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.
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How to compute covariance of brownian motion？
For example, how to compute the covariance of w3 and w2 if Wt means standard Brownian motion.
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1answer
43 views
Simplifying the expectation of the product of two stochastic integrals
Let $f(t, \omega), g(t, \omega)$ be functions that are independent of the increments of the Brownian motion $w(t, \omega)$ in the future. That is, $f(t, \omega), g(t, \omega)$ are independent of $w(t +...
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2answers
839 views
Asymptotic behavior property of geometric Brownian Motion proof
Online I found the asymptotic behavior property of geometric Brownian Motion $X_t$as:
If $\mu$ (drift parameter) is $\ge$ $\sigma^2/2$ where $\sigma$ is the volatility parameter, then $X_t \rightarrow ...
4
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1answer
76 views
Default intensity in Black-Cox model
Consider the model by Black and Cox (Journal of Finance, 1976).
The default intensity function is defined in the usual way: $$h(t) \equiv - \frac{\partial \log P[\tau > t| \mathcal{F}_t]}{\partial ...
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1answer
96 views
What is the expectation of a change in Brownian motion? [closed]
I know $E[W_T-W_t]=0$ but I have a solution which implies this is wrong.
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49 views
How to show the following conditional expectation relation holds for a Brownian motion?
Suppose that $B_k$ stands for a standard Brownian motion process.
\begin{equation}
\mathbb{E}\Big(e^{-w\int_{t}^{S}B_k dk\, -uB_T}\Big| B_t = x\Big)
\end{equation}
where $w$ and $u$ are constants, and ...
1
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1answer
49 views
How to prove that the following is still a Brownian motion [closed]
Given a Brownian motion $B_t$ on a filtered probability space, how can I prove that $W_t=B_t+\alpha t$ is still a Brownian motion, with $\alpha \in \mathbb{R}$? Is it always true? Do I need necessarly ...
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0answers
52 views
Future price in continous time
I am in the following continuous time market:
$S_t^0 = rS_t^0dt$
$S_t^1 = (\mu - \delta) S_t^1dt + \sigma S_t^1 dB_t$
where $r, \mu, \delta$ and $\sigma$ are constant values in $\mathbb{R}$. $\delta$...
3
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1answer
142 views
Conditional probability of Brownian motion (with drift and scaling) hitting barrier
I am trying to understand the pricing of barrier options, and am considering the Brownian motion $\mathrm{d}X_t=a\mathrm{d}t+b\mathrm{d}W_t$, $a$ and $b$ constant. I am trying to:
derive the ...
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1answer
376 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$?
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2answers
224 views
Calculate Ito integral $\int_0^t W_s^2\text dW_s$ from first principles
I am stuck on the 1st equation of the solution where the Wiener process $W_{t_i}^2$ is expanded so that the Itô integral (in terms of infinite sums) looks like the RHS of the first equation of the ...
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1answer
130 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) ...
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0answers
33 views
Do we model stock prices using non-Markovian processes in continuous setting?
In a continuous setting, is it common to model stock prices using non-Markovian processes ? If so, do you have some examples of models ? Or is Markovianity something "embedded" in the ...
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1answer
121 views
Quasi Monte Carlo and Brownian bridge (how to combine them)
I am trying to understand how quasi Monte Carlo (QMC) and the Brownian bridge (BB) can be combined to price an asset, but I am having a hard time understanding how. I am just considering a European ...
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2answers
54 views
How to calculate expectation and variance of smooth function applied to brownian motion [closed]
I applied a smoothing function to a Brownian equation and obtained a stochastic differential equation by using Ito's lemma. The smoothing function is exp(Bt).
How do I get the expected value and ...
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8answers
5k views
Consensus on Cauchy distribution for stock prices
What is the general consensus for using a Cauchy distribution to model stock prices? I can't find much after researching online and wonder if it has been tried and discarded.
My motivation is to find ...
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0answers
79 views
Non attainable claim - Incomplete market
I am wondering whether there is a standard procedure to find a non attainable (i.e. non replicable) asset in an incomplete market.
As an example, let us have the following market ($B = (B^1, B^2, B^3)$...
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1answer
68 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 ...
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1answer
93 views
How to show that a Brownian motion is normally distributed and that the covariance is zero? [closed]
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{...
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1answer
115 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,...
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0answers
153 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 ...
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1answer
115 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:
$...
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1answer
106 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....
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3answers
665 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
217 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
40 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
53 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
200 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) ] -...
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0answers
76 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
4
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1answer
758 views
negative values in geometric brownian motion
A GBM (Geometric Brownian Motion)
$ \frac{dx}{x} = \mu dt + \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. ...
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vote
2answers
851 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:
...
1
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1answer
268 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 ...
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0answers
38 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
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1answer
71 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 Itô's formula.
6
votes
4answers
580 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,...
2
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2answers
173 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 $\...
6
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1answer
627 views
What is a Brownian motion “under the risk-neutral measure”?
I understand that the risk-neutral measure associated with the money-market Numeraire is one under which the discounted price (acc. to the risk-free rate) of any asset is a martingale.
Brownian motion ...
3
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1answer
205 views
Difference between $W_t$ and $X_t= \sqrt{t}Z$
$W_t$ is a brownian motion and $X_t:= \sqrt{t}Z$, where: $Z\sim N(0,1)$.
How to show that for a bounded continuous $f$ process, $$U_t = \int_0^t (f(W_s))ds$$ and $$V_t = \int_0^t (f(X_s))ds$$ have the ...
3
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1answer
148 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 $W_t = x$?
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2answers
161 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 ...
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0answers
96 views
Estimating constant and local volatility based on passage times
Consider a Brownian motion B_t with constant instantaneous volatility σ and zero drift
where ...
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1answer
193 views
Intuition behind prices modeled by Geometric Brownian Motion
Suppose that we model a price $P_t$ to evolve per
$$\frac{dP_t}{P_t}=\mu dt+\sigma dW_t$$
for $\mu\in\mathbb{R}$ and $\sigma>0$. The unique strong solution to this diffusion is
$$P_t=P_0e^{(\mu-\...
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0answers
39 views
Relationship between risk and return for GBM and riskless bond
Suppose we have $S$, a stock following geometric Brownian motion ($dS_t = S_t (\mu dt + \sigma dZ_t)$ for $Z =$ Brownian motion) and $B$, a zero coupon bond with rate $r$, i.e. $dB_t = rB_t dt$.
In ...
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4answers
741 views
Ito Integral of functions of Brownian motion
How does one show that:
$$ \mathbb{E}\left[ \int f(W_s)dWs \right] = 0 $$
For all $f()$ that are powers of $W(s)$?? I assume that one would have to go via the definition of Ito integral and express ...
1
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2answers
146 views
Simulating artificial asset prices: Random walk vs Brownian motion?
How well can each simulate the real-life behavior of stock prices, and what considerations or (dis-)advantages must we be aware of when deciding to use each:
Random walk with drift
Random walk ...
3
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0answers
181 views
Geometric Brownian Motion and Energy-Efficiency Investments
Suppose the payoff $X$ on an investment follows a Geometric Brownian Motion:
$$
dX/X = \mu dt + \sigma dz\ ,
$$
for $dz$ an increment of a Wiener process. I wish to compute the expected present value ...
3
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1answer
69 views
General Dynamics of a Tradable Asset under the Risk Neutral Measure
Is it true that every tradable asset must have a log-normal dynamics under the risk neutral measure where the drift term is the short rate $r$? I.e., is it true that if $X$ is a tradable asset then
$$\...
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0answers
91 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 ...
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0answers
69 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 ...
2
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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. ...
