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## Option pricing and Esscher transform under regime switching

Elliott, R.; Chan, L.; Siu, T.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
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We consider the option pricing problem when the risky underlying assets are driven by Markov-modulated Geometric Brownian Motion (GBM). That is, the market parameters, for instance, the market interest rate, the appreciation rate and the volatility of the underlying risky asset, depend on unobservable states of the economy which are modelled by a continuous-time Hidden Markov process. The market described by the Markov-modulated GBM model is incomplete in general and, hence, the martingale measure is not unique. We adopt a regime switching random Esscher transform to determine an equivalent martingale pricing measure. As in Miyahara [33], we can justify our pricing result by the minimal entropy martingale measure (MEMM).; Robert J. Elliott, Leunglung Chan and Tak Kuen Siu

## Option pricing for GARCH models with Markov switching

Elliott, R.; Siu, T.; Chan, L.
Fonte: World Scientific Publishing Co Pte Ltd Publicador: World Scientific Publishing Co Pte Ltd
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
49.309204%
In this paper we develop a method for pricing derivatives under a Markov switching version of the Heston-Nandi GARCH (1, 1) model by using a well known tool from actuarial science, namely the Esscher transform. We suppose that the dynamics of the GARCH process switch over time according to one of the regimes described by the states of an observable Markov chain process. By augmenting the conditional Esscher transform with the observable Markov switching process, a Markov switching conditional Esscher transform (MSCET) is developed to identify a martingale measure for option valuation in the incomplete market described by our model. We provide an alternative approach for the derivation of an analytical option valuation formula under the Markov switching Heston-Nandi GARCH (1, 1) model. The use of the MSCET can be justified by considering a utility maximization problem with respect to a power utility function associated with the Markov switching risk-averse parameters.; Robert J. Elliott; Tak Kuen Siu; Leunglung Chan

## A PDE approach for risk measures for derivatives with regime switching

Elliott, R.; Siu, T.; Chan, L.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.546348%
This paper considers a partial differential equation (PDE) approach to evaluate coherent risk measures for derivative instruments when the dynamics of the risky underlying asset are governed by a Markov-modulated geometric Brownian motion (GBM); that is, the appreciation rate and the volatility of the underlying risky asset switch over time according to the state of a continuous-time hidden Markov chain model which describes the state of an economy. The PDE approach provides market practitioners with a flexible and effective way to evaluate risk measures in the Markov-modulated Black–Scholes model. We shall derive the PDEs satisfied by the risk measures for European-style options, barrier options and American-style options.; Robert J. Elliott, Tak Kuen Siu and Leunglung Chan; The original publication can be found at www.springerlink.com

## Esscher transforms and consumption-based models

Badescu, A.; Elliott, R.; Siu, T.
Fonte: Elsevier Science BV Publicador: Elsevier Science BV
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
59.757803%
The Esscher transform is an important tool in actuarial science. Since the pioneering work of Gerber and Shiu (1994), the use of the Esscher transform for option valuation has also been investigated extensively. However, the relationships between the asset pricing model based on the Esscher transform and some fundamental equilibrium-based asset pricing models, such as consumption-based models, have so far not been well-explored. In this paper, we attempt to bridge the gap between consumption-based models and asset pricing models based on Esscher-type transformations in a discrete-time setting. Based on certain assumptions for the distributions of asset returns, changes in aggregate consumptions and returns on the market portfolio, we construct pricing measures that are consistent with those arising from Esscher-type transformations. Explicit relationships between the market price of risk, and the risk preference parameters are derived for some particular cases.; http://www.elsevier.com/wps/find/journaldescription.cws_home/505554/description#description; Alex Badescu, Robert J. Elliott and Tak Kuen Siu

