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1 – 10 of over 2000
Book part
Publication date: 30 August 2019

Md. Nazmul Ahsan and Jean-Marie Dufour

Statistical inference (estimation and testing) for the stochastic volatility (SV) model Taylor (1982, 1986) is challenging, especially likelihood-based methods which are…

Abstract

Statistical inference (estimation and testing) for the stochastic volatility (SV) model Taylor (1982, 1986) is challenging, especially likelihood-based methods which are difficult to apply due to the presence of latent variables. The existing methods are either computationally costly and/or inefficient. In this paper, we propose computationally simple estimators for the SV model, which are at the same time highly efficient. The proposed class of estimators uses a small number of moment equations derived from an ARMA representation associated with the SV model, along with the possibility of using “winsorization” to improve stability and efficiency. We call these ARMA-SV estimators. Closed-form expressions for ARMA-SV estimators are obtained, and no numerical optimization procedure or choice of initial parameter values is required. The asymptotic distributional theory of the proposed estimators is studied. Due to their computational simplicity, the ARMA-SV estimators allow one to make reliable – even exact – simulation-based inference, through the application of Monte Carlo (MC) test or bootstrap methods. We compare them in a simulation experiment with a wide array of alternative estimation methods, in terms of bias, root mean square error and computation time. In addition to confirming the enormous computational advantage of the proposed estimators, the results show that ARMA-SV estimators match (or exceed) alternative estimators in terms of precision, including the widely used Bayesian estimator. The proposed methods are applied to daily observations on the returns for three major stock prices (Coca-Cola, Walmart, Ford) and the S&P Composite Price Index (2000–2017). The results confirm the presence of stochastic volatility with strong persistence.

Details

Topics in Identification, Limited Dependent Variables, Partial Observability, Experimentation, and Flexible Modeling: Part A
Type: Book
ISBN: 978-1-78973-241-2

Keywords

Article
Publication date: 1 March 2002

ROBERT G. TOMPKINS

The depth and breadth of the market for contingent claims, including exotic options, has expanded dramatically. Regulators have expressed concern regarding the risks of…

Abstract

The depth and breadth of the market for contingent claims, including exotic options, has expanded dramatically. Regulators have expressed concern regarding the risks of exotics to the financial system, due to the difficulty of hedging these instruments. Recent literature focuses on the difficulties in hedging exotic options, e.g., liquidity risk and other violations of the standard Black‐Scholes model. This article provides insight into hedging problems associated with exotic options: 1) hedging in discrete versus continuous time, 2) transaction costs, 3) stochastic volatility, and 4) non‐constant correlation. The author applies simulation analysis of these problems to a variety of exotics, including Asian options, barrier options, look‐back options, and quanto options.

Details

The Journal of Risk Finance, vol. 3 no. 4
Type: Research Article
ISSN: 1526-5943

Article
Publication date: 1 February 2003

DIMITRIS PSYCHOYIOS, GEORGE SKIADOPOULOS and PANAYOTIS ALEXAKIS

The volatility of a financial asset is an important input for financial decision‐making in the context of asset allocation, option pricing, and risk management. The…

Abstract

The volatility of a financial asset is an important input for financial decision‐making in the context of asset allocation, option pricing, and risk management. The authors compare and contrast four approaches to stochastic volatility to determine which is most appropriate to each of these various needs.

Details

The Journal of Risk Finance, vol. 4 no. 3
Type: Research Article
ISSN: 1526-5943

Book part
Publication date: 5 July 2012

Jens Carsten Jackwerth and Mark Rubinstein

How do stock prices evolve over time? The standard assumption of geometric Brownian motion, questionable as it has been right along, is even more doubtful in light of the…

Abstract

How do stock prices evolve over time? The standard assumption of geometric Brownian motion, questionable as it has been right along, is even more doubtful in light of the recent stock market crash and the subsequent prices of U.S. index options. With the development of rich and deep markets in these options, it is now possible to use options prices to make inferences about the risk-neutral stochastic process governing the underlying index. We compare the ability of models including Black–Scholes, naïve volatility smile predictions of traders, constant elasticity of variance, displaced diffusion, jump diffusion, stochastic volatility, and implied binomial trees to explain otherwise identical observed option prices that differ by strike prices, times-to-expiration, or times. The latter amounts to examining predictions of future implied volatilities.

Certain naïve predictive models used by traders seem to perform best, although some academic models are not far behind. We find that the better-performing models all incorporate the negative correlation between index level and volatility. Further improvements to the models seem to require predicting the future at-the-money implied volatility. However, an “efficient markets result” makes these forecasts difficult, and improvements to the option-pricing models might then be limited.

