Underlying Price Process Research Articles

Constructing the Optimal Exercise Boundary for American Options by Least-Squares Monte Carlo

The least-squares Monte Carlo method of Longstaff-Schwartz is utilized to construct the optimal exercise boundary (OXB) of an American put option when the underlying follows a geometric Brownian motion (GBM). The optimal exercise price at each time step is obtained by solving numerically the equation of the exercising boundary condition. The set of such exercise prices, along with their “standard deviations,” is then fitted to a smooth, monotonic model of a sum of three exponential functions to approximate the OXB, which turns out to be very close to the exact solution of the boundary. The approach can be efficiently implemented and readily computed in practice, and should be applicable to cases when the underlying price process is not GBM.

Liquidity-Based Estimation of Spot Volatility Under Microstructure Noise

Recent literature on realized volatility suggests that the observed price process of an asset may be decomposed into two parts: the genuine (unobservable) price process and microstructure noise. In this article we present a methodology to estimate stochastic volatility by separating these components. Depending on market liquidity, the source of a move in the transaction price of an asset may be distinguished as a move in the underlying price process or as microstructure noise. To identify the weights of these components we use different (order-based and trade-based) liquidity measures to update expectations on the unobservable price process via Bayesian learning. Depending on these weights we are able to estimate stochastic volatility from noisy data.

Along But Beyond Mean-Variance: Utility Maximization in a Semimartingale Model

It is well known that under certain assumptions the strategy of an investor maximizing his expected utility coincides with the mean-variance optimal strategy. In this paper we show that the two strategies are not equal in general and find the connection between a utility maximizing and a mean-variance optimal strategy in a continuous semimartingale model. That is done by showing that the utility maximizing strategy of a CARA investor can be expressed in terms of expectation and the expected quadratic variation of the underlying price process. It coincides with the mean-variance optimal strategy if the underlying price process is a local martingale. Keywords: mean-variance portfolios, utility maximization, dynamic portfolio selection, quadratic variation JEL classification numbers: G11, C61

Option Pricing for Time-Change Exponential Lévy Model Under Memm

The purpose of this article is to study the rational evaluation of European options price when the underlying price process is described by a time-change Levy process. European option pricing formula is obtained under the minimal entropy martingale measure (MEMM) and applied to several examples of particular time-change Levy processes. It can be seen that the framework in this paper encompasses the Black-Scholes model and almost all of the models proposed in the subordinated market.

Insiders' hedging in a jump diffusion model

In this paper, we formulate the optimal hedging problem when the underlying stock price has jumps, especially for insiders who have more information than the general public. The jumps in the underlying price process depend on another diffusion process, which models a sequence of firm-specific information. This diffusion process is observed only by insiders. Nevertheless, the market is incomplete to insiders as well as to the general public. We use the local risk minimization method to find an optimal hedging strategy for insiders. We also numerically compare the value of the insider's hedging portfolio with the value of an honest trader's hedging portfolio for a simulated sample path of a stock price.

Bounds on the American Option

The essential feature of American style claims lies in the holder's right to time the exercise decision. The value of the claim depends on the information about future prices that the holder will acquire over time. Much of the literature makes restrictive assumptions about information revelation - for example that the underlying price process is Markov. This paper explores the upper bound on the price of an American option, placing no assumptions on the information structure. The analysis provides insight into the processes that make the American feature valuable, and points the way to hedging strategies for American options that are robust to model error.

A Market Model for Stochastic Smiles

We introduce a new approach that allows to construct no-arbitrage market models of implied volatility surfaces. The need for developing such models has long been recognized. The framework presented here makes it possible to generate models of a joint evolution of an arbitrary number of option price processes together with the underlying price process. The key idea of the approach is to take a deterministic smile model as a backbone around which a stochastic smile model can be constructed without violating no-arbitrage constraints.

Integro-differential equations for foreign currency option prices in exponential Lévy models

We investigate the rational evaluation of foreign currency option price when the underlying price processes are described by exponential Lévy processes. The main contribution of our study is that we give the integro-differential equations for foreign currency option prices.

