# Tagged Questions

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### Why should we expect geometric Brownian motion to model asset prices?

Disclaimer: I am a complete ignoramus about finance, so this may be an inappropriate forum for me to ask a question in. I am a mathematician who knows nothing about finance. I heard from a popular ...
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### Meaning of w in SDE

I'm missing meaning of $w$ in typical SDE like $dX_t(w) = f_t(X_t(w)) + \sigma(X_t(w))dW_t$, in context of $w \in F_{xxx}$. Does it mean that both $w$ is one of events that could happen before ...
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$S_t$ is the random variable representing the risky asset price at time $t$. M_t is the riskless asset. They are governed by the equations $\frac{dS_t}{dt}=\mu dt + \sigma dZ_t$ and $dM_t = rM_t ... 1answer 82 views ### Differenced Brownian Motion covariance I am having some difficult showing what the following equals, where$x$and$y$,$x>y$, distinct times:$\mathbb{E}[\Delta W_x \Delta W_y]$where each$\Delta W_t = W_t - W_{t-1}$. I have ... 3answers 237 views ### Geometric Brownian motion - Volatility Interpretation (in the drift term) A Geometric Brownian motion satisfying the SDE$dS_t = rS_t dt+\sigma S_t dW_t$has the analytic solution $$S_t = S_0\exp\left\{\left(r-\frac{\sigma^2}{2}\right)t\right\}\exp\{\sigma W_t\}$$ Recently ... 0answers 41 views ### Index Price Simulation Volatility Bands I am building a simple stochastic model for learning purposes in excel. I took daily data for the SPY since 1/1/1993. I computed the daily log returns and found that the SPY has had an average daily ... 2answers 342 views ### How to compute the Radon-Nikodym derivative? Suppose$B(t)$is a standard Brownian motion, and$B_{1}(t)$is given by$dB_{1}(t)=\mu dt+dB(t)$. Suppose$P$is the Wiener measure induced by$B(t)$on the$C[0,\infty)$, and$P_{1}$is the Law ... 2answers 425 views ### Brownian motion - first passage time Can anyone point me to the expression for the first passage time for a geometric Brownian motion process X(t) as a function of the starting point, threshold, drift and diffusion parameters. I am ... 2answers 660 views ### Simulation of GBM I have a question regarding the simulation of a GBM. I have found similar questions here but nothing which takes reference to my specific problem: Given a GBM of the form$dS(t) = \mu S(t) dt + ...
Let's say stock price follows following process: $$dS(t) = \sigma dW(t)$$ where $W(t)$ is Standard Brownian motion. The initial level for the stock is $S(0)$. Define the average of stock price ...