Securitization, structuring and pricing of longevity risk

Pricing and risk management for longevity risk have increasingly become major challenges for life insurers and pension funds around the world. Risk transfer to financial markets, with their major capacity for efficient risk pooling, is an area of significant development for a successful longevity product market. The structuring and pricing of longevity risk using modern securitization methods, common in financial markets, have yet to be successfully implemented for longevity risk management. There are many issues that remain unresolved for ensuring the successful development of a longevity risk market. This paper considers the securitization of longevity risk focusing on the structuring and pricing of a longevity bond using techniques developed for the financial markets, particularly for mortgages and credit risk. A model based on Australian mortality data and calibrated to insurance risk linked market data is used to assess the structure and market consistent pricing of a longevity bond. Age dependence in the securitized risks is shown to be a critical factor in structuring and pricing longevity linked securitizations.     

Stochastic measures of arbitrage

Empirical research has provided evidence supporting the existence of arbitrage opportunities in real financial markets although market imperfections are often the main reason to explain these empirical deviations. Consequently, recent literature has turned the attention to imperfect markets in order to extend the most significant results on asset pricing. This paper develops several stochastic measures providing relative arbitrage earnings available in a financial market. The measures allow us to take into account different type of frictions. They are introduced by means of several dual pairs of vector optimization problems. Primal problems permit us to characterize the arbitrage absence even in an imperfect market and they also provide optimal arbitrage portfolios if the arbitrage absence fails. Dual ones allow us to extend the risk-neutral valuation methodology for imperfect and noarbitrage free markets and provide new interpretations for the measures in terms of “frictions effect” or “committed errors” in the valuation process. Partially funded by Comunidad Autónoma de Madrid (ref: CAM 07T/0027/2000) and Spanish Ministry of Science and Technology (ref: BEC2000-1388-C04)     

A new look at some basic concepts in arbitrage pricing theory

The notion of No Free Lunch with Vanishing Risk (or NFLVR in short) w.r.t. admissible strategies depends on the choice of numeraire. Yan introduced the notion of allowable strategy and showed that condition of NFLVR w.r.t. allowable strategies is independent of the choice of numeraire and is equivalent to the existence of an equivalent martingale measure for the deflated price process. In this paper we establish a version of the Kramkov's optional decomposition theorem in the setting of equivalent martingale measures. Based on this theorem, we have a new look at some basic concepts in arbitrage pricing theory: superhedging, fair price, attainable contingent claims, complete markets and etc.     

Financial Risk Modelling and Econometric Inference

In finance, the explicit modelling of uncertainty takes on a particularly important role. The values of financial derivatives increase in the return volatility of the underlying security. This notion requires a concept of volatility and hence uncertainty. In addition, the choice between modelling in discrete and continuous time is not arbitrary, since it corresponds to a distinction between incomplete and complete markets, respectively, and this distinction matters for asset pricing, financial risk modelling, and inference. Risk and volatility are closely connected, and implied volatility, volatility forecasting, volatility in term structure models, stochastic volatility, and portfolio analysis are considered and related to a more general interplay between cross-sectional and dynamic aspects in finance. Stocks, bonds, and options are considered and placed in the context of efficiency and separation in inference.     

Pricing early exercise contracts in incomplete markets

We present a utility-based methodology for the valuation of early exercise contracts in incomplete markets. Incompleteness stems from nontraded assets on which the contracts are written. This methodology takes into account the individuals attitude towards risk and yields nonlinear pricing rules. The early exercise indifference prices solve a quasilinear variational inequality with an obstacle term. They are also shown to satisfy an optimal stopping problem with criterion given by their European indifference price counterpart. A class of numerical schemes are developed for the variational inequalities and a general approach for solving numerically nonlinear equations arising in incomplete markets is discussed.Accepted: May 2003, AMS Classification: 93E20, 60G40, 60J75The second author acknowledges partial support from NSF Grants DMS 0102909 and DMS 0091946.     

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.     