Details

Derivative Securities Pricing and Modelling
Type: Book
ISBN: 978-1-78052-616-4

Article
Publication date: 19 January 2015

Thomas Kokholm and Martin Stisen

This paper studies the performance of commonly employed stochastic volatility and jump models in the consistent pricing of The CBOE Volatility Index (VIX) and The S&P 500…

Abstract

Purpose

This paper studies the performance of commonly employed stochastic volatility and jump models in the consistent pricing of The CBOE Volatility Index (VIX) and The S&P 500 Index (SPX) options. With the existence of active markets for volatility derivatives and options on the underlying instrument, the need for models that are able to price these markets consistently has increased. Although pricing formulas for VIX and vanilla options are now available for commonly used models exhibiting stochastic volatility and/or jumps, it remains to be shown whether these are able to price both markets consistently. This paper fills this vacuum.

Design/methodology/approach

In particular, the Heston model, the Heston model with jumps in returns and the Heston model with simultaneous jumps in returns and variance (SVJJ) are jointly calibrated to market quotes on SPX and VIX options together with VIX futures.

Findings

The full flexibility of having jumps in both returns and volatility added to a stochastic volatility model is essential. Moreover, we find that the SVJJ model with the Feller condition imposed and calibrated jointly to SPX and VIX options fits both markets poorly. Relaxing the Feller condition in the calibration improves the performance considerably. Still, the fit is not satisfactory, and we conclude that one needs more flexibility in the model to jointly fit both option markets.

Originality/value

Compared to existing literature, we derive numerically simpler VIX option and futures pricing formulas in the case of the SVJ model. Moreover, the paper is the first to study the pricing performance of three widely used models to SPX options and VIX derivatives.

Details

The Journal of Risk Finance, vol. 16 no. 1
Type: Research Article
ISSN: 1526-5943

Keywords

Open Access
Article
Publication date: 7 September 2021

Freddy H. Marín-Sánchez, Julián A. Pareja-Vasseur and Diego Manzur

The purpose of this article is to propose a detailed methodology to estimate, model and incorporate the non-constant volatility onto a numerical tree scheme, to evaluate a…

Abstract

Purpose

The purpose of this article is to propose a detailed methodology to estimate, model and incorporate the non-constant volatility onto a numerical tree scheme, to evaluate a real option, using a quadrinomial multiplicative recombination.

Design/methodology/approach

This article uses the multiplicative quadrinomial tree numerical method with non-constant volatility, based on stochastic differential equations of the GARCH-diffusion type to value real options when the volatility is stochastic.

Findings

Findings showed that in the proposed method with volatility tends to zero, the multiplicative binomial traditional method is a particular case, and results are comparable between these methodologies, as well as to the exact solution offered by the Black–Scholes model.

Originality/value

The originality of this paper lies in try to model the implicit (conditional) market volatility to assess, based on that, a real option using a quadrinomial tree, including into this valuation the stochastic volatility of the underlying asset. The main contribution is the formal derivation of a risk-neutral valuation as well as the market risk premium associated with volatility, verifying this condition via numerical test on simulated and real data, showing that our proposal is consistent with Black and Scholes formula and multiplicative binomial trees method.

Details

Journal of Economics, Finance and Administrative Science, vol. 26 no. 52
Type: Research Article
ISSN: 2218-0648

Keywords

Article
Publication date: 31 January 2011

Nabil Tahani and Xiaofei Li

The purpose of this paper is to derive semi‐closed‐form solutions to a wide variety of interest rate derivatives prices under stochastic volatility in affine‐term structure models.

2132

Abstract

Purpose

The purpose of this paper is to derive semi‐closed‐form solutions to a wide variety of interest rate derivatives prices under stochastic volatility in affine‐term structure models.

Design/methodology/approach

The paper first derives the Frobenius series solution to the cross‐moment generating function, and then inverts the related characteristic function using the Gauss‐Laguerre quadrature rule for the corresponding cumulative probabilities.

Findings

This paper values options on discount bonds, coupon bond options, swaptions, interest rate caps, floors, and collars, etc. The valuation approach suggested in this paper is found to be both accurate and fast and the approach compares favorably with some alternative methods in the literature.

Research limitations/implications

Future research could extend the approach adopted in this paper to some non‐affine‐term structure models such as quadratic models.

Practical implications

The valuation approach in this study can be used to price mortgage‐backed securities, asset‐backed securities and credit default swaps. The approach can also be used to value derivatives on other assets such as commodities. Finally, the approach in this paper is useful for the risk management of fixed‐income portfolios.