Measuring Integrated Variance with Extreme-Value Based Estimators

Driven by the rise in computational power, it has become popular to measure integrated variance with high-frequency squared returns. Though the squared return is a natural choice as a variance estimate, it is not the most efficient one for a given interval length. Extreme-value based estimators incorporate the information of the first, high, low and last price observation of a price path interval and are several times more efficient than the squared return estimate. Though these estimators have mainly been applied to daily data, they are defined over an arbitrary interval length, hence it is straightforward to apply these estimators to high-frequency data to get a non-parametric integrated variance estimate similar to realized variance. This study compares several extreme-value based estimators and the realized variance estimator of integrated variance within a high-frequency framework in terms of bias and efficiency with an extensive Monte Carlo study for various assumptions about the underlying price process including large drifts, stochastic volatility with leverage effects and also an orthogonal market microstructure noise. The simulation results strongly suggest the use of extreme-value based estimators over the common practice of summed high-frequency squared returns to measure the integrated variance. All extreme-value based estimators are superior both interms of bias and efficiency for any given sampling frequency for all considered price process definitions. The efficiency of the estimators is confirmed empirically on the basis of 10 years of S&P 500 futures data sampled at a 5-minute frequency. A state-space framework reveals the latent stochastic volatility process and allows for a comparison of the error of the considered estimators.

Discrete Sine Transform for Multi-Scales Realized Volatility Measures

In this study new realized volatility measures based on Multi-Scale regression and Discrete Sine Transform (DST) approaches are presented. We show that Multi-Scales estimators similar to that recently proposed by Zhang (2004) can be constructed within a simple regression based approach by exploiting the linear relation existing between the market microstructure bias and the realized volatilities computed at different frequencies. These regression based estimators can be further improved and robustified by using the DST approach to prefilter market microstructure noise. The motivation for the DST approach rests on its ability to diagonalize MA type of processes which arise naturally in discrete time models of tick-by-tick returns with market microstructure noise. Hence, the DST provides a natural orthonormal basis decomposition of observed returns which permits to optimally disentangle the volatility signal of the underlying price process from the market microstructure noise. Robustness of the DST approach with respect to more general dependent structure of the microstructure noise is also analytically shown. Then, the combination of such Multi-Scale regression approach with the DST gives us a Multi-Scales DST realized volatility estimator which is robust against a wide class of noise contaminations and model misspecifications. Thanks to the DST orthogonalization which also allows us to analytically derive closed form expressions for the Cramer-Rao bounds of MA(1)processes, an evaluation of the absolute efficiency of volatility estimators under the i.i.d. noise assumption becomes available, indicating that the Multi-Scales DST estimator possesses a finite sample variance very close to the optimal Cramer-Rao bounds. Monte Carlo simulations based on realistic models for price dynamics and market microstructure effects, show the superiority of DST estimators, compared to alternative volatility proxies for a wide range of noise to signal ratios and different types of noise contaminations. Empirical analysis based on six years of tick-by-tick data for S&P 500 index-future, FIB 30, and 30 years U.S. Treasury Bond future, confirms the accuracy and robustness of DST estimators on different types of real data.

Real option analysis of a technology portfolio

This article contributes to methodology of real options analysis of investments in capital intensive process industries, where relatively homogenous outputs are produced using commonly known production technologies. In addition to capacity expansion, the method can be used for analysis of mergers and acquisitions. Valuation of the real option is based on bid price; i.e., the maximum price the firm is willing to pay. To find such a price, stochastic optimization with an expected utility criterion is used to determine investments in product specific technologies as well as in publicly traded financial instruments (the competing investments). To operationalize the valuation principle, we develop a double binary tree employing Kalman filter for scenario generation. For exponential utility, valuation is carried out by dynamic programming. We extend known methods to allow interdependence of the mill cash flow and return on competing financial investments. For forest industries, we provide an illustration, where the underlying price process in our Kalman filter application is a vector error correction model.

An Aggregate Weibull Approach for Modeling Short-Term System Generating Capacity

Deregulation of electricity markets is occurring all over the world. This trend introduces new risks and uncertainties into the electricity industry, the most significant being price risk. The spot price of electricity is highly volatile, and the ability to price risk management contracts on this commodity is contingent on a robust and realistic model of the underlying price process. One key driver of electricity spot price is the forced outages of generating plants in the system. The current paper describes a system aggregate model of short-term generating capacity that can be adapted to any generating system of interest. After describing the model, we test it using the IEEE Reliability Test System (RTS).

Explicit solutions to European options in a regime-switching economy

We provide closed-form solutions for European option values when the dynamics of both the short rate and volatility of the underlying price process are modulated by a continuous-time Markov chain with a finite number of “economic states”. Extensions involving dividends, currencies and cost of carry are further explored.