## A comparison of pricing kernels for garch option pricing with generalized hyperbolic distributions

Badescu, A.; Elliott, R.; Kulperger, R.; Miettinen, J.; Siu, T.
Fonte: World Scientific Publishing Co Pte Ltd Publicador: World Scientific Publishing Co Pte Ltd
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.546348%
Under discrete-time GARCH models markets are incomplete so there is more than one price kernel for valuing contingent claims. This motivates the quest for selecting an appropriate price kernel. Different methods have been proposed for the choice of a price kernel. Some of them can be justified by economic equilibrium arguments. This paper studies risk-neutral dynamics of various classes of Generalized Hyperbolic GARCH models arising from different price kernels. We discuss the properties of these dynamics and show that for some special cases, some pricing kernels considered here lead to similar risk neutral GARCH dynamics. Real data examples for pricing European options on the S&P 500 index emphasize the importance of the choice of a price kernel.; Alexandru Badescu, Robert J. Elliott, Reg Kulperger, Jarkko Miettinen, Tak Kuen Siu

## Option pricing and filtering with hidden Markov-Modulated pure-jump processes

Elliott, R.; Siu, T.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
48.66176%
This article discusses the pricing of derivatives in a continuous-time, hidden Markov-modulated, pure-jump asset price model. The hidden Markov chain modulating the pure-jump asset price model describes the evolution of the hidden state of an economy over time. The market model is incomplete. We employ a version of the Esscher transform to select a price kernel for valuation. We derive a valuation formula for European options using a Fourier transform and the correlation theorem. This formula depends on the hidden Markov chain. It is then estimated using a robust filter of the chain.; Robert J. Elliott & Tak Kuen Siu

## Asset pricing using trading volumes in a hidden regime-switching environment

Elliott, R.J.; Siu, T.K.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
48.50123%
By utilizing information about prices and trading volumes, we discuss the pricing of European contingent claims in a continuous-time hidden regime-switching environment. Hidden market sentiments described by the states of a continuous-time, finite-state, hidden Markov chain represent a common factor for an asset’s drift and volatility, as well as its trading volumes. Using observations about trading volumes, we present a filtered estimate of the hidden common factor. The asset pricing problem is then considered in a filtered market, where the hidden drift and volatility are replaced by their filtered estimates. We adopt the Esscher transform to select an equivalent martingale measure for pricing and derive a partial-differential integral equation for the option price.; Robert J. Elliot, Tak Kuen Siu

## A Dupire equation for a regime-switching model

Elliott, R.J.; Chan, L.; Siu, T.K.
Fonte: World Scientific Publishing Publicador: World Scientific Publishing
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.546348%
A forward equation, which is also called the Dupire formula, is obtained for European call options when the price dynamics of the underlying risky assets are assumed to follow a regime-switching local volatility model. Using a regime-switching version of the adjoint formula, a system of coupled forward equations is derived for the price of the European call over different states of the economy.; Robert J. Elliott, Leunglung Chan, Tak Kuen Siu

## Importance Sampling and Statistical Romberg Method for L\'evy Processes

Alaya, M. Ben; Hajji, K.; Kebaier, A.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.546348%
An important family of stochastic processes arising in many areas of applied probability is the class of L\'evy processes. Generally, such processes are not simulatable especially for those with infinite activity. In practice, it is common to approximate them by truncating the jumps at some cut-off size $\varepsilon$ ($\varepsilon\searrow 0$). This procedure leads us to consider a simulatable compound Poisson process. This paper first introduces, for this setting, the statistical Romberg method to improve the complexity of the classical Monte Carlo one. Roughly speaking, we use many sample paths with a coarse cut-off $\varepsilon^{\beta},$ $\beta\in(0,1)$, and few additional sample paths with a fine cut-off $\varepsilon$. Central limit theorems of Lindeberg-Feller type for both Monte Carlo and statistical Romberg method for the inferred errors depending on the parameter $\varepsilon$ are proved. This leads to an accurate description of the optimal choice of parameters with explicit limit variances. Afterwards, the authors propose a stochastic approximation method of finding the optimal measure change by Esscher transform for L\'evy processes with Monte Carlo and statistical Romberg importance sampling variance reduction. Furthermore...

