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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Integral of Brownian motion w.r.t. time
Let
$$X_t = \int_0^t W_s \,\mathrm d s$$
where $W_s$ is our usual Brownian motion. My questions are the following:
Expectation?
Variance?
Is it a martingale?
Is it an Ito process or a Riemann ...
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Find a formula for the price of a derivative paying $\max(S_T(S_T-K),0)$
Develop a formula for the price of a derivative paying
$$\max(S_T(S_T-K))$$
in the Black Scholes model.
Apparently the trick to this question is to compute the expectation under the stock measure. So,...
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How to simulate stock prices with a Geometric Brownian Motion?
I want to simulate stock price paths with different stochastic processes. I started with the famous geometric brownian motion. I simulated the values with the following formula:
$$R_i=\frac{S_{i+1}-...
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Two correlated brownian motions
Is it true (see here, footnote 2, p.22 / p.14, without proof) that we can obtain two discretized brownian motions $W_t^1, W_t^2$ with correlation $\rho$ by doing
$$d W_t^1 \sim \mathcal N(0,\sqrt{dt}...
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Variance of time integral of squared Brownian motion
I want to calculate the variance of
$$I = \int_0^t W_s^2 ds$$
I was thinking I could define the function $f(t,W_t) = tW_t^2$ and then apply Ito's lemma so I get
$$f(t,W_t)-f(0,0) = \int_0^t \frac{\...
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Simulation of GBM
I have a question regarding the simulation of a GBM. I have found similar questions here but nothing which takes reference to my specific problem:
Given a GBM of the form
$dS(t) = \mu S(t) dt + \...
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Shortcomings of generalized Brownian motion for asset price modelling
I'm simply interested on hearing some views on which shortcomings arise by using the (multidimensional) SDE $$dS(t)=S(t)\alpha(t,S(t))dt+S(t)\sigma(t,S(t))dW(t)$$
as a model for asset prices.
I know ...
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answer
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Intuition behind prices modeled by Geometric Brownian Motion
Suppose that we model a price $P_t$ to evolve per
$$\frac{dP_t}{P_t}=\mu dt+\sigma dW_t$$
for $\mu\in\mathbb{R}$ and $\sigma>0$. The unique strong solution to this diffusion is
$$P_t=P_0e^{(\mu-\...
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GBM - How to make make annualized dividend reflected in one quarter
I want to simulate the price path of a stock for one quarter using geometric Brownian motion. The stock has a continuous dividend yield of 5% based on the annual dividend yield. However, historically ...
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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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Derivation of Ito's Lemma
My question is rather intuitive than formal and circles around the derivation of Ito's Lemma. I have seen in a variety of textbooks that by applying Ito's Lemma, one can derive the exact solution of a ...
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Expectation of maximum draw down in the Brownian motion case
Let
$$
X_t = \mu t + \sigma B_t
$$
be a linear Brownian motion with drift.
Let
$$
S_t = \max(X_u, u \le t)
$$
denote the process of the running max, then the draw down is given by
$$
DD_t = S_t - ...
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Can I always use quadratic variation to calculate variance?
Suppose we have a Brownian Motion $BM(\mu,\sigma)$ defined as
$X_t=X_0 + \mu ds + \sigma dW_t$
The quadratic variation of $X_t$ can be calculated as
$dX_t dX_t = \sigma^2 dW_tdW_t = \sigma^2 dt$
...
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How are Brownian Bridges used in derivatives pricing in practice?
A similar question has already been asked in the past, unfortunately the 2nd question of the OP was never really addressed.
Most references found on internet on Brownian Bridge and Monte-Carlo ...
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Brownian motion - first passage time
Can anyone point me to the expression for the first passage time for a geometric Brownian motion process X(t) as a function of the starting point, threshold, drift and diffusion parameters.
I am ...
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Ito Integral of functions of Brownian motion
How does one show that:
$$ \mathbb{E}\left[ \int f(W_s)dWs \right] = 0 $$
For all $f()$ that are powers of $W(s)$?? I assume that one would have to go via the definition of Ito integral and express ...
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FX Rate dynamics
Let's suppose USD/EUR price in USD follows a GBM with
$$ dS_t = rS_tdt + \sigma S_tdW_t $$
What process does EUR/USD follow in EUR?
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Geometric Brownian motion - Volatility Interpretation (in the drift term)
A Geometric Brownian motion satisfying the SDE $dS_t = rS_t dt+\sigma S_t dW_t$ has the analytic solution
$$S_t = S_0\exp\left\{\left(r-\frac{\sigma^2}{2}\right)t\right\}\exp\{\sigma W_t\}$$
Recently ...
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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 ...
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Consensus on Cauchy distribution for stock prices
What is the general consensus for using a Cauchy distribution to model stock prices? I can't find much after researching online and wonder if it has been tried and discarded.
My motivation is to find ...
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How to simulate correlated Geometric brownian motion for n assets?
So I'm trying to simulate currency movements for several currencies with a given correlation matrix. I have the initial price, drift and volatility for each of the separate currencies, and I want to ...
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Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative
The problem:
Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \in ...
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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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What is a Brownian motion "under the risk-neutral measure"?
I understand that the risk-neutral measure associated with the money-market Numeraire is one under which the discounted price (acc. to the risk-free rate) of any asset is a martingale.
Brownian motion ...
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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:
http://www.goddardconsulting.ca/matlab-...
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Correlation coeffitiont between two stochastic processes
I want to find correlation coeffitiont between $W_t$ and $\int_{0}^{t}W_s ds$.
I think that these are uncorrelated. But Why?
So thanks
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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}$$
where g is the geometric average, $\mu$ the arithmetic average and $ ...
