Answered step by step
Verified Expert Solution
Link Copied!

Question

1 Approved Answer

We consider the Geometric Brownian Motion model for a stock price: dlogS(t)=(212)dt+dW(t). We then define the log return over the interval [t,t+] r(t,)=logS(t+)logS(t). Integrating the

image text in transcribedimage text in transcribed

We consider the Geometric Brownian Motion model for a stock price: dlogS(t)=(212)dt+dW(t). We then define the log return over the interval [t,t+] r(t,)=logS(t+)logS(t). Integrating the first equation over [t,t+] yields logS(t+)logS(t)=(212)+(W(t+)W(t)). In other words, the log return r can be written as r(t,)=(212)+(W(t+)W(t)). 1. (5 points) What is the distribution of r(t,) ? In particular, give its mean and variance. 2. (65 points) Suppose that we are given a set of daily data for which the above model is a good fit with =0.1 per year and =0.2 per year. Note that =1 day =1/252 years. We wish to estimate . Since the random walk model is stationary, ergodic and has a finite variance, which allows us to apply the Central Limit Theorem, we can safely estimate by computing a time-average. This estimator is also the same as the Maximum Likelihood estimator for this simple model. The convergence rate is /N where N is the number of samples. Unfortunately, obtaining an accurate value for requires very long time Series that are never available in practice. We denote by ^ an estimate of . If one wants to determine a 95% confidence interval of the form [^0.01,^+0.01], how many years of data do you need? Hint: this is a very simple computation based on the rate of convergence given by the Central Limit Theorem. Note that you need to have a consistent time unit throughout the calculation in order to obtain the correct result. We consider the Geometric Brownian Motion model for a stock price: dlogS(t)=(212)dt+dW(t). We then define the log return over the interval [t,t+] r(t,)=logS(t+)logS(t). Integrating the first equation over [t,t+] yields logS(t+)logS(t)=(212)+(W(t+)W(t)). In other words, the log return r can be written as r(t,)=(212)+(W(t+)W(t)). 1. (5 points) What is the distribution of r(t,) ? In particular, give its mean and variance. 2. (65 points) Suppose that we are given a set of daily data for which the above model is a good fit with =0.1 per year and =0.2 per year. Note that =1 day =1/252 years. We wish to estimate . Since the random walk model is stationary, ergodic and has a finite variance, which allows us to apply the Central Limit Theorem, we can safely estimate by computing a time-average. This estimator is also the same as the Maximum Likelihood estimator for this simple model. The convergence rate is /N where N is the number of samples. Unfortunately, obtaining an accurate value for requires very long time Series that are never available in practice. We denote by ^ an estimate of . If one wants to determine a 95% confidence interval of the form [^0.01,^+0.01], how many years of data do you need? Hint: this is a very simple computation based on the rate of convergence given by the Central Limit Theorem. Note that you need to have a consistent time unit throughout the calculation in order to obtain the correct result

Step by Step Solution

There are 3 Steps involved in it

Step: 1

blur-text-image

Get Instant Access to Expert-Tailored Solutions

See step-by-step solutions with expert insights and AI powered tools for academic success

Step: 2

blur-text-image_2

Step: 3

blur-text-image_3

Ace Your Homework with AI

Get the answers you need in no time with our AI-driven, step-by-step assistance

Get Started

Recommended Textbook for

International Finance Theory And Policy

Authors: Paul R. Krugman, Maurice Obstfeld, Marc J Melitz,

11th Edition

013451954X, 9780134519548

More Books

Students also viewed these Finance questions

Question

=+How might you approach the topic differently?

Answered: 1 week ago