A natural generalization of the ARCH (Autoregressive Conditional Heteroskedastic) process introduced in Engle (1982) to allow for past conditional variances in the current conditional variance equation is proposed. Stationarity conditions and autocorrelation structure for this new class of parametric models are derived. Maximum likelihood estimation and testing are also considered. Finally an empirical example relating to the uncertainty of the inflation rate is presented.

The fact that expected payo¤s on assets and call options are in…nite under most log-stable distributions led both Paul Samuelson (as quoted by Smith 1976) and Robert Merton (1976) to conjecture that assets and derivatives could not be reasonably priced under these distributions, despite their attractive feature as limiting distributions under the Generalized Central Limit Theorem. Carr and Wu (2003) are able to price options under log-stable uncertainty, but only by making the extreme assumption of maximally negative skewness. This paper demonstrates that when the observed distribution of prices is log-stable, the Risk Neutral Measure (RNM) under which asset and derivative prices may be computed as expectations is not itself log-stable in the problematic cases. Instead, the RNM is determined by the convolution of two densities, one negatively skewed stable, and the other an exponentially tilted positively skewed stable. The resulting RNM gives …nite expected payo¤s for all parameter values, so that the concerns of Samuelson and Merton were in fact unfounded, while the Carr and Wu restriction is unnecessary. Since the log-stable RNM developed here is expressed in terms of its characteristic function, it enables options on log-stable assets to be computed easily by means of the Fast Fourier Transform (FFT) methodology of Carr and Madan (1999), provided a simple extension of the FFT, introduced here, is employed.

Adaptive Least Squares (ALS), a refinement of the Constant Gain Recursive

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The apparent banking market failure modeled by Diamond and Dybvig [1983] rests on their inconsistently applying their “sequential servicing constraint” to private banks but not to their government deposit insurance agency. Without this inconsistency, banks can provide optimal risk-sharing without tax-based deposit insurance, even when the number of “type 1” agents is stochastic, by employing a “contingent bonus contract.” The threat of disintermediation noted by Jacklin [1987] in the nonstochastic case is still present but can be blocked by contractual trading restrictions. This article complements Wallace [1988], who considers an alternative resolution of this inconsistency.

hile systemic risk—the risk of wholesale failure of banks and other financial institutions—is generally considered to be the primary reason for supervision and regulation of the

A multiperiod model is developed to measure the costs to the guaranty fund for insurers by incorporating asset credit risk, liability catastrophe risk, interest rate risk, timing of failure resolution, and moral hazard. We model the regulatory reality of capital forbearance and the use of liquidation as a typical resolution practice. The guaranty contract is viewed as a put option with a stochastic striking price and an uncertain maturity. In our numerical analysis, we simplify the higher-dimensional simulation problem by first partially solving two stochastic differential equations. Our results show how the fair premium rate is affected by the coverage horizon, regulatory capital forbearance, moral hazard, interest rate uncertainty, and credit and catastrophe risks.

A multiperiod model is developed to measure the costs posed to the guaranty fund in a setting that incorporates risk-based capital regulations, interest rate risk and the possibility of catastrophic losses. The guaranty contract is modeled as a put option on the asset of the insurance company with a stochastic strike price and an uncertain maturity. The impacts of the key factors of this model are examined numerically and shown to make material di#erences in the costs to the guaranty fund.

We construct a simple representative agent model to provide a theoretical framework for the logstable option pricing model. We also implement a new parametric method for estimating the risk neutral measure (RNM) using a generalized two-factor log-stable option pricing model. Under the generalized two-factor log-stable uncertainty assumption, the RNM for the log of price is a convolution of two exponentially tilted stable distributions. Since the RNM for generalized two-factor log-stable uncertainty is expressed in terms of its Fourier Transform, we introduce a simple extension of the Fast Fourier Transform inversion procedure in order to reduce computational errors in option pricing. The generalized two-factor log-stable RNM has a very flexible parametric form for approximating other probability distributions. Thus, this model provides a sufficiently accurate tool for estimating the RNM from the observed option prices even if the log-stable assumption might not be satisfied. We estimate the RNM for the S&P 500 index options and find that the generalized two-factor log-stable model gives better performance than the Black-Scholes model, the finite moment log-stable model (Carr and Wu, 2003), and the orthogonal log-stable model (McCulloch, 2003) in fitting the observed option prices.

Abstract. A method of estimating the spectral representation of a generalized bivariate stable distribution is presented, based on a series of maximum likelihood (ML) estimates of the stable parameters of univariate projections of the data. The corresponding stable spectral density is obtained by solving a quadratic program. The proposed method avoids the often arduous task of computing the multivariate stable density, relying instead on the standard univariate stable density. The paper applies this projection procedure, under the simplifying assumption of symmetry, to simulated data as well as to foreign exchange return data, with favorable results. Kanter projection coefficients governing conditional expectations are computed from the estimated spectral density. For the simulated data these compare well to their known true values. Key words: estimation of bivariate stable spectral representation, projection method, foreign exchange rates, Kanter projection coefficient 1.