Definition A.1.2 (Probability space) Let Ω be a nonempty set, and let F be a σ–algebra of subgroups of Ω. The distribution measure of X is the probability measure μ X that assigns to each Borel subgroup B of measure μ X ( B ) = P { X ∈ B. 23 These processes have memory which is the previous step all information is contained in the probability distribution .
When measuring an observable in Monte Carlo simulations, one must generate a sequence of random numbers that are uniformly distributed over the group manifold. For example, in 2D, the probability of getting points inside S 1 , P in , is approximately 75 percent of the time. To bring the points on the surface of the unit sphere, just divide by the norm of the vector space, i.e. these numbers can be used to construct a randomly uniformly distributed vector in R 2 .
Let us consider 1 random number r that is uniformly distributed on and another uniformly distributed number θ ∈ [ 0, 2π. To construct our uniformly distributed vector of random numbers in R 2 with Gaussian probability distribution, one must consider that. A.30).
Partial differential equations
I N this appendix we give a few details on partial differentil equa-tions in particular on how these are classified
Partial differential equations
- Classification partial differential equations
In this appendix, give some terminology on how the PDE is classified, these concepts are used right through the thesis in main text. Ordinary differential equations can be classified according to their order and whether they are linear or non-linear differential equations. Partial differential equations (PDE) can also be classified, but it is generally more difficult to do so.
Using the discriminant, we classify the PDE into three different classes, elliptic, parabolic and hyperbolic, using the following selection criterion.
PDE =
Option pricing details
I N this appendix we summarize some of the mathematical detailsused in the formulation of the exotic options
More Different Options
- Non-Vanilla Exercise Rights
Bermudan option
Canary option
Capped-style option
Compound option
Shout option
Swing option
C.1.2 ’Exotic’ Options with Standard Exercise Styles
Cross option (or composite option)
Quanto option
Basket option
Rainbow option
Non-vanilla path dependent “exotic” options
Lookback option
Asian option
Russian option
Game option or Israeli option
The payoff of a cumulative Parisian option
The payoff of a standard Parisian option
Barrier option
Double Barrier option
Cumulative Parisian barrier option
Maximum of Brownian motion with drift
Standard Parisian barrier option
Reoption
Binary option (also known as a digital option)
Chooser option
Cliquet option
This theorem and conclusion are used to calculate the knock-in/out barrier option when under the Black–Scholes–Merton model.
The Fokker–Planck Equation
I N this appendix we explicitly derive the Sturm–Liouville equationand explain the connection betrween the classical Fokker–Planck
- The Sturm–Liouville equation
- The connection between the Schroedinger equation and Fokker–Planck equation
- The connection between the Schr¨ oedinger equation and Fokker–Planck equation
- The wave function and the probability density in quantum mechanics
Letting P ( t, T; x, y ) denote the transition, the probability of the Fokker-Planck equation is given by . If we set φ ( x ) and λ as eigenfunctions and eigenvalues of the Fokker-Planck operator together with a transformation on the probability density. then we see this when we apply the Fokker-Planck operator to this transformation. Now we use the previously used eigenfunction and eigenvalue, that is, if φ n ( y ) and λ n are eigenfunctions and eigenvalues of the Fokker-Planck operator, then eigenfunctions.
Therefore, even when transforming the Fokker-Planck operator, we have the following eigenvalue problem. This allows us to note the expansion of the transition probability, in Eq. D.21) can be used to explicitly calculate the Hermitian operator, Eq. D.21) can be used to write the Fokker-Planck Hermitian operator explicitly. The wave function can be defined as the projection of the vector | ψ (t) on the position basis, that is,.
The time evolution of the probability density function of a particle's position and velocity can also be described in the classical world by the Fokker-Planck equation. The Fokker-Planck equation can also be used to calculate the probability density of stochastic differential equations as shown in Eq.
Source code
The R script nasdaq.r
The Perl script strip yahoo. pl
The discretization methods
- The headers file for the discretization code
The latticesize.h file
The real t data.h file generated by the perl script Code E.2
The real file.h file generated by the perl script Code E.2
The num pdf.h file generated by the perl script Code E.2
The pdf para.h file generated by the perl script Code E.2
The main code for the discretization code Sde main. f 90
The subroutine dri f t f unc. f 90
The subroutine di f f usion f unc. f 90
The subroutine wiener. f 90
The subroutine explicit sol. f 90
The subroutine euler. f 90
The subroutine milstein. f 90
The subroutine d f ridr 1. f 90
The subroutine strg 15 taylor. f 90
The subroutine d f ridr. f 90
The subroutine gaussian. f 90
The routines used in the main for different distribu- tions and PDF
The routines used in the main for different distributions and PDF
- The subroutine Gen hyper geo ( x, a, y, dyda, delta ) , GH ( x )
- The subroutine hyper geo ( x, a, y, dyda, delta ) , H ( x )
- The subroutine hist vol , the historical volatility
- The subroutine for the Bessel function
For',file_price(dataset),'with dt=',dt,'historical volatility is: ',sig(dataset),'+-',sig_std(dataset) .
