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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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: ...
14
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3answers
6k views

Is there an intuitive explanation for the Feynman-Kac-Theorem?

The Feynman-Kac theorem states that for an Ito-process of the form $$dX_t = \mu(t, X_t)dt + \sigma(t, X_t)dW_t$$ there is a measurable function $g$ such that $$g_t(t,x) + g_x(t, x) \mu(t,x) + ...
11
votes
1answer
4k views

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 ...
10
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1answer
441 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.
9
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4answers
457 views

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 ...
9
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2answers
436 views

GBM 3d plot with R

I want to plot the density of the GBM in a 3d plot. So I have on one axis the stock price, on the other the time and on the z axis the density. At the end I want to produce this graph. The formula I ...
9
votes
1answer
390 views

Distribution of Geometric Brownian Motion

Please let me know where I have been mistaken! Let the SDE satisfied by the GBM $S(t)$ be $$ \frac{dS(t)}{S(t)} = \mu dt + \sigma dW(t). $$ Then, the underlying BM $X(t)$ will satisfy $$ dX(t) = ...
7
votes
5answers
811 views

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 ...
7
votes
2answers
2k views

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 + ...
7
votes
1answer
592 views

How to compute the Radon-Nikodym derivative?

Suppose $B(t)$ is a standard Brownian motion, and $B_{1}(t)$ is given by $dB_{1}(t)=\mu dt+dB(t)$. Suppose $P$ is the Wiener measure induced by $B(t)$ on the $C[0,\infty)$, and $P_{1}$ is the Law ...
6
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8answers
5k views

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 ...
6
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2answers
1k views

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 ...
6
votes
4answers
119 views

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 ...
6
votes
2answers
1k views

What is the average stock price under the Bachelier model?

Let's say stock price follows following process: $$dS(t) = \sigma dW(t)$$ where $W(t)$ is Standard Brownian motion. The initial level for the stock is $S(0)$. Define the average of stock price ...
6
votes
3answers
894 views

What are the limitations of brownian motion in finance? [duplicate]

What are the limitations of brownian motion in its applications to finance?
6
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2answers
210 views

Is the average of independent Brownian Motions still a Brownian Motion?

If $W$ and $B$ are independent Brownian Motions (BM thereafter), then the average of $W$ and $B$ is $X_t=\frac{1}{2}(W_t+B_t)$. Where do I begin to show that indeed it is still a BM? Also, if both ...
6
votes
1answer
158 views

Estimating the Hurst exponent in short terms in developed markets

In the Proceedings of the Estonian Academy of Sciences, Physics and Mathematics (2003), I saw the following sentence: Surprisingly, in the case of developed markets, short-term $H$ results showed ...
6
votes
1answer
110 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 ...
6
votes
2answers
110 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 ...
5
votes
1answer
7k views

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 ...
5
votes
2answers
488 views

Correlation decay in lognormal distribution

I noticed that if you use two correlated geometric brownian motions, the correlation structure decays in time pretty fast even for really high correlation values. I think that is not replicating ...
5
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3answers
1k views

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 ...
5
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5answers
230 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 ...
5
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2answers
86 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 ...
4
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2answers
144 views

What's the name of this nearly-brownian stochastic process?

1) Does the following algorithm (my question is math, not programming-related): ...
4
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2answers
117 views

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 ...
4
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2answers
175 views

Fractional Brownian motion

In Mandelbrot(1968)'s paper, the fractional brownian motion, denoted by $B_{H}(t,\omega)$,(t>0) is defined by $$B_{H}(0,\omega)=b_{0}$$ ...
4
votes
4answers
622 views

Expected Growth

The model assumption of the Black-Scholes formula has two parameters for the geometric Brownian motion, the volatility $\sigma$ and the expected growth $\mu$ (which disappears in the option formulae). ...
4
votes
5answers
1k views

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 ...
4
votes
1answer
90 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 $ ...
4
votes
1answer
90 views

Discounted risky asset stochastic process problem

$S_t$ is the random variable representing the risky asset price at time $t$. M_t is the riskless asset. They are governed by the equations $\frac{dS_t}{dt}=\mu dt + \sigma dZ_t$ and $dM_t = rM_t ...
4
votes
1answer
81 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 ...
4
votes
2answers
126 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 ...
4
votes
1answer
80 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 ...
4
votes
1answer
117 views

Linear-Boundary Crossing Problem for Brownian Motion

This is a question I came across while reading: $W = (W_t)_{t\geq{0}}$ is a standard BM. Let $\mu\in \mathbb{R}$, and let $\tau_{a}^{\mu}$ = $\inf(t>0;W_t = a + \mu{t})$ be the first passage time ...
4
votes
1answer
133 views

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 - ...
4
votes
1answer
290 views

Covariance of brownian motion and its time average

It's a question pertaining to the correlation of a log asset process (following BM) and its time average, to put it into form, if $$X(t)=\mu t+\sigma W(t)$$ then $$ ...
4
votes
0answers
377 views

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 ...
4
votes
0answers
105 views

Estimating two normal random numbers with one equation

Subtitle: Estimating the correlation of the shocks driving two commodities in two multi-factor models I am fitting two 2-factor models to electricity and gas futures, respectively. In order to ...
3
votes
3answers
280 views

Show that $E[B_t|\mathscr{F}_s] = B_s$

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 $B_t = W_t^3 - 3tW_t$. ...
3
votes
2answers
274 views

Is it really possible to create a robust algorithmic trading strategy for intraday trading?

I'm an engineer doing academic research for my master thesis in the area of quantitative finance, basically the purpose is to study the possibility to create an intraday-trading algorithm. I've tried ...
3
votes
2answers
318 views

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 ...
3
votes
2answers
69 views

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 ...
3
votes
2answers
257 views

Modelling driftless stock price with geometric Brownian motion

I wish to understand some basic fact about the (primitive) simulation of stock prices with geometric Brownian motion. If $S(t)$ is the stock price at time $t$, and the stock price follows geometric ...
3
votes
2answers
340 views

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?
3
votes
1answer
51 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 $M$, ...
3
votes
2answers
75 views

Conditional expectation of a non stochastic process

In an example I was working through it was shown that $W_{t}^{2} - t$ was a martingale with respect to the Brownian motion filtration $\mathcal{F}_{s}^{W}$ with $t>s$. Everything was fine except a ...
3
votes
1answer
551 views

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} & ...
3
votes
1answer
107 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 $X_{t}=c*\beta_{t}$ ...
3
votes
1answer
145 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 ...