Mean-Variance Hedging When There Are Jumps

In this paper, we consider the problem of mean-variance hedging in an incomplete market where the underlying assets are jump diffusion processes which are driven by Brownian motion and doubly stochastic Poisson processes. This problem is formulated as a stochastic control problem, and closed form expressions for the optimal hedging policy are obtained using methods from stochastic control and the theory of backward stochastic differential equations. The results we have obtained show how backward stochastic differential equations can be used to obtain solutions to optimal investment and hedging problems when discontinuities in the underlying price processes are modeled by the arrivals of Poisson processes with stochastic intensities. Applications to the problem of hedging default risk are also discussed.

Pricing foreign equity options under Lévy processes

This article investigates the valuation of a foreign equity option whose value depends on the exchange rate and foreign equity prices. Assuming that these underlying price processes are correlated and driven by a multidimensional Levy process, a method suitable for solving the complex valuation problem is developed. First, to reduce the number of dimensions of the problem, the probability measure is changed to embed some dimensions of the Levy process into the pricing measure. Second, to simplify the integral complexity of the discounted terminal payoff, the valuation problem is transformed to Fourier space. The main contribution of this study is that by combining these two methods, the multivariate valuation problem is significantly simplified, and very accurate results are obtained relatively quickly. This powerful method can also be applied to other multivariate pricing problems involving Levy processes. © 2005 Wiley Periodicals, Inc. Jrl Fut Mark 25:917–944, 2005

Pricing and Hedging Path-Dependent Options Under the CEV Process

Much of the work on path-dependent options assumes that the underlying asset price follows geometric Brownian motion with constant volatility. This paper uses a more general assumption for the asset price process that provides a better fit to the empirical observations. We use the so-called constant elasticity of variance (CEV) diffusion model where the volatility is a function of the underlying asset price. We derive analytical formulae for the prices of important types of path-dependent options under this assumption. We demonstrate that the prices of options, which depend on extrema, such as barrier and lookback options, can be much more sensitive to the specification of the underlying price process than standard call and put options and show that a financial institution that uses the standard geometric Brownian motion assumption is exposed to significant pricing and hedging errors when dealing in path-dependent options.

Nonparametric risk management and implied risk aversion

Typical value-at-risk (VaR) calculations involve the probabilities of extreme dollar losses, based on the statistical distributions of market prices. Such quantities do not account for the fact that the same dollar loss can have two very different economic valuations, depending on business conditions. We propose a nonparametric VaR measure that incorporates economic valuation according to the state-price density associated with the underlying price processes. The state-price density yields VaR values that are adjusted for risk aversion, time preferences, and other variations in economic valuation. In the context of a representative agent equilibrium model, we construct an estimator of the risk-aversion coefficient that is implied by the joint observations on the cross-section of option prices and time-series of underlying assest values.

Pricing multivariate contingent claims using estimated risk–neutral density functions

Many asset price series exhibit time-varying volatility, jumps and other features inconsistent with assumptions about the underlying price process made by standard multivariate contingent claims (MVCC) pricing models. This article develops an interpolative technique for pricing MVCCs — flexible NLS pricing — that involves the estimation of a flexible multivariate risk–neutral density function implied by existing asset prices. As an application, the flexible NLS pricing technique is used to value several bivariate contingent claims dependent on foreign exchange rates in 1993 and 1994. The bivariate flexible risk–neutral density function more accurately prices existing options than the bivariate log-normal density implied by a multivariate geometric Brownian motion. In addition, the bivariate contingent claims analyzed have substantially different prices using the two density functions suggesting flexible NLS pricing may improve accuracy over standard methods.

Non-Parametric Risk Management and Implied Risk Aversion

Typical value-at-risk (VaR) calculations involve the probabilities of extreme dollar losses, based on the statistical distributions of market prices. Such quantities do not account for the fact that the same dollar loss can have two very different economic valuations, depending on business conditions. We propose a nonparametric VaR measure that incorporates economic valuation according to the state-price density associated with the underlying price processes. The state-price density yields VaR values that are adjusted for risk aversion, time preferences, and other variations in economic valuation. In the context of a representative agent equilibrium model, we construct an estimator of the risk-aversion coefficient that is implied by the joint observations on the cross-section of option prices and time-series of underlying asset values.

General Properties of Option Prices

ABSTRACTWhen the underlying price process is a one‐dimensional diffusion, as well as in certain restricted stochastic volatility settings, a contingent claim's delta is bounded by the infimum and supremum of its delta at maturity. Further, if the claim's payoff is convex (concave), the claim's price is a convex (concave) function of the underlying asset's value. However, when volatility is less specialized, or when the underlying process is discontinuous or non‐Markovian, a call's price can be a decreasing, concave function of the underlying price over some range, increasing with the passage of time, and decreasing in the level of interest rates.