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.
288
questions
1
vote
1answer
50 views
What is the annualized realized volatility of simulated Brownian motion paths?
I saw this following question in an exam. My intuition is D, but I wonder if there's a trick. (The question can't be that straightforward?)
Take a Brownian motion simulation with drift 5% and ...
0
votes
0answers
27 views
Negative drift when calibrating GBM parameters
Setup for question:
Consider a basket of $N$ stocks $\{S^1, S^2, \dots, S^N\}$. For fixed strike $K$, each stock in the basket, $S^i$, follows the SDE
$$dS_t^i = \mu^i(t) S_t^i dt + \sigma^i(K, t) ...
1
vote
2answers
264 views
Probability that the price of stock following a brownian motion goes under a certain value
The price of the stock XYZ follows a brownian motion pattern with
starting price = 10, μ = 0 and σ = 20 (on annual basis). What's the probability that in 6 months the price is less or equal to 8?
...
2
votes
1answer
56 views
Valuation of Cash-Or-Nothing option
Studying options pricing, I'm stuck with the following problem:
The price of a stock is described by the dynamic:
$$dS_t = \mu\, dt + \sigma\,dW_t$$
Compute the fair price of a Cash or Nothing ...
1
vote
1answer
36 views
In search of double barrier out option on a BM
We have a BM $X_t$ with $dX_t=\sigma dB_t$ ($X_0$ not necessarily zero!) under the risk neutral measure $\Bbb Q$. Given upper barrier $U$, lower barrier $L$, "strike" $K$ such that $L<X_0<U, L&...
1
vote
1answer
75 views
Proof standard Brownian Motion under change of measure
Let's split the usual time horizon $[0,T]$ like $0=T_{0}<T_{1}<\dots<T_{n}=T$ and consider the bond price $P(t,T_{i})$ for $i=1,...,n$. We assume $$\frac{dP(t,T_{i})}{P(t,_{i})}=r_{t}dt+\xi_{...
1
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0answers
67 views
How to solve these SDE Problems
Quuestion1.
I make a solution $r(t)$ used by Ito's lemma
$r(t)=e^{-a t}r(0)+\int _{0}^{t}e^{a (s-t)}\theta (s)ds+\sigma e^{-a t}\int _{0}^{t}e^{a u}\,dB^{1}(u)$
Is this right?
and I try to make ...
1
vote
1answer
103 views
Brownian Motions theorems
I know that if $W$ and $W′$ are two independent brownian motions, then $dWt \ dWt′$ = 0.
How can I prove/demonstrate this theorem?
Additionaly, how can we prove that if $W$ and $W′$ are dependent, ...
3
votes
1answer
140 views
Correlation between stock prices given correlation between returns
assume I have two stocks with known volatilities and a known correlation coefficient of returns - does anyone know how to determine the correlation between the prices and NOT THE RETURNS
0
votes
0answers
75 views
Predicting time series using Jump Diffusion model and Neural Networks
I am trying to understand the difference between using Jump diffusion model and Neural Networks or more precisely LSTM to predict time series data regardless what that data contains for example a ...
1
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0answers
41 views
Brownian motion from price-series, what is the time step?
If I assume a given empirical price-series is a brownian motion, I can estimate the drift and standard deviation as long as I know what the time step was when the process was 'generated'.
But since ...
3
votes
1answer
436 views
Ito`s Lemma problem
Can someone help me with calculus for this problem.
I have these 3 equations and with Ito`s Lemma I have to find $dXt$.
\begin{cases} dY= μYdt+σYdB
\\ X=\frac{1}{2}cY\\ dc =-aαcdt\end{cases}
0
votes
1answer
80 views
Geometric Brownian Motion - Price Probabilities
I am modeling a stock price that follows Geometric Brownian Motion and have the following:
$E(X)$ = .16 (16%)
$\sigma$ = .24 (24%)
$X_0$ = 95
$T$ = 1 (12 months)
I am trying to find the ...
1
vote
0answers
78 views
The conditional expectation of a geometric brownian motion
In this question it states that
$$\mathbb{E}[e^{\sigma(W_t-W_s)}|\mathcal{F}_s] = \mathbb{E}[e^{\sigma(W_t-W_s)}],$$
and I assume that $0 \leq s \leq t$.
The accepted answer states that this step is ...
4
votes
1answer
108 views
Geometric Brownian Motion unable to model / predict jumps
In my finance course, we were talking about the flaws of modelling Stock Prices with the geometric Brownian Motion. According to my professor:
"Since the geometric Brownian Motion has continous time ...
