Time Changes for Lévy Processes

The goal of this paper is to consider pure jump Lévy processes of finite variation with an infinite arrival rate of jumps as models for the logarithm of asset prices. These processes may be written as time-changed Brownian motion. We exhibit the explicit time change for each of a wide class of Lévy processes and show that the time change is a weighted price move measure of time. Additionally, we present a number of Lévy processes that are analytically tractable, in their characteristic functions and Lévy densities, and hence are relevant for option pricing.     

Stochastic Volatility for Lévy Processes

Three processes reflecting persistence of volatility are initially formulated by evaluating three Lévy processes at a time change given by the integral of a mean-reverting square root process. The model for the mean-reverting time change is then generalized to include non-Gaussian models that are solutions to Ornstein-Uhlenbeck equations driven by one-sided discontinuous Lévy processes permitting correlation with the stock. Positive stock price processes are obtained by exponentiating and mean correcting these processes, or alternatively by stochastically exponentiating these processes. The characteristic functions for the log price can be used to yield option prices via the fast Fourier transform. In general mean-corrected exponentiation performs better than employing the stochastic exponential. It is observed that the mean-corrected exponential model is not a martingale in the filtration in which it is originally defined. This leads us to formulate and investigate the important property of martingale marginals where we seek martingales in altered filtrations consistent with the one-dimensional marginal distributions of the level of the process at each future date.     

AN ECONOMETRIC ANALYSIS OF EQUITY COSTS AND RISK PREMIUMS IN THE ELECTRIC UTILITY INDUSTRY: 1971–1985

This paper develops a varying parameter econometric model that estimates the cost of equity of individual utility firms from 1971 to 1985. The equity costs estimated in this framework can be analyzed in terms of their statistical precision. The paper also examines, theoretically and empirically, the relationship between the econometric estimates of the equity risk premiums and the risk-free interest rates. The data do not support the hypothesis that risk premiums are independent of interest rates. Also, the relationship appears to vary over time. These results invalidate the risk premium approach in which equity costs are estimated by adding a constant, historical average risk premium to the prevailing interest rates.     

Option Pricing With V. G. Martingale Components1

European call options are priced when the uncertainty driving the stock price follows the V. G. stochastic process (Madan and Seneta 1990). the incomplete markets equilibrium change of measure is approximated and identified using the log return mean, variance, and kurtosis. an exact equilibrium interpretation is also provided, allowing inference about relative risk aversion coefficients from option prices. Relative to Black-Scholes, V. G. option values are higher, particularly so for out of the money options with long maturity on stocks with high means, low variances, and high kurtosis.     

Pricing Reinsurance Contracts on FDIC Losses

This paper proposes a pricing model for the FDIC's reinsurance risk. We derive a closed‐form Weibull call option pricing model to price a call‐spread a reinsurer might sell to the FDIC. To obtain the risk‐neutral loss‐density necessary to price this call spread we risk‐neutralize a Weibull distributed FDIC annual losses by a tilting coefficient estimated from the traded call options on the BKX index. An application of the proposed approach yield reasonable reinsurance prices.     

Pricing and the psychology of consumption

Most executives know how pricing influences the demand for a product, but few of them realize how it affects the consumption of a product. In fact, most companies don't even believe they can have an effect on whether customers use products they have already paid for. In this article, the authors argue that the relationship between pricing and consumption lies at the core of customer strategy. The extent to which a customer uses a product during a certain time period often determines whether he or she will buy the product again. So pricing tactics that encourage people to use the products they've paid for help companies build long-term relationships with customers. The link between pricing and consumption is clear: People are more likely to consume a product when they are aware of its cost. But for many executives, the idea that they should draw consumers' attention to the price that was paid for a product or service is counterintuitive. Companies have long sought to mask the costs of their goods and services in order to boost sales. And rightly so--if a company fails to make the initial sale, it won't have to worry about consumption. So to promote sales, health club managers encourage members to get the payment out of the way early; HMOs encourage automatic payroll deductions; and cruise lines bundle small, specific costs into a single, all-inclusive fee. The problem is, by masking how much a buyer has spent on a given product, these pricing tactics decrease the likelihood that the buyer will actually use it. This article offers some new approaches to pricing--how and when to charge for goods and services--that may boost consumption.     

The Variance Gamma Process and Option Pricing

A three parameter stochastic process, termed the variance gamma process, that generalizes Brownian motion is developed as a model for the dynamics of log stock prices. The process is obtained by evaluating Brownian motion with drift at a random time given by a gamma process. The two additional parameters are the drift of the Brownian motion and the volatility of the time change. These additional parameters provide control over the skewness and kurtosis of the return distribution. Closed forms are obtained for the return density and the prices of European options. The statistical and risk neutral densities are estimated for data on the S & P500 Index and the prices of options on this Index. It is observed that the statistical density is symmetric with some kurtosis, while the risk neutral density is negatively skewed with a larger kurtosis. The additional parameters also correct for pricing biases of the Black Scholes model that is a parametric special case of the option pricing model developed here.     

The revival of shelf-registered corporate equity offerings

We report that traditional seasoned equity offerings (SEOs) are no longer firms' preferred choice for raising seasoned public equity. Traditional offerings have recently been surpassed by shelf-registered offerings in terms of both annual frequency and total capital raised. This represents a dramatic shift from the 1980s, during which the overwhelming majority of firms favored traditional over shelf-registered offerings. We find that the growth in shelf use is related to firms increasingly valuing and using the option feature of shelf registration to defer offerings. Moreover, the evidence indicates that the way firms now use shelf offerings resolves the shelf under-certification problem and results in no larger market penalties and significantly lower underwriter fees relative to non-shelf offerings. Finally, firms often use universal shelf filings and choose between debt and equity offerings based on the prevailing relative market conditions.