Hedging, Pareto Optimality, and Good Deals

In this paper, we will describe a framework that allows us to connect the problem of hedging a portfolio in finance to the existence of Pareto optimal allocations in economics. We will show that the solvability of both problems is equivalent to the No Good Deals assumption. We will then analyze the case of co-monotone additive monetary utility functions and risk measures.     

Effects of regulatory delays and uncertainty on pricing decisions

In this paper we introduce a simple decision rule that a single product firm may use for filing for a price change to offset variations of the marginal cost. We consider a regulatory body whose response to the price change request involves a time delay with an exponential distribution. Two possibilities regarding the response of the regulatory body are considered. In one case it is assumed to be a binary approval process in which the rate adjustment is either approved in its entirety or rejected. In the second case we consider a partial approval process with a more general distribution. Decision rules for each case are developed. Finally we derive a multi-stage decision rule in which filing decisions are continuously updated based on temporal variations of the cost function. The multi-stage pricing decision model assumes that marginal cost escalation satisfies a Markovian jump process.This work was completed while the authors were with Bell Laboratories, USA.     

A modification of the secant rule derived from a maximum likelihood principle

An estimate of a zero of a complex function, constructed from ordinate information at distinct abscissae, is found from a Maximum Likelihood estimate relative to a normal probability distribution induced by a weak Gaussian distribution on a related Hilbert space. In the case of two ordinate observations this leads to an estimator structurally similar to the Secant Rule, and asymptotically approaching that rule in certain limiting situations. A correspondingly modified version of Newton's method is also derived, and regional and asymptotic convergence results proved.     

Pavement rehabilitation scheduling and toll pricing under different regulatory regimes

This paper addresses the highway pavement rehabilitation scheduling and toll pricing issues over a planning horizon. In the highway system concerned, two types of agents are considered, namely highway operator and road users. Two models, which account for different highway regulatory regimes (i.e. public and private), are proposed. In the public regulatory model, the government aims to maximize total discounted social welfare of the transportation system over the planning horizon by determining the optimal pavement rehabilitation schedule and toll level. In the private regulatory regime, a profit-driven private operator seeks to optimize the pavement rehabilitation schedule and toll level to maximize its own discounted net profit over the planning horizon. The proposed models treat the interactions between the highway operator and the road users in the system as a bi-level hierarchical problem in which the upper level is a multi-period pavement rehabilitation scheduling and toll pricing problem, while the lower level is a multi-period route choice equilibrium problem. A heuristic solution algorithm that combines a greedy approach and a sensitivity analysis based approach is developed to solve the proposed bi-level multi-period optimization models. An illustrative example is used to show the applications of the proposed models. The findings show that the highway regulatory regime, pavement deterioration parameter and the roughness-induced vehicle operating cost can significantly affect the pavement rehabilitation schedules and the toll level as well as the performance of transportation system in terms of total life-cycle travel demand, net profit and social welfare.     

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.     

Total risk aversion and the pricing of options

The objective of this paper is to derive the price of a call option in a stable market for amarket maker who is totally risk averse in the sense that he has a utility function with infinite index of absolute risk aversion. We prove that in the case when the borrowing rateR is the same as the lending rater the option price must be the Black—Scholes price with rateR = r. In the more interesting caseR > r, we prove that two option pricing functions are necessary and sufficient for making a market in the option. These two functions are the Black—choles prices with rater orR. Which price to use at each time will depend on the circumstances (buying or selling) and the capitalization of the market maker. We determine which price to use at each time and the optimal riskless portfolio.Partially supported by a grant, No. AFOSR-86-0202, from the Air Force Office for Scientific Research and a grant from Loyola University of Chicago.Partially supported by Grant No. AFOSR-86-0202 from the Air Force Office for Scientific Research, a grant from the National Science Foundation, and a grant from Loyola University of Chicago.     

