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.
379
questions
-1
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1answer
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How to show that a Brownian motion is normally distributed and that the covariance is zero?
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{...
5
votes
0answers
96 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 ...
0
votes
1answer
110 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
73 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
596 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
186 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
38 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 ...
0
votes
0answers
46 views
Probability of Hitting time of Brownian motion
Let $B =\{ B(t); t \ge 0\}$ be Brownian motion. What is the probability that $B$
hits state one and then state minus one before time one?
My take: Let $T_x = \inf \{ t\ge 0 : B(t) = x\}$, the first ...
1
vote
2answers
153 views
Integral of the square of Brownian motion using definition of variance
Let $B = \{ B(t); t \ge 0\}$ and let $Z = \{ Z(t); t \ge 0 \}$ where $$Z(t) = \int_0^t B^2(s) ds.$$ How do we find $E[Z(t)]$ and $E[Z^2 (t)]$ in order to get the variance $Var [Z^2(t)] = E[Z^2 (t) ] -...
4
votes
0answers
69 views
Continuous option pricing: Brownian Bridge
I have a question on the proof of the formula of Sup(S) between 2 simulation points.
Do you know how the prove the following formula? Thanks
0
votes
0answers
37 views
Rate of return of a bond price
Assume $B(t, T)$ is a zero coupon bond price and assume that it has dynamics $dB(t, T) = B(t, T)[\mu(t, T)dt + \sigma(t, T)dW_t]$, where $W_t$ is a Brownian motion under $(\Omega ;F; P)$ and $P$ is an ...
0
votes
1answer
63 views
Let $W_t$ denote a standard Brownian motion. Evaluate this integral [closed]
$$
\int_{0}^{t}d(W_{u}^2)
$$
How can I deal with this kind of problem? If there is no function given to apply ItƓ's formula.
1
vote
0answers
89 views
Long-Term Energy Price Modelling: Log Returns, Distributions, Time-Weighting
I wish to forecast energy prices in the long-term (ca. 20 years) for energy-efficiency investments. While I understand that the energy carriers are particularly sensitive to external (geo-political) ...
1
vote
2answers
101 views
Proof of Feller condition for CIR square root process. Any reference?
Could you please give me some reference for the proof of the so-called Feller condition as to a stochastic differential equation of the form:
$$dr_t=a(b-r_t)dt+\sigma\sqrt{r_t}dB_t\tag{1}$$
with $\...
2
votes
0answers
36 views
Relationship between risk and return for GBM and riskless bond
Suppose we have $S$, a stock following geometric Brownian motion ($dS_t = S_t (\mu dt + \sigma dZ_t)$ for $Z =$ Brownian motion) and $B$, a zero coupon bond with rate $r$, i.e. $dB_t = rB_t dt$.
In ...
2
votes
0answers
95 views
Estimating constant and local volatility based on passage times
Consider a Brownian motion B_t with constant instantaneous volatility Ļ and zero drift
where ...
1
vote
1answer
302 views
If $W_t$ is standard Brownian motion, what is $\int_0^T W_t \ln(W_t) dW_t$?
If $W_t$ is standard Brownian motion, what is meant by $\int_0^T W_t dW_t$ in finance?
Furthermore, what then is the meaning of $\int_0^T W_t \ln(W_t) dW_t$?
2
votes
2answers
120 views
Simulating artificial asset prices: Random walk vs Brownian motion?
How well can each simulate the real-life behavior of stock prices, and what considerations or (dis-)advantages must we be aware of when deciding to use each:
Random walk with drift
Random walk ...
3
votes
1answer
64 views
General Dynamics of a Tradable Asset under the Risk Neutral Measure
Is it true that every tradable asset must have a log-normal dynamics under the risk neutral measure where the drift term is the short rate $r$? I.e., is it true that if $X$ is a tradable asset then
$$\...
4
votes
0answers
90 views
Expectation of number of hits by a brownian motion
If we denote $\tau_i$ the sequence of stopping times defined by:
$\tau_i = \inf(t>\tau_{i-1} : |B(t)-B(\tau_{i-1})| > a)$, $\tau_0=0$.
If we denote N the number of stopping times below T.
What ...
