Valuing variable annuity guarantees on multiple assets

Guarantees embedded variable annuity contracts exhibit option-like payoff features and the pricing of such instruments naturally leads to risk neutral valuation techniques. This paper considers the pricing of two types of guarantees; namely, the Guaranteed Minimum Maturity Benefit and the Guaranteed Minimum Death Benefit riders written on several underlying assets whose dynamics are given by affine stochastic processes. Within the standard affine framework for the underlying mortality risk, stochastic volatility and correlation risk, we develop the key ingredients to perform the pricing of such guarantees. The model implies that the corresponding characteristic function for the state variables admits a closed form expression. We illustrate the methodology for two possible payoffs for the guarantees leading to prices that can be obtained through numerical integration. Using typical values for the parameters, an implementation of the model is provided and underlines the significant impact of the assets’ correlation structure on the guarantee prices.     

Joint econometric modeling of spot electricity prices, forwards and options

We propose a joint modeling of spot electricity prices, forwards prices and other derivative prices, using recent developments in discrete time asset pricing methods based on the notions of stochastic discount factor and of Compound Autoregressive (or affine) stochastic processes. We show that this approach provides quasi explicit formulae for forward and option prices, while allowing for a large flexibility in the modeling of dynamics, spikes and seasonality, both in the historical and the risk neutral worlds. We also propose a variety of inference techniques involving inversion methods, the Kalman filter and the Kitagawa?CHamilton filter. Finally, an application based on French spot prices and forward products is proposed.     

On cox processes and credit risky securities

A framework is presented for modeling defaultable securities and credit derivatives which allows for dependence between market risk factors and credit risk. The framework reduces the technical issues of modeling credit risk to the same issues faced when modeling the ordinary term structure of interest rates. It is shown how to generalize a model of Jarrow, Lando and Turnbull (1997) to allow for stochastic transition intensities between rating categories and into default. This generalization can handle contracts with payments explicitly linked to ratings. It is also shown how to obtain a term structure model for all different rating categories simultaneously and how to obtain an affine-like structure. An implementation is given in a simple one factor model in which the affine structure gives closed form solutions.     

Counterparty Risk for Credit Default Swaps: Markov Chain Interacting Intensities Model with Stochastic Intensity

We analyze the counterparty risk for credit default swaps using the Markov chain model of portfolio credit risk of multiple obligors with interacting default intensity processes. The default correlation between the protection seller and underlying entity is modeled by an increment in default intensity upon the occurrence of an external shock event. The arrival of the shock event is a Cox process whose stochastic intensity is assumed to follow an affine diffusion process with jumps. We examine how the correlated default risks between the protection seller and the underlying entity may affect the credit default premium in a credit default swap.     

Regime switching affine processes with applications to finance

We introduce the notion of a regime switching affine process. Informally this is a Markov process that behaves conditionally on each regime as an affine process with specific parameters. To facilitate our analysis, specific restrictions are imposed on these parameters. The regime switches are driven by a Markov chain. We prove that the joint process of the Markov chain and the conditionally affine part is a process with an affine structure on an enlarged state space, conditionally on the starting state of the Markov chain. Like for affine processes, the characteristic function can be expressed in a set of ordinary differential equations that can sometimes be solved analytically. This result unifies several semi-analytical solutions found in the literature for pricing derivatives of specific regime switching processes on smaller state spaces. It also provides a unifying theory that allows us to introduce regime switching to the pricing of many derivatives within the broad class of affine processes. Examples include European options and term structure derivatives with stochastic volatility and default. Essentially, whenever there is a pricing solution based on an affine process, we can extend this to a regime switching affine process without sacrificing the analytical tractability of the affine process.     

An analysis of default correlations and multiple defaults

Evaluating default correlations or the probabilities of defaultby more than one firm is an important task in credit analysis,derivatives pricing, and risk management. However, default correlationscannot be measured directly, multiple-default modeling is technicallydifficult, and most existing credit models cannot be appliedto analyze multiple defaults. This article develops a first-passage-timemodel, providing an analytical formula for calculating defaultcorrelations that is easily implemented and conveniently usedfor a variety of financial applications. The model also providesa theoretical justification for several empirical regularitiesin the credit risk literature.     

