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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Integral of Brownian motion w.r.t. time
Let
$$X_t = \int_0^t W_s \,\mathrm d s$$
where $W_s$ is our usual Brownian motion. My questions are the following:
Expectation?
Variance?
Is it a martingale?
Is it an Ito process or a Riemann ...
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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
$$g_t(t,x) + g_x(t, x) \mu(t,x) + \frac{1}{...
user1157
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How to simulate stock prices with a Geometric Brownian Motion?
I want to simulate stock price paths with different stochastic processes. I started with the famous geometric brownian motion. I simulated the values with the following formula:
$$R_i=\frac{S_{i+1}-...
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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 ...
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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 ...
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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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Why is Brownian motion useful in finance?
The following is an interview question from Mark Joshi et al. Quant Job Interview.
Question: Why is Brownian motion useful in finance?
I am from a Pure Maths PhD background (functional analysis, ...
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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 ...
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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 ...
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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 ...
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Finding distribution of $\int_0 ^T uW_u du$
I would like to find the distribution of $\int_0 ^T uW_u du$ where $(W_u)_{u\geq0}$ is the Brownian motion.
What I have tried:
$$\int_0 ^T uW_u du = \int_0 ^T B_udu - \int_0^T \int_0^tB_sdsdt$$ by ...
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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
$$d W_t^1 \sim \mathcal N(0,\sqrt{dt}...
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GBM 3d plot with R
I want to plot the density of the GBM in a 3d plot. So I have on one axis the stock price, on the other the time and on the z axis the density. At the end I want to produce this graph.
The formula I ...
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Distribution of Geometric Brownian Motion
Please let me know where I have been mistaken!
Let the SDE satisfied by the GBM $S(t)$ be
$$
\frac{dS(t)}{S(t)} = \mu dt + \sigma dW(t).
$$
Then, the underlying BM $X(t)$ will satisfy
$$
dX(t) = \...
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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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What's the correct choice for modeling correlated stock prices?
Let's assume we're happy with simulating $n$ stocks as geometric Brownian motion (GBM). But say we also want the prices to be correlated.
When I searched around for how to construct correlated paths, ...
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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 ...
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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 \in ...
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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 + \...
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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 paths,...
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Basic question on Ito integrals
$Let \space X(t) =\begin{cases}
2, \qquad\text{if} \space 0\le t \le 1 \\
3, \qquad\text{if} \space 1 < t \le 3 \\
-5, \qquad\text{if}\space 3 < t \le 4
\end{cases}
$
or in one forumala $...
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Simulate correlated Geometric Brownian Motion in the R programming language
In response to this question: How to simulate correlated Geometric brownian motion for n assets?
One of the responses provides an implementation in MATLAB:
http://www.goddardconsulting.ca/matlab-...
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On a time integral of Brownian motion up to the hitting time
Just come up with a 'simple' and interesting problem that I've been struggling to deal with for some time. Consider a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\in[0,T]},\...
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Determining Hurst exponent of a Brownian motion
I am trying to determine the Hurst exponent of a simple Brownian motion, however, I seem to get a result that differs from 0.5. I am following the instructions given on the Wikipedia-page, and here is ...
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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 ...
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Variance of time integral of squared Brownian motion
I want to calculate the variance of
$$I = \int_0^t W_s^2 ds$$
I was thinking I could define the function $f(t,W_t) = tW_t^2$ and then apply Ito's lemma so I get
$$f(t,W_t)-f(0,0) = \int_0^t \frac{\...
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What is a Brownian motion "under the risk-neutral measure"?
I understand that the risk-neutral measure associated with the money-market Numeraire is one under which the discounted price (acc. to the risk-free rate) of any asset is a martingale.
Brownian motion ...
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Is the average of independent Brownian Motions still a Brownian Motion?
If $W$ and $B$ are independent Brownian Motions (BM thereafter), then the average of $W$ and $B$ is $X_t=\frac{1}{2}(W_t+B_t)$.
Where do I begin to show that indeed it is still a BM?
Also, if both ...
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Estimating the Hurst exponent in short terms in developed markets
In the Proceedings of the Estonian Academy of Sciences, Physics and Mathematics (2003), I saw the following sentence:
Surprisingly, in the case of developed markets, short-term $H$ results showed ...
