This paper proposes and estimates a more general parametric stochastic variance model of equity index returns than has been previously considered using data from both underlying and options markets. The parameters of the model under both the objective and riskneutral measures are estimated simultaneously. I conclude that the square root stochastic variance model of Heston (1993) and others is incapable of generating realistic returns behavior and find that the data are more accurately represented by a stochastic variance model in the CEV class or a model that allows the price and variance processes to have a time-varying correlation. Specifically, I find that as the level of market variance increases, the volatility of market variance increases rapidly and the correlation between the price and variance processes becomes substantially more negative. The heightened heteroskedasticity in market variance that results generates realistic crash probabilities and dynamics and causes returns to display values of skewness and kurtosis much more consistent with their sample values. While the model dramatically improves the fit of options prices relative to the square root process, it falls short of explaining the implied volatility smile for short-dated options.

This article introduces a new nonparametric test to detect jump arrival times and realized jump sizes in asset prices up to the intra-day level. We demonstrate that the likelihood of misclassification of jumps becomes negligible when we use high-frequency returns. Using our test, we examine jump dynamics and their distributions in the U.S. equity markets. The results show that individual stock jumps are associated with prescheduled earnings announcements and other company-specific news events. Additionally, S&P 500 Index jumps are associated with general market news announcements. This suggests different pricing models for individual equity options versus index options. (JEL G12, G22, G14) Financial markets sometimes generate significant discontinuities, so-called jumps, in financial variables. A number of recent empirical and theoretical studies proved the existence of jumps and their substantial impact on financial management, from portfolio and risk management to option and bond pricing

The Black-Scholes theory of option pricing has been considered for many years as an important but very approximate zeroth-order description of actual market behavior. We generalize the functional form of the diffusion of these systems and also consider multi-factor models including stochastic volatility. Daily Eurodollar futures prices and implied volatilities are fit to determine exponents of functional behavior of diffusions using methods of global optimization, Adaptive Simulated Annealing (ASA), to generate tight fits across moving time windows of Eurodollar contracts. These short-time fitted distributions are then developed into long-time distributions using a robust non-Monte Carlo path-integral algorithm, PATHINT, to generate prices and derivatives commonly used by option traders.

Under a guaranteed annuity option, an insurer guarantees to convert a policyholder’s accumulated funds to a life annuity at a fixed rate when the policy matures. If the annuity rates provided under the guarantee are more beneficial to the policyholder than the prevailing rates in the market the insurer has to make up the difference. Such guarantees are common in many US tax sheltered insurance products. These guarantees were popular in UK retirement savings contracts issued in the 1970’s and 1980’s when long-term interest rates were high. At that time, the options were very far out of the money and insurance companies apparently assumed that interest rates would remain high and thus that the guarantees would never become active. In the 1990’s, as long-term interest rates began to fall, the value of these guarantees rose. Because of the way the guarantee was written, two other factors influenced the cost of these guarantees. First, strong stock market performance meant that the amounts to which the guarantee applied increased significantly. Second, the mortality assumption implicit in the guarantee did not anticipate the improvement in mortality which actually occurred. The emerging liabilities under these guarantees threatened the solvency of some companies 1

One of the most popular investments available in the Canadian market today is a mutual fund with the added feature of a long term maturity guarantee. These types of investments are known as segregated funds. They are similar in many respects to variable annuity savings plans in the United States. Segregated funds have very complex option features. These contracts contain multiple embedded shout options which permit the holder to reset the guarantee level and the maturity date for which it applies many times during the life of the contract. These funds also provide mortality benefits if the investor dies prior to the maturity date. This paper explores the valuation of segregated funds using an approach based on the numerical solution of a set of linear complementarity problems. Our results indicate that the option components of many of these contracts seem to be underpriced, especially for riskier funds with relatively high volatilities. This assumes that investors exercise th...

The economic consequences of a long memory assumption about volatility are documented, by comparing implied volatilities for option prices obtained from short and long memory volatility processes. Numerical results are given for options on the S & P 100 index from 1984 to 1998, with lives up to two years. The long memory assumption is found to have a significant impact upon the term structure of implied volatilities and a relatively minor impact upon smile effects. These conclusions are important because evidence for long memory in volatility has been found in the prices of many assets.

A shout option is a financial contract which allows the holder to change the payoff during the lifetime of the contract. For example, the holder could have the right to set the strike price to the current value of the underlying asset. Complex versions of these options are embedded in financial products which offer various types of maturity guarantees such as segregated funds marketed by Canadian insurance companies. The value of these options can be determined by solving a collection of coupled partial differential equations (PDEs). In this work we develop an extensible, object-oriented framework for valuing these contracts which is capable of exploiting modern, high-performance supercomputing architectures. We use this framework to study and illustrate practical aspects of valuing and hedging these contracts. Keywords: PDE option pricing, shout options, object-oriented, high-performance architecture Running Title: Shout Options Acknowledgement: This work was supported by ...

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This article investigates option models in the encompassing class of stochastic volatility, returnjumps, and volatility-jumps. Relying on individual equity options and method of simulated moments estimation, several major results obtained are, first, that the double-jump process is the least misspecified and the least demanding in fitting the tail-size and tail-asymmetry of individual risk-neutral return distributions; second, the double-jump model improves pricing performance beyond return-jumps absent volatility-jumps, and beyond volatility-jumps absent return-jumps; third, between return-jumps and volatility-jumps, the former is empirically more relevant than the latter for pricing options; fourth, the inverse link between volatility-jumps and return-jumps is instrumental for explaining the valuation of deep out-of-money puts and option dynamics of firms with high kurtosis; fifth, stochastic volatility is not as important for individual equity options as it is for index options. Incremental insights that emerge from individual equity options bring clarity to divergent findings on the role of return-jumps and volatility-jumps.

The Black-Scholes theory of option pricing has been considered for many years as an important but very approximate zeroth-order description of actual market behavior. We generalize the functional form of the diffusion of these systems and also consider multi-factor models including stochastic volatility. We use a previous development of a statistical mechanics of financial markets to model these issues. Daily Eurodollar futures prices and implied volatilities are fit to determine exponents of functional behavior of diffusions using methods of global optimization, Adaptive Simulated Annealing (ASA), to generate tight fits across moving time windows of Eurodollar contracts. These short-time fitted distributions are then developed into long-time distributions using a robust non-Monte Carlo path-integral algorithm, PATHINT, to generate prices and derivatives commonly used by option traders.