We’re rewarding the question askers & reputations are being recalculated! Read more.

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.

Filter by
Sorted by
Tagged with
1
vote
1answer
642 views

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?
1
vote
2answers
373 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 ...
1
vote
1answer
166 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^{...
1
vote
1answer
484 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 ...
1
vote
1answer
43 views

In search of double barrier out option on a BM

We have a BM $X_t$ with $dX_t=\sigma dB_t$ ($X_0$ not necessarily zero!) under the risk neutral measure $\Bbb Q$. Given upper barrier $U$, lower barrier $L$, "strike" $K$ such that $L<X_0<U, L&...
1
vote
1answer
85 views

Proof standard Brownian Motion under change of measure

Let's split the usual time horizon $[0,T]$ like $0=T_{0}<T_{1}<\dots<T_{n}=T$ and consider the bond price $P(t,T_{i})$ for $i=1,...,n$. We assume $$\frac{dP(t,T_{i})}{P(t,_{i})}=r_{t}dt+\xi_{...
1
vote
1answer
37 views

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 ...
1
vote
1answer
65 views

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 ...
1
vote
1answer
161 views

Correlated stock prices and geometric Brownian motion

I have two uncorrelated stocks which follow geometric Brownian motion, as follows $$\begin{aligned} dS_a &= \mu_aS_adt + \sigma_aS_adW\\ dS_b &= \mu_bS_bdt + \sigma_bS_b dW \end{aligned}$$ ...
1
vote
1answer
113 views

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 ...
1
vote
1answer
137 views

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 ...
1
vote
1answer
324 views

Write expectation of brownian motion conditional on filtration as an integral?

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 =\...
1
vote
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 ...
1
vote
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 : ...
1
vote
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 ...
1
vote
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 ...
1
vote
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 (...
1
vote
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 ...
1
vote
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., ...
1
vote
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,...
1
vote
1answer
557 views

What is the distribution assumption of the black scholes model

As per wikipedia the Black Scholes assumption is: (...
1
vote
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}...
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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}} \...
1
vote
0answers
108 views

On quadratic covariation

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 ...
1
vote
0answers
46 views

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 ...
1
vote
0answers
95 views

Correlated GBM and OU processes

I want to model two different stochastic processes, such that: $X_t , V_t$ are correlated with coefficient $\rho$. Where: $\frac{dX_t}{X_t}=\mu_1dt+\sigma_1 dW_{1,t}$ and $dV_t=\theta(\mu_2-V_t)dt+\...
1
vote
0answers
93 views

Brownian bridge with time varying volatility

I have a question to ask about the Brownian bridge for a process with deterministic volatility varying over time. In other words, we have this dynamic: $dS_t = \sigma_{t} * dW_t$. We want to know the ...
1
vote
0answers
86 views

Definition request - Brownian Motion Characterised by Sharpe Ratio

What is the Stochastic Differential Equation for a "Brownian Motion Characterised by Sharpe Ratio"? I saw it in a paper ("Lessons from the Mortician: volatility modulation") and the authors do not ...
1
vote
0answers
61 views

Monte Carlo Pricer for Express Certificate delivers wrong price [Mathematica]

So I wanted to price the following Express Certificate with this specific payout structure: If S1 > S0 -> 105.25 , else -> If S2 > 0.95*S0 -> 110.5 , else -> If S3 > 0.9*S0 -> 115.75 , else -> If ...
1
vote
0answers
98 views

Geometric Brownian Motion: Drawdown as a function of time

Suppose I have a strategy (model it as the usual geometric Brownian motion with a drift). Question is, how does max drawdown grow as a function of duration?
1
vote
1answer
349 views

Stochastic integrals wrt to independent Wiener processes are uncorrelated, but potentially dependent?

In Proof of Proposition 1.2.20 in the following lectures notes http://math.uni-heidelberg.de/studinfo/reiss/sode-lecture.pdf I found following quote " stochastic integrals with respect to ...
1
vote
0answers
123 views

Testing whether a process is a Wiener process [closed]

Ideally I would like links to code implementations (eg. Matlab ) or book/paper references, but I would appreciate suggestions on various methods. Update: I was hoping to attract people who test the ...
1
vote
0answers
60 views

Solving for roots of a stochastic pay-off function

I have a pay-off function for a derivative which is defined by the Heaviside difference between $G$ and $B$ shifted by $-F$. To find the value of $V_{t=0}$, I need to find $\tau$ when $\frac{dV}{dt} = ...
1
vote
0answers
72 views

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), ...
1
vote
1answer
603 views

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 ...
1
vote
0answers
112 views

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 ...
1
vote
0answers
109 views

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 ...
1
vote
0answers
61 views

Can trinomial trees be used to model subdiffusion?

I am modeling a sub-diffusive process where the particles follow geometric Brownian motion (GBM) with movement occurring after randomly distributed waiting times. I have set this up as a simulation ...
1
vote
0answers
488 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 $$ \frac{dS^{1}_{t}}{S^{1}_{t}}=\mu_{1} dt + \sigma_{1} dW^{1}_{t} \quad \quad (1)$$ $$ \frac{dS^{2}_{t}}{S^{...
1
vote
0answers
92 views

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] $...
1
vote
1answer
413 views

Obtaining the drift of a Wiener process formed from a random walk

I'm trying to understand how the equation for Geometric Brownian Motion is formed from a random walk. I'm following the book 'Statistics of Financial Markets' but I'm struggling to follow how the ...
1
vote
0answers
221 views

Monte Carlo simulation of Multifractional Brownian Motion in MATLAB

Code under is taken from http://en.literateprograms.org/Monte_Carlo_simulation_(Matlab) ...
1
vote
1answer
38 views

computation involving independent increments [closed]

One can rather easily show that E[$\sum_{i = 0}^{i = n - 1}W_{t_i}(W_{t_{i + 1}} - W_{t_i})]$ = -T + $W_T^2$. What I'm confused about is why we can't simply say that for each i, $W_{t_{i}}$ is ...
1
vote
0answers
230 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 ...
1
vote
1answer
72 views

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 $ ...