广义δ规避策略下的两值期权定价研究

在离散时间场合和不存在交易成本假设下,提出了期权定价的平均自融资极小方差规避策略,得到了含有残差风险的两值看涨期权价格满足的偏微分方程和相应的两值期权定价公式。通过用数值分析来比较新的期权定价模型与经典的期权定价模型,发现投资者的风险偏好和标度对期权定价有重要影响。由此说明,考虑残差风险对两值期权定价研究具有重要的理论和实际意义。     

广义δ规避策略下的两值期权定价研究

在离散时间场合和不存在交易成本假设下,提出了期权定价的平均自融资极小方差规避策略,得到了含有残差风险的两值看涨期权价格满足的偏微分方程和相应的两值期权定价公式。通过用数值分析来比较新的期权定价模型与经典的期权定价模型,发现投资者的风险偏好和标度对期权定价有重要影响。由此说明,考虑残差风险对两值期权定价研究具有重要的理论和实际意义。     

Good Deals and compatible modification of risk and pricing rule: a regulatory treatment

This work studies Good Deals in a scenario in which a firm uses decision-making tools based on a coherent risk measure, and in which the market prices are determined with a sub-linear pricing rule. The most important observation of this work is that the existence of a Good Deal is equivalent to the incompatibility between the pricing rule and the risk measure. In this paper, we look into this situation from a regulatory point of view to rule out Good Deals with the purpose of stabilizing financial markets. We propose some practical ways of modifying a risk measure so a regulator can set appropriate levels of capital requirements for a financial institution.     

Dynamic risk measure for BSVIE with jumps and semimartingale issues

Risk measure is a fundamental concept in finance and in the insurance industry. It is used to adjust life insurance rates. In this article, we will study dynamic risk measures by means of backward stochastic Volterra integral equations (BSVIEs) with jumps. We prove a comparison theorem for such a type of equations. Since the solution of a BSVIEs is not a semimartingale in general, we will discuss some particular semimartingale issues.     

DG method for numerical pricing of multi-asset Asian options—The case of options with floating strike

Option pricing models are an important part of financial markets worldwide. The PDE formulation of these models leads to analytical solutions only under very strong simplifications. For more general models the option price needs to be evaluated by numerical techniques. First, based on an ideal pure diffusion process for two risky asset prices with an additional path-dependent variable for continuous arithmetic average, we present a general form of PDE for pricing of Asian option contracts on two assets. Further, we focus only on one subclass—Asian options with floating strike—and introduce the concept of the dimensionality reduction with respect to the payoff leading to PDE with two spatial variables. Then the numerical option pricing scheme arising from the discontinuous Galerkin method is developed and some theoretical results are also mentioned. Finally, the aforementioned model is supplemented with numerical results on real market data.     

Contingent claim pricing using probability distortion operators: methods from insurance risk pricing and their relationship to financial theory

This paper considers the pricing of contingent claims using an approach developed and used in insurance pricing. The approach is of interest and significance because of the increased integration of insurance and financial markets and also because insurance-related risks are trading in financial markets as a result of securitization and new contracts on futures exchanges. This approach uses probability distortion functions as the dual of the utility functions used in financial theory. The pricing formula is the same as the Black-Scholes formula for contingent claims when the underlying asset price is log-normal. The paper compares the probability distortion function approach with that based on financial theory. The theory underlying the approaches is set out and limitations on the use of the insurance-based approach are illustrated. The probability distortion approach is extended to the pricing of contingent claims for more general assumptions than those used for Black-Scholes option pricing.     

Pricing policy in dynamic markets and a generalization of the Ramsey rule

Pricing policy in a regulated monopoly industry is usually based on maximizing welfare or some other measure of utility level of return on investment. Previously, the Ramsey pricing policy which states that the percentage deviation of quasi-optimal price from marginal cost for each product must be inversely proportional to its price elasticity of demand, has been developed for a static market. The Ramsey framework assumes instantaneous demand response to price changes; empirical evidence suggests demand changes occur dynamically through time.In this paper an optimum pricing rule for a profit maximizing firm based on a general time varying demand model in a dynamic market is obtained assuming a single price change at the beginning of the planning period. A dynamic market equivalent of the well known inverse elasticity law of the static market is developed. Defining the concept of average price elasticity for dynamic markets we show that the inverse elasticity law of static markets takes an inequality form in dynamic markets. For demand functions which decrease, increase or are constant with time the optimum price markups are greater than, less than, or equal to the inverse of the average price elasticity, respectively.The results are then generalized to the case of a constrained welfare maximizing firm. This leads to the development of a dynamic market generalization of the well known Ramsey pricing rule. A simple rule for making quantitative arguments about the relative size of the optimum price in static and dynamic markets is also derived.This work was completed when the author was with Bell Laboratories, USA.     

Impact of risk aversion and belief heterogeneity on trading of defaultable claims

This paper studies the problem of pricing and trading of defaultable claims among investors with heterogeneous risk preferences and market views. Based on the utility-indifference pricing methodology, we construct the bid-ask spreads for risk-averse buyers and sellers, and show that the spreads widen as risk aversion or trading volume increases. Moreover, we analyze the buyer’s optimal static trading position under various market settings, including (i) when the market pricing rule is linear, and (ii) when the counterparty—single or multiple sellers—may have different nonlinear pricing rules generated by risk aversion and belief heterogeneity. For defaultable bonds and credit default swaps, we provide explicit formulas for the optimal trading positions, and examine the combined effect of risk aversions and beliefs. In particular, we find that belief heterogeneity, rather than the difference in risk aversion, is crucial to trigger a trade.     

