Seydel 的 tools for computational finance 总结导图 |
Tools for Computational Finance
Discrete model
Binomial method
Continuous model
modelling tools
stochastic process
Ito lemma
Monte Carlo method
Generating Random Numbers with Specified Distributions
schemes
dynamic
idea
stochastic Taylor expansion
Euler
Milstein
positivity
replace with or
Runge-Kutta
approximate the derivative
pricing steps
represent the value of derivatives by a integral
simulate the dynamic, and calculate the payoff for each sample pathes
get the confidence interval from obtained payoffs
error analysis
behave as by central limit theorem
variance reduction
antithetic variates
control variates
American option
stopping time
parametric methods
regression methods
sensitivity
estimate how the price V changes when parameters or initial states change
Finite Difference method
explicit
implicit
Von Neumann stability analysis
error analysis
Crank-Nicolson method
unconditional stable
American option
Consider put option
Free boundary problem
Black-Scholes equation
Black-Scholes inequality
'=' holds in continuation region
penalty formulation
advantegeous especially when an analysis of the early-exercise curve is difficult
linear comlementarity problem(LCP)
LCP
discretization
iterative method
reformualte as optimization problem
Cryer problem
Motivation
Let , find x, y s.t.
result
Cryer problem is equivalent to the minimization problem (G is strictly convex)
the minimization problem can be solved by an iterative procedure based on SOR (successive overrelaxation)
direct method
Cryer problem restated
solve s.t.
extend the direct method for solving
accuracy
errors
modelling error
discretization error
error arising from sloving the linear system
rounding error
extrapolation
analytical methods
numerical methods are designed to converge, so in principle any accuracy can be obtained given time and computation power
some analytic formula may be sufficient that delivers medium accuracy at low cost
approximation based on interpolation
find and s.t.
quadratic approximation
analytic method of lines
integral-equation method
criterions for comparison
reliability
range of applicability
amount of information provided by the method
speed
error
Finite Element method
extension of finite difference method, but allow more flexibility
Exotic option
differences
payoff
increase in dimension (multifactor option)
path-dependent options
options depending on several assets
barrier option
analytical method
first passage time
finite difference method
change of boundary condition
Asian option
finite difference method
add a new variable
extended dynamic
PDE
dimension reduction
convection-diffusion problem
why some difference schemes applied to the BS equation exhibit faulty oscillations
consider model problem:
Peclet number: the ratio of convection to diffusion
affect the stability
upwind schemes
consider extreme case : ,
upwind discretization
also called Forward Time Backward Space (FTBS) scheme
von Neumann stability analysis leads Courant–Friedrichs–Lewy (CFL) condition
dispersion
the phenomenon of different modes traveling at different speeds
high-resolution method
penalty method for American option