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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Correlation of Asynchronous Brownian Motion

I am trying to use the closing prices of the S&P 500 and the Nikkei Index to see how they are correlated (assuming they are exactly 12 hours apart). In order to test my method, I have generated ...
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390 views

Distribution of Geometric Brownian Motion

Please let me know where I have been mistaken! Let the SDE satisfied by the GBM $S(t)$ be $$ \frac{dS(t)}{S(t)} = \mu dt + \sigma dW(t). $$ Then, the underlying BM $X(t)$ will satisfy $$ dX(t) = ...
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203 views

Distribution of Brownian Bridge

I know from Karatzas & Shreve (1991) that a Brownian Bridge $B(t)$ from $a$ to $b$ on time interval $[0,T]$ satisfies: $B(t)=a(1-t/T) + b*t/T + [W(t) - W(T)*t/T]$, where $W(t)$ is a standard ...
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How do I artificially generate intraday ticks data from a given input (Open,High,Low,Close,Volume) using Brownian Bridge method?

How do I artificially generate intraday ticks data from a given input (Open,High,Low,Close,Volume) using Brownian Bridge method? https://en.wikipedia.org/wiki/Brownian_bridge P.S: Brownian Bridge ...
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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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49 views

On the reflection of a stochastic integral

Let ${(I_t)}_{t\geq 0}$ be a stochastic integral defined by $$ I_t=\int_{0}^{t}\theta_sdW_t, $$ where $W$ is a standard Brownian motion defined on $(\Omega,\mathcal{F},{(\mathcal{F}_t)}_{t\geq ...
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72 views

Simulate correlated Geometric Brownian Motion in the R programming language

In response to this question: How to simulate correlated Geometric brownian motion for n assets? One of the responses provides an implementation in MATLAB: ...
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128 views

Is Geometric Brownian Model suitable for long term price forecast?

I was thinking of using Geometric Brownian Motion to forecast future prices of timber (say one variable, the stumpage price of sawtimber). I tested the time series with Augmented Dickey-Fuller test ...
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7 views

Polynomial interpolation of corrected lognormal distribution

Can anyone provide a formula for a polynomial interpolation of the corrected lognormal distribution used to model returns traditionally resulting from the wrong Brownian motion generated model? ...
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67 views

SDE for a portfolio of two correlated assets $ Y_{t} = 2 S^{1}_{t} S^{2}_{t}$

I am analysing a problem where I have two correlated stocks described by Brownian motions $$ \frac{dS^{1}_{t}}{S^{1}_{t}}=\mu_{1} dt + \sigma_{1} dW^{1}_{t} \quad \quad (1)$$ $$ ...
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On the construction of a Brownian motion from a Gaussian process

Let $X$ a Gaussian process defined by $$ X_t=\int_{0}^{t}\left(\frac{1}{\sigma}\left(r_s-\frac{\sigma^2}{2}\right)-\rho\sigma_P(s,T)\right)\mathrm{d}s+\sqrt{1-\rho^2}Z_2(t)+\rho Z_1(t);\;\;t\in[0,T] ...
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What is the maximum of a brownian motion with drift over the interval [t_1,t_2]

I am having a problem deriving the equation: $$P(max_{(t_1 \leq t \leq t_2)} S(t) > B | S(t_1),S(t_2))= e^{-\frac{2}{T}ln\bigg{(}\frac{B}{S(t_1)}\bigg{)} ln\bigg{(}\frac{B}{S(t_2)}\bigg{)}}$$ ...
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25 views

Soft: Interpretation Fractional BM in finance

Suppose we are in the BS framework. If we replace the Brownian Motion with a more general fractional Brownian motion therein, how can it be interpreted? That is what is a financial interpretation of ...
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65 views

Monte Carlo simulation of Multifractional Brownian Motion in MATLAB

Code under is taken from http://en.literateprograms.org/Monte_Carlo_simulation_(Matlab) ...
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139 views

Monte Carlo, convexity and Risk-Neutral ZCB Pricing

I've built a simplistic Excel monte carlo model to price a zero-coupon bond, but it came up with a slightly unepxected result so I wanted to confirm whether my maths is just a little rusty or my model ...
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82 views

Convolution of inverse gaussian and power law distributions

I am trying to understand how the first passage time density of Brownian motion with drift is modified by the presence of waiting times that are distributed as a power law In other words, what is the ...
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36 views

SAS code for Brownian Motion

I want to simulate call options using monte carlo algorithm. I am a noob SAS user but i know that i need to: -simulate random stock prices with no dividend in respect to different ...
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48 views

Modelling commodity price uncertainty with brownian motion - time period impacts

background I have two separate models of a metals resources company. Each model produces a series of accounting and cashflows forecast for different assets, and consolidates these to a overall ...