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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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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417 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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221 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
39 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 ...
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231 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 ...
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1answer
73 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 $ ...
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2answers
389 views

Getting the next price of a GBM with reversion

Here is the "twin" question of Getting the next price of a GBM (Geometric Brownian Motion) but for GBM with reversion As in that case, I'd like to write a formula for the next price, as function of: ...
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78 views

Do we have a Brownian motion

Background Information: The process $W = (W_t:t\geq 0)$ is a $\mathbb{P}$-Brownian motion if and only if i) $W_t$ is continuous, and $W_0 = 0$ ii) the value of $W_t$ is distributed, under $\mathbb{...
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3k views

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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4answers
251 views

Correlation of Asynchronous Brownian Motion

I am trying to use the closing prices of the S&P 500 and the Nikkei Index to see how they are correlated (assuming they are exactly 12 hours apart). In order to test my method, I have generated ...
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2answers
94 views

How to show the equality of these two events?

For $X_t$ a brownian motion defined for $t\in[0,T]$, How to show the equality of following events: $$ \{ \displaystyle \max_{[\tau,T]}(X_t)\ge 2u-d, \tau\leq T \}=\{\displaystyle \max_{[0,T]}(X_t)\ge ...
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396 views

How to compute $\mathbb{E} \left[ (W_s + W_t - 2W_0)^2 \right]$?

The solution to the SDE $$dx_t= -kx_t dt + cx_t dW_t$$ is $$x_t = x_0 e^{\left(c - \frac{k^2}{2} \right)t}e^{-k W_t}$$ with mean $$\mathbb{E} \left[ x_t \right] = x_0 e^{\left(c - \frac{k^2}{2}...
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287 views

Generally how to simulate bivariate (or multidimensional) BM sample paths?

A topic I am struggling with is the implementation of a (for the simplest higher dimensional case) bivariate normal distribution simulation for geometric brownian motion. The clearest explanation by ...
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2answers
201 views

Stochastic process and brownian motion

I just read the following and i am having some difficulty to interpret it: We begin our analysis in the standard Black-Scholes world consisting of a bank account process of price denoted by $B_t$, ...
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1answer
2k views

Given $S$ is a Geometric Brownian Motion, how to show that $S^n$ is also a Geometric Brownian Motion?

Suppose that a stock price $S$ follows Geometric Brownian Motion with expected return $\mu$ and volatility $\sigma:$ $$dS = \mu S dt +\sigma S dz$$ How to find out the process followed by variable $...
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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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1answer
274 views

Theoretical distribution of (geometric) Brownian motion (with drift)

I am working on a simulation study which focuses on both the Brownian motion with drift (1) and the geometric Brownian motion (2). I denote them by $X_t$. What are the theoretical distributions of ...
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1answer
79 views

Calculate $Cov(e^ {B_t} ,e^{B_s})$

Let $(B_t)_{t \geq0} $ be a Brownian Motion. Calculate $Cov(e^ {B_t} ,e^{B_s})$ I would verify the following solution which the result looks a bit weird. My solution: let $0 \leq s \leq t$. $$Cov(e^ {...
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111 views

Two Wiener process under same martingale measure Q

Let $W_1,$ $W_2$ be to Wiener processes under the martingale measure $Q$. What can be said about $dW_1*dW_2$? I know that $$(dW_i)^2=dt$$ but what about the case with two different wiener processes?
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79 views

Properties of Brownian motion and filtration, Exercise 6.22, Joshi Concepts and applications to mathematical finance

Let $W_t$ be a Brownian motion, and let $F_t$ be its filtration then for $t > s$ we are asked to compute $$\mathbb{E}\left[W_t^2|F_s\right]$$ We have $$W_t = W_s + (W_t - W_s)$$ and $$W_t^{2} ...
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2answers
187 views

Questions on continuously compounded return vs long term expected return

I have reading a paper from Oliver Grandville on long term expected return. I am trying to reconcile what I am reading in that paper vs what I see under "Application to Stock Market" in Kelly ...
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1answer
205 views

