Guanghua Lian - Academia.edu (original) (raw)

Papers by Guanghua Lian

New Mathematics and Natural Computation, 2018

Convexity correction is a well-known approximation technique used in pricing volatility swaps and... more Convexity correction is a well-known approximation technique used in pricing volatility swaps and VIX futures. However, the accuracy of the technique itself and the validity condition of this approximation have hardly been addressed and discussed in the literature. This paper shows that, through both theoretical analysis and numerical examples, this type of approximations is not necessarily accurate and one should be very careful in using it. We also show that a better accuracy cannot be achieved by extending the convexity correction approximation from a second-order Taylor expansion to third-order or fourth-order Taylor expansions. We then analyze why and when it deteriorates, and provide a validity condition of applying the convexity correction approximation. Finally, we propose a new approximation, which is an extension of the convexity correction approximation, to achieve better accuracies.

Journal of Futures Markets, 2022

SSRN Electronic Journal, 2014

SSRN Electronic Journal, 2012

ABSTRACT In this paper we introduce a consistent framework for pricing options writtenon the S&am... more ABSTRACT In this paper we introduce a consistent framework for pricing options writtenon the S&P500 stock index and volatility index (VIX). Gatheral (2007,2008) proposes a three-factor stochastic volatility model to achieve this goal.However, the non-affine structure of the model leads to analytical intractabilityso closed-form pricing formulae may not exist either for S&P500 options,or for VIX options. The Monte Carlo simulation method adopted in Gatheralis rather inefficient in terms of calculation and model calibration. This studyproposes two analytical asymptotic formulae to efficiently price S&P500 optionsand VIX options, respectively, based on Gatheral’s three-factor stochasticvolatility model. By applying singular perturbation techniques, our formulaeare obtained by solving a set of partial differential equation systems. Wethen rigorously justify the convergence of the asymptotic formulae. In addition,we present some numerical examples to demonstrate that our asymptoticformulae can achieve high efficiency and accuracy for a large class of optionswith relative short tenor.

The Journal of Trading, 2008

As the markets move to a new phase represented by a preponderance of negative news and great spik... more As the markets move to a new phase represented by a preponderance of negative news and great spikes in volatility, there is general discomfort with unfamiliar trading patterns of even the most familiar of names. At the same time, regulatory changes and competition have fueled a steady progression of electronic trading automation and venue choices, adding new complexities and pressures. This article discusses the necessity for adopting new trading practices in a market where the forces of automation and uncertainty are rising in parallel with increasing volatility. It investigates market trends by illustrating changes in commonly used trading metrics.

SSRN Electronic Journal, 2014

SSRN Electronic Journal, 2013

Mathematical Finance, 2010

Journal of Futures Markets, 2012

Decisions in Economics and Finance, 2013

Applied Mathematics and Computation, 2012

Range-measured return contains more information than the traditional scalar-valued return. In thi... more Range-measured return contains more information than the traditional scalar-valued return. In this paper, we propose to model the [low, high] price range as a random interval and suggest an interval-valued GARCH (Int-GARCH) model for the corresponding range-measured return process. Under the general framework of random sets, the model properties are investigated. Parameters are estimated by the maximum likelihood method, and the asymptotic properties are established. Empirical application to stocks and financial indices data sets suggests that our Int-GARCH model overall outperforms the traditional GARCH for both in-sample estimation and out-of-sample prediction of volatility.

arXiv: Methodology, 2019

Range-measured return contains more information than the traditional scalar-valued return. In thi... more Range-measured return contains more information than the traditional scalar-valued return. In this paper, we propose to model the [low, high] price range as a random interval and suggest an interval-valued GARCH (Int-GARCH) model for the corresponding range-measured return process. Under the general framework of random sets, the model properties are investigated. Parameters are estimated by the maximum likelihood method, and the asymptotic properties are established. Empirical application to stocks and financial indices data sets suggests that our Int-GARCH model overall outperforms the traditional GARCH for both in-sample estimation and out-of-sample prediction of volatility.

Journal of Time Series Analysis

SSRN Electronic Journal

This paper contributes a generic probabilistic method to derive explicit exact probability densit... more This paper contributes a generic probabilistic method to derive explicit exact probability densities for stochastic volatility models. Our method is based on a novel application of the exponential measure change in [Z. Palmowski & T. Rolski (2002) A technique for exponential change of measure for Markov processes, Bernoulli 8(6), 767–785]. With this generic approach, we first derive explicit probability densities in terms of model parameters for several stochastic volatility models with nonzero correlations, namely the Heston 1993, 3/2, and a special case of the α-Hypergeometric stochastic volatility models recently proposed by [J. Da Fonseca & C. Martini (2016) The α-Hypergeometric stochastic volatility model, Stochastic Processes and their Applications 126(5), 1472–1502]. Then, we combine our method with a stochastic time change technique to develop explicit formulae for prices of timer options in the Heston model, the 3/2 model and a special case of the α-Hypergeometric model.

