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.
1
vote
1answer
31 views
CDF&density of stock price modeled by standard brownian motion
Assume that the price of the stock follows the model
$S(t) = S(0) exp (
mt −
((σ^2)/2 )
t + σW(t)
)$
, (1)
where W(t) is a standard Brownian motion; σ > 0, S(0) > 0, m are some constants.
Derive the ...
0
votes
1answer
27 views
Expectation and variance of standard brownian motion
Assuming that the price of the stock follows the model
$ S(t) = S(0) exp (
mt −
(σ^2/
2)
t + σW(t)
)
,
$
where W(t) is a standard Brownian motion; σ > 0, S(0) > 0, m are some ...
2
votes
1answer
48 views
If S(t) is geometric Brownian motion, what is the distribution of S(t+h)-S(t)?
Suppose we have a geometric Brownian $S(t)$ which follows a lognormal process. Say
$$
\begin{equation}
dS_t = \mu S_t dt + \sigma S_tdW_t
\end{equation}
$$
My question is what is the distribution of $...
1
vote
2answers
81 views
How to numerically simulate exponential stochastic integral
For example given an integral
$$
\int^t_0 \exp(aW(t'))\,dt', t\in\mathbb R_+
$$
where $W(t')$ is a standard Wiener process.
I've been very confused about stochastic integrals like $\int^t_0 W(t')\,...
4
votes
1answer
115 views
Ito's Lemma for this problem
I'm attempting to prove a lemma from a paper, in the context of optimal contracts.
$r,\rho,\gamma,\alpha,\sigma$ are all known constants.
$dR_t = (\alpha + r)dt + \sigma dZ_t$ where $Z_t$ is a ...
1
vote
1answer
45 views
Correlated stock prices and geometric Brownian motion
I have two uncorrelated stocks which follow geometric Brownian motion, as follows
$$\begin{aligned} dS_a &= \mu_aS_adt + \sigma_aS_adW\\ dS_b &= \mu_bS_bdt + \sigma_bS_b dW \end{aligned}$$
...
1
vote
0answers
92 views
On quadratic covariation
I ran through an equality in a paper I was reading but couldn't check if it is correct.
Let $W^1_t$, $W^2_t$ and $W^3_t$ be three brownian motions, not necessarily independent, is it true that the ...
1
vote
1answer
126 views
Measure of a Brownian motion = normal distribution?
Consider some model where the process increments are normally distributed, e.g. Vasicek:
$$dr(t) = \left(\theta - ar(t)\right)dt + \sigma dW(t).$$
We usually say that $W(t)$ is a Brownian motion ...
3
votes
1answer
120 views
Differential of integral of Wiener process over time
I am trying to compute this quantity:
$\frac{d}{dt}\int_{0}^{t} W_s ds $
Where $W_t$ is a Wiener process. Is there a theorem which tells how this can be computed?
I have tried https://en.wikipedia....
3
votes
1answer
156 views
Why is it more accurate to simulate ln(S) rather than S?
Let's take a process $S$ that satisfies:
\begin{equation}
dS = \mu S dt + \sigma S dz
\end{equation}
with $dz$ a Wiener process, $\sigma$ the volatility of $S$, $\mu$ the expected return of $S$.
From ...
2
votes
1answer
69 views
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 \...
2
votes
1answer
84 views
Dynamical Behavior of Hurst Exponent
I feel that the dynamic of financial market is not really modeled by standard Brownian motion, but fractional Brownian motion or even multifractional Brownian motion.
I have read some references on ...
2
votes
4answers
147 views
Basic book on stochastic calculus, Itô and jump processes and Brownian Motion
I was looking for a good book that explains at beginner-level the basic of stochastic calculus, probability and random variables, Itô and jump processes as well as Brownian Motion.
At university we ...
4
votes
1answer
106 views
Expected payoff at future time
Let $a$, $b$, $c$, and $e$ be constants, $W_1$ and $W_2$ be Brownian motions with correlation $\rho$, and $f(t)$ and $g(t)$ be deterministic functions of time. Let $X$ satisfy $$d(X(t))=(aX(t)+ef(t)g(...
