Markov property &stochastic process Flashcards
what is a stochastic process
describes evolution of a RV over time
- prices, IR, currencies can be model as stochastic processes
- path followed by RV is not deterministic
- paths differ by simulation, but statistically spèaking are equivalent as they seem from same RV
stochastic calculus
natural generalisation of standard calculus on RV
continuous-time formulation processes
parallel econometric theory for estimation and testis of these continuous-time process
Markov Process
type of stochastic process where only current value of variable is relevant to determine future value
past history & way present has emerged from past are irrelevant
probability distribution of price at any pat. future moment is not dependent on part. path followed by price in the past only on price today
markow property on stock price = weak form of market efficiency (stock prices at any given time reflect information in past prices)
if weak form wouldn´t hold - technical analyst could make above normal profits by analysis past charts - but little evidence of this (hence WF holds)
pricing approaches to financial assets
a) absolute- equilibrium
- prices endog. determined by model (CAPM)
b) relative - nonarbitrage
- prices given exogenously, take them and explore conditions these prices must satisfy to avoid AO
a) replicating portfolio: PF of assets should provide same payoffs as original asset
b) absence of AO: 2 assets with equal payoffs and risk mst have same pric
pricing tools for derivatives
- binomial trees
- stochastic methods
- partial differential equations (PDEs)
- Monte Carlo Simulation
- others (Schwarz algorithms etc)
Which pricing tool to choose?
choice depends on some factors:
- accuracy of methods differ
- their speed also changes from one to another
- some methods are easier to implement than others
method chosen depends on circumstances and matter of experience
most important: binomial trees and stochastic calculus
stochastic calculus
generalise binomial framework to different n states
multiple time periods T
deterministic calculus:
- maths behind derivatives assumes - time passes continuously
- new info. os revealed continuously and DM faces instan-. changes in random news
- technical tools for derivative pricing requires handling RV over infinitesimal time intervals
- a bit different from standard calculus
Stochastic Calculus in Finance
- information flows more similar to stochastic than standard calculus
- modeling Random behaviour
- complicated Rv can have simple structure in cont. time (dt)
- differentiate = response of a variable o chang in another
- stochastic integral = sums of random increments
- Taylor series = approximate a function by sing simpler functions
- stochastic D.E. = model the dynamical behaviour of cont. time V
Option Pricing
market price
determined by offer and demand for the option
theoretical value
determined by series of parameters - which according to option pricing theory determine the value of option
Assumptions:
efficient markets –> price = theoretical value
not efficient
price not = theoretical value
option is over/undervalued
possibility for realising risk free profit (arbitrage)
Option pricing
European vs American
American
- represent free boundary problems because of possibility of earlier exercise
- value is determined by numerical methods
binomial, MC etc
European: - only be exercised at maturity - thus it is possible to derive an analytical solution - value of option expressed as a fomurla Black Scholes
Option pricing
Black Scholes model
Assumptions:
- option only european style
- underlying pays no dividends
- there are NO market imperfecions
- risk free & volatility is known and constant
- price of underlying if log-normal distributed and follows Geometric Brownian motion:
dS = uSdt + sigmaS dz
and z - N(0,1)
