Minimal-Variance Hedging in Large Financial Markets: Random Fields Approach

We study a large financial market where the discounted asset prices are modeled by martingale random fields. This approach allows the treatment of both the cases of a market with a countable amount of assets and of a market with a continuum amount. We discuss conditions for these markets to be complete and we study the minimal variance hedging problem both in the case of full and partial information. An explicit representation of the minimal variance hedging portfolio is suggested. Techniques of stochastic differentiation are applied to achieve the main results. Examples of large market models with a countable number of assets are considered according to the literature and an example of market model with a continuum of assets is taken from the bond market.     

Longevity risk,cost of capital and hedging for life insurers under Solvency II

The cost of capital is an important factor determining the premiums charged by life insurers issuing life annuities. This capital cost can be reduced by hedging longevity risk with longevity swaps, a form of reinsurance. We assess the costs of longevity risk management using indemnity based longevity swaps compared to costs of holding capital under Solvency II. We show that, using a reasonable market price of longevity risk, the market cost of hedging longevity risk for earlier ages is lower than the cost of capital required under Solvency II. Longevity swaps covering higher ages, around 90 and above, have higher market hedging costs than the saving in the cost of regulatory capital. The Solvency II capital regulations for longevity risk generates an incentive for life insurers to hold longevity tail risk on their own balance sheets, rather than transferring this to the reinsurance or the capital markets. This aspect of the Solvency II capital requirements is not well understood and raises important policy issues for the management of longevity risk.     

需求风险下套期保值与市场进入分析

需求风险是企业面临的主要风险之一,对企业的生产经营和管理决策具有重要影响。本文考虑由多个风险厌恶企业构成的产品竞争市场,分析了需求风险下企业参与套期保值和市场进入的决策问题。文章首先通过Cournot博弈分析了套期保值对于规避需求风险的作用和意义;然后,探讨了企业参与套期保值和市场进入的决策过程,并给出了三种情形下的市场均衡结构;最后,通过数值实验对结论进行了验证。研究表明:套期保值提高了企业应对需求风险的能力,使企业获得更高的产量和收益;参与套期保值企业数量随着进入市场企业数量的增加而减少;当市场竞争程度或市场费用增加时,将会有更多的企业选择参与套期保值,而选择进入市场的企业会减少。     

Martingale optimal transport and robust hedging in continuous time

The duality between the robust (or equivalently, model independent) hedging of path dependent European options and a martingale optimal transport problem is proved. The financial market is modeled through a risky asset whose price is only assumed to be a continuous function of time. The hedging problem is to construct a minimal super-hedging portfolio that consists of dynamically trading the underlying risky asset and a static position of vanilla options which can be exercised at the given, fixed maturity. The dual is a Monge–Kantorovich type martingale transport problem of maximizing the expected value of the option over all martingale measures that have a given marginal at maturity. In addition to duality, a family of simple, piecewise constant super-replication portfolios that asymptotically achieve the minimal super-replication cost is constructed.     

Risk Minimization for a Filtering Micromovement Model of Asset Price

Abstract

The classical option hedging problems have mostly been studied under continuous-time or equally spaced discrete-time models, which ignore two important components in the actual price: random trading times and market microstructure noise. In this paper, we study optimal hedging strategies for European derivatives based on a filtering micromovement model of asset prices with the two commonly ignored characteristics. We employ the local risk-minimization criterion to develop optimal hedging strategies under full information. Then, we project the hedging strategies on the observed information to obtain hedging strategies under partial information. Furthermore, we develop a related nonlinear filtering technique under the minimal martingale measure for the computation of such hedging strategies.     

Quantile Hedging in a semi-static market with model uncertainty

With model uncertainty characterized by a convex, possibly non-dominated set of probability measures, the agent minimizes the cost of hedging a path dependent contingent claim with given expected success ratio, in a discrete-time, semi-static market of stocks and options. Based on duality results which link quantile hedging to a randomized composite hypothesis test, an arbitrage-free discretization of the market is proposed as an approximation. The discretized market has a dominating measure, which guarantees the existence of the optimal hedging strategy and helps numerical calculation of the quantile hedging price. As the discretization becomes finer, the approximate quantile hedging price converges and the hedging strategy is asymptotically optimal in the original market.     

Optimal partial hedging in a discrete-time market as a knapsack problem

We present a new approach for studying the problem of optimal hedging of a European option in a finite and complete discrete-time market model. We consider partial hedging strategies that maximize the success probability or minimize the expected shortfall under a cost constraint and show that these problems can be treated as so called knapsack problems, which are a widely researched subject in linear programming. This observation gives us better understanding of the problem of optimal hedging in discrete time.     