Journal of Banking & Finance

Simple analytical solutions for the prices of discretely monitored barrier options do not yet exi... more Simple analytical solutions for the prices of discretely monitored barrier options do not yet exist in the literature. This paper presents a semi-analytical and fully explicit solution for pricing discretely monitored barrier options when the underlying asset is driven by a general Levy process. The explicit formula only involves elementary functions, and the Greeks are also explicitly available with little additional computation. By performing a Z-transform, we reduce the valuation problem to an integral equation. This equation is solved analytically with the solution expressed in terms of a Fourier cosine series. We then manage to analytically carry out the Z-transform inversion, and obtain a semi-analytical formula for pricing discrete barrier options. We establish the theoretical error bound and analyze the convergence order of our method. Numerical implementation demonstrates that our numerical results are accurate and efficient, and match up with the results from the benchmark methods in the literature.

SSRN Electronic Journal

In this paper, we propose an efficient approach to the calculation of risk measures for an insure... more In this paper, we propose an efficient approach to the calculation of risk measures for an insurer’s liability from writing a variable annuity with guaranteed benefits. Our approach is based on a novel application of the Hermite series expansions on the transition density of a diffusion process to the insurance setting. We compare our method with existing methods in the literature, including the analytical method, the spectral method and the Green’s function method, and illustrate its substantial advantages in calculating risk measures for variable annuities with different guarantee structures. The gained efficiency makes our method flexible to practical implementation in reporting risk measures on a daily basis. We also conduct sensitivity analysis of the risk measures with respect to key parameters.

Journal of Computational and Applied Mathematics, May 4, 2015

Papers focusing on analytically pricing discretely-sampled volatility swaps are rare in literatur... more Papers focusing on analytically pricing discretely-sampled volatility swaps are rare in literature, mainly due to the inherent difficulty associated with the nonlinearity in the pay-off function. In this paper, we present a closed-form exact solution for the pricing of discretely-sampled volatility swaps, under the framework of Heston (1993) stochastic volatility model, based on the definition of the so-called average of realized volatility. By working out such a closed-form exact solution for discretely-sampled volatility swaps, this work represents a substantial progress in the field of pricing volatility swaps, as it has: (1) significantly reduced the computational time in obtaining numerical values for the discretely-sampled volatility swaps; (2) improved the computational accuracy of discretely-sampled volatility swaps, comparing with the continuous sampling approximation, especially when the time interval between sampling points is large; (3) enabled all the hedging ratios of a volatility swap to be analytically derived.

Physica A: Statistical Mechanics and its Applications, 2016

In this paper, we introduce a robust numerical method, based on the upwind scheme, for the pricin... more In this paper, we introduce a robust numerical method, based on the upwind scheme, for the pricing of American puts under the generalized mixed fractional Brownian motion (GMFBM) model. By using portfolio analysis and applying the Wick–Ito formula, a partial differential equation (PDE) governing the prices of vanilla options under the GMFBM is successfully derived for the first time. Based on this, we formulate the pricing of American puts under the current model as a linear complementarity problem (LCP). Unlike the classical Black–Scholes (B–S) model or the generalized B–S model discussed in Cen and Le (2011), the newly obtained LCP under the GMFBM model is difficult to be solved accurately because of the numerical instability which results from the degeneration of the governing PDE as time approaches zero. To overcome this difficulty, a numerical approach based on the upwind scheme is adopted. It is shown that the coefficient matrix of the current method is an M-matrix, which ensures its stability in the maximum-norm sense. Remarkably, we have managed to provide a sharp theoretic error estimate for the current method, which is further verified numerically. The results of various numerical experiments also suggest that this new approach is quite accurate, and can be easily extended to price other types of financial derivatives with an American-style exercise feature under the GMFBM model.

Applied Mathematical Finance, 2015

Abstract This paper provides a pricing formula for perpetual exchange options, where the dynamics... more Abstract This paper provides a pricing formula for perpetual exchange options, where the dynamics of the underlying assets are driven by jump-diffusion processes. It is an extension of Gerber and Shiu, and also Wong, who have priced perpetual exchange options under the pure-diffusion setting, and that of Gerber and Shiu, who have also considered perpetual options on single assets under jump-diffusion dynamics. It complements the results of Cheang and Chiarella, who derive a probabilistic representation of the American exchange option price under jump-diffusion dynamics.