Determine the conditions for Brownian motion to be a Martingale

Let $W_T$ denote normalised univariate Brownian motion and let $X_t = W_t^2 + \alpha W_t + \beta t + \gamma$ where $\alpha, \beta$ and $\gamma$ are constants. Determine conditions on these constants ...
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108 views

Pricing Secured Barrier Call

A European barrier call with barrier $B = 50$, expiration $T = 31$, and strike $K = 33$ costs $12$. The investor is interested in a product that, unlike this barrier call, offers some protection for ...
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1answer
50 views

$P(S_T > S_u \mid S_v = s_*)$

Let $u < v < T$ and assume $S_t$ follows a lognormal $((\mu - \sigma^2/2)t, \sigma^2 t)$ process. I'm interested in computing the conditional probability $$ P(S_T > S_u \mid S_v = s_*) $$ ...
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2answers
151 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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2answers
189 views

Integral of Wiener process over time

This should hopefully be an easy question to answer, but I am new to Stochastic Calculus and am gapping as to why the following is true, for a brownian motion $W_t$: $$d(\int W_t dt ) = W_t dt$$ I ...
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1k views

Integral of Brownian Motion w.r.t Time: what is wrong with this solution? [duplicate]

My question is about a stochastic integral of brownian motion w.r.t time. Let $W(t)$ the Wiener process (or brownian motion). I want to calculate this: \begin{eqnarray} X(t)=\int_{0}^t dt' W(t'). \...
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1answer
154 views

Asset price simulation under Monte Carlo for option pricing using market data

I am trying to use Monte Carlo to price some exotic options. I have in mind to simulate asset prices under GBM (say S&P prices) using Monte Carlo and price the option accordingly from the payoffs ...
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1answer
97 views

Determining the probability of arriving at a price by a time T

A useful calculation for ascertaining the risk of something might be determining the probability of a realization of a set of stock prices $X$ being greater than or equal to some future price $x$. I ...
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532 views

Should I adjust historical data for dividends when estimating drift?

I'm building a Geometric Brownian Motion model which incorporates future dividends which vary over time. Since these should reduce stock price when paid, I can incorporate that into the model, however,...
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1answer
241 views

Will volatility smoothing effects exist for returns driven by geometric brownian motion?

Say I randomly simulate a one-year pathway of 252 prices, where the underlying price model is driven by geometric brownian motion. where $t = (1 / 252)$, $mu = 5$% and annual $st.dev = 10%$%. My ...
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1answer
142 views

Integral with respect of $(dW_s)^n$

I know $$\int _0^t dW_s=W_t-W_0=W_t$$ Since $ dW_s dW_s=ds$ , so $$\int _0^t( dW_s)^2=\int_0^t ds=t-0=t$$ I Want to know why for $n\ge 3$ we have $$\int _0^t (dW_s)^n=0$$ My try $$(dW_s)^2 dW_s (...
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1answer
186 views

Discrete Time to Continuous Time and Summation of Two Geometric Brownian Motions

Could someone please suggest with detailed steps and/or a reference, 1) How to convert the below discrete time summation to continuous time form and write it as an integral? 2) Any methods to ...
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1answer
246 views

Simulating a GBM with martingale condition - Ito process moving downwards

I want to correctly simulate a $\mathcal{Q}$ - martingale $S$, which is a geometric Brownian motion and an exponential of a process $X$, \begin{equation} X_t = X_0 + \mu t + \sigma B_t = X_{t-\Delta t}...
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70 views

What is $W(t) \cdot dt$ where $W(t)$ is a Wiener Process [closed]

When I derive the stochastic differential equation for a particular case, I came across $W(t) \cdot dt$, what is the result for it? I know $dW(t) \cdot dt = 0$, could any expert give me some advice?
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33 views