Unit-linked life insurance policies: Optimal hedging in partially observable market models

In this paper we investigate the hedging problem of a unit-linked life insurance contract via the local risk-minimization approach, when the insurer has a restricted information on the market. In particular, we consider an endowment insurance contract, that is a combination of a term insurance policy and a pure endowment, whose final value depends on the trend of a stock market where the premia the policyholder pays are invested. To allow for mutual dependence between the financial and the insurance markets, we use the progressive enlargement of filtration approach. We assume that the stock price process dynamics depends on an exogenous unobservable stochastic factor that also influences the mortality rate of the policyholder. We characterize the optimal hedging strategy in terms of the integrand in the Galtchouk–Kunita–Watanabe decomposition of the insurance claim with respect to the minimal martingale measure and the available information flow. We provide an explicit formula by means of predictable projection of the corresponding hedging strategy under full information with respect to the natural filtration of the risky asset price and the minimal martingale measure. Finally, we discuss applications in a Markovian setting via filtering.     

Discrete-time local risk minimization of payment processes and applications to equity-linked life-insurance contracts

We develop a theory of local risk minimization for payment processes in discrete time, and apply this theory to the pricing and hedging of equity-linked life-insurance contracts. Thus, we extend the work of Møller (2001a) in several directions: from risk minimization (which is done under a martingale measure) to local risk minimization (which is done under an arbitrary measure), from single claims to payment processes, from complete financial markets to possibly incomplete financial markets, from a single risky asset to several risky assets, and from finite state spaces to general state spaces.Moreover, we show that, when tradable financial assets are independent of mortality, a locally risk-minimizing hedging strategy for most claims in the combined financial and mortality market (such as those arising from equity-indexed annuities) may be expressed as the product of two simpler locally risk-minimizing hedging strategies: one for a purely financial claim, the other for a traditional (i.e. non-equity-linked) life-insurance claim.Finally, we also show, under general assumptions, that the minimal measure for the combined market is the product of the minimal measure for the financial market and the physical measure for the mortality.     

Pricing Defaultable Bonds in a Markov Modulated Market

We address the problem of pricing defaultable bonds in a Markov modulated market. Using Merton's structural approach we show that various types of defaultable bonds are combination of European type contingent claims. Thus pricing a defaultable bond is tantamount to pricing a contingent claim in a Markov modulated market. Since the market is incomplete, we use the method of quadratic hedging and minimal martingale measure to derive locally risk minimizing derivative prices, hedging strategies and the corresponding residual risks. The price of defaultable bonds are obtained as solutions to a system of PDEs with weak coupling subject to appropriate terminal and boundary conditions. We solve the system of PDEs numerically and carry out a numerical investigation for the defaultable bond prices. We compare their credit spreads with some of the existing models. We observe higher spreads in the Markov modulated market. We show how business cycles can be easily incorporated in the proposed framework. We demonstrate the impact on spreads of the inclusion of rare states that attempt to capture a tight liquidity situation. These states are characterized by low risk-free interest rate, high payout rate and high volatility.     

Option Pricing and Hedging under a Markov Switching Lévy Process Model

In this paper,we consider a Markov switching Lévy process model in which the underlying risky assets are driven by the stochastic exponential of Markov switching Lévy process and then apply the model to option pricing and hedging.In this model,the market interest rate,the volatility of the underlying risky assets and the N-state compensator,depend on unobservable states of the economy which are modeled by a continuous-time Hidden Markov process.We use the MEMM(minimal entropy martingale measure) as the equivalent martingale measure.The option price using this model is obtained by the Fourier transform method.We obtain a closed-form solution for the hedge ratio by applying the local risk minimizing hedging.     

Super-replication under proportional transaction costs: From discrete to continuous-time models

In this paper, we study the problem of finding the minimal initial capital (i.e. super-replication value) needed in order to hedge (without risk) European contingent claims in a Markov setting under proportional transaction costs. The main result is that the cheapest (trivial) buy-and-hold strategy is optimal. Such a negative result has been derived previously in different contexts. First, we focus on discrete-time binomial models. We prove that the continuous-time limit of the super-replication value is the cost of the cheapest buy-and-hold strategy. Then, the result is proved in a multivariate continuous-time model with Brownian filtration. As a direct consequence, we obtain an explicit characterization of the hedging set, i.e. the set of all initial positions in the market assets from which the contingent claim can be hedged through some admissible portfolio strategy.     

