Bridge Copula Model for Option Pricing

We aim at accommodating the existing affine jump-diffusion and quadratic models under the same roof, namely the linear-quadratic jump-diffusion (LQJD) class. We give a complete characterization of the dynamics underlying this class of models as well as identification constraints, and compute standard and extended transforms relevant to asset pricing. We also show that the LQJD class can be embedded into the affine class through use of an augmented state vector. We further establish that an equivalence relationship holds between both classes in terms of transform analysis. An option pricing application to multifactor stochastic volatility models reveals that adding nonlinearity into the model significantly reduces pricing errors, and further addition of a jump component in the stock price largely improves goodness-of-fit for in-the-money calls but less for out-of-the-money ones.

Social Science Research Network, 2014

We derived a model-free analytical approximation of the price of a multi-asset option defined over an arbitrary multivariate process, applying a semi-parametric expansion of the unknown risk-neutral density with the moments. The analytical expansion termed as the Multivariate Generalised Edgeworth Expansion (MGEE) is an infinite series over the derivatives of the known continuous time density. The expected value of the density expansion is calculated to approximate the option price. The expansion could be used to enhance a Monte Carlo pricing methodology incorporating the information about moments of the risk-neutral distribution. The numerical efficiency of the approximation is tested over a jump-diffusion density. For the known density, we tested the multivariate lognormal (MVLN), even though arbitrary densities could be used, and we provided its derivatives until the fourth-order. The MGEE relates two densities and isolates the effects of multivariate moments over the option prices. Results show that a calibrated approximation provides a good fit when the difference between the moments of the risk-neutral density and the auxiliary density are small relative to the density function of the former, and the uncalibrated approximation has immediate implications over risk management and hedging theory. The possibility to select the auxiliary density provides an advantage over classical Gram-Charlier A, B and C series approximations. The density approximation and the methodology can be applied to other fields of finance like asset pricing, econometrics, and areas of statistical nature.

SSRN Electronic Journal, 2000

In this paper we suggest the adoption of copula functions in order to price multivariate contingent claims. Copulas enable us to imbed the marginal distributions extracted from vertical spreads in the options markets in a multivariate pricing kernel. We prove that such kernel is a copula fucntion, and that its super-replication strategy is represented by the Fréchet bounds. As applications, we provide prices for binary digital options, options on the minimum and options to exchange one asset for another. For each of these products, we provide no-arbitrage pricing bounds, as well as the values consistent with independence of the underlying assets. As a …nal reference value, we use a copula function calibrated on historical data.

2017

In finance, dependence structure between assets is of great importance. For example, pricing options involving many assets, one must make preassumption about the dependence structure between assets or one important issue in risk management is to find out the dependence structure when calculating VaR. The aim of this paper is to explore the dynamic properties of a multidimensional Variance Gamma process, which has non Gaussian marginal features and non linear dependence structure. We use copula functions to specify the dependence structure of underlying assets. We study the effect of different choices for the dependence functions to the prices of a set of multi-asset equity options. The analysis is conducted using 5-dimensional baskets that consist of Jakarta Stock Exchange Composite Index (IHSG) and four other Asian Indices, Hang Seng, Nikkei, KOSPI, Straits Times Index (STI) and a standard payoff functions for multi-asset options. The results show that the different choices of depe...

International Journal of Bonds and Derivatives, 2018

The diversity of exotic-option contracts available in the market has increased significantly in recent years, and has aroused the interest to develop alternative valuation methodological approaches. Among the most interesting innovations, copulas analysis represents a major contribution to improve valuation methodologies. This paper explores the pricing of a bivariate call option on the better-of-two-markets: Mexico's Stock Exchange index, and the Standard & Poor's 500. The approach consists of a GARCH process that combines with copulas analysis and the Black and Scholes classical European call option valuation model. Copulas from the elliptical and Archimedean families provide the dependence structure among the underlying assets, and the estimated prices prove significantly different from those obtained using a static dependence assumption. The study concludes that dynamic copulas produce more robust prices than static dependence models.

Applied Mathematical Finance, 2002

The adoption of copula functions is suggested in order to price bivariate contingent claims. Copulas enable the marginal distributions extracted from vertical spreads in the options markets to be imbedded in a multivariate pricing kernel. It is proved that such a kernel is a copula function, and that its super-replication strategy is represented by the Frechet bounds. Applications provided include

Chaos, Solitons & Fractals, 2016

We derived a model-free analytical approximation of the price of a multi-asset option defined over an arbitrary multivariate process, applying a semi-parametric expansion of the unknown risk-neutral density with the moments. The analytical expansion termed as the Multivariate Generalised Edgeworth Expansion (MGEE) is an infinite series over the derivatives of an auxiliary continuous time density. The expansion could be used to enhance a Monte Carlo pricing methodology incorporating the information about moments of the risk-neutral distribution. The efficiency of the approximation is tested over a jump-diffusion and a q-Gaussian diffusion. For the known density, we tested the multivariate lognormal (MVLN), even though arbitrary densities could be used. The MGEE relates two densities and isolates the effects of multivariate moments over the option prices. Results show that a calibrated approximation provides a good fit when the difference between the moments of the risk-neutral density and the auxiliary density are small relative to the density function of the former, and the uncalibrated approximation has immediate implications over risk management and hedging theory. The possibility to select the auxiliary density provides an advantage over classical Gram-Charlier A, B and C series approximations.

