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ON VALUATION AND RISK MANAGEMENT AT THE INTERFACE OF INSURANCE AND FINANCE
"... This paper reviews methods for hedging and valuation of insurance claims with an inherent financial risk, with special emphasis on quadratic hedging approaches and indifference pricing principles and their applications in insurance. It thus addresses aspects of the interplay between finance and insu ..."
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This paper reviews methods for hedging and valuation of insurance claims with an inherent financial risk, with special emphasis on quadratic hedging approaches and indifference pricing principles and their applications in insurance. It thus addresses aspects of the interplay between finance and insurance, an area which has gained considerable attention during the past years, in practice as well as in theory. Products combining insurance risk and financial risk have gained considerable market shares. Special attention is paid to unitlinked life insurance contracts, and it is demonstrated how these contracts can be valuated and hedged by using traditional methods as well as more recent methods from incomplete financial markets such as riskminimization, meanvariance hedging, superreplication and indifference pricing with meanvariance utility functions.
From Financial Economics to Fair Valuation
, 2001
"... In this paper we address the issue of how to establish the fair value of an insurancelinked liability. This is done by considering the introduction into a simple, oneperiod market model of a new and quite general security (which, amongst other things, could be such a liability). We investigate the ..."
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In this paper we address the issue of how to establish the fair value of an insurancelinked liability. This is done by considering the introduction into a simple, oneperiod market model of a new and quite general security (which, amongst other things, could be such a liability). We investigate the impact of this new security on the market and attempt to predict the price (the fair value) at which it will enter the market, assuming a liquid market.
Pricing Contingent Claims in Incomplete Markets When the Holder Can Choose Among Different Payoffs
"... We suggest a valuation principle to price general claims giving the holder the right to choose (in a predefined way) among several random payoffs in an incomplete financial market. Examples are socalled "chooser options" and American options with finitely many possible exertion times but also some ..."
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We suggest a valuation principle to price general claims giving the holder the right to choose (in a predefined way) among several random payoffs in an incomplete financial market. Examples are socalled "chooser options" and American options with finitely many possible exertion times but also some life insurance contracts. Our premium is defined by the minimal amount the writer must receive at time zero such that for all possible decision functions of the holder, the writer's utility is at least as big as the utility he would have if he did not offer this contingent claim. The valuation principle is consistent with noarbitrage and can be interpreted as a generalization of Schweizer's indi erence principle [17]. We show that in a complete financial market or, in general, if the writer has an exponential utility function, our premium is the supremum over all "utilityindifference premiums" related to all fixed random payo s we get by xing the decision function of the holder. In general, our premium can be even larger than this supremum as we show by an example.
Hedging with a Correlated Asset: Solution of a Nonlinear Pricing PDE ∗
, 2005
"... Hedging a contingent claim with an asset which is not perfectly correlated with the underlying asset results in unhedgeable residual risk. Even if the residual risk is considered diversifiable, the option writer is faced with the problem of uncertainty in the estimation of the drift rates of the und ..."
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Hedging a contingent claim with an asset which is not perfectly correlated with the underlying asset results in unhedgeable residual risk. Even if the residual risk is considered diversifiable, the option writer is faced with the problem of uncertainty in the estimation of the drift rates of the underlying and the hedging instrument. If the residual risk is not considered diversifible, then this risk can be priced using an actuarial standard deviation principle in infinitesmal time. In both cases, these models result in the same nonlinear partial differential equation (PDE). A fully implicit, monotone discretization method is developed for solution of this pricing PDE. This method is shown to converge to the viscosity solution. Certain grid conditions are required to guarantee monotonicity. An algorithm is derived which, given an initial grid, inserts a finite number of nodes in the grid to ensure that the monotonicity condition is satisfied. At each timestep, the nonlinear discretized algebraic equations are solved using an iterative algorithm, which is shown to be globally convergent. Monte Carlo hedging examples are given to illustrate the standard deviation of the profit and loss distribution at the expiry of the option.
Valuation of Mortality Risk via the Instantaneous Sharpe Ratio: Applications to Life Annuities
, 2008
"... Applications to Life Annuities Abstract: We develop a theory for valuing nondiversifiable mortality risk in an incomplete market. We do this by assuming that the company issuing a mortalitycontingent claim requires compensation for this risk in the form of a prespecified instantaneous Sharpe rati ..."
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Applications to Life Annuities Abstract: We develop a theory for valuing nondiversifiable mortality risk in an incomplete market. We do this by assuming that the company issuing a mortalitycontingent claim requires compensation for this risk in the form of a prespecified instantaneous Sharpe ratio. We apply our method to value life annuities. One result of our paper is that the value of the life annuity is identical to the upper good deal bound of Cochrane and SaáRequejo (2000) and of Björk and Slinko (2006) applied to our setting. A second result of our 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 value as an expectation with respect to an equivalent martingale measure (as in BlanchetScalliet, El Karoui, and Martellini (2005)), and from this representation, one can interpret the instantaneous Sharpe ratio as an annuity market’s price of mortality risk.
