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.
3
votes
2answers
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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 $$\frac{dS_t}{S_t}=\mu dt+...
0
votes
0answers
44 views
Euler scheme to simulate trajectories + Python Code
Question: Euler scheme to simulate trajectories of S + Python code to get independent trajectories of S?
For solving a problem I have the following assumptions:
stochastic basis (Ω,FT,(Ft)t≥0,P) ...
5
votes
1answer
59 views
Distribution of time integral of Brownian motion squared (where the Brownian motion occurs in square root time)?
Let $I_t = \int_0^t W_{\sqrt{u}}^2du$. What is the distribution of $I$?
If I recall correctly, if the Brownian motion were instead $W_u$, then it would be $I_t \sim N\left(\frac{t^2}{2},\frac{t^4}{3}\...
4
votes
1answer
48 views
Discounted asset price is martingale in BS model
I want to verify that the discounted stock price process $\mathrm{e}^{-r(T-t)}V(S_t,t)$ is a martingale in the BS-model. Using Ito's formula and the BS-PDE I get that
$$
\mathrm{d}\mathrm{e}^{-r(T-t)}...
3
votes
2answers
84 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
75 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
56 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
46 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
118 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:
$$\...
1
vote
0answers
32 views
Model of asset substitution/risk shifting in continuous time
Consider a firm with cash flows $X_t$, which under a risk-neutral probability measure, follows a geometric brownian motion:
$$dX_t = X_t[(r-\beta)dt + \sigma dZ_t]$$
where $r>0$ is the risk-free ...
1
vote
1answer
89 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
63 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$...
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votes
2answers
63 views
0
votes
1answer
68 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
88 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$.
0
votes
0answers
48 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
141 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
212 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
63 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
83 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
135 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
221 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
78 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
0answers
25 views
About a Trivariate Density of Brownian motion, its local and occupation times
I need help on a technical detail in the article by Ioannis Karatzas and Steven E. Shreve (1984) titled: " Trivial density of Brownian motion, its local time and occupation,with application to ...
0
votes
0answers
22 views
Combining two returns forecasts of different lengths
Hi Quantitative Finance stack exchange,
I'm analyzing a price process. I've made two forecasts, namely
$Ret_{\text{10min}}$ = the return after 10 mins and $Ret_{\text{20min}}$ = the return after 20 ...
0
votes
0answers
25 views
About a formula concerning the occupation time of a Brownian motion (The arc-sine formula)
Let the process $Z_t \buildrel\textstyle\over={W_t}+ {\lambda}t $, $t\geq 0$ a brownian Motion with Drift and $A_{T}^{+k, Z}$ his occupation time above the barrier $k$ defined by
$$A_{T}^{+k, Z}=\...
0
votes
1answer
73 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
32 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
72 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
180 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
155 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 ...
3
votes
1answer
97 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
51 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 $\...
-4
votes
1answer
42 views
Which expression of $S_t$ to use under the Black-Scholes model?
I am currently looking at example exam questions relating to the evolution of a stock price under the Black-Scholes model. However, I am confused by seemingly inconsistent expressions used for the ...
2
votes
1answer
141 views
Calculating the stochastic integral of $\exp(-rt)S_t$
I am currently reading lecture notes which aim to show that if
$$
S_t = S_0 \exp (\mu t + \sigma W_t)
$$
then, under the probability measure $\tilde{\mathbb{P}}$ with density
$$
\gamma_T = \exp (c W_T ...
1
vote
1answer
138 views
Question about the Cameron-Martin-Girsanov (CMG) theorem
Within my lecture notes, the following definition of the CMG theorem is given:
Under the probability measure $\mathbb{\tilde{P}}$ with density $\gamma_T = \exp(cW_T - \frac{c^2}{2}T)$, the process $...
2
votes
0answers
168 views
Applying Ito's formula to complex functions
Within my lecture notes, the following definition is given:
We say that the stochastic process $X_t$ has stochastic differential
$$
dX_t = b_t dt + \sigma_t dW_t
$$
if and only if
$$
X_t = ...
1
vote
0answers
56 views
Definition request - Brownian Motion Characterised by Sharpe Ratio
What is the Stochastic Differential Equation for a "Brownian Motion Characterised by Sharpe Ratio"?
I saw it in a paper ("Lessons from the Mortician: volatility modulation") and the authors do not ...
3
votes
0answers
103 views
Prove the Markov property for the stochastic process $Y^x_t$
Prove the Markov property for the stochastic process $Y^x_t=xe^{at+bW_t}$
Given a function $u(t,x)=\mathbb{E}[f(Y^*_t)]$ with $Y^*_0=x$. For $s<t$ we have $\mathbb{E}[f(Y^*_t)]=u(t-s,Y^*_s)$ by ...
8
votes
3answers
256 views
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 ...
0
votes
1answer
72 views
Calculate $Cov(e^ {B_t} ,e^{B_s})$
Let $(B_t)_{t \geq0} $ be a Brownian Motion. Calculate $Cov(e^ {B_t} ,e^{B_s})$
I would verify the following solution which the result looks a bit weird.
My solution: let $0 \leq s \leq t$.
$$Cov(e^ {...
1
vote
0answers
53 views
Monte Carlo Pricer for Express Certificate delivers wrong price [Mathematica]
So I wanted to price the following Express Certificate with this specific payout structure:
If S1 > S0 -> 105.25 , else ->
If S2 > 0.95*S0 -> 110.5 , else ->
If S3 > 0.9*S0 -> 115.75 , else ->
If ...
1
vote
3answers
117 views
Browian motion: $P(B_1<4 | B_2 =1)$
I want to calculate $P(B_1<4 | B_2 =1)$ for the B.M.
What I tried:
$P(B_1<4 | B_2 =1)=P(B_1 - B_2 < 3- B_2 | B_2 =1)$
but I cant use any independence to calculate further.
3
votes
2answers
254 views
About SDE of Geometric Brownian Motion
It's known that most of the financial assets are subject to Geometric Brownian Motion, which satisfies the following equations:
$\frac{dS}{S}=\mu dt + \sigma dX$ (1)
$S_t = S_0 e^{(\mu + \frac{1}{2}...
3
votes
2answers
152 views
Isn't GBM the equivalent of adding infinitessimally small normally distributed returns?
The classic treatment of GBM for asset pricing leads to a point where eventually one gets a solution that is the same as assuming an underlying arithmetic Brownian motion, $X_t$, which has (over unit ...
4
votes
1answer
795 views
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 ...
0
votes
0answers
34 views
Continuous returns in BS-market
Let's assume we are in a Black-Scholes market. The price processes in
the BS-market are given by $dP_0(t)=P_0(t)rdt, P_0(0)=1$
$dP_i(t)=P_i(t)\cdot(\mu_i dt + \sigma_idW(t))=P_i(t)\cdot \left(\...
2
votes
0answers
114 views
What is a stochastic processes which reasonably captures commodity price dynamics?
I ran into a stumbling block earlier when I tried to price stochastic annuities (see Asian options). This is actually technically an acturial problem, but is well adapted to the techniques of quant ...
3
votes
0answers
463 views
Properties of Geometric Brownian Motion Integrated w.r.t. Time (i.e., distribution of a Yor Process)
Let $S_t$ be a process which follows a Geometric Brownian Motion:
$\frac{dS_\tau}{S_\tau} = \mu \,d\tau + \sigma \,dW_\tau$
By Ito's lemma, we have:
$S_T = S_t e^{(\mu-{\sigma^2 \over 2})(T-t) + \...