## Pricing Currency Derivatives with Markov-modulated Levy Dynamics

Swishchuk, Anatoliy; Tertychnyi, Maksym; Elliott, Robert
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.546348%
Using a Levy process we generalize formulas in Bo et al.(2010) for the Esscher transform parameters for the log-normal distribution which ensure the martingale condition holds for the discounted foreign exchange rate. Using these values of the parameters we find a risk-neural measure and provide new formulas for the distribution of jumps, the mean jump size, and the Poisson process intensity with respect to to this measure. The formulas for a European call foreign exchange option are also derived. We apply these formulas to the case of the log-double exponential distribution of jumps. We provide numerical simulations for the European call foreign exchange option prices with different parameters.; Comment: 25 pages, 9 figures

## Currency Derivatives Pricing for Markov-modulated Merton Jump-diffusion Spot Forex Rate

Swishchuk, Anatoliy; Tertychnyi, Maksym; Hoang, Winsor
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.546348%
We derived similar to Bo et al. (2010) results but in the case when the dynamics of the FX rate is driven by a general Merton jump-diffusion process. The main results of our paper are as follows: 1) formulas for the Esscher transform parameters which ensure that the martingale condition for the discounted foreign exchange rate is a martingale for a general Merton jump--diffusion process are derived; using the values of these parameters we proceeded to a risk-neural measure and provide new formulas for the distribution of jumps, the mean jump size, and the Poisson process intensity with respect to the measure; pricing formulas for European call foreign exchange options have been given as well; 2) obtained formulas are applied to the case of the exponential processes; 3) numerical simulations of European call foreign exchange option prices for different parameters are also provided; 4) codes for Matlab functions used in numerical simulations of option prices are given.; Comment: 17 pages, 3 figures. arXiv admin note: substantial text overlap with arXiv:1402.1953

## Option Pricing in a Dynamic Variance-Gamma Model

Mercuri, Lorenzo; Bellini, Fabio
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
27.546348%
We present a discrete time stochastic volatility model in which the conditional distribution of the logreturns is a Variance-Gamma, that is a normal variance-mean mixture with Gamma mixing density. We assume that the Gamma mixing density is time varying and follows an affine Garch model, trying to capture persistence of volatility shocks and also higher order conditional dynamics in a parsimonious way. We select an equivalent martingale measure by means of the conditional Esscher transform as in Buhlmann et al. (1996) and show that this change of measure leads to a similar dynamics of the mixing distribution. The model admits a recursive procedure for the computation of the characteristic function of the terminal logprice, thus allowing semianalytical pricing as in Heston and Nandi (2000). From an empirical point of view, we check the ability of this model to calibrate SPX option data and we compare it with the Heston and Nandi (2000) model and with the Christoffersen, Heston and Jacobs (2006) model, that is based on Inverse Gaussian innovations. Moreover, we provide a detailed comparison with several variants of the Heston and Nandi model that shows the superiority of the Variance-Gamma innovations also from the point of view of historical MLE estimation.

## Unconstrained Recursive Importance Sampling

Lemaire, Vincent; Pagès, Gilles
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
27.546348%
We propose an unconstrained stochastic approximation method of finding the optimal measure change (in an a priori parametric family) for Monte Carlo simulations. We consider different parametric families based on the Girsanov theorem and the Esscher transform (or exponential-tilting). In a multidimensional Gaussian framework, Arouna uses a projected Robbins-Monro procedure to select the parameter minimizing the variance. In our approach, the parameter (scalar or process) is selected by a classical Robbins-Monro procedure without projection or truncation. To obtain this unconstrained algorithm we intensively use the regularity of the density of the law without assume smoothness of the payoff. We prove the convergence for a large class of multidimensional distributions and diffusion processes. We illustrate the effectiveness of our algorithm via pricing a Basket payoff under a multidimensional NIG distribution, and pricing a barrier options in different markets.; Comment: 30p