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Variance of a time integral with respect to a Brownian Motion function
Let process
$$I_t = \int_0^t f(s) W_s \,\mathrm d s $$
where $W_s$ is standard Brownian motion. My question are the following:
We know that $\mathbb{E} (I_{t})=0$ for all $t$ and $f$ a integrable ...
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Why do we usually use normal distribution and not Laplace distribution to generate stochastic process?
When working with a stochastic process based on brownian motion, the increments have normal (gaussian) distribution.
However, it seems that a Laplace distribution, with density:
$$f(t) = \frac{\...
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Distribution of time integral of Brownian motion squared (where the Brownian motion occurs in square root time)?
Let $I_t = \int_0^t W_{\sqrt{u}}^2du$. What is the distribution of $I$?
If I recall correctly, if the Brownian motion were instead $W_u$, then it would be $I_t \sim N\left(\frac{t^2}{2},\frac{t^4}{3}\...
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Geometric brownian motion vs. Ornstein Uhlenbeck
I'm looking at the SDE of Geometric brownian motion(*):
$$d X(t) = \sigma X(t) d B(t) + \mu X(t) d t$$
(with analytic solution $X(t) = X(0) e^{(\mu - \sigma^2 / 2) t + \sigma B(t)}$)
and the SDE of ...
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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 $W_t$,
$$...
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Geometric Brownian Motion: percentage returns vs log-returns
In classical calculus, we know that the limit of percentage return (ie $dS/S$) equals that of the log return (ie. $dln(S)$ ).
With uncertainty, we rely on Ito Lemma to draw a relationship between the ...
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Random Walk with normal increments and n time periods why is the increment $\sqrt{(t/n)}$?
Question is basically in the title. I have found several sources stating that $R_i = \sqrt{\frac{t}{n}}$, but I couldn't find the intuition behind taking the square root. And it seems to be crucial ...
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Expected payoff at future time
Let $a$, $b$, $c$, and $e$ be constants, $W_1$ and $W_2$ be Brownian motions with correlation $\rho$, and $f(t)$ and $g(t)$ be deterministic functions of time. Let $X$ satisfy $$d(X(t))=(aX(t)+ef(t)g(...
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Limits of integration when applying stochastic Fubini theorem to Brownian motion
I'm looking at the solution below from Quantuple, it's a nice, succinct solution but I'm confused about how the limits of the integrals in the second line come from. Could someone please elaborate on ...
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Correlation between stock prices given correlation between returns
assume I have two stocks with known volatilities and a known correlation coefficient of returns - does anyone know how to determine the correlation between the prices and NOT THE RETURNS
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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(\...
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Comparison of Brownian Motion Expected Drawdown and simulated results
Can anyone tell me whether results as predicted by Brownian Motion for a given mean and std, match what you get by measuring actual drawdown from simulated results over a number of iterations?
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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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Variance of Brownian Motion
Can someone point me into the right direction to calculate this one:
$E(B^4_t)=3t^2$
I had tried using the following property with no luck:
$E(B^4_t)=E(B^2_tB^2_t)=E(\int B^2 dt )E(\int B^2 dt )=[E(\...
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probability question about brownian motion
Assume $W_{t}$ is a standard Brownian Motion, calculate the the probability that $W_{t}*W_{2t}$ is negative, i.e., $P(W_{t}*W_{2t}<0)$. I find it tricky to calculate the probability.Thank you.
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Show that $E[B_t|\mathscr{F}_s] = B_s$ for $B_t = W_t^3 - 3 t W_t$
Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$
Let $(B_t)_{t \geq 0}$ where $B_t = W_t^3 - 3tW_t$. ...
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Geometric Brownian Motion - increasing simulations or smaller step size
I am running Monte Carlo simulations to estimate future share prices of some stocks.
For stock A, I need 1 share price exactly one year from now.
For stock B, I need daily prices for each trading ...
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Asymptotic behavior property of geometric Brownian Motion proof
Online I found the asymptotic behavior property of geometric Brownian Motion $X_t$as:
If $\mu$ (drift parameter) is $\ge$ $\sigma^2/2$ where $\sigma$ is the volatility parameter, then $X_t \rightarrow ...
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How to express the volatility of two correlated Ito processes $Wt_1, Wt_2$ expressed in terms of $W_t$?
Having two correlated Ito processes
($W_t^1$ and $W_t^2$ are correlated Brownian motions with correlation $\rho$)
$dX_{t} =\mu_{1} dt + \sigma_1 dWt_1 $
$dY_{t} = \mu_{2} dt + \sigma_2 dWt_2 $
...
3
votes
2
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How to compute the conditional expected value of a geometric brownian motion?
I'm working on a project, and I have to use the cumulative and conditional expected value of the variations of a stock following a Geometric Brownian Motion.
I know that the cumulative is as follows :...
3
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3
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Unique risk neutral measure for Brownian Motion
For a standard geometric Brownian motion model of stock prices:
$$ dS = a S dt + \sigma S dZ$$
we can transform the process to be under risk neutral measure:
$$ dS = r S dt + \sigma S d \tilde{Z}$$
...
3
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Covariance matrix and Cholesky decomposition
I am simulating a spread option with stochastic volatility using Monte Carlo simulation. I have the positive-definite covariance matrix
$$
\rho = \left( \begin{array}{cccc}
1 & \rho_{1,2} & \...
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votes
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How to find the transition distribution functions of these two processes?
What are the transition distribution (or density) functions of processes defined by
$dX_t=\mu dt +\sigma dW_t$
and
$dX_t= \theta(\mu-X_t) dt +\sigma dW_t,$
where $\theta>0$, $\mu$ is a real ...