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• 7,458
Accepted

What is the difference between standard deviation, volatility and quadratic variation?

Using only words and no equations: Knowing the Variance (or standard deviation) of a Brownian Motion we can calculate the uncertainty in the future position of a particle. Knowing $\sigma^2$ and ...
• 9,935

Are two stochastic processes independent if the Wiener processes inside are uncorrelated

 My "answer" below is not a really an answer for I have completely misinterpreted your original question. I thought you asked about the covariance of 2 processes over a given time horizon (i.e. ...
• 14.3k
Accepted

What are the units of the variables appearing in a standard stochastic differential equation for a Wiener process?

$\sigma S$ is in units of dollars per square root of a unit of time. $\sigma$ is usually quoted as an annual or daily percentage. $dX ^2$ is in units of time, as $E[(dX)^2] = dt$. Here ...
• 1,335

Bounded Stochastic discrete process

I believe that the process you postulate has a Beta conditional distribution. If my memory serves me well, I have encountered it in the book by Liptser and Shiryayev "Statistics of Random Processes" ...
• 4,247

• 5,503
1 vote
Accepted

Regression of stochastic integral on Wiener process

By definition, $${\mathbb Cov}(M_t,W_T) = {\mathbb E}[M_t W_T] - {\mathbb E}[M_t] {\mathbb E}[W_T] = {\mathbb E}[M_t W_T]$$ since ${\mathbb E}[M_t] = {\mathbb E}[W_T] = 0$. We now consider the ...
1 vote

Differentiability of solutions of a stochastic differential equation

Rather non rigorously, $\frac{W(t)}{t} \sim N(0,\frac{1}{t})$ if $t \to 0$ , we can see the variance goes to infinity. Hence Ito process is not differentiable.
• 351
1 vote

How to Evaluate Expected Value powered 4 of a Wiener Process?

You state $X(t_j) - X(t_{j-1}) \backsim \mathcal{N}(0, \frac{t}{n})$. Thus: $$X(t_j) - X(t_{j-1}) = \sqrt{\frac{t}{n}} Z ,$$ where $Z \backsim \mathcal{N}(0, 1)$. Note that:...
1 vote

Can anyone explain to how Hull get from stock returns to continuously compounded stock returns?

The key is on the left hand side. Recall that the differential of log of x is: $d \ln x =\frac{1}{x}dx$ So you get: $\ln x_t-\ln x_0=at$ Which you will need to exponentiate to get rid of the log: ...
1 vote
Accepted

Interpretation of IV and its use in stock movement prediction

The Bachelier model which assumes that price follows a normal distribution is a correct approximation for the Black-Scholes one for short times t. When time is short it's fine to ignore drift because ...
• 2,137

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