3
votes
1answer
65 views
The conditional mean of a product of standard Brownian motions [closed]
Suppose $\{W_t, t>=0\}$ is a Standard Brownian Motion. How to compute
$ \mathbb{E} \left[ W_2 W_3 \vert W_1 =0 \right]$? We know $ W_2 \vert W_1 = 0 \sim N(0,1)$ and $ W_3 \vert W_1 = 0 \sim N(0,...
1
vote
1answer
60 views
How to calculate mean and volatility parameters for Geometric Brownian motion?
Say I have a time series $S_K$ for monthly asset prices for the last 30 years. I want to run a monte carlo simulation using geometric brownian motion
$$S_t = S_0\exp\left(\left(\mu - \frac{\sigma^2}{...
0
votes
1answer
88 views
negative values in geometric brownian motion
A GBM
$ \frac{dx}{x} = \mu dx + \sigma dW $
solves to
$x_t = x_o e^{(\mu - \sigma^2)t + \sigma W_t}$
From the solution, it is clear that $x_t$ cannot become negative. However, it is not so clear ...
3
votes
1answer
81 views
Limits of integration when applying stochastic Fubini theorem to Brownian motion
I'm looking at the solution below from Quantuple, it's a nice, succinct solution but I'm confused about how the limits of the integrals in the second line come from. Could someone please elaborate on ...
3
votes
2answers
204 views
What is the Brownian motion in the model for the return of a stock price trying to capture?
I have read that in the derivation of the Black-Scholes PDE, we assume that the return of a stock $S$ is given by
$$\frac{dS}{S}=\mu dt+\sigma dB$$
where $\mu$ is the average growth of $S$, $\sigma$ ...
4
votes
0answers
85 views
Reference request for research on the maximum drawdown **ratio** (NOT value)
Let's suppose the asset price process follows a Geometric Brownian motion $S_t \sim GBM(\mu, \sigma),\,t\ge 0$, and define the two process:
$$
\begin{align}
\text{MSF}_t &:= \max_{\tau\in[0,t]} S_\...
3
votes
2answers
44 views
What are the underlying events that the random variables map to the real line in the derivation of the Black-Scholes PDE?
When we first try and set up a model for the evolution of S, the value of the underlying stock, I have seen in a lot of textbooks that they model the evolution by the formula $$\frac{dS_t}{S_t}=\mu dt+...
5
votes
1answer
88 views
Distribution of time integral of Brownian motion squared (where the Brownian motion occurs in square root time)?
Let $I_t = \int_0^t W_{\sqrt{u}}^2du$. What is the distribution of $I$?
If I recall correctly, if the Brownian motion were instead $W_u$, then it would be $I_t \sim N\left(\frac{t^2}{2},\frac{t^4}{3}\...
4
votes
1answer
58 views
Discounted asset price is martingale in BS model
I want to verify that the discounted stock price process $\mathrm{e}^{-r(T-t)}V(S_t,t)$ is a martingale in the BS-model. Using Ito's formula and the BS-PDE I get that
$$
\mathrm{d}\mathrm{e}^{-r(T-t)}...
3
votes
2answers
93 views
Random Walk with normal increments and n time periods why is the increment $\sqrt{(t/n)}$?
Question is basically in the title. I have found several sources stating that $R_i = \sqrt{\frac{t}{n}}$, but I couldn't find the intuition behind taking the square root. And it seems to be crucial ...
0
votes
2answers
110 views
Integral of Wiener process over time
This should hopefully be an easy question to answer, but I am new to Stochastic Calculus and am gapping as to why the following is true, for a brownian motion $W_t$:
$$d(\int W_t dt ) = W_t dt$$
I ...
3
votes
1answer
56 views
Autocovariance of increments of a semimartingale
Say that $X_t$ is an Itō process with
\begin{equation}
dX_t = \mu_t dt + \sigma_t dW_t
\end{equation}
where $\mu_t$ and $\sigma_t$ are adapted processes.
Is it always true that
\begin{equation}
E[...