## A pricing measure to explain the risk premium in power markets

Tipo: Artigo de Revista Científica
Relevância na Pesquisa
28.50123%
In electricity markets, it is sensible to use a two-factor model with mean reversion for spot prices. One of the factors is an Ornstein-Uhlenbeck (OU) process driven by a Brownian motion and accounts for the small variations. The other factor is an OU process driven by a pure jump L\'evy process and models the characteristic spikes observed in such markets. When it comes to pricing, a popular choice of pricing measure is given by the Esscher transform that preserves the probabilistic structure of the driving L\'evy processes, while changing the levels of mean reversion. Using this choice one can generate stochastic risk premiums (in geometric spot models) but with (deterministically) changing sign. In this paper we introduce a pricing change of measure, which is an extension of the Esscher transform. With this new change of measure we also can slow down the speed of mean reversion and generate stochastic risk premiums with stochastic non constant sign, even in arithmetic spot models. In particular, we can generate risk profiles with positive values in the short end of the forward curve and negative values in the long end. Finally, our pricing measure allows us to have a stationary spot dynamics while still having randomly fluctuating forward prices for contracts far from maturity.; Comment: 37 pages...

## A L\'evy-driven rainfall model with applications to futures pricing

Noven, Ragnhild C.; Veraart, Almut E. D.; Gandy, Axel
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
27.546348%
We propose a parsimonious stochastic model for characterising the distributional and temporal properties of rainfall. The model is based on an integrated Ornstein-Uhlenbeck process driven by the Hougaard L\'evy process. We derive properties of this process and propose an extended model which generalises the Ornstein-Uhlenbeck process to the class of continuous-time ARMA (CARMA) processes. The model is illustrated by fitting it to empirical rainfall data on both daily and hourly time scales. It is shown that the model is sufficiently flexible to capture important features of the rainfall process across locations and time scales. Finally we study an application to the pricing of rainfall derivatives which introduces the market price of risk via the Esscher transform. We first give a result specifying the risk-neutral expectation of a general moving average process. Then we illustrate the pricing method by calculating futures prices based on empirical daily rainfall data, where the rainfall process is specified by our model.; Comment: 28 pages, 9 figures

## Esscher transform and the duality principle for multidimensional semimartingales

Eberlein, Ernst; Papapantoleon, Antonis; Shiryaev, Albert N.
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
38.194768%
The duality principle in option pricing aims at simplifying valuation problems that depend on several variables by associating them to the corresponding dual option pricing problem. Here, we analyze the duality principle for options that depend on several assets. The asset price processes are driven by general semimartingales, and the dual measures are constructed via an Esscher transformation. As an application, we can relate swap and quanto options to standard call and put options. Explicit calculations for jump models are also provided.; Comment: Published in at http://dx.doi.org/10.1214/09-AAP600 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

## On the Esscher transforms and other equivalent martingale measures for Barndorff-Nielsen and Shephard stochastic volatility models with jumps

Hubalek, Friedrich; Sgarra, Carlo
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
28.84088%
We compute and discuss the Esscher martingale transform for exponential processes, the Esscher martingale transform for linear processes, the minimal martingale measure, the class of structure preserving martingale measures, and the minimum entropy martingale measure for stochastic volatility models of Ornstein-Uhlenbeck type as introduced by Barndorff-Nielsen and Shephard. We show, that in the model with leverage, with jumps both in the volatility and in the returns, all those measures are different, whereas in the model without leverage, with jumps in the volatility only and a continuous return process, several measures coincide, some simplifications can be made and the results are more explicit. We illustrate our results with parametric examples used in the literature.

## A sharp Abelian theorem for the Laplace transform

Biret, Maeva; Broniatowski, Michel; Cao, Zhansheng
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
27.85281%
This paper states asymptotic equivalents for the three first moments of the Eescher transform of a distribution on R with smooth density in the upper tail. As a by product if provides a tail approximation for its moment generating function, and shows that the Esscher transforms have a Gaussian behavior for large values of the parameter.; Comment: To appear in M. Hallin, D. Mason, D. Pfeifer, and J. Steinebach Eds, Mathematical Statistics and Limit Theorems: Festschrift in Honor of Paul Deheuvels. Springer, 20 pages

## Financial and actuarial valuation of insurance derivatives.

Murmann, Alexander
Fonte: London School of Economics and Political Science Thesis Publicador: London School of Economics and Political Science Thesis
Tipo: Thesis; NonPeerReviewed Formato: application/pdf