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Risk neutrality correction for Monte Carlo Bootstrapping according to PRIIP regulation for products of category III
The PRIIP (packaged products) regulation prescribes Monte Carlo bootstrapping simulation for calculation of VaR for products of category III (non-linearly leveraged products). The idea is based on ...
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Expectation of maximum draw down in the Brownian motion case
Let
$$
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 - ...
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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 ...
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Find a formula for the price of a derivative paying $\max(S_T(S_T-K),0)$
Develop a formula for the price of a derivative paying
$$\max(S_T(S_T-K))$$
in the Black Scholes model.
Apparently the trick to this question is to compute the expectation under the stock measure. So,...
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More questions about integral of Brownian Motion w.r.t time
A similar question have been posted earlier but one part has remained unanswered. Let us define:
$$X_t = \int_0^t W_s ds,$$
where $W_t$ is a standard Brownian Motion. Is $X_t$ an Itô process or a ...
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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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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)...
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Can I always use quadratic variation to calculate variance?
Suppose we have a Brownian Motion $BM(\mu,\sigma)$ defined as
$X_t=X_0 + \mu ds + \sigma dW_t$
The quadratic variation of $X_t$ can be calculated as
$dX_t dX_t = \sigma^2 dW_tdW_t = \sigma^2 dt$
...
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What are the limitations of brownian motion in finance? [duplicate]
What are the limitations of brownian motion in its applications to finance?
user693
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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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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 ...
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Why does the diffusion term remain the same when we change pricing measure?
Consider some Itô process $dS(t)=\mu(t)dt+\sigma(t)dW^{\mathbb P}_{t}$ under the measure $\mathbb P$, where $W^{\mathbb P}$ is a $\mathbb P$-Brownian motion
In plenty of interest rate examples, I have ...
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Does Black Scholes need to assume no arbitrage?
Since Girsanov's theorem guarantees a risk neutral measure for Geometric Brownian motion, by the fundamental theorem of asset pricing there can be no arbitrage. So, why does the model assume no ...
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Integral of Wiener process w.r.t. time
I have a doubt with regards to the calculation of the below integral-
$\int_0^t W_sds$
where $W_s$ is the Wiener Process.
This has been solved very ably in the following page. It turns out to be a ...
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Option pricing with Brownian Bridge
Say I have an asset following arithmetic Brownian motion
$$
dX(t) = \sigma dW^\bot (t)
$$
with $\sigma$ constant, and I have prices of vanilla options on $X$.
I introduce a Brownian bridge
$$
dY(t) = ...
user34971
7
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1
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Modelling EUR/USD with Ornstein-Uhlenbeck + jumps?
I'm trying to simulate a process as close as possible to EUR/USD of the ten past years.
I've used a Ornstein-Uhlenbeck process:
$$d X_t = -\theta (X_t - \mu) d t + \sigma d B_t$$
with the parameters $\...
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Likelihood ratio and pathwise sensitivity method for coupled SDEs
I have two coupled SDEs
\begin{align*}
dS_t=rS_tdt+V_tdW_t^{(1)},\\
dV_t=aV_tdt+b(V_t)dW_t^{(2)},\\
\end{align*}
where $W_t^{(1)}$ and $W_t^{(2)}$ are independent Brownian motions, initial input data ...
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Non attainable claim - Incomplete market
I am wondering whether there is a standard procedure to find a non attainable (i.e. non replicable) asset in an incomplete market.
As an example, let us have the following market ($B = (B^1, B^2, B^3)$...
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Expectation of exponential of 3 correlated Brownian Motion
Consider,
are correlated Brownian motions with a given
I want to calculate the,
,
I can't think of a way to solve this although I have solved an expectation question with only a single exponential ...
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2
answers
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Variance of a time integral with respect to a Brownian Motion function
Let process
$$I_t = \int_0^t f(s) W_s \,\mathrm d s $$
where $W_s$ is standard Brownian motion. My question are the following:
We know that $\mathbb{E} (I_{t})=0$ for all $t$ and $f$ a integrable ...
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1
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How to get the probability of exercise call option in Black-Scholes model?
From Black-Scholes model, I'm trying to prove:
$p(S_t>K) = N(d_2)$
No luck yet!
Can anyone suggest a reference showing that how to obtain this equation?
All I get is:
$S_t = S_0e^{ (\mu-0.5 \...
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