New Mathematics and Natural Computation, 2018

Convexity correction is a well-known approximation technique used in pricing volatility swaps and... more Convexity correction is a well-known approximation technique used in pricing volatility swaps and VIX futures. However, the accuracy of the technique itself and the validity condition of this approximation have hardly been addressed and discussed in the literature. This paper shows that, through both theoretical analysis and numerical examples, this type of approximations is not necessarily accurate and one should be very careful in using it. We also show that a better accuracy cannot be achieved by extending the convexity correction approximation from a second-order Taylor expansion to third-order or fourth-order Taylor expansions. We then analyze why and when it deteriorates, and provide a validity condition of applying the convexity correction approximation. Finally, we propose a new approximation, which is an extension of the convexity correction approximation, to achieve better accuracies.

Journal of Futures Markets, 2022

SSRN Electronic Journal, 2014

SSRN Electronic Journal, 2012

ABSTRACT In this paper we introduce a consistent framework for pricing options writtenon the S&am... more ABSTRACT In this paper we introduce a consistent framework for pricing options writtenon the S&P500 stock index and volatility index (VIX). Gatheral (2007,2008) proposes a three-factor stochastic volatility model to achieve this goal.However, the non-affine structure of the model leads to analytical intractabilityso closed-form pricing formulae may not exist either for S&P500 options,or for VIX options. The Monte Carlo simulation method adopted in Gatheralis rather inefficient in terms of calculation and model calibration. This studyproposes two analytical asymptotic formulae to efficiently price S&P500 optionsand VIX options, respectively, based on Gatheral’s three-factor stochasticvolatility model. By applying singular perturbation techniques, our formulaeare obtained by solving a set of partial differential equation systems. Wethen rigorously justify the convergence of the asymptotic formulae. In addition,we present some numerical examples to demonstrate that our asymptoticformulae can achieve high efficiency and accuracy for a large class of optionswith relative short tenor.

The Journal of Trading, 2008

As the markets move to a new phase represented by a preponderance of negative news and great spik... more As the markets move to a new phase represented by a preponderance of negative news and great spikes in volatility, there is general discomfort with unfamiliar trading patterns of even the most familiar of names. At the same time, regulatory changes and competition have fueled a steady progression of electronic trading automation and venue choices, adding new complexities and pressures. This article discusses the necessity for adopting new trading practices in a market where the forces of automation and uncertainty are rising in parallel with increasing volatility. It investigates market trends by illustrating changes in commonly used trading metrics.

SSRN Electronic Journal, 2014

SSRN Electronic Journal, 2013

Mathematical Finance, 2010

Journal of Futures Markets, 2012

Decisions in Economics and Finance, 2013

Applied Mathematics and Computation, 2012

Range-measured return contains more information than the traditional scalar-valued return. In thi... more Range-measured return contains more information than the traditional scalar-valued return. In this paper, we propose to model the [low, high] price range as a random interval and suggest an interval-valued GARCH (Int-GARCH) model for the corresponding range-measured return process. Under the general framework of random sets, the model properties are investigated. Parameters are estimated by the maximum likelihood method, and the asymptotic properties are established. Empirical application to stocks and financial indices data sets suggests that our Int-GARCH model overall outperforms the traditional GARCH for both in-sample estimation and out-of-sample prediction of volatility.

arXiv: Methodology, 2019

Range-measured return contains more information than the traditional scalar-valued return. In thi... more Range-measured return contains more information than the traditional scalar-valued return. In this paper, we propose to model the [low, high] price range as a random interval and suggest an interval-valued GARCH (Int-GARCH) model for the corresponding range-measured return process. Under the general framework of random sets, the model properties are investigated. Parameters are estimated by the maximum likelihood method, and the asymptotic properties are established. Empirical application to stocks and financial indices data sets suggests that our Int-GARCH model overall outperforms the traditional GARCH for both in-sample estimation and out-of-sample prediction of volatility.

Journal of Time Series Analysis

SSRN Electronic Journal

This paper contributes a generic probabilistic method to derive explicit exact probability densit... more This paper contributes a generic probabilistic method to derive explicit exact probability densities for stochastic volatility models. Our method is based on a novel application of the exponential measure change in [Z. Palmowski & T. Rolski (2002) A technique for exponential change of measure for Markov processes, Bernoulli 8(6), 767–785]. With this generic approach, we first derive explicit probability densities in terms of model parameters for several stochastic volatility models with nonzero correlations, namely the Heston 1993, 3/2, and a special case of the α-Hypergeometric stochastic volatility models recently proposed by [J. Da Fonseca & C. Martini (2016) The α-Hypergeometric stochastic volatility model, Stochastic Processes and their Applications 126(5), 1472–1502]. Then, we combine our method with a stochastic time change technique to develop explicit formulae for prices of timer options in the Heston model, the 3/2 model and a special case of the α-Hypergeometric model.

