Expected value of the stock price BS model

[kobebry]

I understand where this BS equation comes from:
dSt = µSt dt + σSt dW
This reflects the change of the stock price. I want to find the expected value of the stock price and variance of the stock price when t=1/4. However, I am not sure how to do it. Should I just take the integral of the both sides and take the expected values? I saw some replicating methods but I did not understand anything.

 

[IntoDarkness]

pretty much but you need to solve the sde first before integration. Also does the question ask for expected value under p measure or pricing measure? If it’s later, you need girsanov

 

[kobebry]

pretty much but you need to solve the sde first before integration. Also does the question ask for expected value under p measure or pricing measure? If it’s later, you need girsanov

It asks under p measure. dSt = 0.06*Stdt + 0.4St*dWt This is my equation. S0=100
and the interest rate of continuous compounding is r = 0.05. It asks me to find the expected value and the variance of the stock price after 3 months from t=0 so basically at t=1/4.
Is solution of the SDE equals to s0*e^(drift*t)+standard deviation*Wt ?
What should I do afterwards? Or could you solve for me because I have really problems.

 

[Daniel Duffy]

Is this homework? Do you read books?

 

[kobebry]

Is this homework? Do you read books?

I am studying for my final exam. I am studying from the exercises of the book but I could not even find the solution for this simple question.

 

[Daniel Duffy]

Which book?

Have you tried Wikipedia?

 

[kobebry]

Which book?

Have you tried Wikipedia?

Introduction to Stochastic Calculus Applied to Finance bok. I tried I explored all the websites to find a similar question but I only found problems regarding how to price call option using black scholes. However, my question is about expected value of a stock not about the price of the call option.

 

[Daniel Duffy]

Then you are not looking hard enough.

 

[Qui-Gon]

The moment generating function of a normal random variable is what? This is all you need to know once you have the closed form solution for S_t. Finding this closed form is a standard application of Itô’s Lemma.

 

[PepeQuant]

[math]dlogS=\frac{1}{S}dS-\frac{1}{2}\frac{1}{S^2}(dS)^2=(\mu-\frac{1}{2}\sigma^2)dt+\sigma dW[/math]
[math]E=E[e^{logS}]=E[e^X]=e^{(\mu-\frac{1}{2}\sigma^2)t+\frac{1}{2}\sigma^2 t} = e^{\mu t}[/math]

 

[PepeQuant]

Introduction to Stochastic Calculus Applied to Finance bok. I tried I explored all the websites to find a similar question but I only found problems regarding how to price call option using black scholes. However, my question is about expected value of a stock not about the price of the call option.

well. if you know how to price call under black sholes, you know the forward price is Se^rT. that is due to the change of meausre with the bank numeraire where drift is r instead of \mu.

by simply using the same analogy, then u should know under physical measure it is Se^\mu T.

probably study these whole thing again. if u apply for a pricing quant job you would definitely fail the interview

 

[Qui-Gon]

[math]dlogS=\frac{1}{S}dS-\frac{1}{2}\frac{1}{S^2}(dS)^2=(\mu-\frac{1}{2}\sigma^2)dt+\sigma dW[/math]
[math]E=E[e^{logS}]=E[e^X]=e^{(\mu-\frac{1}{2}\sigma^2)t+\frac{1}{2}\sigma^2 t} = e^{\mu t}[/math]

Simply giving him/her the answer is not useful...