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## Option pricing and Esscher transform under regime switching

Fonte: Springer-Verlag
Publicador: Springer-Verlag

Tipo: Artigo de Revista Científica

Publicado em //2005
Português

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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

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## Option pricing for GARCH models with Markov switching

Fonte: World Scientific Publishing Co Pte Ltd
Publicador: World Scientific Publishing Co Pte Ltd

Tipo: Artigo de Revista Científica

Publicado em //2006
Português

Relevância na Pesquisa

49.309204%

#Markov switching conditional Esscher transform#Markov switching Heston-Nandi’s GARCH model#recursive formula#analytical option valuation.

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

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

Fonte: Springer-Verlag
Publicador: Springer-Verlag

Tipo: Artigo de Revista Científica

Publicado em //2008
Português

Relevância na Pesquisa

27.546348%

#Risk measures#Regime-switching PDE#Regime-switching HJB equation#Stochastic optimal control#Esscher transform#Delta-neutral hedging#Jump risk#American options#Exotic options

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

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## Esscher transforms and consumption-based models

Fonte: Elsevier Science BV
Publicador: Elsevier Science BV

Tipo: Artigo de Revista Científica

Publicado em //2009
Português

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59.757803%

#Esscher transform#Esscher–Girsanov transform#Consumption-based model#Stochastic discount factor#Exponential affine form#Euler equation#Radon–Nikodym derivative#Utility function

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

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

Fonte: World Scientific Publishing Co Pte Ltd
Publicador: World Scientific Publishing Co Pte Ltd

Tipo: Artigo de Revista Científica

Publicado em //2011
Português

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#Option pricing#risk neutral valuation#Generalized Hyperbolic GARCH#extended Girsanov principle#Esscher transform#mean correcting martingale measure#Radon-Nikodym derivative

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

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

Fonte: Routledge
Publicador: Routledge

Tipo: Artigo de Revista Científica

Publicado em //2013
Português

Relevância na Pesquisa

48.66176%

#Option pricing#hidden Markov-modulated pure-jump processes#Esscher transform#Laplace cumulant process#robust filters#integral representation

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

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

Fonte: Springer
Publicador: Springer

Tipo: Artigo de Revista Científica

Publicado em //2015
Português

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

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## A Dupire equation for a regime-switching model

Fonte: World Scientific Publishing
Publicador: World Scientific Publishing

Tipo: Artigo de Revista Científica

Publicado em //2015
Português

Relevância na Pesquisa

27.546348%

#Regime-switching local volatility model#Esscher transform#forward equations#regime-switching adjoint formula

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/08/2014
Português

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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...

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## Pricing Currency Derivatives with Markov-modulated Levy Dynamics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/02/2014
Português

Relevância na Pesquisa

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#Quantitative Finance - Computational Finance#Quantitative Finance - Pricing of Securities#91B70, 60H10, 60F25

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/02/2014
Português

Relevância na Pesquisa

27.546348%

#Quantitative Finance - Computational Finance#Quantitative Finance - Pricing of Securities#91B70, 60H10, 60F25

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

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## Option Pricing in a Dynamic Variance-Gamma Model

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 28/05/2014
Português

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.

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## Unconstrained Recursive Importance Sampling

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/08/2013
Português

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...

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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

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## Esscher transform and the duality principle for multidimensional semimartingales

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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)

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 08/07/2008
Português

Relevância na Pesquisa

28.84088%

#Quantitative Finance - Computational Finance#Mathematics - Probability#Mathematics - Statistics Theory#91B70#91B28

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.

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## A sharp Abelian theorem for the Laplace transform

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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

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## Financial and actuarial valuation of insurance derivatives.

Fonte: London School of Economics and Political Science Thesis
Publicador: London School of Economics and Political Science Thesis

Tipo: Thesis; NonPeerReviewed
Formato: application/pdf

Publicado em //2002
Português

Relevância na Pesquisa

28.013337%

This dissertation looks into the interplay of financial and insurance markets that is created by securitization of insurance related risks. It comprises four chapters on both the common ground and different nature of actuarial and financial risk valuation. The first chapter investigates the market for catastrophe insurance derivatives that has been established at the Chicago Board of Trade in 1992. Modeling the underlying index as a compound Poisson process the set of financial derivative prices that exclude arbitrage opportunities is characterized by the market prices of frequency and jump size risk. Fourier analysis leads to a representation of price processes that separates the underlying stochastic structure from the contract's payoff and allows derivation of the inverse Fourier transform of price processes in closed form. In a market with a representative investor, market prices of frequency and jump size risk are uniquely determined by the agent's coefficient of absolute risk aversion which consequently fixes the price process on the basis of excluding arbitrage strategies. The second chapter analyzes a model for a price index of insurance stocks that is based on the Cramer-Lundberg model used in classical risk theory. It is shown that price processes of basic securities and derivatives can be expressed in terms of the market prices of risk. This parameterization leads to formulae in closed form for the inverse Fourier transform of prices and the conditional probability distribution. Financial spreads are examined in more detail as their structure resembles the characteristics of stop loss reinsurance treaties. The equivalence between a representative agent approach and the Esscher transform is shown and the financial price process that is robust to these two selection criteria is determined. Finally...

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## Pricing and hedging in an incomplete interest rate market: Applications of the Laplace transform.

Fonte: London School of Economics and Political Science Thesis
Publicador: London School of Economics and Political Science Thesis

Tipo: Thesis; NonPeerReviewed
Formato: application/pdf

Publicado em //2008
Português

Relevância na Pesquisa

38.112124%

This thesis explores pricing models for interest rate markets. The model used to describe the short rate is based on the discontinuous shot noise process. As a consequence the market is incomplete, meaning that not all securities contingent on the short rate can be replicated perfectly with a dynamically adjusted portfolio of a bond and cash. This framework is still consistent with the absence of arbitrage as evidenced by the existence of an equivalent martingale measure. This measure is not unique, however, due to the incompleteness of the market. Two approaches to pricing contingent claims are pursued. The first, risk-neutral pricing, evaluates the expected value of the pay-off at expiration under an equivalent martingale measure. A parameterized class of martingales, based on the Esscher transform, allows for the definition of a flexible set of equivalent martingale measures and results in a formula for the conditional joint Laplace transform of the short rate and its time-integral. The pricing formula for a discount bond follows trivially from these results. A method for pricing a European call option is also proposed, requiring numerical inversion of the aforementioned Laplace transform. The second approach, mean-variance hedging...

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