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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188 views

example Hamilton-Jacobi-Bellman Equation - clarification of $dX_t$ derivation using $\pi_t$, $\Pi_t$

I have a market with safe rate r and risky asset S $$ \frac{dS_t}{S_t}=(r+Y_t)dt+\sigma dW_t \quad \quad (1)$$ $$ dY_t = - \lambda Y_t +dB_t \quad \quad (2)$$ where W, B are Brownian Motions with ...
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2answers
303 views

Ito calculus problem

given $S^1$ satifying the SDE $\quad dS_{t}^{1}=S_{t}^{1}((r+\mu)dt + \sigma dW_t), \quad S_{0}^{1}=1 $ and the safe asset $S_{t}^{0}$ $\quad S_{t}^{0}:=e^{rt} \quad for \quad r\geq 0$ Q1. how ...
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176 views

Multivariate Ito problem $M_t=\frac{X_t}{Y_t}$

I am analyzing a problem given in the lecture slides published here (Slide 7-8 Example of Multivariate Ito’s Lemma). Can anybody explain how the $M_t$ was calculated out of the Ito formula. I ...
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1answer
93 views

investor terminal value of portfolio with two risky assets 1) correlated 2)not correlated $\phi_t^1=S^{2}_{t}, \ \phi_t^2=S^{1}_{t}$

I am analyzing a problem where I have two stocks described by the equations $$ \frac{dS^{1}_{t}}{S^{1}_{t}}=\mu_{1} dt + \sigma_{1} dW^{1}_{t}$$ $$ \frac{dS^{2}_{t}}{S^{2}_{t}}=\mu_{2} dt + \sigma_{2}...
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1answer
202 views

2 Ito processes - $d(X_{t} + X^{'}_{t})^2 = (Y_t + Y^{'}_{t})^2 dt$ why it is true?

Having two Ito processes $dX_{t} =z_{1} dt + Y_{t} dB_t $ $dX^{'}_{t} =z^{'}_{1} dt + Y^{'}_{t} dB_t $ I am analyzing a proof of the product rule $d(X_t X_t^{'})=X_t dX_t^{'}+ X_t^{'} dX_t + Y_t ...
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669 views

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 $ ...
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543 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^{...
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94 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] $...
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What is the name of all 1-day movements, 2-day movements etc

When looking at historical data (index or stock), one can find all 1-day differences/movements, all 2-day, all 3-day etc and graph the extremes of each of these. This gives two line graphs forming a ...
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1answer
269 views

Lookback option to find stock price

Consider the payoff equation for the lookback option $\psi(T)= max(S_t-S_T)$, where $t\in[0,T]$ and $S_t$ is modeled by the geometric Brownian motion with constant parameters. Find the price of stock ...
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1answer
346 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 (...
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1answer
731 views

Processes used in quant finance

What are the main stochastic processes (and their SDE) used in quant finance? For example to model currency prices, stock prices, etc.
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203 views

Simulations of (standard, one-dimensional) Brownian motion

Consider the following two proposed simulations of paths of standard, one-dimensional Brownian motion between time $0$ and time $1$. Normal Increments Roll out a large sequence of, say $M$, ...
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4answers
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Stochastic process with non-independent increments

All stochastic process I see always have independent increments. It is true for: standard brownian motion geometric brownian motion (?) Ornstein Uhlenbeck (?) in general, Levy process etc. What are ...
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165 views

How to arrive at expectation of negative utility function via Taylor series expansion

I'm attempting to follow an author's steps in an argument and having trouble seeing how Taylor series expansion can be applied to give the stated result. The scenario is as follows. The mid price ...
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1answer
427 views

Modelling EUR/USD with Ornstein-Uhlenbeck + jumps?

I'm trying to simulate a process as close as possible to EUR/USD of the ten past years. I've used a Ornstein-Uhlenbeck process: $$d X_t = -\theta (X_t - \mu) d t + \sigma d B_t$$ with the ...
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What's the name of this nearly-brownian stochastic process?

1) Does the following algorithm (my question is math, not programming-related): ...
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2k views

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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Question about the martingale property of stochastic integral

Let $W_{t}$ be a Wiener process, and let $$X_{t} = \int^{t}_{0}W_{\tau}d\tau$$ Is $X_{t}$ a martingale? We can rewrite in differential form as $$dX_{t} = W_{t}dt$$ ,which means $X_{t}$ is a diffusion ...
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1k views

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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1answer
484 views

Black Scholes Geometric Brownian Motion Option Pricing

I'm doing a past paper for one of my masters modules and I'm stuck on this and I have no idea how to tackle such a thing. It's worth 30% of the exam so would be great if anyone here has any ...
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394 views

Constructing a Brownian motion from a Simple Random Walk

I'm trying to get my head around how a Brownian motion is formed from a simple random walk. I've seen two similar methods used: Why has one approach used $\frac{1}{\sqrt{k}}$ and the other hasn't? ...
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2answers
419 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 ...
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1answer
456 views

Solving a backwards heat equation using stochastic calculus

Given the PDE $$\frac{\partial F}{\partial t} + \frac{1}{2}\sigma^2 \frac{\partial^2 F}{\partial x^2} = 0$$ with condition $F(T,x) = x^2$, one can use the Feynman-Kac formula to arrive at $$F(t,x) =...
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1answer
69 views

Can the differential operator be removed to get the mean/variance of an Ito process?

