I have these two problems on hand and I don't think I have the basic idea
Consider the generalized Wiener Process
dx=0.1dt+0.3dz
suppose the initial value f x is x0=1
a)calculate the expected value of x after 2 years
b)calculate the standard deviation of x in 2 years
c)determine the 95%confidence interval for x after 2 months.
d. consider a European call option on this asset with an exercise price of $2 and expiry date in 3 years. What is the probability that this option will be exercised at expiry? show your work.
e) what is the probability that a European put option put option on this asset with the same maturity and exercise price will be exercised?
My work
E(dx)=0.1*2=0.2 Ex=1+0.2=1.2
variance E(dx^2)=0.3^2*2=0.18
SD=sqrt0.18
and I am not sure about the rest
second question
consider the GBM process:
dS1 = 0.1S1dt+0.3S1dz
suppose the initial value of S1 after 2 years.
b.Calculate the stadard deviation of S1 after 2 years
c. what can you say about the distribution of X1=log(S1) after 2 years?
what are its expected value and standard deviation?
d. Find a 95% confidence interval fr X1 after 2 years. Use your result to calculate a 95% confidence interval for S1
I don't get it, what is the difference btw wiener and GBM?I cannot find any source that tell me the exact difference
I don't know how to solve these questions, and I am at my wits end.
Help!