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Title:

Proxy simulation schemes using likelihood ratio weighted Monte Carlo for generic robust Monte-Carlo sensitivities and high accuracy drift approximation (with applications to the LIBOR Market Model)

Description:

We consider a generic framework for generating likelihood ratio weighted Monte Carlo simulation paths, where we use one simulation scheme K° (proxy scheme) to generate realizations and then reinterpret them as realizations of another scheme K* (target scheme) by adjusting measure (via likelihood ratio) to match the distribution of K° such that E...

We consider a generic framework for generating likelihood ratio weighted Monte Carlo simulation paths, where we use one simulation scheme K° (proxy scheme) to generate realizations and then reinterpret them as realizations of another scheme K* (target scheme) by adjusting measure (via likelihood ratio) to match the distribution of K° such that E( f(K*) | F_t ) = E( f(K°) w | F_t ). This is done numerically in every time step, on every path. This makes the approach independent of the product (the function f) and even of the model, it only depends on the numerical scheme. The approach is essentially a numerical version of the likelihood ratio method [Broadie & Glasserman, 1996] and Malliavin's Calculus [Fournie et al., 1999; Malliavin, 1997] reconsidered on the level of the discrete numerical simulation scheme. Since the numerical scheme represents a time discrete stochastic process sampled on a discrete probability space the essence of the method may be motivated without a deeper mathematical understanding of the time continuous theory (e.g. Malliavin's Calculus). The framework is completely generic and may be used for high accuracy drift approximations and the robust calculation of partial derivatives of expectations w.r.t. model parameters (i.e. sensitivities, aka. Greeks) by applying finite differences by reevaluating the expectation with a model with shifted parameters. We present numerical results using a Monte-Carlo simulation of the LIBOR Market Model for benchmarking. ; Monte-Carlo, Likelihood Ratio, Malliavin Calculus, Sensitivities, Greeks Minimize

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preprint

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Title:

PERTURBATION STABLE CONDITIONAL ANALYTIC MONTE-CARLO PRICING SCHEME FOR AUTO-CALLABLE PRODUCTS

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In this paper, we present a generic method for the Monte-Carlo pricing of (generalized) auto-callable products (aka. trigger products), i.e., products for which the payout function features a discontinuity with a (possibly) stochastic location (the trigger) and value (the payout).The Monte-Carlo pricing of products with discontinuous payout is k...

In this paper, we present a generic method for the Monte-Carlo pricing of (generalized) auto-callable products (aka. trigger products), i.e., products for which the payout function features a discontinuity with a (possibly) stochastic location (the trigger) and value (the payout).The Monte-Carlo pricing of products with discontinuous payout is known to come with a high Monte-Carlo error. The numerical calculation of sensitivities (i.e., partial derivatives) of such prices by finite differences gives very noisy results, since the Monte-Carlo approximation (being a finite sum of discontinuous functions) is not smooth. Additionally, the Monte-Carlo error of the finite-difference approximation explodes as the shift size tends to zero.Our method combines a product specific modification of the underlying numerical scheme, which is to some extent similar to an importance sampling and/or partial proxy simulation scheme and a reformulation of the payoff function into an equivalent smooth payout.From the financial product we merely require that hitting of the stochastic trigger will result in an conditionally analytic value. Many complex derivatives can be written in this form. A class of products where this property is usually encountered are the so called auto-callables, where a trigger hit results in cancellation of all future payments except for one redemption payment, which can be valued analytically, conditionally on the trigger hit.From the model we require that its numerical implementation allows for a calculation of the transition probability of survival (i.e., non-trigger hit). Many models allows this, e.g., Euler schemes of Itô processes, where the trigger is a model primitive.The method presented is effective across a large range of cases where other methods fail, e.g. small finite difference shift sizes or short time to trigger reset (approaching maturity); this means that a practitioner can use this method and be confident that it will work consistently.The method itself can be viewed as a generalization of the method proposed by Glasserman and Staum (2001), both with respect to the type (and shape) of the boundaries, as well as, with respect to the class of products considered. In addition we explicitly consider the calculation of sensitivities. ; Monte-Carlo simulation, pricing, greeks, variance reduction, auto-callable, trigger product, target redemption note Minimize

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article

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Title:

Markov Functional Modeling of Equity, Commodity and other Assets

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In this short note we show how to setup a one dimensional single asset model, e.g. equity model, which calibrates to a full (two dimensional) implied volatility surface. We show that the efficient calibration procedure used in LIBOR Markov functional models may be applied here too. In a addition to the calibration to a full volatility surface th...