Swap rate in the annuity measure and Martingale Representation Theorem

As we know, swap rate evolves as a martingale in the appropriate annuity measure. Martingale representation theorem says if I can find a Brownian motion in the annuity measure and the swap rate is ...
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0answers
35 views

Risk neutral measure in the binomial approximation of geometric Brownian motion

Suppose an asset is described by geometric Brownian motion with a drift, i.e. $$dS_t = S_t\mu dt + S_t \sigma dW_t$$ for a Wiener process $W_t$ and $S_0=1$. By some consequence of Girsanov's theorem (...
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33 views

Negative drift when calibrating GBM parameters

Setup for question: Consider a basket of $N$ stocks $\{S^1, S^2, \dots, S^N\}$. For fixed strike $K$, each stock in the basket, $S^i$, follows the SDE $$dS_t^i = \mu^i(t) S_t^i dt + \sigma^i(K, t) ...
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0answers
93 views

Predicting time series using Jump Diffusion model and Neural Networks

I am trying to understand the difference between using Jump diffusion model and Neural Networks or more precisely LSTM to predict time series data regardless what that data contains for example a ...
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0answers
42 views

Does it make sense to simulate from the multidimensional GBM?

Suppose I have times series data on 3 assets and I do $N$ simulations (GBM) first for each of assets individually and then from a multidimensional GBM since their log-returns are correlated (I use ...
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180 views

Interpretation of drift parameter $\mu$ in GBM

Currently studying Ito's calculus. Looking on the GBM model: $ \frac{d S_t}{S_t} = μ dt + \sigma d B_t$ we end up on the expected stock price at time t: $E[S_t]=s_0 e^{\mu t}$.What does actually $\mu$ ...
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616 views

Proof that integral of Brownian motion wrt time is not a martingale

Let $X_t=\int_0^t W_s ds$ where $W_s$ is Brownian motion, so $E[W_s]=0$. Then $E[X_t]=\int_0^t E[W_s] ds=\int_0^t 0 ds=0$. So $E[X_t|{\cal F}_s]=0\neq X_s$, almost everywhere. So by previous ...
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1answer
356 views

Probability of Brownian motion particle touching barrier given path starts at $X_0$ and ends at a known $X_t$

I have been reading Su and Rieger's paper on barriers and from there have been able to work out the unconditional probability of the process $dXt = μ dt + σ dWt$ touching a down barrier $α$ to be $\...
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3answers
201 views

Does the Ito correction term in GBM result in 'real money', or is it illusory?

There are two ways to think about investment returns and randomness. First is sort of like 'bank interest', with randomness. Suppose we invest 100 units of currency. Suppose each year there is a ...
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1answer
291 views

Mark Joshi, The concepts and practice of mathematical finance chapter 6 exercise 6 [duplicate]

Suppose a stock allows a geometric Brownian motion in a Black-Scholes world. Develop an expression for the price of an option that pays $S^2 - K$ if $S^2 > K$ and zero otherwise. What PDE will this ...
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2answers
77 views

Fourth moment of a itos integral

$I(t)=\int_0^t \sqrt sdW_s$ What is $E(I(t)^4)$
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1answer
645 views

Bond price and its process

Suppose that x is the yield to maturity with continuous compounding on a discount bond that pays off $1 at time T. Assume that the x follows the process $dx=a(x_0-x)dt + sxdz$ where $a, x_0$ and $s$ ...
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1answer
219 views

Expectation of the product of two Brownian motions [closed]

Could you please let me know the steps to follow to get to the solution?
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1answer
975 views

Probability distribution and Stock Price Movement [closed]

How can we use normal distribution for finding the probability of a stock price offer where current price offer depends upon the last price offer. The price offer on some day can go 10% above (at the ...
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1answer
70 views

Differential product Correlated processes

I am trying to derive the differential of the product of two processes, but I got stuck. This is what I have until now: We have the following two stochastic processes: $dX_t= \mu_t dt +\sigma_t dW_t$...