Journal of Mathematical Finance, 2011

An alternative option pricing model is proposed, in which the asset prices follow the jump-diffusion model with square root stochastic volatility. The stochastic volatility follows the jump-diffusion with square root and mean reverting. We find a formulation for the European-style option in terms of characteristic functions of tail probabilities.

We derived a model-free analytical approximation of the price of a multi-asset option defined over an arbitrary multivariate process, applying a semi-parametric expansion of the unknown risk-neutral density with the moments. The analytical expansion termed as the Multivariate Generalised Edgeworth Expansion (MGEE) is an infinite series over the derivatives of the known continuous time density. The expected value of the density expansion is calculated to approximate the option price. The expansion could be used to enhance a Monte Carlo pricing methodology incorporating the information about moments of the risk-neutral distribution. The numerical efficiency of the approximation is tested over a jump-diffusion density. For the known density, we tested the multivariate lognormal (MVLN), even though arbitrary densities could be used, and we provided its derivatives until the fourth-order. The MGEE relates two densities and isolates the effects of multivariate moments over the option prices. Results show that a calibrated approximation provides a good fit when the difference between the moments of the risk-neutral density and the auxiliary density are small relative to the density function of the former, and the uncalibrated approximation has immediate implications over risk management and hedging theory. The possibility to select the auxiliary density provides an advantage over classical Gram–Charlier A, B and C series approximations. The density approximation and the methodology can be applied to other fields of finance like asset pricing, econometrics, and areas of statistical nature.

Communications in Nonlinear Science and Numerical Simulation, 2019

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MATEC Web of Conferences

In financial mathematics, option pricing models are vital tools whose usefulness cannot be overemphasized. Modern approaches and modelling of financial derivatives are therefore required in option pricing and valuation settings. In this paper, we derive via the application of Ito lemma, a pricing model referred to as Generalized Squared Gaussian Diffusion Model (GSGDM) for option pricing and valuation. Same approach can be considered via Stratonovich stochastic dynamics. We also show that the classical Black-Scholes, and the square root constant elasticity of variance models are special cases of the GSGDM. In addition, general solution of the GSGDM is obtained using modified variational iterative method (MVIM).

Journal of Applied Mathematics

This paper shows how to value multiasset options analytically in a modeling framework that combines both continuous and discontinuous variations in the underlying equity or foreign exchange processes and a stochastic, two-factor yield curve. All correlations are taken into account, between the factors driving the yield curve, between fixed income and equity as asset classes, and between the individual equity assets themselves. The valuation method is applied to three of the most popular two-asset options.

Journal of Statistical …, 2008

We study the pricing problem for a European call option when the volatility of the underlying asset is random and follows the exponential Ornstein-Uhlenbeck model. The random diffusion model proposed is a two-dimensional market process that takes a log-Brownian motion to describe price dynamics and an Ornstein-Uhlenbeck subordinated process describing the randomness of the log-volatility. We derive an approximate option price that is valid when (i) the fluctuations of the volatility are larger than its normal level, (ii) the volatility presents a slow driving force toward its normal level and, finally, (iii) the market price of risk is a linear function of the log-volatility.

Journal of …, 2006

The multivariate modeling of default risk is a crucial aspect of the pricing of credit derivative products referencing a portfolio of underlying assets, and the evaluation of Value at Risk of such portfolios. This paper proposes a model for the joint dynamics of credit ratings of several firms. Namely, individual credit ratings are modeled by univariate continuous time Markov chain, while their joint dynamic is modeled using copulas. A by-product of the method is the joint laws of the default times of all the firms in the portfolio. The use of copulas allows us to incorporate our knowledge of the modeling of univariate processes, into a multivariate framework. The Normal and Student copulas commonly used in the literature as well as by practitioners do not produce very different estimates of default risk prices. We show that this result is restricted to these two two basic copulas. That is, for any other family of copula, the choice of the copula greatly affects the pricing of default risk.

SSRN Electronic Journal, 2000

We aim at accommodating the existing affine jump-diffusion and quadratic models under the same roof, namely the linear-quadratic jump-diffusion (LQJD) class. We give a complete characterization of the dynamics underlying this class of models as well as identification constraints, and compute standard and extended transforms relevant to asset pricing. We also show that the LQJD class can be embedded into the affine class through use of an augmented state vector. We further establish that an equivalence relationship holds between both classes in terms of transform analysis. An option pricing application to multifactor stochastic volatility models reveals that adding nonlinearity into the model significantly reduces pricing errors, and further addition of a jump component in the stock price largely improves goodness-of-fit for in-the-money calls but less for out-of-the-money ones.