Sabtu, 27 Desember 2014

The Markovian Wiener process embedding dimension



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Mathematical Notation
The mathematics behind Monte Carlo methods
A few basic terms in probability and statistics
Monte Carlo simulations
Monte Carlo supremacy
Multidimensional integration
Some common distributions
Kolmogorov's strong law
The central limit theorem
The continuous mapping theorem
Error estimation for Monte Carlo methods
The FeynmanKac theorem
The MoorePenrose pseudoinverse
Stochastic dynamics
Brownian motion
Ito's lemma
Normal processes
Lognormal processes
The Markovian Wiener process embedding dimension
Bessel processes
Constant elasticity of variance processes
Displaced diffusion

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Process driven sampling
Strong versus weak convergence
Numerical
The Euler scheme
The Milstein scheme
Transformations
Predictor-Corrector
Spurious paths
Strong convergence for Euler and Milstein
Correlation and comovement
Measures for codependence
Copulae
The Gaussian copula
The t-copula
Archimedean copulae
Salvaging a linear correlation matrix
Hypersphere decomposition
Spectral decomposition
Angular decomposition of lower triangular form
Angular coordinates on a hypersphere of unit radius
Pseudorandom numbers and Chaos
The midsquare method
Congruential generation
Ran to Ran
The Mersenne twister
Which one to use?

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Low-discrepancy numbers
Discrepancy
Halton numbers
Sobol' numbers
Primitive polynomials modulo two
The construction of Sobol' numbers
The Gray code
The initialisation of Sobol' numbers
Niederreiter numbers
Pairwise projections
Empirical discrepancies
The number of iterations
Explicit formula for the L-norm discrepancy on the unit hypercube
Expected L-norm discrepancy of truly random numbers
Non-uniform variates
Inversion of the cumulative probability function
Using a sampler density
Importance sampling
Rejection sampling
Normal variates
The BoxMuller method
The Neave effect
Simulating multivariate copula draws
Variance reduction techniques
Antithetic sampling
Variate recycling
Control variates
Stratified sampling
Importance sampling
Moment matching
Latin hypercube sampling

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Path construction
Incremental
Spectral
The Brownian bridge
A comparison of path construction methods
Multivariate path construction
Eigenvalues and eigenvectors of a discretetime covariance matr
The conditional distribution of the Brownian bridge
Greeks
Importance of Greeks
An Up-Out-Call option
Finite differencing with path recycling
Finite differencing with importance sampling
Pathwise differentiation
The likelihood ratio method
Comparative figures
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The likelihood ratio formula for Vega
The likelihood ratio formula for Rho
Monte Carlo in the BGM/J framework
The BraceGatarekMusielanamshidian market model
Factorisation
Bermudan swaptions
Calibration to European swaptions
The PredictorCorrector scheme
Heuristics of the exercise boundary
Exercise boundary parametrisation
The algorithm
Numerical results
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Below are some topics which important for your thesis/dissertation: vii
Nonrecombining trees
Evolving the forward rates
Optimal simplex alignment
Implementation
Convergence performance
Variance matching
Exact martingale conditioning
Clustering
A simple example
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Miscellaneaous
Interpolation of the term structure of implied volatility
Watch your cpu usage
Numerical overflow and underflow
A single number or a convergence diagram?
Embedded path creation
How slow is exp ?
Parallel computing and multithreading
Bibliography

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Products and Markets-Equities, Commodities, Exchange Rates, i
Forwards and Futures
Equities
Dividends
Stock splits
Commodities
Currencies
Indices
The time value of money
Fed-income securities
Inflationproof bonds
Forwards and futures
A first example of no arbitrage
More about futures
Commodity futures
FX futures
Index futures
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Derivatives
Options
Definition of common terms
Payoff diagrams
Other representations of value
Writing options
Margin
Market conventions
The value of the option before expiry
Factors affecting derivative prices

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