I use a new technique to derive a closed-form solution for the price of a European call option on an asset with stochastic volatility. The model allows arbitrary correlation between volatility and spotasset returns. I introduce stochastic interest rates and show how to apply the model to bond options and foreign currency options. Simulations show that correlation between volatility and the spot asset’s price is important for explaining return skewness and strike-price biases in the BlackScholes (1973) model. The solution technique is based on characteristi c functions and can be applied to other problems.

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A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options

Monte Carlo methods are revolutionizing the on-line analysis of data in fields as diverse as financial modeling, target tracking and computer vision. These methods, appearing under the names of bootstrap filters, condensation, optimal Monte Carlo filters, particle filters and survival of the fittest, have made it possible to solve numerically many complex, non-standard problems that were previously intractable. This book presents the first comprehensive treatment of these techniques, including convergence results and applications to tracking, guidance, automated target recognition, aircraft navigation, robot navigation, econometrics, financial modeling, neural networks, optimal control, optimal filtering, communications, reinforcement learning, signal enhancement, model averaging and selection, computer vision, semiconductor design, population biology, dynamic Bayesian networks, and time series analysis. This will be of great value to students, researchers and practitioners, who have some basic knowledge of probability. Arnaud Doucet received the Ph. D. degree from the University of Paris-XI Orsay in 1997. From 1998 to 2000, he conducted research at the Signal Processing Group of Cambridge University, UK. He is currently an assistant professor at the Department of Electrical Engineering of Melbourne University, Australia. His research interests include Bayesian statistics, dynamic models and Monte Carlo methods. Nando de Freitas obtained a Ph.D. degree in information engineering from Cambridge University in 1999. He is presently a research associate with the artificial intelligence group of the University of California at Berkeley. His main research interests are in Bayesian statistics and the application of on-line and batch Monte Carlo methods to machine learning. Neil Gordon obtained a Ph.D. in Statistics from Imperial College, University of London in 1993. He is with the Pattern and Information Processing group at the Defence Evaluation and Research Agency in the United Kingdom. His research interests are in time series, statistical data analysis, and pattern recognition with a particular emphasis on target tracking and missile guidance.

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Sequential Monte Carlo methods in practice

A voluminous literature has emerged for modeling the temporal dependencies in financial market volatility using ARCH and stochastic volatility models. While most of these studies have documented highly significant in-sample parameter estimates and pronounced intertemporal volatility persistence, traditional ex-post forecast evaluation criteria suggest that the models provide seemingly poor volatility forecasts. Contrary to this contention, we show that volatility models produce strikingly accurate interdaily forecasts for the latent volatility factor that would be of interest in most financial applications. New methods for improved ex-post interdaily volatility measurements based on high-frequency intradaily data are also discussed.

https://faculty.washington.edu/ezivot/econ589//////AndersenBollerslevIER1998.pdf

Answering the skeptics: yes, standard volatility models do provide accurate forecasts*

This paper defines the news impact curve which measures how new information is incorporated into volatility estimates. Various new and existing ARCH models including a partially nonparametric one are compared and estimated with daily Japanese stock return data. New diagnostic tests are presented which emphasize the asymmetry of the volatility response to news. Our results suggest that the model by Glosten, Jagannathan, and Runkle is the best parametric model. The EGARCH also can capture most of the asymmetry; however, there is evidence that the variability of the conditional variance implied by the EGARCH is too high.

/pdf/measuring-and-testing-the-impact-of-news-on-volatility-533bchskmf.pdf

Measuring and Testing the Impact of News on Volatility

This book concerns the use of concepts from statistical physics in the description of financial systems. The authors illustrate the scaling concepts used in probability theory, critical phenomena, and fully developed turbulent fluids. These concepts are then applied to financial time series. The authors also present a stochastic model that displays several of the statistical properties observed in empirical data. Statistical physics concepts such as stochastic dynamics, short- and long-range correlations, self-similarity and scaling permit an understanding of the global behaviour of economic systems without first having to work out a detailed microscopic description of the system. Physicists will find the application of statistical physics concepts to economic systems interesting. Economists and workers in the financial world will find useful the presentation of empirical analysis methods and well-formulated theoretical tools that might help describe systems composed of a huge number of interacting subsystems.