2
votes
0answers
66 views
Theoretical Expected Maximum Drawdown vs Empirical Maximum Drawdown
I have been looking at the approach for calculating the expected maximum drawdown of a Brownian Motion [1] and the corresponding function maxddStats in the fBasics package in R [2].
I do not ...
3
votes
0answers
180 views
Geometric Brownian Motion and Energy-Efficiency Investments
Suppose the payoff $X$ on an investment follows a Geometric Brownian Motion:
$$
dX/X = \mu dt + \sigma dz\ ,
$$
for $dz$ an increment of a Wiener process. I wish to compute the expected present value ...
2
votes
0answers
78 views
law of absolute of max of brownian motion
What is the law of $\max\left(|B_t|\right)$ for $t$ in $[0,T]$ and $B_t$ is a Brownian motion?
Any references for properties of this process?
2
votes
0answers
70 views
Brownian function and Clark's formula
I was reading a paper (link) from Richard Bass about Brownian functionals, and came across the following passage :
Let $X_t$ be a Brownian motion, $F_t$ its filtration and $g$ a real-valued function. ...
0
votes
0answers
66 views
Alternatives to Lognormality for negative Prices
If I would want to use a different type of distributions (i.e. to allow for negative prices) f.e. a beta distribution how would I have to start to proceed to apply it f.e. to a SDE of the type of a ...
0
votes
0answers
32 views
Modelling Power Prices
Since electricity prices involve strong seasonality, jump components as well as negative prices and can not be modelled by the GBM, what models/distributions exist, which would allow for modelling ...
0
votes
1answer
75 views
Infinitesimal generator - Is it obtained from a stochastic process or It can construct the process
We can see here that the generator is an operator which can be determined for a stochastic process. But, in the answers and comments here we can see that the brownian motion on sphere can be ...
2
votes
4answers
604 views
Ito Integral of functions of Brownian motion
How does one show that:
$$ \mathbb{E}\left[ \int f(W_s)dWs \right] = 0 $$
For all $f()$ that are powers of $W(s)$?? I assume that one would have to go via the definition of Ito integral and express ...
0
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0answers
42 views
Simulating correlated stock paths to calculate VaR
So I wanted to generate a Monte Carlo simulation for two correlated assets to derive then the VaR as a quantile of the generated distributions. My code is the following, where the input parameters are ...
0
votes
1answer
99 views
Instantaneous correlation in the 2 factor Hull White model
I'm trying to understand which parameter controls the instantaneous correlation in the 2 F HW model. As in, correlation b/w 2 rates observed at the same time. My thinking is as follows:
$$Rate(1)=P(t,...
0
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0answers
59 views
Summary of Pricing Options of Log-Normal Claims Using Black's Formula
Cross posted from here.
Let $B$ be a $Q$-Brownian motion and $X^{s,x}$ given by
$$dX_t = X_t(\mu_t dt + \sigma_t dB_t),\quad X_s = x$$
for $\mu, \sigma$ deterministic. Let $\mu_{s,t}=\int_s^t \mu_u du$...
1
vote
2answers
523 views
How to fix my Ornstein-Uhlenbeck parameter MLE in Python?
I am trying to fit time-series data into an Ornstein-Uhlenbeck process.
Here is my code so far:
...
6
votes
4answers
511 views
Find a formula for the price of a derivative paying $\max(S_T(S_T-K),0)$
Develop a formula for the price of a derivative paying
$$\max(S_T(S_T-K))$$
in the Black Scholes model.
Apparently the trick to this question is to compute the expectation under the stock measure. So,...
4
votes
4answers
270 views
Price of Call Option with or without jumps
Suppose two assets in the Black Scholes world have the same volatility, but different drifts and that one has downward jumps at random times. How does this affect the option prices?
I would have ...
1
vote
0answers
59 views
How to price a down-and-out leveraged barrier call option using Brownian motion?
I am trying to price a type of leveraged down-and-out (LDAO) barrier call option, using geometric Brownian motion.
My python script is below. I am not sure how to correctly model the increasing ...
3
votes
1answer
406 views
Simulate stock prices with Geometric Brownian Motion motion with mu and signa based on 'normal' or continuous compounding?
I have written a simple script for modelling stock prices using Geometric Brownian Motion. The time series I am downloading are daily adjusted closing prices. My aim is to be able to change the ...
