A dynamic stochastic programming model for international portfolio management

We develop a multi-stage stochastic programming model for international portfolio management in a dynamic setting. We model uncertainty in asset prices and exchange rates in terms of scenario trees that reflect the empirical distributions implied by market data. The model takes a holistic view of the problem. It considers portfolio rebalancing decisions over multiple periods in accordance with the contingencies of the scenario tree. The solution jointly determines capital allocations to international markets, the selection of assets within each market, and appropriate currency hedging levels. We investigate the performance of alternative hedging strategies through extensive numerical tests with real market data. We show that appropriate selection of currency forward contracts materially reduces risk in international portfolios. We further find that multi-stage models consistently outperform single-stage models. Our results demonstrate that the stochastic programming framework provides a flexible and effective decision support tool for international portfolio management.     

Capacity expansion and forward contracting in a duopolistic power sector

The surge in demand for electricity in recent years requires that power companies expand generation capacity sufficiently. Yet, at the same time, energy demand is subject to seasonal variations and peak-hour factors that cause it to be extremely volatile and unpredictable, thereby complicating the decision-making process. We investigate how power companies can optimise their capacity-expansion decisions while facing uncertainty and examine how expansion and forward contracts can be used as suitable tools for hedging against risk under market power. The problem is solved through a mixed-complementarity approach. Scenario-specific numerical results are analysed, and conclusions are drawn on how risk aversion, competition, and uncertainty interact in hedging, generation, and expansion decisions of a power company. We find that forward markets not only provide an effective means of risk hedging but also improve market efficiency with higher power output and lower prices. Power producers with higher levels of risk aversion tend to engage less in capacity expansion with the result that together with the option to sell in forward markets, very risk-averse producers generate at a level that hardly varies with scenarios.     

Open versus closed loop capacity equilibria in electricity markets under perfect and oligopolistic competition

We consider two game-theoretic models of the generation capacity expansion problem in liberalized electricity markets. The first is an open loop equilibrium model, where generation companies simultaneously choose capacities and quantities to maximize their individual profit. The second is a closed loop model, in which companies first choose capacities maximizing their profit anticipating the market equilibrium outcomes in the second stage. The latter problem is an equilibrium problem with equilibrium constraints. In both models, the intensity of competition among producers in the energy market is frequently represented using conjectural variations. Considering one load period, we show that for any choice of conjectural variations ranging from perfect competition to Cournot, the closed loop equilibrium coincides with the Cournot open loop equilibrium, thereby obtaining a ‘Kreps and Scheinkman’-like result and extending it to arbitrary strategic behavior. When expanding the model framework to multiple load periods, the closed loop equilibria for different conjectural variations can diverge from each other and from open loop equilibria. We also present and analyze alternative conjectured price response models with switching conjectures. Surprisingly, the rank ordering of the closed loop equilibria in terms of consumer surplus and market efficiency (as measured by total social welfare) is ambiguous. Thus, regulatory approaches that force marginal cost-based bidding in spot markets may diminish market efficiency and consumer welfare by dampening incentives for investment. We also show that the closed loop capacity yielded by a conjectured price response second stage competition can be less or equal to the closed loop Cournot capacity, and that the former capacity cannot exceed the latter when there are symmetric agents and two load periods.     

Hedging electricity swaptions using partial integro-differential equations

The basic contracts traded on energy exchanges are swaps involving the delivery of electricity for fixed-rate payments over a certain period of time. The main objective of this article is to solve the quadratic hedging problem for European options on these swaps, known as electricity swaptions. We consider a general class of Hilbert space valued exponential jump-diffusion models. Since the forward curve is an infinite-dimensional object, but only a finite set of traded contracts are available for hedging, the market is inherently incomplete. We derive the optimization problem for the quadratic hedging problem under the risk neutral measure and state a representation of its solution, which is the starting point for numerical algorithms.     