/pdf/introduction-to-econophysics-correlations-and-complexity-in-4a1wviggsu.pdf

Introduction to Econophysics: Correlations and Complexity in Finance

One option-pricing problem that has hitherto been unsolved is the pricing of a European call on an asset that has a stochastic volatility. This paper examines this problem. The option price is determined in series form for the case in which the stochastic volatility is independent of the stock price. Numerical solutions are also produced for the case in which the volatility is correlated with the stock price. It is found that the Black-Scholes price frequently overprices options and that the degree of overpricing increases with the time to maturity. ONE OPTION-PRICING PROBLEM that has hitherto remained unsolved is the pricing of a European call on a stock that has a stochastic volatility. From the work of Merton [12], Garman [6], and Cox, Ingersoll, and Ross [3], the differential equation that the option must satisfy is known. The solution of this differential equation is independent of risk preferences if (a) the volatility is a traded asset or (b) the volatility is uncorrelated with aggregate consumption. If either of these conditions holds, the risk-neutral valuation arguments of Cox and Ross [4] can be used in a straightfoward way. This paper produces a solution in series form for the situation in which the stock price is instantaneously uncorrelated with the volatility. We do not assume that the volatility is a traded asset. Also, a constant correlation between the instantaneous rate of change of the volatility and the rate of change of aggregate consumption can be accommodated. The option price is lower than the BlackScholes (B-S) [1] price when the option is close to being at the money and higher when it is deep in or deep out of the money. The exercise prices for which overpricing by B-S takes place are within about ten percent of the security price. This is the range of exercise prices over which most option trading takes place, so we may, in general, expect the B-S price to overprice options. This effect is exaggerated as the time to maturity increases. One of the most surprising implications of this is that, if the B-S equation is used to determine the implied volatility of a near-the-money option, the longer the time to maturity the lower the implied volatility. Numerical solutions for the case in which the volatility is correlated with the stock price are also examined. The stochastic volatility problem has been examined by Merton [13], Geske [7], Johnson [10], Johnson and Shanno [11], Eisenberg [5], Wiggins [16], and

https://onlinelibrary.wiley.com/doi/pdf/10.1111/j.1540-6261.1987.tb02568.x

The Pricing of Options on Assets with Stochastic Volatilities

This article shows that the one-state-variableinterest-rate models of Vasicek (1977) and Cox,Ingersoll, and Ross (1985b) can be extended sothat they are consistent with both the current termstructure of interest rates and either the currentvolatilities of all spot interest rates or the currentvolatilities of all forward interest rates. Theextended Vasicek model is shown to be very tracta-ble analytically. The article compares option pricesobtained using the extended Vasicek model withthose obtained using a number of other models.

/pdf/pricing-interest-rate-derivative-securities-38plaxysft.pdf

Pricing Interest-Rate-Derivative Securities

A company’s credit default swap spread is the cost per annum for protection against a default by the company. In this paper we analyze data on credit default swap spreads collected by a credit derivatives broker. We first examine the relationship between credit default spreads and bond yields and reach conclusions on the benchmark risk-free rate used by participants in the credit derivatives market. We then carry out a series of tests to explore the extent to which credit rating announcements by Moody’s are anticipated by participants in the credit default swap market.

https://papers.ssrn.com/sol3/Delivery.cfm/SSRN_ID2173171_code1557616.pdf?abstractid=2173171&mirid=1&type=2

The Relationship between Credit Default Swap Spreads, Bond Yields, and Credit Rating Announcements

/pdf/the-relationship-between-credit-default-swap-spreads-bond-6z7clzujxz.pdf

The relationship between credit default swap spreads, bond yields, and credit rating announcements

Many of the new credit derivative products are based on default experience for a portfolio of financial instruments. These include collateralized debt obligations (CDOs) and similar tranched credit products, and “n-th to default swaps.” Devising good default risk models for single-name credits has been challenging enough, but applying them to credit portfolios introduces much greater complexity, because of the critical importance of correlation. The most common valuation technology is Monte Carlo simulation, but with many bonds, each of which is subject to both correlated and idiosyncratic risk factors, the simulation is time-consuming and limited in scope. In this article, Hull and White offer two straightforward approximation techniques for evaluating default risk within the industry-standard “copula“ model that eliminate simulation of the idiosyncratic risks. Their approach greatly accelerates the solution while still allowing a large degree of flexibility in the choice of factor correlation structure and probability distributions. For example, Student-t distributed shocks that have fatter tails than the normal are easily accommodated.

Alan White

Papers

The Pricing of Options on Assets with Stochastic Volatilities

Pricing Interest-Rate-Derivative Securities

The Relationship between Credit Default Swap Spreads, Bond Yields, and Credit Rating Announcements

The relationship between credit default swap spreads, bond yields, and credit rating announcements

Valuation of a CDO and an n-th to Default CDS Without Monte Carlo Simulation