Pricing Weather Derivatives Using the Indifference Pricing Approach

Abstract

This paper adopts an incomplete market pricing model–the indifference pricing approach–to analyze valuation of weather derivatives and the viability of the weather derivatives market in a hedging context. It incorporates price risk, weather/quantity risk, and other risks in the financial market. In a mean-variance framework, the relationship between the actuarial price and the indifference price of weather derivatives is analyzed, and conditions are obtained concerning when the actuarial price does not provide an appropriate valuation for weather derivatives. Conditions for the viability of the weather derivatives market are examined. This paper also analyzes the effects of partial hedging, natural hedges, basis risk, quantity risk, and price risk on investors’ indifference prices by examining the distributional impacts of the stochastic variables involved.     

A Fair Pricing Approach to Weather Derivatives

This paper proposes a consistent approach to the pricing of weather derivatives. Since weather derivatives are traded in an incomplete market setting, standard hedging based pricing methods cannot be applied. The growth optimal portfolio, which is interpreted as a world stock index, is used as a benchmark or numeraire such that all benchmarked derivative price processes are martingales. No measure transformation is needed for the proposed fair pricing. For weather derivative payoffs that are independent of the value of the growth optimal portfolio, it is shown that the classical actuarial pricing methodology is a particular case of the fair pricing concept. A discrete time model is constructed to approximate historical weather characteristics. The fair prices of some particular weather derivatives are derived using historical and Gaussian residuals. The question of weather risk as diversifiable risk is also discussed. 1991 Mathematics Subject Classification: primary 90A12; secondary 60G30; 62P20 JEL Classification: C16, G10, G13     

Pricing Interest Rate Derivatives: A General Approach

The relationship between affine stochastic processes and bondpricing equations in exponential term structure models has beenwell established. We connect this result to the pricing of interestrate derivatives. If the term structure model is exponentialaffine, then there is a linkage between the bond pricing solutionand the prices of many widely traded interest rate derivativesecurities. Our results apply to m-factor processes with n diffusionsand l jump processes. The pricing solutions require at mosta single numerical integral, making the model easy to implement.We discuss many options that yield solutions using the methodsof the article.     

A Pricing Framework for Real Estate Derivatives

New methods are developed here for pricing the main real estate derivatives — futures and forward contracts, total return swaps, and options. Accounting for the incompleteness of this market, a suitable modelling framework is outlined that can produce exact formulae, assuming that the market price of risk is known. This framework can accommodate econometric properties of real estate indices such as predictability due to autocorrelations. The term structure of the market price of risk is calibrated from futures market prices on the Investment Property Databank index. The evolution of the market price of risk associated with all five futures curves during 2009 is discussed.     

Survivor Derivatives: A Consistent Pricing Framework

Survivorship risk is a significant factor in the provision of retirement income. Survivor derivatives are in their early stages and offer potentially significant welfare benefits to society. This article applies the approach developed by Dowd et al. (2006) , Olivier and Jeffery (2004) , Smith (2005) , and Cairns (2007) to derive a consistent framework for pricing a wide range of linear survivor derivatives, such as forwards, basis swaps, forward swaps, and futures. It then shows how a recent option pricing model set out by Dawson et al. (2009) can be used to price nonlinear survivor derivatives, such as survivor swaptions, caps, floors, and combined option products. It concludes by considering applications of these products to a pension fund that wishes to hedge its survivorship risks.     

A Simple Econometric Approach for Utility-Based Asset Pricing Models

Utility-based models of asset pricing may be estimated with or without assuming a distribution for security returns; both approaches are developed and compared here. The chief strength of a parametric estimator lies in its computational simplicity and statistical efficiency when the added distributional assumption is true. In contrast, the nonparametric estimator is robust to departures from any particular distribution, and it is more consistent with the spirit underlying utility-based asset pricing models since the distribution of asset returns remains unspecified even in the empirical work. The nonparametric approach turns out to be easy to implement with precision nearly indistinguishable from its parametric counterpart in this particular application. The application shows that log utility is consistent with the data over the period 1926–1981.     

A Semiautoregression Approach to the Arbitrage Pricing Theory

This paper developes a semiautoregression (SAR) approach to estimate factors of the arbitrage pricing theory (APT) that has the advantage of providing a simple asymptotic variance-covariance matrix for the factor estimates, which makes it easy to adjust for measurement errors. Using the extracted factors, I confirm the finding that the APT describes asset returns slightly better than the CAPM, although there is still some mispricing in the APT model. I find that not only are the factors “priced” by the market, but the factor premiums move over time in relation to business cycle variables.     

