Quantitative Finance Stack Exchange
Quantitative Finance Stack Exchange

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.

The standard Wiener process $W_t$ is characterized by four facts:

  1. $W_0$ = 0
  2. $W_t$ is almost surely continuous.
  3. $W_t$ has independent increments.The condition that it has independent increments means that if $0\leq s_1\leq t_1\leq s_2\leq t_2$ then $W_{t_1}-W_{s_1}$ and $W_{t_2}-W_{s_2}$ are independent random variables.
  4. $W_{t}-W_{s}\tilde{} N(0,t-s)$
478 questions
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Estimation of Geometric Brownian Motion drift

One can find many papers about estimators of the historical volatility of a geometric Brownian motion (GBM). I'm interested in the estimation of the drift of such a process. Any link on this topic would be very helpful.
Juergen
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What are the limitations of brownian motion in finance?

What are the limitations of brownian motion in its applications to finance?
user693
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Mathematical proof of $g = \mu - \frac{\sigma^2}{2}$ relationship between CAGR and average returns

I found in a paper the relation between the CAGR and the arithmetic average of returns to be $$g \sim \mu - \frac{\sigma^2}{2}$$ where g is the geometric average, $\mu$ the arithmetic average and $ \sigma^2$ the variance of the returns. I cannot…
emanuele
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Math background required to understand geometric brownian motion

What mathematical concepts are required before I can understand what exactly is a Geometric Brownian motion as applicable to stock prices? I mean which branches of probability, calculus, statistics etc. are needed to understand GBM? By 'understand',…
Victor123
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Questions about exponential Brownian motion

Let $(\Omega,\mathcal{F},P)$ be a probability space, equipped with a filtration $(\mathcal{F})_{0 \leq t \leq T}$ that is the natural filtration of a standard Brownian motion $(W_{t})_{0 \leq t \leq T}$. Let $X=\exp(W_{T/2}+W_{T})$. Find the…
oamc
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Covariance of (fractional) Brownian motions with different Hurst parameters

I'd like to calculate the covariance function for fractional Brownian motions $$ E_t \left[ dW^H(t) dW^{H'}(t) \right] $$ but where the Hurst parameters are not equal: $H \neq H'$. My first idea would be to look at the Riemann-Liouville…
user34971
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Distribution of portfolio values with constant spending rate

If your portfolio is invested in an asset that follows a geometric Brownian motion, and you withdraw a constant dollar amount at the beginning of each year, is there an approximate analytical distribution for the portfolio value after N years and…
Fortranner
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GBM probability of hitting barrier

I tried using the brownian bridge approach to determine the probability $$P(S_t<\beta,t\in [0,T]|S_0,S_T)$$ where $S_t$ is a GBM in the usual Black Scholes setup. We know that for a BM $W_t$, $$P(W_t<\beta,t\in…
user128836
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The conditional mean of a product of standard Brownian motions

Suppose $\{W_t, t>=0\}$ is a Standard Brownian Motion. How to compute $ \mathbb{E} \left[ W_2 W_3 \vert W_1 =0 \right]$? We know $ W_2 \vert W_1 = 0 \sim N(0,1)$ and $ W_3 \vert W_1 = 0 \sim N(0,2)$. Thank you so much.
MathMan12
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Brownian bridge density and risk neutral density for derivative pricing

The book The Volatility Surface by Gatheral (2006) introduces the Brownian bridge like density $q(x_t,t;x_T,T)$ of $x_t$ conditional on $x_T = log(K)$. Can we use $q(x_t,t;x_T,T)$ as the risk neutral measure? Why $q(x_t,t;x_T,T)$ can easily exceed…
Smirk
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Drift irrelevance on high frequency data

Let's assume that price of a certain asset follows Brownian Semimartingale process with a drift term and a Brownian-driven continuous part (no jumps for simplicity). In literature it is often stated that on high frequency data (i.e. order of seconds…
Always Right Never Left
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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 and found the data series as non-stationary which…
Umesh Chaudhari
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Fractional Brownian motion references

Does anyone know any good references to understand the fractional Brownian motion and its numerical simulation, preferably applied to finance.Thank you.
Mehdi
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Expectation of an Integral of a function of a Brownian Motion

I would really appreciate some guidance on how to calculate the expectation of an integral of a function of a Brownian Motion. Let $B(t)$ be a Brownian motion with drift $\mu$ and standard deviation $\sigma$. At time $t$, $e^{-kt}$ represents time…
Effie Stavrulaki
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Browian motion: $P(B_1<4 | B_2 =1)$

I want to calculate $P(B_1<4 | B_2 =1)$ for the B.M. What I tried: $P(B_1<4 | B_2 =1)=P(B_1 - B_2 < 3- B_2 | B_2 =1)$ but I cant use any independence to calculate further.
Focus
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