金融危机前后国际信用衍生产品市场的发展及启示

信用衍生产品自问世以来在分散金融机构信用风险、完善信用风险定价机制、提高债券市场流动性等方面发挥了积极的作用。该文介绍了信用衍生产品的主要功能及其在本次金融危机前、中、后三个时期的发展特点,肯定了基础信用衍生产品对金融市场的重要作用,指出中国应推动金融创新,建立有中国特色的信用衍生产品市场。     

Covered interest arbitrage profits: The role of liquidity and credit risk

We study the profitability of Covered Interest Parity (CIP) arbitrage violations and their relationship with market liquidity and credit risk using a novel and unique dataset of tick-by-tick firm quotes for all financial instruments involved in the arbitrage strategy. The empirical analysis shows that positive CIP arbitrage deviations include a compensation for liquidity and credit risk. Once these risk premia are taken into account, small arbitrage profits only accrue to traders who are able to negotiate low trading costs. The results are robust to stale pricing and the nonsynchronous trading occurring in the markets involved in the arbitrage strategy.     

Affine Models for Credit Risk Analysis

Continuous-time affine models have been recently introducedin the theoretical financial literature on credit risk. Theyprovide a coherent modeling, rather easy to implement, but havenot yet encountered the expected success among practitionersand regulators. This is likely due to a lack of flexibilityof these models, which often implied poor fit, especially comparedto more ad hoc approaches proposed by the industry. The aimof this article is to explain that this lack of flexibilityis mainly due to the continuous-time assumption. We developa discrete-time affine analysis of credit risk, explain howdifferent types of factors can be introduced to capture separatelythe term structure of default correlation, default heterogeneity,correlation between default, and loss-given-default; we alsoexplain why the factor dynamics are less constrained in discretetime and are able to reproduce complicated cycle effects. Thesemodels are finally used to derive a credit-VaR and various decompositionsof the spreads for corporate bonds or first-to-default basket.     

Subprime mortgage funding and liquidity risk

In this article, we use actuarial methods to solve a nonlinear stochastic optimal liquidity risk management problem for subprime originators with deposit inflow rates and marketable securities allocation as controls. The main objective is to minimize liquidity risk in the form of funding and credit crunch risk in an incomplete market. In order to accomplish this, we construct a stochastic model that incorporates originator mortgage and deposit reference processes. Finally, numerical examples that illustrate the main modeling and optimization features of the article are provided.     

How Integrated are Credit and Equity Markets? Evidence from Index Options

We study the extent to which credit index (CDX) options are priced consistent with S&P 500 (SPX) equity index options. We derive analytical expressions for CDX and SPX options within a structural credit-risk model with stochastic volatility and jumps using new results for pricing compound options via multivariate affine transform analysis. The model captures many aspects of the joint dynamics of CDX and SPX options. However, it cannot reconcile the relative levels of option prices, suggesting that credit and equity markets are not fully integrated. A strategy of selling CDX volatility yields significantly higher excess returns than selling SPX volatility.     

Pricing structured products with economic covariates

We introduce a top-down no-arbitrage model for pricing structured products. Losses are described by Cox processes whose intensities depend on economic variables. The model provides economic insight into the impact of structured products on financial institutions’ risk exposure and systemic risk. We estimate the model using CDO data and find that spreads decrease with higher interest rates and increase with volatility and leverage. Volatility is the primary determinant of variation in tranche spreads. Leverage and interest rates are more closely associated with rare credit events. Model-implied risk premiums and the probabilities of tranche losses increase substantially during the financial crisis.     

Implied migration rates from credit barrier models

The risk-neutral credit migration process captures quantitative information which is relevant to the pricing theory and risk management of credit derivatives. In this article, we derive implied migration rates by means of a recently introduced credit barrier model which is calibrated on the basis of aggregate information such as credit migration rates and credit spread curves. The model is characterized by an underlying stochastic process that represents credit quality, and default events are associated to barrier crossings. The stochastic process has state dependent volatility and jumps which are estimated by using empirical migration and default rates. A risk-neutralizing drift and forward liquidity spreads are estimated to consistently match the average spread curves corresponding to all the various ratings. The implied migration rates obtained with our credit barrier model are then compared with those obtained via the Kijima–Komoribayashi model.     

