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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Geometric brownian motion small timesteps high volatility
I'm trying to generate some sample geometric brownian motion paths for an asset which is traded 24/7 without interruption and is highly volatile (upwards to 150% implied volatility on options markets)....
2
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1answer
58 views
Equivalent local martingale measure vs. equvalent martingale measure in a Brownian setup
Assume you have the standard financial market built up of a Brownian motion. I have seen some books say that an equivalent local martingale measure imples no arbitrage, and some say that an equivalent ...
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0answers
26 views
Generate a Fractional Gaussian Noise
I am trying to simulate a Fractional Gaussian Noise using Fast Fourrier algorithm.However,I couldn't even if I could retrieve my original covariance matrix such :
$E\left[ X(t)X(s) \right] = \frac{1}{...
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1answer
87 views
Second variation of a Brownian motion under jump-diffusion process
I am trying to solve exercise 15.3 from the book The concepts and practice of mathematical finance where it is asked
Suppose the $\log S_t$ follows a Brownian motion over the period $[0, 1]$ except ...
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2answers
153 views
Ito's lemma $f(t,W_t^2)$
Let $f$ be a function of $t$ and $W_t^2$.
a)Find a function $f$ such that $f(t,W_t^2)$ is a $F_{t^-}$ martingale, with $F$ the Brownian filtration.
b)Use Ito's lemma to show that $f(t,W_t^2)$ is a ...
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2answers
123 views
Sampling change in the driving brownian motion of a CIR process
I have volatility driven by a CIR process:
$$\mathrm{d}v_t = \kappa (\bar{v}-v_t)\mathrm{d}t + \omega \sqrt{v_t}\mathrm{d}W_v\text{.}\tag{1}$$
I am working with several (complicated) approximations of ...
1
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0answers
58 views
Finding the PDE and replicating strategy of a european contigent claim [duplicate]
Suppose that we have the Black and Scholes model where the interest rate and the volatility are time varying:
$dB(t)=r(t)B(t)dt$ and
$dS(t)=S(t)b(t)dt+S(t)\sigma(t)dW(t), S(0)=s>0$
where $r,b,\...
2
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1answer
104 views
Solving an SDE using Ito's Lemma
Suppose that
$Z(t)=e^{-\int_0^t \theta'(s)dW(s)-\frac{1}{2}\int_0^t ||\theta(s)||^2ds}$
with $\theta()=\sigma^{-1}()[b()-r()]$, $\sigma()>0$ and invertable and $W()$ a Wiener process
There is also ...
3
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1answer
148 views
Brownian Bridge general case
The SDE for the Brownian bridge is the following:
$dY_t=\frac{b-Y(t)}{1-t}dt+dW(t)$
with solution:
$Y(t)=Y(0)(1-t)+bt+(1-t)\int_0^t \dfrac{dW(s)}{1-s}$
Can someone help me on proving that $$\lim_{t\...
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0answers
192 views
On a time integral of Brownian motion up to the hitting time
Just come up with a 'simple' and interesting problem that I've been struggling to deal with for some time. Consider a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\in[0,T]},\...
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0answers
95 views
MGF of Generalised Itô Integral
The following derivation produces a moment closure problem - I would appreciate any insight. It may seem trivial at first glance, but the key aspect is the integrand dependence on $t$.
Consider $W_t$ ...
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0answers
47 views
Distribution and Analytical solution of a GBM with stochastic interest rate?
We model the exchange rate $S_t$ with a geometric Brownian motion and the USD and EUR interest rates $r_u$ and $r_e$ each according to the Vasicek model. Under the domestic equivalent martingale ...
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2answers
82 views
Drift Term in Black-Scholes Model Martingale
How would I prove that a Black-Scholes Model is not a Martingale if it has drift. In many cases it is just stated as a fact (without proof).
For instance if Im looking at:
$$dS_{t} = \mu S_{t} + \...
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1answer
99 views
Sum of discretely sampled BM
If an underlying follows lognormal GM with no drift $dS_t = \sigma S_t dW_t $ and $A_N = \Sigma_{i=1}^{N} S_{t_i}$. How to compute variance of $A_N$?
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1answer
97 views
Deriving Law of Motion by Ito's Lemma
I've been trying to derive the law of motion for the stochastic process above using Ito's Lemma, given Geometric Brownian Motion with it's law of motion shown below:
I've managed to take the partial ...
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2answers
172 views
Conditional expectation of integral of brownian motion
I am trying to calculate
$$\mathbb{E}\biggl[\biggl(\int_s^t W_u du\biggl)^2 \biggl|W_s=x, W_t=y\biggl] $$
where $W$ is a Standard Brownian Motion and $s\leq u \leq t$.
Any help or tips would be ...
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1answer
83 views
Monte carlo simulations giving biased output [closed]
I wrote code to simulate the stock price using geometric brownian motion.
My code is as follows:
...
