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# 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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### 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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### 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 ...
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### CDF&density of stock price modeled by standard brownian motion

Assume that the price of the stock follows the model $S(t) = S(0) exp ( mt − ((σ^2)/2 ) t + σW(t) )$ , (1) where W(t) is a standard Brownian motion; σ > 0, S(0) > 0, m are some constants. Derive the ...
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### Expectation and variance of standard brownian motion

Assuming that the price of the stock follows the model $S(t) = S(0) exp ( mt − (σ^2/ 2) t + σW(t) ) ,$ where W(t) is a standard Brownian motion; σ > 0, S(0) > 0, m are some ...
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### Correlated stock prices and geometric Brownian motion

I have two uncorrelated stocks which follow geometric Brownian motion, as follows \begin{aligned} dS_a &= \mu_aS_adt + \sigma_aS_adW\\ dS_b &= \mu_bS_bdt + \sigma_bS_b dW \end{aligned} ...
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### Does GBM stock price model have E[S(t)] unaffected by volatility?

Many an author claims that, if you model stock prices through GBM, $E[S(t)]=e^{\mu t}$, and the expectation is thus not related to volatility. I keep running around in circles on this one. First ...
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### Ignore the difference between normal and log-normal distributions

I am trying to solve the following problem from a Quant exam (abridged): You have 1000 USD. You can only invest in two (independent) stocks, A and B, with the annualized expected returns and ...
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Let $W_t$ be a Brownian motion, so $W_t=z_t \sqrt{t}$ where $z_t \in N(0,1)$ and the pdf of $z$ is $f(z)=\frac{e^{-\frac{z^2}{2}}}{\sqrt{2\pi}}$. So $$E(W_t)=\int_{-\infty}^{\infty} W_t f(z) dz =\... 1answer 367 views ### Vector of differences of Brownian motion integrals is multivariate normal Given a 2-dimensional Wiener process (W_{1},W_{2}) with correlation \rho. Let \begin{equation*} X(t):= F(t) + \int_{0}^{t} f(s) dW_{1}(s) + \int_{0}^{t} g(s) dW_{2}(s)\end{equation*} for some ... 1answer 177 views ### Perform scipy Kolmogorov-Smirnov Test for lognormal distribution in GBM I am simulating asset prices for n days using GMB with Euler scheme, calculate returns and then perform Kolmogorov-Smirnov test on simulated returns. Code for simulating GBM : ... 1answer 303 views ### Brownian motion simulation - scaling issue I'm trying to simulate some BM for 500 observations. I got correlated increments as I needed and they are not exactly N(0,1), so I standardize them (x-mean(x))/sd(x). But then the resulting Brownian ... 1answer 196 views ### Probability that return exceeds a certain level before a certain time (Black-Scholes) I am self studying for an actuarial exam on financial economics. I encountered the following problem and solution. It seems to me that the author intended to mean what is the probability that the ... 1answer 278 views ### Geometric Brownian Motion: d(S) vs. d(ln(S)) I am quoting from "Tools for Computational Finance, 5th Edition" [Seydel]. I wonder whether the histogram of simulations of the first (yellow) SDE makes sense... especially given that Seydel (... 2answers 378 views ### European call down and out option (geometric Brownian motion, Monte Carlo, Euler) I need to estimate the expected value and the Greeks of an European call down and out option, assuming geometrical Brownian motion of the asset, with Monte Carlo simulation employing Euler ... 1answer 321 views ### What Exactly is Expected Return Consider the following plot, courtesy of this page: Regarding the y-axis, how does this "expected return" relate to the "instantaneous expected return" in a geometric Brownian motion (GBM)? E.g., ... 1answer 101 views ### Geometric Brownian Motion in a general interval [t_1,t_2] I know that the Geomtric Brownian Motion, with the expresion dX_t = v X_t dt + \sigma X_t dW_t has the next solution$$X_t = X_0 e^{\sigma W_t+ (v-\frac{\sigma ^2}{2})t}$$on the interval [0,t]. But,... 1answer 557 views ### What is the distribution assumption of the black scholes model As per wikipedia the Black Scholes assumption is: (... 1answer 148 views ### Bivariate Black-Sholes Model Let us propose bivariate Black-Sholes Model. Assume, we have an arbitrage-free complete market. r_{f} is risk-free rate. Under real-world measure P: dS_{1} (t)=S_{1} (t) [\mu_{1}dt+\sigma_{1}... 1answer 2k views ### Cholesky Decomposition on Correlation Matrix for Correlated Asset Paths I found a matlab example for modelling correlated asset paths: http://www.goddardconsulting.ca/matlab-monte-carlo-assetpaths-corr.html In this model the author uses the matlab code chol() in order to ... 0answers 74 views ### How to solve these SDE Problems Quuestion1. I make a solution r(t) used by Ito's lemma r(t)=e^{-a t}r(0)+\int _{0}^{t}e^{a (s-t)}\theta (s)ds+\sigma e^{-a t}\int _{0}^{t}e^{a u}\,dB^{1}(u) Is this right? and I try to make ... 0answers 45 views ### Brownian motion from price-series, what is the time step? If I assume a given empirical price-series is a brownian motion, I can estimate the drift and standard deviation as long as I know what the time step was when the process was 'generated'. But since ... 0answers 86 views ### The conditional expectation of a geometric brownian motion In this question it states that$$\mathbb{E}[e^{\sigma(W_t-W_s)}|\mathcal{F}_s] = \mathbb{E}[e^{\sigma(W_t-W_s)}],$$and I assume that 0 \leq s \leq t. The accepted answer states that this step is ... 0answers 64 views ### Brownian motion for modelling future asset values Assume that an asset price S is given by a Brownian motion. Argue from the definition why it is not possible to predict future values of the asset based on the past values of S. I am not sure ... 0answers 167 views ### Geometric Brownian Motion with Dividends I am working on a problem and had a quick question. I understand that for Geometric Brownian Motion we use the formula:$$X_{t_n} = X_{t_{n-1}} + \mu X_{t_{n-1}} \Delta t + \sigma X_{t_{n-1}} \...
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I ran through an equality in a paper I was reading but couldn't check if it is correct. Let $W^1_t$, $W^2_t$ and $W^3_t$ be three brownian motions, not necessarily independent, is it true that the ...
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### Model of asset substitution/risk shifting in continuous time

Consider a firm with cash flows $X_t$, which under a risk-neutral probability measure, follows a geometric brownian motion: $$dX_t = X_t[(r-\beta)dt + \sigma dZ_t]$$ where $r>0$ is the risk-free ...
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### Pricing defaultable asset with finite maturity

Assume a stochastic process $X_0 = 0$ and $X_t = \nu t + \sigma W_t$ where $W_t$ is standard Brownian motion and $\nu$ is a drift (can have $\nu \leq 0$ if necessary, but prefer it to be general), ...
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### Matlab implementation for modelling stock price process

I am trying to model the stock's price process. Let's assume volatility and risk-free rate is given. I've come up with the code below to try and model the price process with the geometrical Brownian ...
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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 ...
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### Smooth ornstein uhlenbeck process

I want to simulate paths for a commodity price. I use the historic data in the following way: $X_t$ is the price. $\ln\left(\frac{X_t}{X_{t-1}}\right)$ is the daily return. I calculate the slope of ...