Anomalies and financial distress

This paper explores commonalities across asset pricing anomalies. In particular, we assess implications of financial distress for the profitability of anomaly-based trading strategies. Strategies based on price momentum, earnings momentum, credit risk, dispersion, idiosyncratic volatility, and capital investments derive their profitability from taking short positions in high credit risk firms that experience deteriorating credit conditions. In contrast, the value-based strategy derives most of its profitability from taking long positions in high credit risk firms that survive financial distress and subsequently realize high returns. The accruals anomaly is an exception. It is robust among high and low credit risk firms in all credit conditions.     

Financial statement recasting and credit risk assessment

This article examines the importance of adjustments to corporate financial statements for credit risk assessment. Prior research has tended to examine individual adjustments one at a time. As correlations among adjustments and control variables may bias inferences when researchers examine a single adjustment and ignore other adjustments, our results provide important new information about previous research by documenting whether or not such bias exists. We find that financial statement recasting adjustments – which aim to better reflect firms' indebtedness, financing costs and recurring earnings than reported financial numbers – are reflected in bond yield spreads and have an economically significant impact on credit pricing and loss forecasting. Among individual adjustment categories, we find that those for off‐balance‐sheet leases, defined benefit pensions and securitized debt have an economically significant impact on credit pricing and loss forecasting.     

The ordered qualitative model for credit rating transitions

Information on the expected changes in credit quality of obligors is contained in credit migration matrices which trace out the movements of firms across ratings categories in a given period of time and in a given group of bond issuers. The rating matrices provided by Moody's, Standard & Poor's and Fitch became crucial inputs to many applications, including the assessment of risk on corporate credit portfolios (CreditVar) and credit derivatives pricing. We propose a factor probit model for modeling and prediction of credit rating matrices that are assumed to be stochastic and driven by a latent factor. The filtered latent factor path reveals the effect of the economic cycle on corporate credit ratings, and provides evidence in support of the PIT (point-in-time) rating philosophy. The factor probit model also yields the estimates of cross-sectional correlations in rating transitions that are documented empirically but not fully accounted for in the literature and in the regulatory rules established by the Basle Committee.     

Polynomial diffusions and applications in finance

This paper provides the mathematical foundation for polynomial diffusions. They play an important role in a growing range of applications in finance, including financial market models for interest rates, credit risk, stochastic volatility, commodities and electricity. Uniqueness of polynomial diffusions is established via moment determinacy in combination with pathwise uniqueness. Existence boils down to a stochastic invariance problem that we solve for semialgebraic state spaces. Examples include the unit ball, the product of the unit cube and nonnegative orthant, and the unit simplex.     

后危机时代信用衍生市场的发展方向

信用衍生市场能够促进信用风险的定价、分散和转移,并有助于金融稳定。遗憾的是,此次金融危机的发生,导致各方对信用衍生品产生很多误解。本文试图对此作一澄清,通过分析后危机时代信用衍生市场的新特点和改革路径,找出其未来的发展方向,并对发展中国的信用衍生市场提出建议。     

Option Pricing under Stochastic Interest Rates: An Empirical Investigation

Using daily data of the Nikkei 225 index, call option prices and call money rates of the Japanese financial market,a comparison is made of the pricing performance of stock option pricing modelsunder several stochastic interest rate processes proposedby the existing term structure literature.The results show that (1) one option pricing modelunder a specific stochastic interest ratedoes not significantly outperformanother option pricing model under an alternative stochasticinterest rate, and (2) incorporating stochastic interest ratesinto stock option pricing does not contribute to the performanceimprovement of the original Black–Scholes pricing formula.     

Pricing Bonds and Bond Options with Default Risk

The pricing of bonds and bond options with default risk is analysed in the general equilibrium model of Cox, Ingersoll, and Ross (1985). This model is extended by means of an additional parameter in order to deal with financial and credit risk simultaneously. The estimation of such a parameter, which can be considered as the market equivalent of an agencies' bond rating, allows to extract from current quotes the market perceptions of firm's credit risk. The general pricing model for defaultable zero-coupon bond is first derived in a simple discrete-time setting and then in continuous-time. The availability of an integrated model allows for the pricing of default-free options written on defaultable bonds and of vulnerable options written either on default-free bonds or defaultable bonds. A comparison between our results and those given by Jarrow and Turnbull (1995) is also presented.