Asset prices in an ambiguous economy

Models with ambiguity averse preferences have the potential to explain some pricing anomalies on financial markets. However, the models used in applications make additional assumptions, beyond ambiguity aversion, on the structure of the investor’s preferences. Therefore, it is not clear how to disentangle the effect of ambiguity aversion from other features of preferences on equilibrium prices. This paper offers a general theory of asset pricing assuming only ambiguity aversion. Price indeterminacy may result in equilibrium when preferences are not smooth. A set of priors, which is identifiable in all the models used in applications, contains the relevant information to price assets. Ambiguity enriches the standard pricing formula by an additional stochastic discount factor and we calculate its explicit form for various models.     

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.     

基于鞅测度的流动性风险溢价的测算

研究了在一般市场条件下流动性风险的定价问题.首先借助金融数学和金融工程的无套利思想在鞅测度下对市场风险和流动性风险进行定价,通过等价测度变换,使可交易资产的贴现价值过程转化为鞅过程,得到了市场风险和流动性风险的市场价格,进而给出了流动性风险溢价的计算公式.得到的风险的市场价格在同一市场中对于所有可交易资产都是相同的,并且这一价格对于所有投资者也都是相同的,不会因投资者的风险厌恶水平的不同而不同.     

On the Price of Risk of the Underlying Markov Chain in a Regime-Switching Exponential Lévy Model

Regime-switching models (RSM) have been recently used in the literature as alternatives to the Black-Scholes model. Several authors favor RSM as being more realistic since, by construction, they model those exogenous macroeconomic cycles against which asset prices evolve. In the context of derivatives pricing, these models lead to incomplete markets and therefore there exist multiple Equivalent Martingale Measures (EMM) yielding different pricing rules. A fair amount of literature (Buffington and Elliott, Int J Theor Appl Finance 40:267–282, 2002; Elliott et al., Ann Finance 1(4):23–432, 2005) focuses on conveniently choosing a family of EMM leading to closed-form formulas for option prices. These studies often make the assumption that the risk associated with the Markov chain is not priced. Recently, Siu and Yang (Acta Math Appl Sin Engl Ser 25(3):339–388, 2009), proposed an EMM kernel that takes into account all risk components of a regime-switching Black-Scholes model. In this paper, we extend the results and observations made in Siu and Yang (Acta Math Appl Sin Engl Ser 25(3):339–388, 2009) in order to include more general Lévy regime-switching models that allow us to assess the influence of jumps on the price of risk. In particular, numerical results are given for Regime-switching Jump-Diffusion and Variance-Gamma models. Also, we carry out a comparative analysis of the resulting option price formulas with existing regime-switching models such as Naik (J Financ 48:1969–1984, 1993) and Boyle and Draviam (Insur Math Econ 40:267–282, 2007).     

带基差风险和交易费用的不完全市场下的期权定价方法

本文研究了同时带有基差风险和交易费用的不安全市场中的权证定价方法。把[1]的模型推广到了考虑基差风险的情况[2]。期权的价格以一个三维自由边界问题的解给出,并含有两个相关的股票价格变量的相关系数。     

Optimal dynamic marketing-mix policies for frequently purchased products and services versus consumer durable goods: A generalized analytic approach

This paper deals with the qualitative characterization of optimal pricing and advertising policies together with the optimal ratio of the advertising elasticity of demand to its price elasticity over time. The problem is studied for frequently purchased products and services (FPS) as well as consumer durable goods (CDG) in both monopolistic and duopolistic markets. Demand dynamics, cost learning and discounting of future profits are taken into consideration. In addition, both the open-loop and feedback methodologies are pursued to characterize and compare the derived optimal policies.The paper uses an analytical approach to characterize the optimal dynamic policies in a general setting as is mathematically tractable, followed by the analysis of more specific models to gain additional managerial insights while maintaining a certain degree of generality. Optimal FPS marketing-mix policies are shown to be different from their CDG counterparts for both monopolistic and duopolistic markets. While the ratio of advertising elasticity to price elasticity appears to have been governed by similar set of rules for FPS and CDG, the direction of change of such ratio over time looks different from each other. Managerial implications and directions for future research are also discussed.