Questions tagged [brownian-motion]

In mathematics, Brownian motion is described by the Wiener process; a continuous-time stochastic process named in honor of Norbert Wiener.

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8
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3answers
4k views

Usage of Brownian Bridge?

I was recommended to read something about Brownian Bridge. Could someone familiar with BB give some recommendation? It was mentioned that BB benefits in 2 places BB could reduce the simulation paths,...
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1answer
132 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 ...
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2answers
381 views

Simple question about expected value of brownian motion

I would appreciate some help with the math in this paper : High Frequency Trading in a Limit Order Book Specifically, I would like to understand how the authors calculated the expected value of price ...
4
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1answer
265 views

Linear-Boundary Crossing Problem for Brownian Motion

This is a question I came across while reading: $W = (W_t)_{t\geq{0}}$ is a standard BM. Let $\mu\in \mathbb{R}$, and let $\tau_{a}^{\mu}$ = $\inf(t>0;W_t = a + \mu{t})$ be the first passage time ...
4
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2answers
887 views

Shortcomings of generalized Brownian motion for asset price modelling

I'm simply interested on hearing some views on which shortcomings arise by using the (multidimensional) SDE $$dS(t)=S(t)\alpha(t,S(t))dt+S(t)\sigma(t,S(t))dW(t)$$ as a model for asset prices. I know ...
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1answer
1k views

Are my estimates of parameters of geometric brownian motion correct?

I wrote a simulation of a geometric Brownian motion which works like this: ${ t }_{ i }-{ t }_{ i-1 } \sim Exp(\lambda )$ ${ Z }_{ i }\sim N(0,1)$ ${ Y }_{ i }\sim { e }^{ \sigma \sqrt { { t }_{ i }-{...
3
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2answers
142 views

What is the difference between these two equations for GBMs?

The two equations commonly found online for GBM are: $\begin{matrix} S_{ t }=S_{ 0 }\exp\left( \left( \mu -\frac { \sigma ^{ 2 } }{ 2 } \right) t+\sigma W_{ t } \right) \\ S_{ t }=S_{ 0 }\exp\left(\...
1
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1answer
168 views

Scaling Intervals in Diffusion Process

I know this is a very elementary question but... when modeling asset prices through a stochastic process as in $$dS_t=S_t μ dt+S_t σdW_t,$$ where the following is a wiener process $$dW_t=σN(0,1)dt^{...
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1answer
525 views

Basics about the scaling property of volatility

It is a usual practice to calculate realized volatility $\sigma$ using the square root of the usual variance estimator $\hat{{\sigma}²}$. This is done using the stock log returns (practitioners ...
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2answers
2k 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 ...
5
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1answer
423 views

Covariance of brownian motion and its time average

It's a question pertaining to the correlation of a log asset process (following BM) and its time average, to put it into form, if $$X(t)=\mu t+\sigma W(t)$$ then $$ \bar{X}(t):=\frac{1}{t}\int_0^tX(...
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2answers
2k views

What is the average stock price under the Bachelier model?

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 $Z(t)...
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2answers
175 views

Statistics of difference between two GBMs

if I have two asset prices modeled separately as geometric brownian motions. How do i go about calculating the expected statistics of their difference? Like given the sigmas and mus of both processes, ...
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2answers
589 views

Correlation decay in lognormal distribution

I noticed that if you use two correlated geometric brownian motions, the correlation structure decays in time pretty fast even for really high correlation values. I think that is not replicating ...
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0answers
123 views

Estimating two normal random numbers with one equation

Subtitle: Estimating the correlation of the shocks driving two commodities in two multi-factor models I am fitting two 2-factor models to electricity and gas futures, respectively. In order to ...
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2answers
588 views

GBM 3d plot with R

I want to plot the density of the GBM in a 3d plot. So I have on one axis the stock price, on the other the time and on the z axis the density. At the end I want to produce this graph. The formula I ...
7
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3answers
2k views

What are the limitations of brownian motion in finance? [duplicate]

What are the limitations of brownian motion in its applications to finance?
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4answers
879 views

Expected Growth

The model assumption of the Black-Scholes formula has two parameters for the geometric Brownian motion, the volatility $\sigma$ and the expected growth $\mu$ (which disappears in the option formulae). ...

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