In this short note we show how to setup a one dimensional single asset model, e.g. equity model, which calibrates to a full (two dimensional) implied volatility surface. We show that the efficient calibration procedure used in LIBOR Markov functional models may be applied here too. In a addition to the calibration to a full volatility surface the model allows the calibration of the joint asset-interest rate movement (i.e. local interest rates) and forward volatility. The latter allows the calibration of compound or Bermudan options. The Markov functional modeling approach consists of a Markovian driver process x and a mapping functional representing the asset states S(t) as a function of x(t). It was originally developed in the context of interest rate models, see [7]. Our approach however is similar to the setup of the hybrid Markov functional model in spot measure, as considered in [5]. For equity models it is common to use a deterministic Numéraire, e.g. the bank account with deterministic interest rates. In our approach we will choose the asset itself as Numéraire. This is a subtle, but crucial difference to other approaches considering Markov functional modeling. Choosing the asset itself as Numéraire will allow for a very efficient numerically calibration Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2012-04-09

Source:

http://www.christian-fries.de/finmath/markovfunctionaleqmodel/MarkovFunctionalEQModel.pdf

http://www.christian-fries.de/finmath/markovfunctionaleqmodel/MarkovFunctionalEQModel.pdf Minimize

Document Type:

text

Language:

en

DDC:

330 Economics *(computed)*

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Metadata may be used without restrictions as long as the oai identifier remains attached to it.

Metadata may be used without restrictions as long as the oai identifier remains attached to it. Minimize

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Title:

Abstract Localized Proxy Simulation Schemes for Generic and Robust Monte-Carlo Greeks

Description:

For the numerical calculation of partial derivatives (aka. sensitivites or greeks) from a Monte-Carlo simulation there are essentially two possible approaches: The pathwise method and the likelihood ratio method. Both methods have their shortcomings: While the pathwise method works very well for smooth payouts it fails for discontinuous payouts....

For the numerical calculation of partial derivatives (aka. sensitivites or greeks) from a Monte-Carlo simulation there are essentially two possible approaches: The pathwise method and the likelihood ratio method. Both methods have their shortcomings: While the pathwise method works very well for smooth payouts it fails for discontinuous payouts. On the other hand, the likelihood ratio gives much better results on discontinuous payouts, but falls short of the pathwise method if smooth payouts are considered. In this paper, we present a modification to the (partial) proxy simulation scheme framework, resulting in a per-path selection of either the pathwise method or the likelihood ratio method. This allows us to chose the optimal simulation method on a path-by-path basis. Since the method is implemented as a proxy simulation scheme as well, the sensitivities can be calculated from simple finite differences applied to the pricing engine. Minimize

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The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2008-07-01

Source:

http://www.christian-fries.de/finmath/proxyscheme/Fries-LocalizedProxySchemeForGreeks.pdf

Document Type:

text

Language:

en

DDC:

518 Numerical analysis *(computed)*

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Metadata may be used without restrictions as long as the oai identifier remains attached to it.

Metadata may be used without restrictions as long as the oai identifier remains attached to it. Minimize

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Title:

Foresight Bias and Suboptimality Correction in Monte-Carlo Pricing of Options with Early Exercise: Classification, Calculation & Removal

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In this paper we investigate the so called foresight bias that may appear in the Monte-Carlo pricing of Bermudan and compound options if the exercise criteria is calculated by the same Monte-Carlo simulation as the exercise values. The standard approach to remove the foresight bias is to use two independent Monte-Carlo simulations: One simulatio...