Journal of Banking & Finance

Simple analytical solutions for the prices of discretely monitored barrier options do not yet exi... more Simple analytical solutions for the prices of discretely monitored barrier options do not yet exist in the literature. This paper presents a semi-analytical and fully explicit solution for pricing discretely monitored barrier options when the underlying asset is driven by a general Levy process. The explicit formula only involves elementary functions, and the Greeks are also explicitly available with little additional computation. By performing a Z-transform, we reduce the valuation problem to an integral equation. This equation is solved analytically with the solution expressed in terms of a Fourier cosine series. We then manage to analytically carry out the Z-transform inversion, and obtain a semi-analytical formula for pricing discrete barrier options. We establish the theoretical error bound and analyze the convergence order of our method. Numerical implementation demonstrates that our numerical results are accurate and efficient, and match up with the results from the benchmark methods in the literature.

SSRN Electronic Journal

In this paper, we propose an efficient approach to the calculation of risk measures for an insure... more In this paper, we propose an efficient approach to the calculation of risk measures for an insurer’s liability from writing a variable annuity with guaranteed benefits. Our approach is based on a novel application of the Hermite series expansions on the transition density of a diffusion process to the insurance setting. We compare our method with existing methods in the literature, including the analytical method, the spectral method and the Green’s function method, and illustrate its substantial advantages in calculating risk measures for variable annuities with different guarantee structures. The gained efficiency makes our method flexible to practical implementation in reporting risk measures on a daily basis. We also conduct sensitivity analysis of the risk measures with respect to key parameters.

Journal of Computational and Applied Mathematics, May 4, 2015

Papers focusing on analytically pricing discretely-sampled volatility swaps are rare in literatur... more Papers focusing on analytically pricing discretely-sampled volatility swaps are rare in literature, mainly due to the inherent difficulty associated with the nonlinearity in the pay-off function. In this paper, we present a closed-form exact solution for the pricing of discretely-sampled volatility swaps, under the framework of Heston (1993) stochastic volatility model, based on the definition of the so-called average of realized volatility. By working out such a closed-form exact solution for discretely-sampled volatility swaps, this work represents a substantial progress in the field of pricing volatility swaps, as it has: (1) significantly reduced the computational time in obtaining numerical values for the discretely-sampled volatility swaps; (2) improved the computational accuracy of discretely-sampled volatility swaps, comparing with the continuous sampling approximation, especially when the time interval between sampling points is large; (3) enabled all the hedging ratios of a volatility swap to be analytically derived.

Physica A: Statistical Mechanics and its Applications, 2016

In this paper, we introduce a robust numerical method, based on the upwind scheme, for the pricin... more In this paper, we introduce a robust numerical method, based on the upwind scheme, for the pricing of American puts under the generalized mixed fractional Brownian motion (GMFBM) model. By using portfolio analysis and applying the Wick–Ito formula, a partial differential equation (PDE) governing the prices of vanilla options under the GMFBM is successfully derived for the first time. Based on this, we formulate the pricing of American puts under the current model as a linear complementarity problem (LCP). Unlike the classical Black–Scholes (B–S) model or the generalized B–S model discussed in Cen and Le (2011), the newly obtained LCP under the GMFBM model is difficult to be solved accurately because of the numerical instability which results from the degeneration of the governing PDE as time approaches zero. To overcome this difficulty, a numerical approach based on the upwind scheme is adopted. It is shown that the coefficient matrix of the current method is an M-matrix, which ensures its stability in the maximum-norm sense. Remarkably, we have managed to provide a sharp theoretic error estimate for the current method, which is further verified numerically. The results of various numerical experiments also suggest that this new approach is quite accurate, and can be easily extended to price other types of financial derivatives with an American-style exercise feature under the GMFBM model.

Applied Mathematical Finance, 2015

Abstract This paper provides a pricing formula for perpetual exchange options, where the dynamics... more Abstract This paper provides a pricing formula for perpetual exchange options, where the dynamics of the underlying assets are driven by jump-diffusion processes. It is an extension of Gerber and Shiu, and also Wong, who have priced perpetual exchange options under the pure-diffusion setting, and that of Gerber and Shiu, who have also considered perpetual options on single assets under jump-diffusion dynamics. It complements the results of Cheang and Chiarella, who derive a probabilistic representation of the American exchange option price under jump-diffusion dynamics.