基于风险约束的套期保值模型及沪深300股指实证研究

提出利用风险价值VaR建立套期保值资产组合的风险约束.以套期保值资产组合收益最大为目标,以控制套期保值资产组合风险为约束,建立了基于风险约束的套期保值模型.该模型在有效控制风险的基础上,可以大幅提高套期保值资产组合的收益.对沪深300股指现货和期货的数据进行了实证分析,对比了现有研究的最小二乘((OLS)、向量自回归(VAR)、向量误差修正(VEC)三种模型以及本文建立的基于风险约束的期货套期保值模型.样本内检验结果表明,本模型比现有研究模型的收益有大幅提高,平均增加81.6%.同时并没有失去对风险的控制,与现有研究模型只有5.32%的差别.对于样本外检验,模型在控制风险和提高收益两个方面都要优于现有研究模型.模型比现有研究模型平均可提高收益21.4%,平均降低风险3.61%.     

均衡的再保险市场

本文讨论了再保险市场达到均衡状态的定义、性质以及与 Pareto最优的关系 .     

The role of index bonds in universal currency hedging

This study examines the demand for index bonds and their role in hedging risky asset returns against currency risks in a complete market where equity is not hedged against inflation risk. Avellaneda's uncertain volatility model with non-constant coefficients to describe equity price variation, forward price variation, index bond price variation and rate of inflation, together with Merton's intertemporal portfolio choice model, are utilized to enable an investor to choose an optimal portfolio consisting of equity, nominal bonds and index bonds when the rate of inflation is uncertain. A hedge ratio is universal if investors in different countries hedge against currency risk to the same extent. Three universal hedge ratios (UHRs) are defined with respect to the investor's total demand for index bonds, hedging risky asset returns (i.e. equity and nominal bonds) against currency risk, which are not held for hedging purposes. These UHRs are hedge positions in foreign index bond portfolios, stated as a fraction of the national market portfolio. At equilibrium all the three UHRs are comparable to Black's corrected equilibrium hedging ratio. The Cameron-Martin-Girsanov theorem is applied to show that the Radon-Nikodym derivative given under a P -martingale, the investor's exchange rate (product of the two currencies) is a martingale. Therefore the investors can agree on a common hedging strategy to trade exchange rate risk irrespective of investor nationality. This makes the choice of the measurement currency irrelevant and the hedge ratio universal without affecting their values.     

Optimal investment strategies for participating contracts

Participating contracts are popular insurance policies, in which the payoff to a policyholder is linked to the performance of a portfolio managed by the insurer. We consider the portfolio selection problem of an insurer that offers participating contracts and has an S-shaped utility function. Applying the martingale approach, closed-form solutions are obtained. The resulting optimal strategies are compared with portfolio insurance hedging strategies (CPPI and OBPI). We also study numerical solutions of the portfolio selection problem with constraints on the portfolio weights.     

Risk-minimization for life insurance liabilities with basis risk

In this paper we study the hedging of typical life insurance payment processes in a general setting by means of the well-known risk-minimization approach. We find the optimal risk-minimizing strategy in a financial market where we allow for investments in a hedging instrument based on a longevity index, representing the systematic mortality risk. Thereby we take into account and model the basis risk that arises due to the fact that the insurance company cannot perfectly hedge its exposure by investing in a hedging instrument that is based on the longevity index, not on the insurance portfolio itself. We also provide a detailed example within the context of unit-linked life insurance products where the dependency between the index and the insurance portfolio is described by means of an affine mean-reverting diffusion process with stochastic drift.     

Optimal strategies for hedging portfolios of unit-linked life insurance contracts with minimum death guarantee

In this paper, we are interested in hedging strategies which allow the insurer to reduce the risk to their portfolio of unit-linked life insurance contracts with minimum death guarantee. Hedging strategies are developed in the Black and Scholes model and in the Merton jump-diffusion model. According to the new frameworks (IFRS, Solvency II and MCEV), risk premium is integrated into our valuations. We will study the optimality of hedging strategies by comparing risk indicators (Expected loss, volatility, VaR and CTE) in relation to transaction costs and costs generated by the re-hedging error. We will analyze the robustness of hedging strategies by stress-testing the effect of a sharp rise in future mortality rates and a severe depreciation in the price of the underlying asset.     