Developments in Insurance Mathematics
"... Insurance mathematics in the 1990s has been influenced firstly, by the increase in catastrophic claims which had already become apparent during the early 1970s and 1980s and required new mathematical and statistical methods, and, secondly, by a fast increasing financial market that is interested in ..."
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Insurance mathematics in the 1990s has been influenced firstly, by the increase in catastrophic claims which had already become apparent during the early 1970s and 1980s and required new mathematical and statistical methods, and, secondly, by a fast increasing financial market that is interested in new investment possibilities. Ideas from extremevalue theory and mathematical finance have been introduced into insurance mathematics and enriched classical insurance methods. But the exchange is not only from mathematical finance to insurance mathematics. The continuing occurrence of crashes in the financial market has led to a new development in mathematical finance: models and tools from insurance mathematics have entered the world of finance. This paper presents examples, from both the insurance and the financial worlds. The choice of topics is guided by personal taste and my own work.
Harvard Business School
, 2001
"... Under the new Capital Accord banks can choose between different type of risk management systems. Using a stylized model of risk management systems which differ in quality and by modelling the relationship between the bank board and the risk manager, we consider the incentives for the adoption of a p ..."
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Under the new Capital Accord banks can choose between different type of risk management systems. Using a stylized model of risk management systems which differ in quality and by modelling the relationship between the bank board and the risk manager, we consider the incentives for the adoption of a particular system. We show that in some cases banks may adversely adopt an unsophisticated risk management system in order to evade regulation. JEL G18 ∗We are grateful for the detailed and helpful comments of a anonymous referee, to
NAAJ Paper # 494 MARTINGALE VALUATION OF CASHFLOWS FOR INSURANCE AND INTEREST MODELS
"... Using a pricing axiom from financial economics, a martingale valuation method is presented with the properties of numéraire invariance and noarbitrage. The pricing method is then applied to cashflows in actuarial models like loans, bonds, insurances, annuities, reserves and surplus processes. Spec ..."
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Using a pricing axiom from financial economics, a martingale valuation method is presented with the properties of numéraire invariance and noarbitrage. The pricing method is then applied to cashflows in actuarial models like loans, bonds, insurances, annuities, reserves and surplus processes. Special emphasis is given to portfolios of defaultable bonds, where new modelling results are given. Also, an optimal repayment analysis of a common loan arrangement reveals that the book and market interest rates have to be equal. 1 Key Words: Semimartingales, Lévy processes, noarbitrage, numéraire, measure, optimal stopping, loans, book values, defaultable bonds, life insurance and annuities. This paper is based on an article presented at the 2003 Stochastic Modelling Symposium hosted by the Canadian Institute of Actuaries, where it won a best paper award.
Financial Services Forum
, 2004
"... This paper has been prepared for the Institute of Actuaries of Australia’s (IAAust) Financial Services Forum 2004. The IAAust Council wishes it to be understood that opinions put forward herein are not necessarily those of the IAAust and the Council is not responsible for those opinions. ..."
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This paper has been prepared for the Institute of Actuaries of Australia’s (IAAust) Financial Services Forum 2004. The IAAust Council wishes it to be understood that opinions put forward herein are not necessarily those of the IAAust and the Council is not responsible for those opinions.
Pricing Options in Incomplete Equity Markets via the Instantaneous Sharpe Ratio
, 2007
"... Abstract: We use a continuous version of the standard deviation premium principle for pricing in incomplete equity markets by assuming that the investor issuing an unhedgeable derivative security requires compensation for this risk in the form of a prespecified instantaneous Sharpe ratio. First, we ..."
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Abstract: We use a continuous version of the standard deviation premium principle for pricing in incomplete equity markets by assuming that the investor issuing an unhedgeable derivative security requires compensation for this risk in the form of a prespecified instantaneous Sharpe ratio. First, we apply our method to price options on nontraded assets for which there is a traded asset that is correlated to the nontraded asset. Our main contribution to this particular problem is to show that our seller/buyer prices are the upper/lower good deal bounds of Cochrane and SaáRequejo (2000) and of Björk and Slinko (2006) and to determine the analytical properties of these prices. Second, we apply our method to price options in the presence of stochastic volatility. Our main contribution to this problem is to show that the instantaneous Sharpe ratio, an integral ingredient in our methodology, is the negative of the market price of volatility risk, as defined in Fouque, Papanicolaou, and Sircar (2000).