In this paper we investigate the so called foresight bias that may appear in the Monte-Carlo pricing of Bermudan and compound options if the exercise criteria is calculated by the same Monte-Carlo simulation as the exercise values. The standard approach to remove the foresight bias is to use two independent Monte-Carlo simulations: One simulation is used to estimate the exercise criteria (as a function of some state variable), the other is used to calculate the exercise price based on this exercise criteria. We shall call this the numerical removal of the foresight bias. In this paper we give an exact definition of the foresight bias in closed form and show how to apply an analytical correction for the foresight bias. Our numerical results show that the analytical removal of the foresight bias gives similar results as the standard numerical removal of the foresight bias. The analytical correction allows for a simpler coding and faster pricing, compared to a numerical removal of the foresight bias. Our analysis may also be used as an indication of when to neglect the foresight bias removal altogether. While this is sometimes possible, neglecting foresight bias will break the possibility Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2013-07-25

Source:

http://www.christian-fries.de/finmath/foresightbias/Fries_ForesightBias.pdf

Document Type:

text

Language:

en

DDC:

650 Management & auxiliary services *(computed)*

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Metadata may be used without restrictions as long as the oai identifier remains attached to it.

Metadata may be used without restrictions as long as the oai identifier remains attached to it. Minimize

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Title:

The Foresight Bias in Monte-Carlo Pricing of Options with Early Exercise: Classification, Calculation & Removal

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In this paper we investigate the so called foresight bias that may appear in the Monte-Carlo pricing of Bermudan and compound options if the exercise criteria is calculated by the same Monte-Carlo simulation as the exercise values. The standard approach to remove the foresight bias is to use two independent Monte-Carlo simulations: One simulatio...

In this paper we investigate the so called foresight bias that may appear in the Monte-Carlo pricing of Bermudan and compound options if the exercise criteria is calculated by the same Monte-Carlo simulation as the exercise values. The standard approach to remove the foresight bias is to use two independent Monte-Carlo simulations: One simulation is used to estimate the exercise criteria (as a function of some state variable), the other is used to calculate the exercise price based on this exercise criteria. We shall call this the numerical removal of the foresight bias. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-19

Source:

http://www.christian-fries.de/finmath/foresightbias/Fries_ForesightBias.pdf

Document Type:

text

Language:

en

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Title:

A recursive formula for the Kurtosis of an approximation to the distribution of share prices

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Introduction Given the recursive approximation to the distribution of share prices (see [1, 2]) we will derive a recursive formula for Kurtosis. The Kutosis of a distribution consisting of samples x 1 . . . x N with mean and standard deviation s is given by K = x j . (1) Recursive calculation of Kurtosis We fix notation as follows: Assuming that...

Introduction Given the recursive approximation to the distribution of share prices (see [1, 2]) we will derive a recursive formula for Kurtosis. The Kutosis of a distribution consisting of samples x 1 . . . x N with mean and standard deviation s is given by K = x j . (1) Recursive calculation of Kurtosis We fix notation as follows: Assuming that a distributioin of prices p j ( j = 1, . . . , n) occuring at a volume v j is given (say at time t n ). At later time (say at time t n+1 ) the distribution changed to . one price p n+1 occuring at volume v n+1 . the prices p 1 , . . . , p n occuring at volumes v 1 , . . . , v n , respectively (where N = j=1 v j = j=1 v j +v n+1 is the total volume of shares floating). The Kurtosis of the distribution at time t n is thus given by K n = and the Kurtosis of the distribution at time t n+1 is given by K n+1 = v i where v i = N v i for i = 1, . . . , n a Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-18

Source:

http://www.christian-fries.de/evwma/PDF/fries_evwma_kurtosis_2001.pdf

Document Type:

text

Language:

en

DDC:

519 Probabilities & applied mathematics *(computed)*

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Title:

Autonomous Robot Motion Planning in Diverse Terrain Using Genetic Algorithms

Description:

Optimal motion planning is critical for the successful operation of an autonomous mobile robot. Many proposed approaches use either fuzzy logic or genetic algorithms (GAs), however, most approaches offer only path planning or only trajectory planning, but not both. In addition, few approaches attempt to address the impact of varying terrain cond...