The natural hedge of a gas-fired power plant

Electricity industries worldwide have been restructured in order to introduce competition. As a result, decision makers are exposed to volatile electricity prices, which are positively correlated with those of natural gas in markets with price-setting gas-fired power plants. Consequently, gas-fired plants are said to enjoy a “natural hedge.” We explore the properties of such a built-in hedge for a gas-fired power plant via a stochastic programming approach, which enables characterisation of uncertainty in both electricity and gas prices in deriving optimal hedging and generation decisions. The producer engages in financial hedging by signing forward contracts at the beginning of the month while anticipating uncertainty in spot prices. Using UK energy price data from 2006 to 2011 and daily aggregated dispatch decisions of a typical gas-fired power plant, we find that such a producer does, in fact, enjoy a natural hedge, i.e., it is better off facing uncertain spot prices rather than locking in its generation cost. However, the natural hedge is not a perfect hedge, i.e., even modest risk aversion makes it optimal to use gas forwards partially. Furthermore, greater operational flexibility enhances this natural hedge as generation decisions provide a countervailing response to uncertainty. Conversely, higher energy-conversion efficiency reduces the natural hedge by decreasing the importance of natural gas price volatility and, thus, its correlation with the electricity price.     

期权组合最优套期保值模型及实证研究

根据实际投资中投资者可以选择不同到期日、不同敲定价格的期权组合进行套期保值的现实,本文建立了二次效用函数下期权组合最优动态套期保值模型,证明了该模型最优解存在的唯一性,并在协方差矩阵可逆和不可逆两种情形下分别给出了期权最优头寸的显式表达式。在50ETF价格先升后降、先降后升、下降和上升四种情形下,对上证50ETF期权的多种期权组合套期保值问题进行实证分析。研究结果表明:不同到期日不同敲定价格的看跌期权组合具有较好的套期保值效果。本文的研究为选择期权组合进行套期保值和解决展期期权套期保值问题提供了借鉴。     

Hedging pure endowments with mortality derivatives

In recent years, a market for mortality derivatives began developing as a way to handle systematic mortality risk, which is inherent in life insurance and annuity contracts. Systematic mortality risk is due to the uncertain development of future mortality intensities, or hazard rates. In this paper, we develop a theory for pricing pure endowments when hedging with a mortality forward is allowed. The hazard rate associated with the pure endowment and the reference hazard rate for the mortality forward are correlated and are modeled by diffusion processes. We price the pure endowment by assuming that the issuing company hedges its contract with the mortality forward and requires compensation for the unhedgeable part of the mortality risk in the form of a pre-specified instantaneous Sharpe ratio. The major result of this paper is that the value per contract solves a linear partial differential equation as the number of contracts approaches infinity. One can represent the limiting price as an expectation under an equivalent martingale measure. Another important result is that hedging with the mortality forward may raise or lower the price of this pure endowment comparing to its price without hedging, as determined in Bayraktar et al. (2009). The market price of the reference mortality risk and the correlation between the two portfolios jointly determine the cost of hedging. We demonstrate our results using numerical examples.     

Risky asset allocation and consumption rule in the presence of background risk and insurance markets

This study examines joint decisions regarding risky asset allocation and consumption rate for a representative agent in the presence of background risk and insurance markets. Contrary to the conclusion of the “mutual fund separation theorem”, we show that the optimal risky asset mix will reflect an agent’s risk attitude as long as background risk is not independent of investment risk. This result can, however, be used to solve the “riskyasset allocation puzzle”. We also unveil that optimal insurance to shift background risk is determined through establishing a hedging portfolio against investment risk and is an arrangement maintaining the balance between growth and volatility of expected consumption. Because the optimal insurance we obtain generally leads to a smoother consumption path, it may plausibly explain the “equity premium puzzle” in the financial literature.     

Pricing and hedging equity-linked life insurance contracts beyond the classical paradigm: The principle of equivalent forward preferences

By applying the principle of equivalent forward preferences, this paper revisits the pricing and hedging problems for equity-linked life insurance contracts. The equity-linked contingent claim depends on, not only the future lifetime of the policyholder, but also the performance of the reference portfolio in the financial market for the segregated account of the policyholder. For both zero volatility and non-zero volatility forward utility preferences, prices and hedging strategies of the contract are represented by solutions of random horizon backward stochastic differential equations. Numerical illustration is provided for the zero volatility case. The derived prices and hedging strategies are also compared with classical results in the literature.     

