Abstract not found

We present a model for pricing and hedging derivative securities and option portfolios in an environment where the volatility is not known precisely, but is assumed instead to lie between two extreme values oe min and oe max . These bounds could be inferred from extreme values of the implied volatilities of liquid options, or from high-low peaks in historical stock- or option-implied volatilities. They can be viewed as defining a confidence interval for future volatility values. We show that the extremal non-arbitrageable prices for the derivative asset which arise as the volatility paths vary in such a band can be described by a non-linear PDE, which we call the Black-Scholes-Barenblatt equation. In this equation, the "pricing" volatility is selected dynamically from the two extreme values oe min ,oe max , according to the convexity of the valuefunction. A simple algorithm for solving the equation by finite-differencing or a trinomial tree is presented. We show that this model capture...

pp.567-611. Contents

Simulation has proved to be a valuable tool for estimating security prices for which simple closed form solutions do not exist. In this paper we present two direct methods, a pathwise method and a likelihood ratio method, for estimating derivatives of security prices using simulation. With the direct methods, the information from a single simulation can be used to estimate multiple derivatives along with a security’s price. The main advantage of the direct methods over re-simulation is increased computational speed. Another advantage is that the direct methods give unbiased estimates of derivatives, whereas the estimates obtained by resimulation are biased. Computational results are given for both direct methods and comparisons are made to the standard method of re-simulation to estimate derivatives. The methods are illustrated for a path independent model (European options), a path dependent model (Asian options), and a model with multiple state variables (options with stochastic volatility).

The paper proposes an original class of models for the continuous time price process of a financial security with non-constant volatility. The idea is to define instantaneous volatility in terms of exponentially-weighted moments of historic log-price. The instantaneous volatility is therefore driven by the same stochastic factors as the price process, so that unlike many other models of non-constant volatility, it is not necessary to introduce additional sources of randomness. Thus the market is complete and there are unique, preference-independent options prices. We find a partial differential equation for the price of a European Call Option. Smiles and skews are found in the resulting plots of implied volatility. Keywords: Option pricing, stochastic volatility, complete markets, smiles. Acknowledgement. It is a pleasure to thank the referees of an earlier draft of this paper whose perceptive comments have resulted in many improvements. 1 Research supported in part by Record Treasu...

There are two strands of literature on option pricing under stochastic volatility - the bivariate diffusion and GARCH approaches. These two strands of models are unified in this article by a convergence result. The existing bivariate diffusion option pricing models are shown to be the limits of the GARCH option pricing model. Theoretically, the limit result not only unifies these two strands of literature but also provides new insights into the existing bivariate diffusion option pricing models. Operationally, the limit result gives rise to the possibility of interchanging the GARCH and bivariate diffusion models for purposes of estimation and option valuation.

We show a class of stochastic volatility price models for which the most natural candidates for martingale measures are only strictly local martingale measures, contrary to what is usually assumed in the finance literature. We also show the existence of martingale measures, however, and give explicit examples. 1 Introduction In the mathematical finance literature there is a growing interest in the study of incomplete markets, and among the efforts in understanding their properties the stochastic volatility models have become increasingly popular: see for example Hull and White (87), Scott (87), Wiggins (87), Johnson and Shanno (87), Stein and Stein (91), Heston (93), Dupire (92), Hofmann, Platen and Schweizer (92) among others. They all start by imposing certain dynamics for a "volatility process" and construct asset prices as stochastic-exponentials of integrals of this volatility with respect to a Brownian motion. If we want to apply results from arbitrage-pricing theory, which char...

Characterizing asset return dynamics using volatility models is an important part of empirical finance. The existing literature on GARCH models favors some rather complex volatility specifications whose relative performance is usually assessed through their likelihood based on a time-series of asset returns. This paper compares a range of GARCH models along a different dimension, using option prices and returns under the risk-neutral as well as the physical probability measure. We judge the relative performance of various models by evaluating an objective function based on option prices. In contrast with returns-based inference, we find that our option-based objective function favors a relatively parsimonious model. Specifically, when evaluated out-of-sample, our analysis favors a model that besides volatility clustering only allows for a standard leverage effect. JEL Classification: G12

The manufacturing environment is becoming increasingly dynamic with upsurges in electronic-commerce, supply chain management, forecasting, and procurement and resource planning. It also includes trends toward more process data acquisition and analysis, shorter production runs, and more stringent quality requirements. These drivers lead to an opportunity for companies to collect and use information to identify changes that will affect their manufacturing systems. In conjunction with an industry partner who produces home fashion products, we developed a case-study that highlights four major manufacturing transitions: new product introduction; moving a product from research and development (R&D) to commercialization; new plant location; and starting or restarting production of existing products. These types of changes cross many levels of the operation - including the product level, plant level, and organizational level - and typically present significant operational challenges. We use this case-study to motivate the theoretical and applied research needed to support a real option framework for system changes in manufacturing. The key elements of our framework are to quantify manufacturing changes, develop a real option model for these activities, value the options to identify the best scenarios, and integrate these elements so that we can monitor and manage the overall process. The advantage of this approach is that it allows us to directly incorporate a market driven perspective, tying the manufacturing operations with the organizational economic goals.

Stochastic volatility (SV) is the main concept used in the fields of financial economics and mathematical finance to deal with the endemic time-varying volatility and codependence found in financial markets. Such dependence has been known for a long time, early comments include Mandelbrot (1963) and Officer (1973). It was also clear to the founding fathers of modern continuous time finance that homogeneity was an unrealistic if convenient simplification, e.g. Black and Scholes (1972, p. 416) wrote “... there is evidence of non-stationarity in the variance. More work must be done to predict variances using the information available. ” Heterogeneity has deep implications for the theory and practice of financial economics and econometrics. In particular, asset pricing theory is dominated by the idea that higher rewards may be expected when we face higher risks, but these risks change through time in complicated ways. Some of the changes in the level of risk can be modelled stochastically, where the level of volatility and degree of codependence between assets is allowed to change over time. Such models allow us to explain, for example, empirically observed departures from Black-Scholes-Merton prices for options and understand why we should expect to see occasional dramatic moves in financial markets. The outline of this article is as follows. In section 2 I will trace the origins of SV and provide links with the basic models used today in the literature. In section 3 I will briefly discuss some of the innovations in the second generation of SV models. In section 4 I will briefly discuss the literature on conducting inference for SV models. In section 5 I will talk about the use of SV to price options. In section 6 I will consider the connection of SV with realised volatility. A extensive reviews of this literature is given in Shephard (2005). 2 The origin of SV models The origins of SV are messy, I will give five accounts, which attribute the subject to different sets of people.