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.
301
questions
2
votes
2answers
176 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 ...
7
votes
2answers
261 views
Stochastic Integral Graph
As we can represent the integration of $f(x)$ on $[a,b]$ with the graph below,
I was wondering how to represent the following integral with $X(t)$ a Brownian motion, $f(t)$ any function and $t_j = ...
-3
votes
0answers
67 views
Integrals with respect to Brownian motion
I have two questions:
Let $(B_t)$ be a Brownian motion.
Find all constants $a$ and $b$ such that
$X_t =\int_a^t (a +b\frac{u}{t}) \mathrm{d}B_u$ is also a Brownian motion.
Find all constants $a$, $b$...
3
votes
2answers
816 views
probability question about brownian motion
Assume $W_{t}$ is a standard Brownian Motion, calculate the the probability that $W_{t}*W_{2t}$ is negative, i.e., $P(W_{t}*W_{2t}<0)$. I find it tricky to calculate the probability.Thank you.
0
votes
0answers
28 views
Swap rate in the annuity measure and Martingale Representation Theorem
As we know, swap rate evolves as a martingale in the appropriate annuity measure. Martingale representation theorem says if I can find a Brownian motion in the annuity measure and the swap rate is ...
5
votes
2answers
187 views
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 ...
1
vote
2answers
139 views
What is the stock price expectation?
The Hull textbook (and accompanying technical note) says that the expected stock price $\mathbb{E}[S_T]=S_0 \exp(\mu T)$. However, the answers to a British actuarial examination (Q4 for September 2018)...
1
vote
3answers
139 views
How to calculate standard deviation of continuously compounded four-year stock returns?
Currently I am preparing for quant interview and I encounter the following question in Heard on the street.
Question: If the standard deviation of continuously compounded annual stock returns is $...
28
votes
4answers
19k views
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 ...
4
votes
1answer
115 views
Invariance Scaling of Brownian Motion
Prove $\frac{1}{\sqrt{t}}\log\left(\int_0^t \exp(B_s)\mathrm{d}s\right)$ converges to $\sup\limits_{t\in [0,1]}B_t$ in distribution as $t\to\infty$. I have a sense to use scaling invariance, but no ...
1
vote
1answer
72 views
Mathematical proof of $g = \mu - \frac{\sigma^2}{2}$ relationship between CAGR and average returns
I found in a paper the relation between the CAGR and the arithmetic average of returns to be
$$g \sim \mu - \frac{\sigma^2}{2}$$
where g is the geometric average, $\mu$ the arithmetic average and $ ...
1
vote
1answer
74 views
Integrating Brownian Motion [closed]
I just wonder how to integrate standard Brownian motion on time interval $(t, T)$.
Let $Z$ be a standard Brownian motion with mean $0$ and standard deviation $1$, with $dZ^2 = dt$. How to derive the ...
1
vote
1answer
37 views
Moments of discrete Asset Price Model
Say if B is standard Brownian motion then:
$S(t) = S0e^{((𝜇- σ^2)/2)t+σB(t)}$
The mean of this SDE would be
$𝐄[𝑆(𝑡)]=𝑆_0𝑒^{𝜇𝑡}$
I know to do this you use the density function and ...
1
vote
2answers
137 views
Understanding $N(d_1)$ and $N(d_2)$
Firstly, if the solution to geometric Brownian motion is $S_t = S_0 \exp((r-\sigma^2)t + \sigma W_t$ then if I have a payment that is not necessarily a full call option e.g. if the exercise price $K$ ...
0
votes
0answers
30 views
Risk neutral measure in the binomial approximation of geometric Brownian motion
Suppose an asset is described by geometric Brownian motion with a drift, i.e.
$$dS_t = S_t\mu dt + S_t \sigma dW_t$$
for a Wiener process $W_t$ and $S_0=1$. By some consequence of Girsanov's theorem (...
5
votes
1answer
84 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[...
2
votes
0answers
39 views
How to calculate the multiple integrals where the integral domain is based on the sum of normal distribution random variables?
The integral is shown below:
And how to use python to calculate pi (better if we don't need to code for each pi)?
2
votes
1answer
86 views
What is the annualized realized volatility of simulated Brownian motion paths?
I saw this following question in an exam.
Take a Brownian motion simulation with drift 5% and annualized volatility of 20% for a period of 1 year. Then the annualized realized volatility of the ...
0
votes
0answers
32 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
$$dS_t^i = \mu^i(t) S_t^i dt + \sigma^i(K, t) ...
6
votes
2answers
841 views
Risk neutrality correction for Monte Carlo Bootstrapping according to PRIIP regulation for products of category III
The PRIIP (packaged products) regulation prescribes Monte Carlo bootstrapping simulation for calculation of VaR for products of category III (non-linearly leveraged products). The idea is based on ...
