ch14 wiener process and Itos lemma

Question 1

Variables that follows a stochastic process are…

Answer 1

variables whose value changes over time in an uncertain way

Question 2

What is a discrete-time stochastic process?

Answer 2

It is one where the value of the variable can change only at certain fixed points in time

Question 3

What is a continuous-time stochastic process?

Answer 3

It is one where changes can take place at any time

Question 4

What is a continuous-variable process?

Answer 4

the underlying variable can take any value within a certain range

Question 5

What is a discrete-variable process?

Answer 5

only certain discrete values are possible

Question 6

How do we observe stock prices?

Answer 6

Stock prices are restricted to discrete values (e.g., multiples of a cent) and changes can be observed only when the exchange is open for trading.

Question 7

What is the Marcov process?

Answer 7

A Markov process is a particular type of stochastic process where only the current value of a variable is relevant for predicting the future. The past history of the variable and the way that the present has emerged from the past are irrelevant.

Question 8

What kind of market efficiency does the Markov property have?

Answer 8

The Markov property of stock prices is consistent with the weak form of market efficiency. This states that the present price of a stock impounds all the information contained in a record of past prices.

Question 9

What kind of market efficiency does the Markov property have?

Answer 9

The Markov property of stock prices is consistent with the weak form of market efficiency. This states that the present price of a stock impounds all the information contained in a record of past prices.

Question 10

What if the Markov property didn’t have weak form of market efficiency?

Answer 10

If the weak form of market efficiency were not true, technical analysts could make above-average returns by interpreting charts of the past history of stock prices.

Question 11

Consider a variable that follows a Markov stochastic process. Suppose that its current value is 10 and that the change in its value during a year is f(0, 1), where f(m, v), denotes a probability distribution that is normally distributed with mean m and variance v.2 What is the probability distribution of the change in the value of the variable during 2 years?

Answer 11

The change in 2 years is the sum of two normal distributions, each of which has a mean of zero and variance of 1.0. Because the variable is Markov, the two probability distributions are independent. When we add two independent normal distributions, the result is a normal distribution where the mean is the sum of the means and the variance is the sum of the variances. The mean of the change during 2 years in the variable we are considering is, therefore, zero and the variance of this change is 2.0. Hence, the change in the variable over 2 years has the distribution f(0, 2). The standard deviation of the change is sqrt(2).

Question 12

What is a wiener process?

Answer 12

The process followed by the variable we have been considering is known as a Wiener process. It is a particular type of Markov stochastic process with a mean change of zero and a variance rate of 1.0 per year (sometimes referred to as Brownian motion).

Question 13

property 1 of wiener process

Answer 13

delta z=e sqrt (∆t) where e has a standard normal distribution fi(0, 1).

Question 14

property 2 of wiener process

Answer 14

The values of ∆z for any two different short intervals of time, ∆t, are independent.

mean of ∆z = 0
standard deviation of∆z= sqrt(∆t)
variance of ∆z = ∆t

Question 15

A generalized Wiener process for a variable x can be defined in terms of dz as

Answer 15

dx = adt + bdz

Question 16

What is the Itô Process

Answer 16

A further type of stochastic process, known as an Itô process, can be defined. This is a generalized Wiener process in which the parameters a and b are functions of the value of the underlying variable x and time t.

dx=a(x,t)dt+b(x,t)dz

Question 17

What is a Monte Carlo simulation?

Answer 17

A Monte Carlo simulation of a stochastic process is a procedure for sampling random outcomes for the process.

Question 18

μ

Answer 18

expected return (annualized) earned by an investor in a short period of time. value of μ should depend on the risk of the return from the stock. It should also depend on the level of interest rates in the economy. The higher the level of interest rates, the higher the expected return required on any given stock.

Question 19

σ

Answer 19

the stock price volatility, is critically important to the determination of the value of many derivatives. The standard deviation of the proportional change in the stock price in a small interval of time Δt is σ sqrt(t). As a rough approximation, the standard deviation of the proportional change in the stock price over a relatively long period of time ΔT is σ sqrt(T).

Question 20

What is the Wiener process?

Answer 20

The Wiener process, also known as Brownian motion, is a mathematical model used to describe the random and continuous movements of the movements of financial assets in a market.

Question 21

What is the Itô integral?

Answer 21

The Itô integral is a stochastic integral used to evaluate the integral of a stochastic process with respect to a Wiener process, and is a key tool in the stochastic calculus.

Question 22

What is the Itô formula?

Answer 22

The Itô formula is a generalization of the chain rule from classical calculus, and is used to compute the derivative of a function of a stochastic process with respect to time.

Question 23

What is the diffusion process?

Answer 23

The diffusion process is a type of stochastic process that describes the continuous and random movement of financial assets, and is commonly modeled using the Wiener process.

Question 24

What is the Brownian motion?

Answer 24

The Brownian motion, named after Robert Brown, is a type of Wiener process that models the random and continuous movement of financial assets

Question 25

What is the geometric Brownian motion?

Answer 25

The geometric Brownian motion is a type of Wiener process that is commonly used to model the random and continuous movement of financial assets, such as stock prices, and takes into account the effects of drift and volatility.

Question 26

What is the martingale property?

Answer 26

The martingale property is a key concept in stochastic calculus, and refers to a stochastic process for which the expected future value, given the current information, is equal to the current value. This property is important in options pricing, as it allows for the creation of a risk-neutral measure for pricing options.