If $X_t$ is an Ito process, such that: $dX_t = \mu(t, X_t)dt + \sigma(t, Xt)dW_t$ where $W_t$ is a standard brownian motion. Then we can say that: $E(dX_t) = \mu(t, X_t)dt$ and that $Var(dX_t) = \...
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1answer
226 views

Brownian motion. Solve stoc. integral by using Ito's lemma

I want to show that following statement is true by using Ito's lemma to solve stochastic integrals: I define the functions in Ito's model: a()=0, b()= (2wt-2)^2. f(t)=Integrate[(2wt-2)^2] Then df=(b^...
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Is there a python code for estimating the parameters of geometric brownian motion?

I was trying to find the parameters of GBM but could not find a python code for the same.
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382 views

How is fundamental data taken into account when modelling stock prices with a Geometric Brownian Motion?

I have a basic understanding of the principles behind Geometric Brownian Motion and how it can be used to model stock prices, however I am confused as to how it is used in practice. In particular, how ...
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1answer
459 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 ...
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0answers
232 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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1answer
109 views

What is this ratio: expected returns on stock divided by risk free rate?

So this ratio has come up in some work I'm doing and I can't seem to figure out if it is attested in the literature. Here's the setting: Given a risk free rate $r(t)$ and a stock price which follows ...
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5answers
647 views

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 ...
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1answer
372 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., ...
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2answers
2k views

How to discretize a GBM under P- and Q-measures?

Under the P-measure, a geometric Brownian motion can be specified using the following SDE: $$dS_t=\mu S_tdt+\sigma S_tdW_t^P$$ and its Euler discretization is $$S_{t+\Delta t}=S_t + \mu S_t \Delta ...
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1answer
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Geometric Brownian Motion - Why Sqrt(dt)? [closed]

I was going to simulate a geometric brownian motion in matlab, when I recognized that I didnt fully understand the underlying Wiener process. Following the instuctions here I am starting from the ...
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1answer
103 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,...
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206 views

How to price an European Call/Put Option of a jump difussion Process?

Lets have the next jump difussion Stochastic Process: $$S_t = S_0 e^{\sigma W_t + (v-\frac{\sigma ^2}{2})t}\prod_{i=1}^{N_t}(1+J_i)$$ where $W_t$ is the Brownian Motion, hence $G_t \equiv e^{\sigma ...
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2answers
748 views

The Distribution of Future Stock Price

In Hull, we are presented that $$\frac{\Delta S}{S_{0}}=\mu \Delta t+\sigma\sqrt{\Delta t}\cdot \varepsilon.$$ Following some algebra, $$ \begin{align*} \frac{\Delta S}{S_{0}} &=\mu \Delta t+\...
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3answers
708 views

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: $$ dS = r S dt + \sigma S d \tilde{Z}$$ ...
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Confidence Intervals of Stock Following a Geometric Brownian Motion

In preparation for my Options, Future's and Risk Management examination next week, I have been presented with a series of questions and their answers. Unfortunately, my lecturer, one of the less ...
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1answer
284 views

Distribution of Black Scholes call option price at time 0<t <T

Does anyone know how to find the probability law (distribution) under P* of a Black Scholes Call Option price $C_t$ for $0 < t < T $? (Under P*, $ dC_t = \frac{\partial c}{\partial s}\sigma S_t ...
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2answers
969 views

Forecasting problem with Geometric Brownian Motion in Wolfram Mathematica

I'm a full time undergraduate student from Peru, and I'm trying to use the Geometric Brownian Motion example used in the help section from Wolfram Mathematica in order to forecast future stock prices,...
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1answer
621 views

What is the distribution assumption of the black scholes model

As per wikipedia the Black Scholes assumption is: (...
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2answers
721 views

Is the Brownian motion multiplication rule a definition or is it a theorem?

Is the Brownian motion multiplication rule a definition or is it a theorem? Refer to the highlight part of http://i.stack.imgur.com/doQuT.png where $dw_1(t)dw_1(t)=dt$
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1answer
174 views

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}...
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485 views

Fractional Brownian motion

In Mandelbrot(1968)'s paper, the fractional brownian motion, denoted by $B_{H}(t,\omega)$,(t>0) is defined by $$B_{H}(0,\omega)=b_{0}$$ $$B_{H}(t,\omega)-B_{H}(0,\omega)=\frac{1}{\Gamma(H+\frac{1}{2})...
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1answer
302 views

Why Variations of order higher than two vanish for Brownian motion?

Let $W_{t}$ be a Brownian Motion. Verify that variations of Brownian Motion of higher order, say, of order three, vanishes. I try to prove that $\lim_{n\rightarrow\infty}\sum^{n}_{i=1}(W_{t_{i}}-W_{...
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1answer
162 views

stochastic calculus - brownian motion

I don't know how to prove this : let be $X_t = \int_{0}^{t}\sigma_{u}dW_{u}$ where $\sigma_{t}$ is a predictable process. If $|\sigma_{t}| = c$ a.s. how can I prove that $X_{t}=c*\beta_{t}$ (...
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4answers
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Why is Brownian motion merely 'almost surely' continuous?

Why is Brownian motion required to be merely almost surely continuous instead of continuous? For example, this is stated as condition 2 in this article in section 1, Characterizations of the Wiener ...
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1answer
399 views

Stochastic Differentials - Ito's formula for a self-financing portfolio

Suppose I have a portfolio of stocks $(S)$ and savings account ($\beta_t$) then, the value is $$V = a_t S_t + b_t \beta_t$$ and for this portfolio to be self replicating, we need by Ito's lemma $$dV ...

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