Penera

# Timeline

By June 15, finish part III: financial markets and products

1. Book: Chapter 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 26
2. Notes III

By July 31, finish part IV: valuation and risk models

— Thank you my dear for bringing this baby home. Cheer UP

# Ito Process

It is a generalized Wiener process in which the parameters a and b are functions of the variable x and t: $dx = a\times (x,t) dt + b\times (x,t) dz$

This is Markov because the value of x only depends on time t. Both the drift rate and volatility rate are changing overtime.

For the change $\Delta x$ in small time interval $\Delta t$,

$\Delta x = a\times (x,t)\Delta t + b\times (x,t) \sqrt{\Delta t}$

# Generalized Wiener Process

If the change $\Delta z$ in a small period of time $\Delta t$ is independent, with a standardized normal distribution (0,1), then z follows the basic Wiener process (Brownian motion). $\Delta z = normal(0, \Delta t)$; for any period T, z = normal(0,T)

A generalized Wiener process of variable x: $dx = a\times dt + b\times dz$, where a and b are constants

In a $\Delta t$, the change

$\Delta x = a\times \Delta t + b\times \sqrt{\Delta t}$

$\Delta x$ has a normal distribution with an expected drift rate of a and a variance of $b^2$.

x in any time interval T is normally distributed with mean of change in x = aT, and variance of change in x = $b^2\times T$