# Tag Info

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 ...

### Integrated Brownian motion

Your derivation is correct. Even if we fix your obvious typo the formula $$\textstyle\int_t^TW_s\,ds=\int_t^T(T-s)\,dW_s$$ is wrong. There is no doubt that \begin{align} TW_T&=\textstyle\int_0^...

Accepted

### Expectation on a function of Wiener Process

The proof uses the martingale property of the Ito integral. For an adapted stochastic process $X_t$ such that $$\mathbb{E}\int_0^{t}|X_s|^2ds <\infty$$ we have $$\mathbb{E}\int_0^{t}X_sdW_s =0$$ ...

### Arbitrage portfolio example

You can choose an arbitrage for the classical 1-dimensional Black-Scholes model, and not use $S_2$ at all. Such an arbitrage is e.g. presented in Example 3.5 in "Fractional Processes As Models In ...
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.
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: \begin{equation} X(t_j) - X(t_{j-1}) = \sqrt{\frac{t}{n}} Z , \end{equation} 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 ...
1 vote
Accepted

### Solving a backwards heat equation using stochastic calculus

Based on the form of your equation, we can consider the SDE \begin{align*} dX_t = \sigma dW_t, \end{align*} where $W$ is a standard Brownian motion. Since, for $0 \leq t \leq T$, \begin{align*} X_T =...

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