2
votes
0answers
54 views
Novikov condition for Vasicek process
Suppose that we have a money account $S^{(0)}$ with dynamics
\begin{align}
dS^{(0)}_{t} = r_{t} S^{(0)}_{t}\, dt,
\end{align}
where
\begin{align}
dr_t = a(b-r_t)\, dt + \sigma_{r} \, dW_t^{(0)}.
\...
1
vote
1answer
120 views
Bitcoin dynamics - C++ Simulation
I would like perform a simulation of Bitcoin future prices given a sample of the 4 past years (2014-2018). My problem is that I do not know what model to use! For common stocks I used the geometric ...
5
votes
0answers
80 views
Distribution of portfolio values with constant spending rate
If your portfolio is invested in an asset that follows a geometric Brownian motion, and you withdraw a constant dollar amount at the beginning of each year, is there an approximate analytical ...
3
votes
2answers
125 views
Find the brownian motion associated to a linear combination of dependant brownian motions
I have $N$ correlated standard one-dimensional Brownian motions $W_1,\ldots,W_N$ with correlation matrix $\rho$ and I consider the process $Z_t \equiv \sum_{i=1}^N \mu_i (t) W_t$ where the $\mu_i$ are ...
0
votes
0answers
58 views
Brownian motion for modelling future asset values
Assume that an asset price $S$ is given by a Brownian motion. Argue from the definition why it is not possible to predict future values of the asset based on the past values of $S$.
I am not sure ...
1
vote
0answers
95 views
Geometric Brownian Motion with Dividends
I am working on a problem and had a quick question. I understand that for Geometric Brownian Motion we use the formula:
$$X_{t_n} = X_{t_{n-1}} + \mu X_{t_{n-1}} \Delta t + \sigma X_{t_{n-1}} \...
2
votes
0answers
78 views
For an Ito Process, $d\ln{X} \neq \frac{dX}{X}$ and $(d\ln{X})^2 = (\frac{dX}{X})^2$, but $d\ln{X} \neq \pm \frac{dX}{X}$
In normal calculus we can write $d\ln{x} = \frac{dx}{x}$ since there is no quadratic variation to deal with. This isn't true for stochastic processes, and Ito's Lemma is used to calculate $d\ln{X}$. ...
1
vote
1answer
37 views
CDF&density of stock price modeled by standard brownian motion
Assume that the price of the stock follows the model
$S(t) = S(0) exp (
mt −
((σ^2)/2 )
t + σW(t)
)$
, (1)
where W(t) is a standard Brownian motion; σ > 0, S(0) > 0, m are some constants.
Derive the ...
1
vote
1answer
54 views
Expectation and variance of standard brownian motion
Assuming that the price of the stock follows the model
$ S(t) = S(0) exp (
mt −
(σ^2/
2)
t + σW(t)
)
,
$
where W(t) is a standard Brownian motion; σ > 0, S(0) > 0, m are some ...
2
votes
1answer
68 views
If S(t) is geometric Brownian motion, what is the distribution of S(t+h)-S(t)?
Suppose we have a geometric Brownian $S(t)$ which follows a lognormal process. Say
$$
\begin{equation}
dS_t = \mu S_t dt + \sigma S_tdW_t
\end{equation}
$$
My question is what is the distribution of $...
1
vote
2answers
106 views
How to numerically simulate exponential stochastic integral
For example given an integral
$$
\int^t_0 \exp(aW(t'))\,dt', t\in\mathbb R_+
$$
where $W(t')$ is a standard Wiener process.
I've been very confused about stochastic integrals like $\int^t_0 W(t')\,...
4
votes
1answer
167 views
Ito's Lemma for this problem
I'm attempting to prove a lemma from a paper, in the context of optimal contracts.
$r,\rho,\gamma,\alpha,\sigma$ are all known constants.
$dR_t = (\alpha + r)dt + \sigma dZ_t$ where $Z_t$ is a ...
1
vote
1answer
136 views
Correlated stock prices and geometric Brownian motion
I have two uncorrelated stocks which follow geometric Brownian motion, as follows
$$\begin{aligned} dS_a &= \mu_aS_adt + \sigma_aS_adW\\ dS_b &= \mu_bS_bdt + \sigma_bS_b dW \end{aligned}$$
...
1
vote
0answers
107 views
On quadratic covariation
I ran through an equality in a paper I was reading but couldn't check if it is correct.
Let $W^1_t$, $W^2_t$ and $W^3_t$ be three brownian motions, not necessarily independent, is it true that the ...