Data breaches: Goodness of fit,pricing, and risk measurement

Some research on cyber risk has been conducted in the field of information technology, but virtually no research exists in the actuarial domain. As a first step toward a more profound actuarial discussion, we use multidimensional scaling and goodness-of-fit tests to analyze the distribution of data breach information. Our results show that different types of data breaches need to be modeled as distinct risk categories. For severity modeling, the log-skew-normal distribution provides promising results. The findings add to the recent discussion on the use of skewed distributions in actuarial modeling (Vernic, 2006; Bolancé et al., 2008; Eling, 2012). Moreover, they provide useful insights for actuaries working on the implementation of cyber insurance policies. We illustrate the usefulness of our results in two applications on risk measurement and pricing.     

Determination of risk pricing measures from market prices of risk

A new insurance provider or a regulatory agency may be interested in determining a risk measure consistent with observed market prices of a collection of risks. Using a relationship between distorted coherent risk measures and spectral risk measures, we provide a method for reconstructing distortion functions from the observed prices of risk. The technique is based on an appropriate application of the method of maximum entropy in the mean, which builds upon the classical method of maximum entropy.     

On a modification of the classical risk process

In this paper, we present the classical risk process with two-step premium function. This means that the gross risk premium rate changes if the insurer’s surplus reaches a certain threshold level. The formula for the infinite-time ruin probability is obtained. The asymptotic behaviour of the ruin probability in the case where the claim size distribution has a light tail is considered as well.     

On the robustness of longevity risk pricing

For longevity bond pricing, the most popular methods contain the risk-neutral method, the Wang transform and the Sharpe ratio rule. This paper studies robustness of these three methods and investigates connections and differences among them through theoretic analysis and numerical illustrations. We adopt the dynamic mortality models with jumps to capture the permanent effects caused by unexpected factors and allow the correlation between mortality and interest rate be nonzero. The analysis is based on four typical mortality models, including the mean-reverting models and the non mean-reverting ones. Our work may provide a guidance for participants on choice of pricing methods.     

A modification of Armijo's step-size rule for negative curvature

Armijo's step-size procedure for function minimization is modified to include second derivative information. Accumulation points using this procedure are shown to be stationary points with positive semi-definite Hessian matrices.This research was supported by the United States Army Research Office.     

Option pricing under residual risk and imperfect hedging

This paper is concerned in the option pricing in a discrete time incomplete market. We emphasize the interplay between option pricing and residual risk as well as imperfect hedging. It has been shown that the value of a European option satisfies a hyperbolic, rather than parabolic, partial differential equation. The closed-form solution for this hyperbolic equation has been obtained, which will collapse to the Black–Scholes formula as the time scaling converges to zero.     

Deregulation in electricity: understanding strategic and regulatory risk

This paper is motivated by our experiences since 1990 with developing system simulation models to help UK companies in the restructured electricity industry understand the radically different market within which they must become competitive. When public utilities such as electricity have been restructured, deregulated and/or privatised, the process has often been associated with a major change in the competitive environment. As a consequence, the strategic and regulatory uncertainties ahead for these companies are unprecedented. In such a market there has been no historical evolution and all the participants including the regulatory institutions have very little understanding of how it will operate in the short term and evolve in the future. In this situation, the use of system dynamic models appears to offer an attractive way of gaining insights into how aspects of the competitive market might evolve. In the absence of real experience and relevant analogies, learning from models assumes a key role. Such models cannot be validated empirically, but can be developed to represent how the system is designed to operate. From such a prototypical basis, sensitivity analysis can generate insights on the strategic opportunities created by failings in the market design, or its potential instability to shocks and market imperfections.     

Intrafirm trade, arm’s-length transfer pricing rule, and coordination failure

This paper demonstrates that uniform imposition of the arm’s-length principle on transfer pricing leads to coordination failure among countries in terms of economic welfare if the countries trade products in the form of intrafirm transactions by multinational firms (MNFs). To highlight this implication, we first show that imposition of the arm’s-length principle on an MNF induces it to transfer a product among subordinate divisions at marginal cost, i.e., the competitive price, which is consistent with the purpose of the principle. Nonetheless, if regulators in each country impose the principle on MNFs, all of the following economic welfare measures decrease compared with the situation where the principle is not imposed: (1) consumer welfare in each of the trading countries, (2) profit of each MNF, and thus (3) total world economic welfare. This result indicates that it is possible that enforcement of the principle has no positive effect at all in the world because economic welfare of all economic agents deteriorates when the principle is imposed. A numerical analysis demonstrates that this possibility arises in a broad range of circumstances, even including the situation where a giant economic world power and a small underdeveloped country mutually trade products. In these circumstances, an agreement among trading countries that no country imposes the arm’s-length principle may encourage Pareto improvement of the world economy.     

