problem of mean-variance hedging with a random horizon T ^ tau , where T is a deterministic constant and is a jump time of the underlying asset price process. We rst formulate this problem as a stochastic control problem and relate it to a system of BSDEs with jumps. We then provide a veri cation theorem which gives the optimal strategy for the mean-variance hedging using the solution of the previous system of BSDEs. Finally, we prove that this system of BSDEs admits a solution via a decomposition approach coming from ltration enlargement theory.

We use mean–variance hedging in discrete time in order to value an insurance liability. The prediction of the insurance liability is decomposed into claims development results, that is, yearly deteriorations in its conditional expected values until the liability is finally settled. We assume the existence of a tradeable derivative with binary pay-off written on the claims development result and available in each development period. General valuation formulas are stated and, under additional assumptions, these valuation formulas simplify to resemble familiar regulatory cost-of-capital-based formulas. However, adoption of the mean–variance framework improves upon the regulatory approach by allowing for potential calibration to observed market prices, inclusion of other tradeable assets, and consistent extension to multiple periods. Furthermore, it is shown that the hedging strategy can also lead to increased capital efficiency.

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

In this paper we consider the mean-variance hedging problem of a jump diffusion continuous state space financial model with the re-balancing strategies for the hedging portfolio taken at discrete times, a situation that more closely reflects real market conditions. A direct expression based on some change of measures, not depending on any recursions, is derived for the optimal self-financing mean-variance hedging strategy problem as well as for the ¿fair hedging price¿, considering any given payoff. For the case of a European call option these expressions can be evaluated in a closed form.

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 present paper we give some preliminary results for option pricing and hedging in the framework of the Bates model based on quadratic risk minimization. We provide an explicit expression of the mean-variance hedging strategy in the martingale case and study the Minimal Martingale measure in the general case.

We use mean-variance hedging in discrete time, in order to value a terminal insurance liability. The prediction of the liability is decomposed into claims development results, that is, yearly deteriorations in its conditional expected value. We assume the existence of a tradeable derivative with binary pay-off, written on the claims development result and available in each period. In simple scenarios, the resulting valuation formulas become very similar to regulatory cost-of-capital-based formulas. However, adoption of the mean-variance framework improves upon the regulatory approach, by allowing for potential calibration to observed market prices, inclusion of other tradeable assets, and consistent extension to multiple periods. Furthermore, it is shown that the hedging strategy can also lead to increased capital efficiency and consistency of market valuation with Euler-type capital allocations.

We apply the local risk-minimization approach to defaultable claims and we compare it with intensity-based evaluation formulas and the mean-variance hedging. We solve analytically the problem of finding respectively the hedging strategy and the associated portfolio for the three methods in the case of a default put option with random recovery at maturity.