Pricing Derivatives using the Asymptotic Expansion Approach: Credit Migration Models with Stochastic Credit Spreads

The purpose of this paper is to evaluate the asymptotic approximation formulas for the price of contingent claims with credit risk, such as credit default swaps and options on defaultable bonds, in a Markovian credit migration model. Often the generator matrix of a credit migration process is assumed to be deterministic; however, a stochastically varying generator matrix is used in this paper. To apply such a model to the valuation of options on defaultable bonds, the small disturbance asymptotic expansion approach of Kunitomo and Takahashi is used in this study.     

First-Order Approach to Principal-Agent Problems: A Generalization

The first-order approach (FOA) to principal agent problems is very convenient and mathematically tractable. However, existing results show that the FOA is valid only for additively separable utility functions. This is somewhat limiting. In this article sufficient conditions are identified that extend the validity of the FOA to nonseparable cases. The additional conditions involve restrictions on the agent's preferences, particularly interactions between action and the wage contract. These conditions imply that leisure is normal and the agent's absolute risk aversion increases with action. Comparative static results regarding the wage contract and its gradient are also discussed.     

A New Approach to International Arbitrage Pricing

This paper uses a nonlinear arbitrage-pricing model, a conditional linear model, and an unconditional linear model to price international equities, bonds, and forward currency contracts. Unlike linear models, the nonlinear arbitrage-pricing model requires no restrictions on the payoff space, allowing it to price payoffs of options, forward contracts, and other derivative securities. Only the nonlinear arbitrage-pricing model does an adequate job of explaining the time series behavior of a cross section of international returns.     

A Unified Approach for Pricing Contingent Claims on Multiple Term Structures

This paper provides a unified approach for pricing contingent claims on multiple term structures using a foreign currency analogy. All existing option pricing applications are seen to be special cases of this unified approach. This approach is used to price options on financial securities subject to credit risk.     

Pricing Models: A Bayesian DSGE Approach for the U.S. Economy

This paper compares and estimates three pricing mechanisms in the context of a small DSGE model of the U.S. economy. We interpret our results as favoring the pricing mechanism presented in Wolman (1999 Wolman model) over the New Keynesian model with indexation ( Gali and Gertler 1999 , Smets and Wouters 2004a ) and the sticky information model of Mankiw and Reis (2002) . The key factor that explains the performance of the Wolman model is that the data reject the key assumption of the New Keynesian model that the firm's probability of price change is constant over time and independent of the contract's vintage. Our results also show that incorporating indexation in the New Keynesian model represents a poor expedient in matching the autocorrelation function of the inflation process over the last 20 years.     

No Arbitrage and Arbitrage Pricing: A New Approach

We argue that arbitrage-pricing theories (APT) imply the existence of a low-dimensional nonnegative nonlinear pricing kernel. In contrast to standard constructs of the APT, we do not assume a linear factor structure on the payoffs. This allows us to price both primitive and derivative securities. Semi-nonparametric techniques are used to estimate the pricing kernel and test the theory. Empirical results using size-based portfolio returns and yields on bonds reject the nested capital asset-pricing model and linear APT and support the nonlinear APT. Diagnostics show that the nonlinear model is more capable of explaining variations in small firm returns.     

Interest Rate Derivatives in a Duffie and Kan Model with Stochastic Volatility: An Arrow-Debreu Pricing Approach

Simple analytical pricing formulae have been derived, by different authors and for several derivatives, under the Gaussian Langetieg (1980) model. The purpose of this paper is to use such exact Gaussian solutions in order to obtain approximate analytical pricing formulas under the most general stochastic volatility specification of the Duffie and Kan (1996) model. Using Gaussian Arrow-Debreu state prices, first order stochastic volatility approximate pricing solutions will be derived only involving one integral with respect to the time-to-maturity of the contingent claim under valuation. Such approximations will be shown to be much faster than the existing exact numerical solutions, as well as accurate.     

天气衍生品的运作机制与精算定价

天气衍生品是为了规避天气风险给天气敏感行业带来收入的不稳定性而兴起的创新型风险管理工具,其实质是通过衍生合约对天气风险进行分割、重组和交易的证券化产品。不同于传统金融衍生品,天气衍生品的价值取决于温度、湿度或降雨量等天气指数。本文在分析天气衍生品市场发展的基础上,重点探讨了最常见的天气期货和天气期权的运作机制及其精算定价。     

On the Robustness of Least-Squares Monte Carlo (LSM) for Pricing American Derivatives

This paper analyses the robustness of Least-Squares Monte Carlo, a technique proposed by Longstaff and Schwartz (2001) for pricing American options. This method is based on least-squares regressions in which the explanatory variables are certain polynomial functions. We analyze the impact of different basis functions on option prices. Numerical results for American put options show that this approach is quite robust to the choice of basis functions. For more complex derivatives, this choice can slightly affect option prices. This revised version was published online in June 2006 with corrections to the Cover Date.