-1
votes
1answer
80 views
Expectation of the product of two Brownian motions [closed]
Could you please let me know the steps to follow to get to the solution?
1
vote
1answer
60 views
Does GBM stock price model have E[S(t)] unaffected by volatility?
Many an author claims that, if you model stock prices through GBM, $E[S(t)]=e^{\mu t}$, and the expectation is thus not related to volatility.
I keep running around in circles on this one. First ...
2
votes
1answer
129 views
How to calculate the covariance between two stochastic integrals?
How to calculate the covariance between the integral of a Brownian motion at different times:
$$\text{Cov}\left(\int^{t_1}_0\sigma(t)dW_t,\int^{t_2}_0\sigma(t)dW_t\right)\ ?$$
I know the answer is:
$$\...
1
vote
0answers
37 views
Model of asset substitution/risk shifting in continuous time
Consider a firm with cash flows $X_t$, which under a risk-neutral probability measure, follows a geometric brownian motion:
$$dX_t = X_t[(r-\beta)dt + \sigma dZ_t]$$
where $r>0$ is the risk-free ...
1
vote
1answer
101 views
Fractional Brownian motion references
Does anyone know any good references to understand the fractional Brownian motion and its numerical simulation, preferably applied to finance.Thank you.
-1
votes
1answer
66 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:
$dX_t= \mu_t dt +\sigma_t dW_t$...
-2
votes
2answers
64 views
0
votes
1answer
99 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 ...
3
votes
1answer
142 views
Quadratic variation of an integral of a function of a Brownian motion
I'm asked to find the quadratic variation of the integral
$\int_{0}^{t} W_s^2 ds$.
1
vote
0answers
62 views
Correlated GBM and OU processes
I want to model two different stochastic processes, such that:
$X_t , V_t$ are correlated with coefficient $\rho$. Where:
$\frac{dX_t}{X_t}=\mu_1dt+\sigma_1 dW_{1,t}$ and $dV_t=\theta(\mu_2-V_t)dt+\...
3
votes
1answer
228 views
Calculate drift of Brownian Motion using Euler method
I am working on a project to approximate numerically the solution $X_t$ of a stochastic differential equation (SDE) using the Euler method. I have do to this for the Brownian motion with drift. I am ...
0
votes
2answers
351 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').
\...
1
vote
0answers
64 views
Brownian bridge with time varying volatility
I have a question to ask about the Brownian bridge for a process with deterministic volatility varying over time.
In other words, we have this dynamic: $dS_t = \sigma_{t} * dW_t$. We want to know the ...
0
votes
1answer
107 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 ...
3
votes
1answer
154 views
Conditional Probability - Geometric Brownian Motion
Background
I am trying to find a way to price a variant of a gap option by using closed-end expressions. What makes this option a bit tricky is that it can be exercised at four predetermined dates (t=...
5
votes
2answers
276 views
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 ...
3
votes
1answer
84 views
Using Geometric Brownian Motion for Index Options
As far as I understand, in most of the cases we derive the option valuation assuming that the log-return of the asset is partly driven by its own Brownian motion, and we use Geometric Brownian motion (...
0
votes
1answer
77 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 ...
0
votes
0answers
37 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 ...
1
vote
1answer
85 views
GBM in R giving negative numbers?
I was under the impression that simulations involving geometric brownian motion are not supposed to yield negative numbers. However, I was trying the following Monte Carlo simulation in R for a GBM, ...
2
votes
1answer
227 views
Expectation of an Integral of a function of a Brownian Motion
I would really appreciate some guidance on how to calculate the expectation of an integral of a function of a Brownian Motion.
Let $B(t)$ be a Brownian motion with drift $\mu$ and standard ...
-1
votes
3answers
166 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 ...
5
votes
1answer
131 views
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 $...
1
vote
1answer
53 views
How might I answer this past exam question relating to the value of a European option under the BMS market model?
The following question, which is not homework, was taken from a past paper for a module I will soon be sitting:
Consider a Black-Merton-Scholes stochastic market with drift $\mu = 1$, volatility $\...
