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 uncertain mortality intensity framework: Pricing and hedging unit-linked life insurance contracts

We study the valuation and hedging of unit-linked life insurance contracts in a setting where mortality intensity is governed by a stochastic process. We focus on model risk arising from different specifications for the mortality intensity. To do so we assume that the mortality intensity is almost surely bounded under the statistical measure. Further, we restrict the equivalent martingale measures and apply the same bounds to the mortality intensity under these measures. For this setting we derive upper and lower price bounds for unit-linked life insurance contracts using stochastic control techniques. We also show that the induced hedging strategies indeed produce a dynamic superhedge and subhedge under the statistical measure in the limit when the number of contracts increases. This justifies the bounds for the mortality intensity under the pricing measures. We provide numerical examples investigating fixed-term, endowment insurance contracts and their combinations including various guarantee features. The pricing partial differential equation for the upper and lower price bounds is solved by finite difference methods. For our contracts and choice of parameters the pricing and hedging is fairly robust with respect to misspecification of the mortality intensity. The model risk resulting from the uncertain mortality intensity is of minor importance.     

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.     

Quantile hedging for equity-linked contracts

We consider an equity-linked contract whose payoff depends on the lifetime of the policy holder and the stock price. We provide the best strategy for an insurance company assuming limited capital for the hedging. The main idea of the proof consists in reducing the construction of such strategies for a given claim to a problem of superhedging for a modified claim, which is the solution to a static optimization problem of the Neyman-Pearson type. This modified claim is given via some sets constructed in an iterative way. Some numerical examples are also given.     

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.     

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.     

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.     

离散时间单位连结人寿保险合同的局部风险最小对冲策略

单位连结人寿保险合同是保险利益依赖于某特定股票的价格的保险合同 .当保险公司发行这样的保险合同后 ,保险公司将面临金融和被保险人死亡率两类风险 .因此这样的保险合同相当于不完全金融市场上的或有索取权 ,不能利用自我融资交易策略复制出 .本文提出利用不完全市场的局部风险最小对冲方法对冲保险者的风险 .我们在离散时间的框架下给出了局部风险最小对冲策略 .     

离散时间单位连结人寿保险合同的局部风险最小对冲策略

单位连结人寿保险合同是保险利益依赖于某特定股票的价格的保险合同。当保险公司发行这样的保险合同后，保险公司将面临金融和被保险人死亡率两类风险。因此这样的保险合同相当对不完全金融市场上的或有索取权，不能利用自我融资交易策略复制出。本提出利用不完全市场的局部风险最小对冲方法对冲保险的风险，我们在离散时间的框架下给出了局部风险最小对冲策略。     

Risk Minimizing Option Pricing in a Regime Switching Market

Abstract

We study option pricing in a regime switching market where the risk free interest rate, growth rate and the volatility of a stock depends on a finite state Markov chain. Using a minimal martingale measure we show that the risk minimizing option price satisfies a system of Black–Scholes partial differential equations with weak coupling.     

用等效用原理对马氏链市场的UL生存合约定价

利用效用无差异原理,根据动态规划原则,最大化财富的期望指数效用,在马氏链驱动的市场下,导出HJB方程,给出unit-linked(UL)生存合约在简单Poisson市场下的保费方程,并给出它的数值模拟．这个结果推广了Brown运动驱动的市场下的保费方程,使得UL生存合约在联接到纯跳的市场时,可以用效用无差异原理定价．     

Risk minimizing hedging for a partially observed high frequency data model

Risk-minimizing hedging strategies for contingent claims are studied in a general model for intraday stock price movements in the case of partial information. The dynamics of the risky asset price is described throught a marked point process Y, whose local characteristics depend on some unobservable hidden state variable X. In the model presented the processes Y and X may have common jump times, which means that the trading activity may affect the law of X and could be also related to the presence of catastrophic events. The hedger is restricted to observing past asset prices. Thus, we are in presence not only of an incomplete market situation but also of partial information. Considering the case where the price of the risky asset is modeled directly under a martingale measure, the computation of the risk-minimizing hedging strategy under this partial information is obtained by using a projection result (M. Schweizer, Risk minimizing hedging strategies under restricted information, Mathematical Finance 4 (1994) 327–342). This approach leads to a filtering problem with marked point process observations whose solution, obtained via the Kushner-Stratonovich equation, allows us to provide a complete solution to the heding problem.     

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.     

Utility-Based Valuation and Hedging of Basis Risk With Partial Information

Abstract

We analyse the valuation and hedging of a claim on a non-traded asset using a correlated traded asset under a partial information scenario, when the asset drifts are unknown constants. Using a Kalman filter and a Gaussian prior distribution for the unknown parameters, a full information model with random drifts is obtained. This is subjected to exponential indifference valuation. An expression for the optimal hedging strategy is derived. An asymptotic expansion for small values of risk aversion is obtained via partial differentiation equation (PDE) methods, following on from payoff decompositions and a price representation equation. Analytic and semi-analytic formulae for the terms in the expansion are obtained when the minimal entropy measure coincides with the minimal martingale measure. Simulation experiments are carried out which indicate that the filtering procedure can be beneficial in hedging, but sometimes needs to be augmented with the increased option premium, which takes into account parameter uncertainty in order to be effective. Empirical examples are presented which conform to these conclusions.     

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.     

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.     

The Mean-Variance Hedging of a Defaultable Option with Partial Information

Abstract

We consider the mean-variance hedging of a defaultable claim in a general stochastic volatility model. By introducing a new measure Q ⁰, we derive the martingale representation theorem with respect to the investors' filtration . We present an explicit form of the optimal-variance martingale measure by means of a stochastic Riccati equation (SRE). For a general contingent claim, we represent the optimal strategy and the optimal cost of the mean-variance hedging by means of another backward stochastic differential equation (BSDE). For the defaultable option, especially when there exists a random recovery rate we give an explicit form of the solution of the BSDE.     

Pricing and hedging of variable annuities with state-dependent fees

We investigate the problem of pricing and hedging variable annuity contracts for which the fee deducted from the policyholder’s account depends on the account value. It is believed that state-dependent fees are beneficial to policyholders and insurers since they reduce policyholders’ incentives to lapse the policies and match the costs incurred by policyholders with the pay-offs received from embedded guarantees. We consider an incomplete financial market which consists of two risky assets modelled with a two-dimensional Lévy process. One of the assets is a security which can be traded by the insurer, and the second asset is a security which is the underlying fund for the variable annuity contract. In our model we derive an equation from which the fee for the guaranteed benefit can be calculated and we characterize a strategy which allows the insurer to hedge the benefit. To solve the pricing and hedging problem in an incomplete financial market we apply a quadratic objective.     

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.     

Pricing life insurance under stochastic mortality via the instantaneous Sharpe ratio

We develop a pricing rule for life insurance under stochastic mortality in an incomplete market by assuming that the insurance company requires compensation for its risk in the form of a pre-specified instantaneous Sharpe ratio. Our valuation formula satisfies a number of desirable properties, many of which it shares with the standard deviation premium principle. The major result of the paper is that the price per contract solves a linear partial differential equation as the number of contracts approaches infinity. One can represent the limiting price as an expectation with respect to an equivalent martingale measure. Via this representation, one can interpret the instantaneous Sharpe ratio as a market price of mortality risk. Another important result is that if the hazard rate is stochastic, then the risk-adjusted premium is greater than the net premium, even as the number of contracts approaches infinity. Thus, the price reflects the fact that systematic mortality risk cannot be eliminated by selling more life insurance policies. We present a numerical example to illustrate our results, along with the corresponding algorithms.