3
votes
1answer
77 views
EMM for Bachelier model
The stock price is assumed to evolve as $S_{t}=S_{0}+\mu t+\sigma B_{t}$, where $S_{0}>0, \mu>0$ and the process $B_{t}$ is Brownian motion.
The saving account is assumed to be $\beta_{t}=e^{r t}...
4
votes
1answer
189 views
Covariance of two Brownian Motions
During revision, I came across the following question in a past paper:
Suppose $(B_t, t\geq0)$ is a standard Brownian motion. Compute for $0<s<t$ the covariance $$cov(tB_{3t}-B_{2t}+5, B_s-1).$$
...
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1answer
153 views
How is the formula of Quadratic Variation of Brownian Motion derived? [closed]
This is a follow up on this question on quant SE:
The question mentions for a Brownian motion :
$X_t = X_0 + \int_0^t\mu ds + \int_0^t\sigma dW_t $
, the quadratic variation is calculated as
$dX_t ...
3
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2answers
152 views
Covariance between integral of brownian motion and brownian motion
Let
$$
I = \int_0^1W_tdt,
$$
where $W_t$ is a Brownian motion.
From Integral of Brownian motion w.r.t. time we have that
$$
\mathbb{E}[I]=0,
$$
by Fubini's theorem. And that
$$
\mathbb{V}\text{ar}[I] =...
2
votes
1answer
109 views
Calculating futures price
Consider a world as follows:
$$\frac{dB}{B} = r_tdt$$
$$\frac{dS}{S} = r_tdt - 0.05dW_1 + 0.5dW_2$$
$$dr_t = 0.2 dW_1$$
where $r_0=0$. The Wiener processes $W_1$ and $W_2$ are independent. The price ...
3
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0answers
87 views
Fractional Brownian Motion's Covariance Proof
Let's have the non independent Brownian motion such :
$B_{H}(r)=\frac{1}{A(H)} \int_{R}\left[\left\{(r-s)_{+}\right\}^{H-1 / 2}-\left\{(-s)_{+}\right\}^{H-1 / 2}\right] \mathrm{d} B(s), \quad r \in R$
...
2
votes
1answer
85 views
Integral of Brownian motion w.r.t. time and integral not starting at zero
I'm new to stochastic calculus and try to calculate (1) mean and (2) variance of
$$\int_s^t W_u du$$
where $W_u$ is a Brownian motion. I already found this helpful answer, where it was shown that $\...
1
vote
1answer
95 views
Hermite polynomials as martingales [closed]
Let $\left\{W_{t}: t \geq 0\right\}$ be a standard B.M. on the filtered probability space $\left(\Omega, \mathcal{F},\left\{\mathcal{F}_{t}\right\}_{t \geq 0}, \mathbb{P}\right)$. Define the Hermite ...
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votes
1answer
92 views
Mutual variation of Brownian motions
Let $\{W^1\}_{t\geq0}$ and $\{W^2\}_{t\geq0}$ be two Brownian motions with correlation coefficient $\rho \in [0, 1]$, i.e., $\mathbb{E}[(W^1(t)-W^1(s))(W^2(t)-W^2(s))]=\rho(t-s)$ for all $t,s \geq 0$. ...
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1answer
165 views
What does it mean to “compute” an Itô integral?
I'm reading Shreve's Stochastic Calculus for Finance II. On page 191, Exercise 4.6, we are given the problem
Exercise 4.6. Let $S(t)=S(0)\exp\Big \{\sigma W(t)+(\alpha-\frac{1}{2}\sigma^2)t\Big\}$ be ...
3
votes
1answer
180 views
Help on solving a stochastic differential equation
I am trying to solve the following SDE
$$dX(t)=rdt+aX(t)dW(t),\ t>0$$
$$X(0)=x$$
where W() is a Wiener process and r,a and x real numbers. I have proceeded by using the integrating factor
$$F(t)=...
2
votes
2answers
209 views
Proving that a stochastic process is a martingale using Ito's Lemma
Assume a Wiener process W and a bounded F-adjusted stochastic process a. Show that the following process is a martingale on F
$$X(t)=(\int_{0}^{t}a(s)dW(s))^{2}-\int_{0}^{t}a^{2}(s)ds,\ t\geq0$$
Can ...
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votes
1answer
135 views
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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1answer
60 views
Reason why a European binary call should be worth half of its American counterpart when driftless and out-of-the-money
Exercise 11 of chapter 8 of Mark Joshi's "The concepts and practice of mathematical finance", asks to compare prices of an American and a European digital (binary) calls when out-of-the-...
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1answer
60 views
Simplifying the expectation of the product of two stochastic integrals
Let $f(t, \omega), g(t, \omega)$ be functions that are independent of the increments of the Brownian motion $w(t, \omega)$ in the future. That is, $f(t, \omega), g(t, \omega)$ are independent of $w(t +...