2
votes
1answer
65 views
Valuation of Cash-Or-Nothing option
Studying options pricing, I'm stuck with the following problem:
The price of a stock is described by the dynamic:
$$dS_t = \mu\, dt + \sigma\,dW_t$$
Compute the fair price of a Cash or Nothing ...
1
vote
2answers
280 views
Probability that the price of stock following a brownian motion goes under a certain value
The price of the stock XYZ follows a brownian motion pattern with
starting price = 10, μ = 0 and σ = 20 (on annual basis). What's the probability that in 6 months the price is less or equal to 8?
...
1
vote
1answer
43 views
In search of double barrier out option on a BM
We have a BM $X_t$ with $dX_t=\sigma dB_t$ ($X_0$ not necessarily zero!) under the risk neutral measure $\Bbb Q$. Given upper barrier $U$, lower barrier $L$, "strike" $K$ such that $L<X_0<U, L&...
1
vote
1answer
85 views
Proof standard Brownian Motion under change of measure
Let's split the usual time horizon $[0,T]$ like $0=T_{0}<T_{1}<\dots<T_{n}=T$ and consider the bond price $P(t,T_{i})$ for $i=1,...,n$. We assume $$\frac{dP(t,T_{i})}{P(t,_{i})}=r_{t}dt+\xi_{...
2
votes
2answers
259 views
What are some examples of non-solvable SDE where Monte Carlo discretization is necessary
Reading Glasserman - "Monte Carlo Methods in Finance" it says in the introduction to Chapter 6 - Discretization Methods, that moste models arising in derivatives pricing can be simulated only ...
1
vote
0answers
74 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 ...
3
votes
1answer
148 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
1
vote
1answer
110 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, ...
4
votes
1answer
459 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
0answers
92 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
vote
0answers
45 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 ...
7
votes
1answer
167 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
0answers
86 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 ...
3
votes
2answers
129 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 ...
4
votes
1answer
116 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
56 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
128 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 ...
1
vote
1answer
65 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 ...
1
vote
0answers
64 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
167 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:
$$X_{t_n} = X_{t_{n-1}} + \mu X_{t_{n-1}} \Delta t + \sigma X_{t_{n-1}} \...
3
votes
1answer
219 views
Why is it more accurate to simulate ln(S) rather than S?
Let's take a process $S$ that satisfies:
\begin{equation}
dS = \mu S dt + \sigma S dz
\end{equation}
with $dz$ a Wiener process, $\sigma$ the volatility of $S$, $\mu$ the expected return of $S$.
From ...
2
votes
0answers
103 views
For an Ito Process, $d\ln{X} \neq \frac{dX}{X}$ and $(d\ln{X})^2 = (\frac{dX}{X})^2$, but $d\ln{X} \neq \pm \frac{dX}{X}$
In normal calculus we can write $d\ln{x} = \frac{dx}{x}$ since there is no quadratic variation to deal with. This isn't true for stochastic processes, and Ito's Lemma is used to calculate $d\ln{X}$. ...
1
vote
1answer
37 views
CDF&density of stock price modeled by standard brownian motion
Assume that the price of the stock follows the model
$S(t) = S(0) exp (
mt −
((σ^2)/2 )
t + σW(t)
)$
, (1)
where W(t) is a standard Brownian motion; σ > 0, S(0) > 0, m are some constants.
Derive the ...
1
vote
2answers
113 views
How to numerically simulate exponential stochastic integral
For example given an integral
$$
\int^t_0 \exp(aW(t'))\,dt', t\in\mathbb R_+
$$
where $W(t')$ is a standard Wiener process.
I've been very confused about stochastic integrals like $\int^t_0 W(t')\,...
2
votes
1answer
81 views
If S(t) is geometric Brownian motion, what is the distribution of S(t+h)-S(t)?
Suppose we have a geometric Brownian $S(t)$ which follows a lognormal process. Say
$$
\begin{equation}
dS_t = \mu S_t dt + \sigma S_tdW_t
\end{equation}
$$
My question is what is the distribution of $...
8
votes
2answers
9k views
Two correlated brownian motions
Is it true (see here, footnote 2, p.22 / p.14, without proof) that we can obtain two discretized brownian motions $W_t^1, W_t^2$ with correlation $\rho$ by doing
$$d W_t^1 \sim \mathcal N(0,\sqrt{dt}...
1
vote
1answer
161 views
Correlated stock prices and geometric Brownian motion
I have two uncorrelated stocks which follow geometric Brownian motion, as follows
$$\begin{aligned} dS_a &= \mu_aS_adt + \sigma_aS_adW\\ dS_b &= \mu_bS_bdt + \sigma_bS_b dW \end{aligned}$$
...
4
votes
1answer
181 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 ...
10
votes
4answers
5k 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 ...