Optimal motion planning is critical for the successful operation of an autonomous mobile robot. Many proposed approaches use either fuzzy logic or genetic algorithms (GAs), however, most approaches offer only path planning or only trajectory planning, but not both. In addition, few approaches attempt to address the impact of varying terrain conditions on the optimal path. This paper presents a fuzzy-genetic approach that provides both path and trajectory planning, and has the advantage of considering diverse terrain conditions when determining the optimal path. The terrain conditions are modeled using fuzzy linguistic variables to allow for the imprecision and uncertainty of the terrain data. Although a number of methods have been proposed using GAs, few are appropriate for a dynamic environment or provide response in real-time. The method proposed in this paper is robust, allowing the robot to adapt to dynamic conditions in the environment. Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2010-02-17

Source:

http://www.cs.bham.ac.uk/~wbl/biblio/gecco2005lbp/papers/62-fries.pdf

http://www.cs.bham.ac.uk/~wbl/biblio/gecco2005lbp/papers/62-fries.pdf Minimize

Document Type:

text

Language:

en

Subjects:

Methodologies ; Robotics – autonomous vehicles ; I.2.8 [Computing Methodologies ; Problem Solving ; Control Methods ; and Search – plan execution ; formation ; and generation ; General Terms Algorithms ; design Keywords Genetic algorithms ; fuzzy sets ; autonomous robots ; motion planning ; robot

Methodologies ; Robotics – autonomous vehicles ; I.2.8 [Computing Methodologies ; Problem Solving ; Control Methods ; and Search – plan execution ; formation ; and generation ; General Terms Algorithms ; design Keywords Genetic algorithms ; fuzzy sets ; autonomous robots ; motion planning ; robot Minimize

DDC:

629 Other branches of engineering *(computed)*

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Title:

A recursive formula for the Kurtosis of an approximation to the distribution of share prices

Description:

Given the recursive approximation to the distribution of share prices (see [1, 2]) we will derive a recursive formula for Kurtosis. The Kutosis of a distribution consisting of samples x 1 . x N with mean and standard deviation s is given by K = x j . (1) Recursive calculation of Kurtosis We fix notation as follows: Assuming that a distribution o...

Given the recursive approximation to the distribution of share prices (see [1, 2]) we will derive a recursive formula for Kurtosis. The Kutosis of a distribution consisting of samples x 1 . x N with mean and standard deviation s is given by K = x j . (1) Recursive calculation of Kurtosis We fix notation as follows: Assuming that a distribution of prices p j ( j = 1, . . . , n) occuring at a volume v j is given (say at time t n ). At later time (say at time t n+1 ) the distribution changed to . one price p n+1 occuring at volume v n+1 . the prices p 1 , ., p n occuring at volumes v 1 , ., v n , respectively (where N = j=1 v j = j=1 v j +v n+1 is the total volume of shares floating). The Kurtosis of the distribution at time t n is thus given by K n = and the Kurtosis of the distribution at time t n+1 is given by K n+1 = v i where v i = N v i for i = 1, ., n a Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-17

Source:

http://www.spacelike.com/fries/evwma/fries_evwma_kurtosis_2001.pdf

Document Type:

text

Language:

en

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Title:

The Distribution of Share Prices and Elastic Time and Volume Weighted Moving Averages

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this article we consider "the distribution of prices paid per share" and derive simple approximation formulas for its mean and other statistical characteristica. The formulas we derive are approximations since the data about the distribution of prices paid by each shareholder for each share is not available. After going through the derivation of...

this article we consider "the distribution of prices paid per share" and derive simple approximation formulas for its mean and other statistical characteristica. The formulas we derive are approximations since the data about the distribution of prices paid by each shareholder for each share is not available. After going through the derivation of the formula we will end up with recursive definitions of time series which could be viewed as random coefficient autoregressive (RCA) time series (see [6, 5]) Minimize

Contributors:

The Pennsylvania State University CiteSeerX Archives

Year of Publication:

2009-04-17

Source:

http://www.spacelike.com/fries/evwma/fries_evwma_preprint_2002.pdf

Document Type:

text

Language:

en

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