1
vote
1answer
150 views
Measure of a Brownian motion = normal distribution?
Consider some model where the process increments are normally distributed, e.g. Vasicek:
$$dr(t) = \left(\theta - ar(t)\right)dt + \sigma dW(t).$$
We usually say that $W(t)$ is a Brownian motion ...
3
votes
1answer
156 views
Differential of integral of Wiener process over time
I am trying to compute this quantity:
$\frac{d}{dt}\int_{0}^{t} W_s ds $
Where $W_t$ is a Wiener process. Is there a theorem which tells how this can be computed?
I have tried https://en.wikipedia....
3
votes
1answer
214 views
Why is it more accurate to simulate ln(S) rather than S?
Let's take a process $S$ that satisfies:
\begin{equation}
dS = \mu S dt + \sigma S dz
\end{equation}
with $dz$ a Wiener process, $\sigma$ the volatility of $S$, $\mu$ the expected return of $S$.
From ...
2
votes
1answer
170 views
How to get the probability of exercise call option in Black-Scholes model?
From Black-Scholes model, I'm trying to prove:
$p(S_t>K) = N(d_2)$
No luck yet!
Can anyone suggest a reference showing that how to obtain this equation?
All I get is:
$S_t = S_0e^{ (\mu-0.5 \...
2
votes
1answer
112 views
Dynamical Behavior of Hurst Exponent
I feel that the dynamic of financial market is not really modeled by standard Brownian motion, but fractional Brownian motion or even multifractional Brownian motion.
I have read some references on ...
2
votes
4answers
186 views
Basic book on stochastic calculus, Itô and jump processes and Brownian Motion
I was looking for a good book that explains at beginner-level the basic of stochastic calculus, probability and random variables, Itô and jump processes as well as Brownian Motion.
At university we ...
4
votes
1answer
115 views
Expected payoff at future time
Let $a$, $b$, $c$, and $e$ be constants, $W_1$ and $W_2$ be Brownian motions with correlation $\rho$, and $f(t)$ and $g(t)$ be deterministic functions of time. Let $X$ satisfy $$d(X(t))=(aX(t)+ef(t)g(...
3
votes
1answer
78 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
138 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
$$S_t = S_0\exp\left(\left(\mu - \frac{\sigma^2}{...
1
vote
1answer
213 views
negative values in geometric brownian motion
A GBM
$ \frac{dx}{x} = \mu dx + \sigma dW $
solves to
$x_t = x_o e^{(\mu - \sigma^2)t + \sigma W_t}$
From the solution, it is clear that $x_t$ cannot become negative. However, it is not so clear ...
3
votes
1answer
113 views
Limits of integration when applying stochastic Fubini theorem to Brownian motion
I'm looking at the solution below from Quantuple, it's a nice, succinct solution but I'm confused about how the limits of the integrals in the second line come from. Could someone please elaborate on ...
3
votes
2answers
264 views
What is the Brownian motion in the model for the return of a stock price trying to capture?
I have read that in the derivation of the Black-Scholes PDE, we assume that the return of a stock $S$ is given by
$$\frac{dS}{S}=\mu dt+\sigma dB$$
where $\mu$ is the average growth of $S$, $\sigma$ ...
4
votes
0answers
90 views
Reference request for research on the maximum drawdown **ratio** (NOT value)
Let's suppose the asset price process follows a Geometric Brownian motion $S_t \sim GBM(\mu, \sigma),\,t\ge 0$, and define the two process:
$$
\begin{align}
\text{MSF}_t &:= \max_{\tau\in[0,t]} S_\...
3
votes
2answers
46 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 $$\frac{dS_t}{S_t}=\mu dt+...
5
votes
1answer
226 views
Distribution of time integral of Brownian motion squared (where the Brownian motion occurs in square root time)?
Let $I_t = \int_0^t W_{\sqrt{u}}^2du$. What is the distribution of $I$?
If I recall correctly, if the Brownian motion were instead $W_u$, then it would be $I_t \sim N\left(\frac{t^2}{2},\frac{t^4}{3}\...
4
votes
1answer
91 views
Discounted asset price is martingale in BS model
I want to verify that the discounted stock price process $\mathrm{e}^{-r(T-t)}V(S_t,t)$ is a martingale in the BS-model. Using Ito's formula and the BS-PDE I get that
$$
\mathrm{d}\mathrm{e}^{-r(T-t)}...
3
votes
2answers
102 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
167 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
66 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
144 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
105 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 ...