Originality/value

This paper utilizes a new approach to value many of the most commonly traded interest rate derivatives in a stochastic volatility framework.

Details

Managerial Finance, vol. 37 no. 1
Type: Research Article
ISSN: 0307-4358

Keywords

Article
Publication date: 1 April 2000

ALVIN KURUC

The development of standardized measures of institution‐wide volatility exposures has so far lagged that for measures of asset price and interest‐rate exposure—largely…

Abstract

The development of standardized measures of institution‐wide volatility exposures has so far lagged that for measures of asset price and interest‐rate exposure—largely because it is difficult to reconcile the various mathematical models used to value options. Recent mathematical results, however, can be used to construct standardized measures of volatility exposure. We consider here techniques for reconciling “vegas” for financial options valued using stochastic models that may be mathematically inconsistent with each other.

Details

The Journal of Risk Finance, vol. 2 no. 1
Type: Research Article
ISSN: 1526-5943

Article
Publication date: 1 January 2013

Ning Rong and Farzad Alavi Fard

The purpose of this paper is to propose a model for ruin‐contingent life annuity (RCLA) contracts under a jump diffusion model, where the dynamics of volatility is…

Abstract

Purpose

The purpose of this paper is to propose a model for ruin‐contingent life annuity (RCLA) contracts under a jump diffusion model, where the dynamics of volatility is governed by the Heston stochastic volatility framework. The paper aims to illustrate that the proposed jump diffusion process, for both asset price and stochastic volatility, will provide a more realistic pricing model for RCLA contracts in comparison to existing models.

Design/methodology/approach

Under the assumption of the deterministic withdrawals, the authors use a partial integro differential equation (PIDE) approach to develop the pricing scheme for the fair value of the lump sum charges of RCLA contracts. Consequently, the authors employ an elegant numerical scheme, finite difference method, for solving the PIDEs for the reference portfolio, as well as the volatility. The findings show that a different pricing behaviour of the RCLA contracts under the authors' model parameters is obtained compared to that in the existing literature.

Findings

RCLA pricing in the complete market often underestimates the jump risk and the persistent factor in the volatility process. The authors' generalized model shows how these two random sources of risks can be precisely characterized.

Research limitations/implications

The parameter values used in the numerical analysis require more empirical evidence. Hence, in order for more precise pricing practice, the calibration from real data is needed.

Practical implications

The model, as adopted in this study, for pricing of RCLA contracts should provide a general guideline for the commercialization of this product by insurance companies.

Social implications

The demand for RCLA contracts as an alternative to the commonly‐practised annuitization option has recently increased, rapidly, among the soon‐to‐retire baby boomers. This paper investigates the fair value of this particular product, which could be beneficial to researchers for a better understanding of the product design.

Originality/value

This is the first research paper which prices the RCLA contracts in the incomplete market. The gap between RCLA contract pricing and studies of jump diffusion models for derivative pricing, in the literature, is therefore filled.

Book part
Publication date: 30 November 2011

Diep Duong and Norman R. Swanson

The topic of volatility measurement and estimation is central to financial and more generally time-series econometrics. In this chapter, we begin by surveying models of…

Abstract

The topic of volatility measurement and estimation is central to financial and more generally time-series econometrics. In this chapter, we begin by surveying models of volatility, both discrete and continuous, and then we summarize some selected empirical findings from the literature. In particular, in the first sections of this chapter, we discuss important developments in volatility models, with focus on time-varying and stochastic volatility as well as nonparametric volatility estimation. The models discussed share the common feature that volatilities are unobserved and belong to the class of missing variables. We then provide empirical evidence on “small” and “large” jumps from the perspective of their contribution to overall realized variation, using high-frequency price return data on 25 stocks in the DOW 30. Our “small” and “large” jump variations are constructed at three truncation levels, using extant methodology of Barndorff-Nielsen and Shephard (2006), Andersen, Bollerslev, and Diebold (2007), and Aït-Sahalia and Jacod (2009a, 2009b, 2009c). Evidence of jumps is found in around 22.8% of the days during the 1993–2000 period, much higher than the corresponding figure of 9.4% during the 2001–2008 period. Although the overall role of jumps is lessening, the role of large jumps has not decreased, and indeed, the relative role of large jumps, as a proportion of overall jumps, has actually increased in the 2000s.

Details

Missing Data Methods: Time-Series Methods and Applications
Type: Book
ISBN: 978-1-78052-526-6

Keywords

1 – 10 of over 2000