00fAsianOptions-package: EBM and Asian Option Valuation

In fAsianOptions: Rmetrics - EBM and Asian Option Valuation

EBM and Asian Option Valuation

Description

The Rmetrics "fAsianOptions" package is a collection of functions to analyze and model Exponential Brownian Motion and to valuate Asian options.

Details

        Package:    \tab fAsianOptions\cr
        Type:       \tab Package\cr
        Version:    \tab R 3.0.1\cr
        Date:       \tab 2014\cr
        License:    \tab GPL Version 2 or later\cr
        Copyright:  \tab (c) 1999-2014 Rmetrics Association\cr
        Repository: \tab R-FORGE\cr
        URL:        \tab \url{https://www.rmetrics.org}
    

1 Introduction

The fAsianOptions package provides functions for pricing and valuating Asian Options together with tools for analyzing and modeling Exponential Brownian Motion (EBM).

2 Asian Option Pricing

This is a collection of functions used in the theory of exponential Brownian Motion, EBM, and in the valuation of Asian options.

    MomentMatchedAsianOption     valuate moment matched option prices
    - "LN"                       Log-Normal Approx of Levy, Turnbull and Wakeman
    - "RG"                       Reciprocal-Gamma Approx of Milevski and Posner
    - "JI"                       Johnson Type I Approx of Posner and Milevsky
    MomentMatchedAsianDensity    valuate moment matched option densities
    - "LN"                       Log-Normal Approximation
    - "RG"                       Reciprocal-Gamma Approximatio
    - "JI"                       Johnson Type I Approximation
    GramCharlierAsianOption      calculates Gram-Charlier option prices
    

    AsianOptionMoments           methods to calculate Asian Moments
    - "A"                        using moments from Abrahamson's Formula
    - "D"                        using moments from Dufresne's Formula
    - "TW"                       using first 2 Moments from Turnbull-Wakeman
    - "T"                        including asymptotic Behavior after Tolmatz
    

    ZhangAsianOption             Asian option price by Zhang's 1D PDE
    VecerAsianOption             Asian option price by Vecer's 1D PDE
    

    gGemanYor                   Function to be Laplace inverted
    GemanYorAsianOption         Asian option price by Laplace Inversion
    gLinetzky                   Function to be integrated
    LinetzkyAsianOption         Asian option price by Spectral Expansion
    

    BoundsOnAsianOption          using lower and upper bonds on Asian calls
    CurranThompsonAsianOption    using Thompson's continuous limit
    RogerShiThompsonAsianOption  using Thompson's single integral formula
    ThompsonAsianOption          using Thompson's upper bound
    TolmatzAsianOption           using lower bound from Tolmatz' asymptotics
    

    CallPutParityAsianOption     using Call-Put parity relation
    WithDividendsAsianOption     adding dividends to Asian option formula
    

    FuMadanWangTable             returns table from Fu, Madan and Wang's paper
    FusaiTaglianiTable           returns able from Fusai und tagliani's paper
    GemanTable                   returns table from Geman's paper
    LinetzkyTable                returns table from Linetzky's paper
    ZhangTable                   returns table from Zhang's paper
    ZhangLongTable               returns long table from Zhang's paper
    ZhangShortTable              returns short table from Zhang's paper
    

2 EBM Distribution

In this section we summarize distributions and related functions which are useful in the theory of exponential Brownian motion and Asian option valuation. The functions compute densities and probabilities for the log-Normal distribution, the Gamma distribution, the Reciprocal-Gamma distribution, and the Johnson Type-I distribution. Functions are made available for the compution of moments including the Normal, the log-Normal, the Reciprocal-Gamma, and the Asian-Option Density. In addition a function is given to compute numerically first and second derivatives of a given function.

    dlognorm        returns the log-Normal density and derivatives
    plognorm        returns the log-Normal, a synonyme for R's plnorm
    

    dgam            returns the Gamma density, a synonyme for R's dgamma
    pgam            returns the Gamma probability, a synonyme for R's pgamma
    

    drgam           returns the Reciprocal-Gamma density
    prgam           returns the Reciprocal-Gamma probability
    

    djohnson        returns the Johnson Type I density
    pjohnson        returns the Johnson Type I probability
    

    mnorm           returns the moments of Normal density
    mlognorm        returns the moments of log-Normal density
    mrgam           returns the moments of reciprocal-Gamma density
    masian          returns the moments of Asian Option density
    derivative      returns the first and second numerical derivative
    

3 Bessel Functions

This section offers special mathematical functions which compute the modified Bessel functions of integer order of the first and second kind as well as their derivatives.

    BesselI         computes modified Bessel function of the 1st kind
    BesselDI        computes its derivative
    BesselK         computes the modified Bessel function of the 3nd kind
    BesselDK        computes its derivative
    

4 Gamma Function

This section offers special mathematical functions which compute the Gamma function and related functions. The functions include the error function, the Psi function, the incomplete Gamma function, the Gamma function for complex argument, and the Pochhammer symbol. The Gamma function the logarithm of the Gamma function, their first four derivatives, and the Beta function and the logarithm of the Beta function are part of R's base package (marked by an asterisk). For example, these functions are required to valuate Asian Options based on the theory of exponential Brownian motion.

    erf             computes the Error function
    gamma*          computes the Gamma function
    lgamma*         the logarithm of the Gamma function
    digamma*        the first derivative of the Log Gamma function
    trigamma*       the second derivative of the Log Gamma function
    tetragamma*     the third derivative of the Log Gamma function
    pentagamma*     the fourth derivative of the Log Gammafunction
    beta*           the Beta function
    lbeta*          the logarithm of the Beta function
    Psi             computes the Psi or Digamma function
    igamma          computes the incomplete Gamma function
    cgamma          computes the Gamma function for complex argument
    Pochhammer      returns the Pochhammer symbol
    

5 Hypergeometric Function

This section offers special mathematical functions which compute the confluent hypergeometric and related functions. These functions are required to valuate Asian Options based on the theory of exponential Brownian motion.

    kummerM         the Confluent Hypergeometric Function of the 1st kind
    kummerU         the Confluent Hypergeometric Function of the 2nd kind
    whittakerM      the Whittaker M Function
    whittakerW      the Whittaker W Function
    hermiteH        the Hermite Polynomials
    

About Rmetrics

The fOptions Rmetrics package is written for educational support in teaching "Computational Finance and Financial Engineering" and licensed under the GPL.

fAsianOptions documentation built on Sept. 9, 2022, 3:08 p.m.