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: $$R_i=\frac{S_{i+1}-...
15
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3answers
7k 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) + \frac{1}{...
11
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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 ...
11
votes
2answers
449 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) = \...
10
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1answer
450 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.
10
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4answers
507 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 ...
10
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2answers
441 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 ...
8
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5answers
863 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
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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 ...
7
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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
601 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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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
125 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 $Z(t)...
6
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3answers
916 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
215 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
168 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
112 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
112 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
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2answers
491 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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4answers
628 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). ...
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
votes
1answer
86 views

Given $\mathbb Q$ and $X_t$ is $\mathbb Q$-Brownian, find $\frac{d\mathbb Q}{d\mathbb P}$ / Uniqueness of Brownian or Radon-Nikodym derivative

The problem: Let $T >0$, and let $(\Omega, \mathscr F, \{ \mathscr F_t \}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \mathscr F_t^W$ where $W = \{W_t\}_{t \...
5
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5answers
244 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
88 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
146 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
132 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 N(0,\sqrt{dt}...
4
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2answers
177 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})...
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
94 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
92 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
85 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
143 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 T}...
4
votes
1answer
124 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
137 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
299 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 $$ \bar{X}(t):=\frac{1}{t}\int_0^tX(...
4
votes
0answers
23 views

Polynomial interpolation of corrected lognormal distribution

Can anyone provide a formula for a polynomial interpolation of the corrected lognormal distribution used to model returns traditionally resulting from the wrong Brownian motion generated model? ...
4
votes
1answer
201 views

Monte Carlo, convexity and Risk-Neutral ZCB Pricing

I've built a simplistic Excel monte carlo model to price a zero-coupon bond, but it came up with a slightly unepxected result so I wanted to confirm whether my maths is just a little rusty or my model ...
4
votes
0answers
403 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
1answer
233 views

What is the distribution of Brownian Bridge over a given time interval?

I know from Karatzas & Shreve (1991) that a Brownian Bridge $B(t)$ from $a$ to $b$ on time interval $[0,T]$ satisfies: $$B(t)=a(1-t/T) + b*t/T + [W(t) - W(T)*t/T]$$ where $W(t)$ is a standard ...
4
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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
435 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
1answer
78 views

Is this a poorly written example, or could volatility in fact be negative?

I'm self-studying and I encountered the following example. It seems to suggest that volatility is negative in this example. I was under the impression that volatility can never be negative, both from ...
3
votes
2answers
334 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
72 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
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2answers
274 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
345 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?