Valuation and hedging strategy of currency options under regime-switching jump-diffusion model

The main purpose of this thesis is in analyzing and empirically simulating risk minimizing European foreign exchange option pricing and hedging strategy when the spot foreign exchange rate is governed by a Markov-modulated jump-diffusion model. The domestic and foreign money market interest rates, the drift and the volatility of the exchange rate dynamics all depend on a continuous-time hidden Markov chain which can be interpreted as the states of a macro-economy. In this paper, we will provide a practical lognormal diffusion dynamic of the spot foreign exchange rate for market practitioners. We employing the minimal martingale measure to demonstrate a system of coupled partial-differential-integral equations satisfied by the currency option price and attain the corresponding hedging schemes and the residual risk. Numerical simulations of the double exponential jump diffusion regime-switching model are used to illustrate the different effects of the various parameters on currency option prices.     

Risk Minimizing Option Pricing for a Class of Exotic Options in a Markov-Modulated Market

We address risk minimizing option pricing in a regime switching market where the floating interest rate depends on a finite state Markov process. The growth rate and the volatility of the stock also depend on the Markov process. Using the minimal martingale measure, we show that the locally risk minimizing prices for certain exotic options satisfy a system of Black-Scholes partial differential equations with appropriate boundary conditions. We find the corresponding hedging strategies and the residual risk. We develop suitable numerical methods to compute option prices.     

Data-driven hedging of stock index options via deep learning

We develop deep learning models to learn the hedge ratio for S&P500 index options from options data. We compare different combinations of features and show that with sufficient training data, a feedforward neural network model with time to maturity, the Black-Scholes delta and market sentiment as inputs performs the best in the out-of-sample test under daily hedging. This model significantly outperforms delta hedging and a data-driven hedging model. Our results also demonstrate the importance of market sentiment for hedging.     

Utility indifference pricing and hedging for structured contracts in energy markets

In this paper we study the pricing and hedging of structured products in energy markets, such as swing and virtual gas storage, using the exponential utility indifference pricing approach in a general incomplete multivariate market model driven by finitely many stochastic factors. The buyer of such contracts is allowed to trade in the forward market in order to hedge the risk of his position. We fully characterize the buyer’s utility indifference price of a given product in terms of continuous viscosity solutions of suitable nonlinear PDEs. This gives a way to identify reasonable candidates for the optimal exercise strategy for the structured product as well as for the corresponding hedging strategy. Moreover, in a model with two correlated assets, one traded and one nontraded, we obtain a representation of the price as the value function of an auxiliary simpler optimization problem under a risk neutral probability, that can be viewed as a perturbation of the minimal entropy martingale measure. Finally, numerical results are provided.     

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.     

The Application of backward stochastic differential equation with stopping time in hedging American contingent claims

We consider a more general wealth process with a drift coefficient which is Lipschitz continuous and the portfolio process with convex constraint. We convert the problem of hedging American contingent claims into the problem of minimal solution of backward stochastic differential equation with stopping time. We adopt the penalization method for constructing the minimal solution of stochastic differential equations and obtain the upper hedging price of American contingent claims.     

带固定比例交易费率的跳扩散欧式期权的定价

考虑市场存在交易费率的跳扩散欧式期权的定价问题.由于交易费的存在使得传统的对冲方法不适用,我们将该问题转化为两元的随机控制问题.证明了带固定比例交易费率的跳扩散欧式期权的价格是对应的积分微分不等方程的约束粘性解,并通过马尔科夫链对变分问题进行离散,证明了在粘性意义下离散方法的收敛性.最后给出了数值结果.     

Optimal partial hedging of an American option: shifting the focus to the expiration date

As a main contribution we present a new approach for studying the problem of optimal partial hedging of an American contingent claim in a finite and complete discrete-time market. We assume that at an early exercise time the investor can borrow the amount she has to pay for the option holder by entering a short position in the numéraire asset and that this loan in turn will mature at the expiration date. We model and solve a partial hedging problem, where the investor’s purpose is to find a minimal amount at which she can hedge the above-mentioned loan with a given probability, while the potential shortfall is bounded above by a certain number of numéraire assets. A knapsack problem approach and greedy algorithm are used in solving the problem. To get a wider view of the subject and to make interesting comparisons, we treat also a closely related European case as well as an American case where a barrier condition is applied.