2
votes
2answers
152 views
Itos Lemma Derivation notation
So in Hull (2012) the main point is that $\Delta x^2 = b^2 \epsilon ^2 \Delta t + $higher order terms$ $ has a term of order $\Delta t$ and can not be ignored as the Brownian motion exhibits the ...
4
votes
0answers
84 views
mixing fractional Brownian motions
Given two Brownian motions $W_t^1, W_t^2$, we can have them correlated by
$$W_t^1 = \rho W_t^2+\sqrt{1-\rho^2}Z_t$$
where $W_t^{2}$ and $Z_t$ are independent of each other.
My question then: is there ...
0
votes
1answer
95 views
Sampling from SDE
In the case of the classic Geometric Brownian motion
$$dS_t = \mu S_t dt + \sigma S_tdW_t$$
we solve it as
$$ S_t = S_0 \exp\left[ \left(\mu - \frac{\sigma^2}{2}\right)t + \sigma dW_t\right] $$
and ...
1
vote
0answers
30 views
How to expand lognormal approximation of Brownian motion
How can we expand this sum? $\sum_{i=1}^n (e^{rt_i-\frac{1}{2}\sigma^2t_i+\sigma w_{t_i}})^2$ where: $w_{t_i}$ is a standard Brownian motion.
If we let $m_t=e^{-\frac{1}{2}\sigma^2t_i+\sigma w_{t_i}}$...
1
vote
1answer
82 views
Ito's lemma for a Forward
I'm trying to understand the derivation of Ito's process with respect to a Forward $F$ on a stock $S$ that pays a constant dividend yield, say $y$. Stock follows brownian motion $\\$
$dS_{t} = S_{t}(\...
1
vote
1answer
90 views
true or false: the risk-neutral measure is useless in this situation
Example 2 of this Wiki article on the risk-measure describes how a stock price $S_t$ that is modeled with Geometric Brownian motion with drift $\mu$
$$
dS_t = \mu S_t dt + \sigma S_t dW_t
$$
can be ...
0
votes
0answers
46 views
How to find the derivative for a multi-factor geometric brownian motion model
Does anyone know how to find the derivative for a multi-factor geometric brownian motion model $ \frac { dS_{i}}{S_{i}} $. I have seen solutions for the standard GBM model however I suspect that the ...
1
vote
1answer
69 views
Properties of integrated GBM
(I asked this question on MSE but I think it might have more success here)
Good day,
I was going over some exercises and I stumbled upon a question that, for its solution, requires me to find/...
2
votes
1answer
91 views
Brownian motion and Stochastic Integration
I have two questions relating stochastic integration which perhaps could be answered together.
First question:
First of all, I don't really understand why we can't use Riemann-Stieltjes integration ...
0
votes
0answers
24 views
Minimal bounds to enclose most sample paths of a GBM (Geometric Brownian Motion)
For a (generalized) Brownian motion $Y = F(t,W)$, starting at $InitialValue$ and running for a total of $T$ time, if I want to "enclose" (in a visual way) "most" of the possible sample paths, I could ...
1
vote
1answer
63 views
Sample path simulation using two random variables
I was wondering if there is a way of generating a sample path of a Geometric Brownian Motion using two independent standard normal random variables instead of just one.
The exact scheme that uses ...
0
votes
0answers
52 views
Compo/Quanto Adjustment & Multivariate Ito
Related to the issue that I have raised here, I am facing another question. As the rule here is 1 question / 1 post, I take the opportunity to ask it below:
By exploring StackExchange, I noticed the ...
3
votes
1answer
143 views
Conditional distribution of $X_t = \int_0^t W_s \mathrm{d}s$
What is the conditional distribution of $$X_t = \int_0^t W_s \mathrm{d}s$$with respect to $W_t = x$?
2
votes
1answer
98 views
Exact solution stock price with Vasicek interest rate model
Define two correlated stock price- and interest rate (Vasicek) processes, governed by the Wiener processes $W^{S}(t)$ and $W^{r}(t)$
$$dS(t)=r(t)S(t)dt+\sigma S(t)dW^{S}(t)$$
$$dr(t)=\kappa(\theta-r(...