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.     

Reverse mortgage pricing and risk analysis allowing for idiosyncratic house price risk and longevity risk

Reverse mortgages provide an alternative source of funding for retirement income and health care costs. The two main risks that reverse mortgage providers face are house price risk and longevity risk. Recent real estate literature has shown that the idiosyncratic component of house price risk is large. We analyse the combined impact of house price risk and longevity risk on the pricing and risk profile of reverse mortgage loans in a stochastic multi-period model. The model incorporates a new hybrid hedonic–repeat-sales pricing model for houses with specific characteristics, as well as a stochastic mortality model for mortality improvements along the cohort direction (the Wills–Sherris model). Our results show that pricing based on an aggregate house price index does not accurately assess the risks underwritten by reverse mortgage lenders, and that failing to take into account cohort trends in mortality improvements substantially underestimates the longevity risk involved in reverse mortgage loans.     

Longevity bond premiums: The extreme value approach and risk cubic pricing

The purpose of this study is to analyze the securitization of longevity risk with an emphasis on longevity risk modeling and longevity bond premium pricing. Various longevity derivatives have been proposed, and the capital market has experienced one unsuccessful attempt by the European Investment Bank (EIB) in 2004. After carefully analyzing the pros and cons of previous securitizations, we present our proposed longevity bonds, whose payoffs are structured as a series of put option spreads. We utilize a random walk model with drift to fit small variations of mortality improvements and employ extreme value theory to model rare longevity events. Our method is a new approach in longevity risk securitization, which has the advantage of both capturing mortality improvements within sample and extrapolating rare, out-of- sample longevity events. We demonstrate that the risk cubic model developed for pricing catastrophe bonds can be applied to mortality and longevity bond pricing and use the model to calculate risk premiums for longevity bonds.     

The effect of risk aversion on a supply chain with postponed pricing

We consider a supply chain consisting of a supplier and a risk-averse retailer operating under endogenous demand in retail pricing. The demand potential is uncertain and is revealed at the beginning of the selling season when it is too late to order products. The product price, on the other hand, is not determined in advance and can be postponed until the demand is revealed. The goal is to study the effect of risk-aversion and postponed pricing on both the retailer’s decisions and the overall supply chain. We find that the risk-averse retailer does not necessarily order less than the risk-neutral one and may introduce a bias by choosing a specific demand distribution. We contrast two specific choices. One is symmetric (balanced) with respect to the mean demand potential. The other is skewed (pessimistic) with most observations expected below the mean demand potential. Our numerical results show that the binding downside risk constraint deteriorates the supply chain performance when the forecast is balanced and improves it when the forecast is pessimistic.     

Optimal pricing for a heterogeneous portfolio for a given risk factor and convex distance measure

Consider a portfolio containing heterogeneous risks. The premiums of the policyholders might not cover the amount of the payments which an insurance company pays the policyholders. When setting the premium, this risk has to be taken into consideration. On the other hand the premium that the insured pays has to be fair. This fairness is measured by a function of the difference between the risk and the premium paid—we call this function a distance function. For a given small probability of insolvency, we find the premium for each class, such that the distance function is minimized. Next we formulate and solve the dual problem, which is minimizing the insolvency probability under the constraint that the distance function does not exceed a given level. This paper generalizes a previous paper [Zaks, Y., Frostig, E., Levikson, B., 2006. Optimal pricing of a heterogeneous portfolio for a given risk level. Astin Bull. 36 (1), 161–185] where only a square distance function was considered.     

Asymptotic expansions for the statistic and risk function of a Bayesian classification rule

Asymptotic expansions for the stastistic and risk function of a Bayesian classification rule for the problem of assigning an observation to one of the two one-dimensional populations with unknown parameters are constructed. Translated fromStatisticheskie Metody Otsenivaniya i Proverki Gipotez, pp. 69–77, Perm, 1991.