Optimal hedging strategies for multi-period guarantees in the presence of transaction costs: A stochastic programming approach

Multi-period guarantees are often embedded in life insurance contracts. In this paper we consider the problem of hedging these multi-period guarantees in the presence of transaction costs. We derive the hedging strategies for the cheapest hedge portfolio for a multi-period guarantee that with certainty makes the insurance company able to meet the obligations from the insurance policies it has issued. We find that by imposing transaction costs, the insurance company reduces the rebalancing of the hedge portfolio. The cost of establishing the hedge portfolio also increases as the transaction cost increases. For the multi-period guarantee there is a rather large rebalancing of the hedge portfolio as we go from one period to the next. By introducing transaction costs we find the size of this rebalancing to be reduced. Transaction costs may therefore be one possible explanation for why we do not see the insurance companies performing a large rebalancing of their investment portfolio at the end of each year.     

Hedging Large Portfolios of Options in Discrete Time*

The problem studied is that of hedging a portfolio of options in discrete time where underlying security prices are driven by a combination of idiosyncratic and systematic risk factors. It is shown that despite the market incompleteness introduced by the discrete time assumption, large portfolios of options have a unique price and can be hedged without risk. The nature of the hedge portfolio in the limit of large portfolio size is substantially different from its continuous time counterpart. Instead of linearly hedging the total risk of each option separately, the correct portfolio hedge in discrete time eliminates linear as well as second and higher order exposures to the systematic risk factors only. The idiosyncratic risks need not be hedged, but disappear through diversification. Hedging portfolios of options in discrete time thus entails a trade‐off between dynamic and cross‐sectional hedging errors. Some computations are provided on the outcome of this trade‐off in a discrete‐time Black–Scholes world.     

考虑结算风险的套期保值研究

期货市场的风险转移功能主要通过套期保值策略来实现,期货市场套期保值的关键问题是套期保值比率的确定。现有套期保值研究侧重于规避价格风险,忽略了期货市场另一个重要的风险因素-结算风险。本文通过建立考虑结算风险的期货套期保值决策模型,有效地平衡了套期保值过程中的价格风险与结算风险。具体特色一是将套保者的结算风险厌恶态度直接反映到套期比的计算中,体现了结算风险对套期保值决策的影响;二是在一定条件下,本模型的套期比趋近于最小方差套期比;三是利用ARMA时间序列方法预测期货与现货的价格走势,有效地反映了期货价格一阶平稳和季节性变化规律,使估计的套期比更加精确可靠。     

Evaluating the impact of average cost based contracts on the industrial sector in the European emission trading scheme

The inception of the emission trading scheme in Europe has contributed to power price increases. Energy intensive industries have reacted by arguing that this may affect their competitiveness and will induce them to leave Europe. Taking up a proposal of these industrial sectors, we explore the possible application of special contracts, where electricity is sold at average generation cost to mitigate the impact of CO₂ cost on power prices. The model supposes fixed generation capacities. We first consider a reference model representing a perfectly competitive market where all consumers (industries and the rest of the market) are price-takers and buy electricity at short-run marginal cost. We then change the market design by assuming that energy intensive industries pay power either at a regional or at a zonal average cost price. The analysis is conducted with simulation models applied to the Central Western European power market. The models are implemented in GAMS/PATH. This work has been financially supported by the Chair Lhoist Berghmans in Environmental Economics and Management and by the Italian project PRIN 2006, Generalized monotonicity: models and applications, whose national responsible is Prof. Elisabetta Allevi.     

A locally risk-minimizing hedging strategy for unit-linked life insurance contracts in a Lévy process financial market

In [Riesner, M., 2006. Hedging life insurance contracts in a Lévy process financial market. Insurance Math. Econom. 38, 599–608] the (locally) risk-minimizing hedging strategy for unit-linked life insurance contracts is determined in an incomplete financial market driven by a Lévy process. The considered risky asset is not a martingale under the original measure and therefore, a change of measure to the minimal martingale measure is performed.The goal of this paper is to show that the risk-minimizing hedging strategy under the new martingale measure which is found in the paper cited above is not the locally risk-minimizing strategy under the original measure. Finally, the real locally risk-minimizing strategy is derived and a relationship between the number of risky assets held in the proposed portfolio cited in the above-mentioned paper and the one proposed here is given.