The Bessel of first kind
The Bessel of second kind
The Bessel of third kind
The modified Bessel of third kind
The first modified Bessel of third kind
- The subroutine student ( x, a, y, dyda, delta ) , f ( x )
- The subroutine gauss ( x, a, y, dyda, delta ) , f ( x )
- The subroutine f unc, f unc test v and f unc improper v test func- tions
- The subroutine get pd f ()
- The subroutine f it pd f . f 90
- The subroutine mrqmin. f 90
- The subroutine qromo md. f 90
- The subroutine midexp md. f 90
- The subroutine trapzd. f 90
- Some functions used in the evaluation of functions and options
- The subroutine black scholes , the Black–Scholes model and the Greeks
- The cum norm ( x ) function
- The Er f ( x ) function
- The Er f c ( x ) function
- The log ( Γ ( x )) function
- The minority game
- The headers file for the minority game code
REAL(SP), PURPOSE(IN) :: x REAL(SP) :: function FUNCTION FUNC FUNC INTERFACE END FUNCTION dfridr_1 end interface. SUBROUTINE mrqmin(x,y,sig,a,mask,covar,alpha,chisq,funcs,alamda,delta) USAGE nrtype; USE nrutil, ONLY : assert_eq,diagmult.
The real t data.h file
The minority source code
The routine that initialises the real data
The routine real. f 90
The code that calculates the Bubble in section 7.3
Maple output for the non–Gaussian model
I N this appendix we give the Maple output for the evaluation ofthe Chapman–Kolmogorov for the non–Gaussian model discussed
Maple output for the path intgeral when N = 2
Maple output for the path intgeral when N = 3
Asset Pricing According to Endogenous Expectations in an Artificial Stock Market, The Economy as a Complex Evolving System, II, Vol. Gobay, Sceaux, 1995 (reprinted in P. Cootner, Ed., The Random Character of Stock Market Prices, MIT Press, Cambridge, MA), pp.
Bibliography
Stationarity of the GARCH process and some non-negative time series, Journal of Econometrics, 52, p. implementation of equations and stability issues, Archives of Journal of Computational and Applied Mathematics, 125, pp.
Evaluating derivative security prices in the hjm framework as path integrals using fast fourier transform techniques, Journal of Financial Engineering, 6, pp. Evaluating US option prices in a path integral framework using fourier-hermite series expansions, Journal of Economics Dynamics. Maximum likelihood from incomplete data using the em algorithm, Journal of the Royal Statistical Society of London B, 39, pp.
Bridging the gap between realized volatility distribution (ECU) and ARCH (of EURO) modeling: The GARCH-NIG model, Journal of Applied Econometrics, 17, p. Journal of Finance, 48, pp. A discussion of the inclusion of the Black and Scholes option pricing model in a quantum physics setting, Physica A, 304, pp.
A survey of recent theoretical results for time series models with garch errors, Journal of Economics Survey, 16, pp. Power ARCH modeling of commodity futures data on the London Metal Exchange, The European Journal of Finance, 7, pp. Quantum Field Theory and Critical Phenomena (The International Series of Monographs on Physics, 113), Oxford, New York.
Index
He completed high school here in Adelaide and began studying at the University of Adelaide in 1993 for a Bachelor of Mathematics and Computer Science. The project was in lattice quantum chromodynamics, the project looked in particular at various aspects on how to reduce the discretization errors on the lattice. Using these improved actions, we were able to gain further insights into the structure of the QCD vacuum, and how different algorithms relate to each other and also learn more about the structure of the gluon propagator in the infrared region.
After completing a postdoctoral year in North America working on the pion form factor at the Jefferson Laboratory, he decided to return to Adelaide University and join the complex systems group at the Center of Biomedical Engineering to study econophysics and start a new Ph. D. The goal of the current project is to come up with an alternative approach to pricing options using more realistic models that incorporate non-Gaussian distributions.