4
votes
1answer
100 views
Default intensity in Black-Cox model
Consider the model by Black and Cox (Journal of Finance, 1976).
The default intensity function is defined in the usual way: $$h(t) \equiv - \frac{\partial \log P[\tau > t| \mathcal{F}_t]}{\partial ...
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1answer
102 views
What is the expectation of a change in Brownian motion? [closed]
I know $E[W_T-W_t]=0$ but I have a solution which implies this is wrong.
Question
Answer
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0answers
52 views
How to show the following conditional expectation relation holds for a Brownian motion?
Suppose that $B_k$ stands for a standard Brownian motion process.
\begin{equation}
\mathbb{E}\Big(e^{-w\int_{t}^{S}B_k dk\, -uB_T}\Big| B_t = x\Big)
\end{equation}
where $w$ and $u$ are constants, and ...
1
vote
1answer
62 views
How to prove that the following is still a Brownian motion [closed]
Given a Brownian motion $B_t$ on a filtered probability space, how can I prove that $W_t=B_t+\alpha t$ is still a Brownian motion, with $\alpha \in \mathbb{R}$? Is it always true? Do I need necessarly ...
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0answers
57 views
Future price in continous time
I am in the following continuous time market:
$S_t^0 = rS_t^0dt$
$S_t^1 = (\mu - \delta) S_t^1dt + \sigma S_t^1 dB_t$
where $r, \mu, \delta$ and $\sigma$ are constant values in $\mathbb{R}$. $\delta$...
3
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1answer
165 views
Conditional probability of Brownian motion (with drift and scaling) hitting barrier
I am trying to understand the pricing of barrier options, and am considering the Brownian motion $\mathrm{d}X_t=a\mathrm{d}t+b\mathrm{d}W_t$, $a$ and $b$ constant. I am trying to:
derive the ...
3
votes
2answers
252 views
Calculate Ito integral $\int_0^t W_s^2\text dW_s$ from first principles
I am stuck on the 1st equation of the solution where the Wiener process $W_{t_i}^2$ is expanded so that the Itô integral (in terms of infinite sums) looks like the RHS of the first equation of the ...
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0answers
45 views
Do we model stock prices using non-Markovian processes in continuous setting?
In a continuous setting, is it common to model stock prices using non-Markovian processes ? If so, do you have some examples of models ? Or is Markovianity something "embedded" in the ...
2
votes
1answer
146 views
Quasi Monte Carlo and Brownian bridge (how to combine them)
I am trying to understand how quasi Monte Carlo (QMC) and the Brownian bridge (BB) can be combined to price an asset, but I am having a hard time understanding how. I am just considering a European ...
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votes
2answers
58 views
How to calculate expectation and variance of smooth function applied to brownian motion [closed]
I applied a smoothing function to a Brownian equation and obtained a stochastic differential equation by using Ito's lemma. The smoothing function is exp(Bt).
How do I get the expected value and ...
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0answers
91 views
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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1answer
95 views
How to show that a Brownian motion is normally distributed and that the covariance is zero? [closed]
I need help under standing this question. So i have the following given the logarithm of the price of a share of stock is given by
\begin{align*}
p(t)=p(0)+\mu t+\sigma W(t), \quad t \in[0, T]
\end{...
4
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0answers
160 views
Summary of Stochastic Derivatives, Integrals, Expectations, and Variances
I wanted to make a summary table of stochastic functions to improve my understanding. Maybe the following should be a wiki page on this site so others can add functions and examples? Does the ...
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1answer
121 views
Black Scholes price of exotic claim
Given a time horizon N, I want to know the time-$t$ Black-Scholes fair price of $$\int_0^T S_u du$$ where $S_u$ denotes the time-$u$ stock price. I have used the formula I have been given as follows:
$...
-4
votes
1answer
164 views
Using geometric brownian motion for stock price forecasting [closed]
I am doing a dissertation in finance on a maths degree. I wanted to forecast stock prices using artifcial neural networks but none of my tutors are able to supervise so I'm having to do something else....
6
votes
3answers
737 views
Expectation of exponential of 3 correlated Brownian Motion
Consider,
are correlated Brownian motions with a given
I want to calculate the,
,
I can't think of a way to solve this although I have solved an expectation question with only a single exponential ...
4
votes
1answer
227 views
Expectation of $\int_0^t \frac{1}{1+W_s^2} \text dW_s$ [duplicate]
I am trying to calculate the expectation of
$$\int\limits_0^t \frac{1}{1+W_s^2} \text dW_s,$$
where $(W_t)$ is a Wiener process.
I was told that the value of this expectation is zero. Can someone ...
0
votes
1answer
42 views
Multiple underlying brownian motions
I'm trying to find a way to price a triple product forward with payoff XYZ at time T using risk-neutral pricing.
But I don't really have a math background and I have trouble finding a way to account ...
