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.
394
questions
3
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1answer
79 views
Help on solving a stochastic differential equation
I am trying to solve the following SDE
$$dX(t)=rdt+aX(t)dW(t),\ t>0$$
$$X(0)=x$$
where W() is a Wiener process and r,a and x real numbers. I have proceeded by using the integrating factor
$$F(t)=...
2
votes
2answers
114 views
Proving that a stochastic process is a martingale using Ito's Lemma
Assume a Wiener process W and a bounded F-adjusted stochastic process a. Show that the following process is a martingale on F
$$X(t)=(\int_{0}^{t}a(s)dW(s))^{2}-\int_{0}^{t}a^{2}(s)ds,\ t\geq0$$
Can ...
2
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1answer
105 views
Likelihood ratio and pathwise sensitivity method for coupled SDEs
I have two coupled SDEs
\begin{align*}
dS_t=rS_tdt+V_tdW_t^{(1)},\\
dV_t=aV_tdt+b(V_t)dW_t^{(2)},\\
\end{align*}
where $W_t^{(1)}$ and $W_t^{(2)}$ are independent Brownian motions, initial input data ...
0
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1answer
33 views
Reason why a European binary call should be worth half of its American counterpart when driftless and out-of-the-money
Exercise 11 of chapter 8 of Mark Joshi's "The concepts and practice of mathematical finance", asks to compare prices of an American and a European digital (binary) calls when out-of-the-...
2
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1answer
52 views
Simplifying the expectation of the product of two stochastic integrals
Let $f(t, \omega), g(t, \omega)$ be functions that are independent of the increments of the Brownian motion $w(t, \omega)$ in the future. That is, $f(t, \omega), g(t, \omega)$ are independent of $w(t +...
4
votes
1answer
83 views
Default intensity in Black-Cox model
Consider the model by Black and Cox (Journal of Finance, 1976).
The default intensity function is defined in the usual way: $$h(t) \equiv - \frac{\partial \log P[\tau > t| \mathcal{F}_t]}{\partial ...
0
votes
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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0answers
50 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
vote
1answer
53 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
votes
1answer
143 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 ...
3
votes
2answers
232 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 ...
1
vote
0answers
36 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 ...
2
votes
1answer
122 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 ...
-1
votes
2answers
55 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 ...
5
votes
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 ($B = (B^1, B^2, B^3)$...
-1
votes
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{...
4
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0answers
154 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
117 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
112 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
668 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
218 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
203 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
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
0
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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.
1
vote
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) ...
2
votes
2answers
177 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 $\...
1
vote
0answers
40 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
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 ...
1
vote
1answer
377 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 $\int_0^T W_t \ln(W_t) dW_t$?
1
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2answers
149 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
70 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
$$\...
4
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0answers
91 views
Expectation of number of hits by a brownian motion
If we denote $\tau_i$ the sequence of stopping times defined by:
$\tau_i = \inf(t>\tau_{i-1} : |B(t)-B(\tau_{i-1})| > a)$, $\tau_0=0$.
If we denote N the number of stopping times below T.
What ...
2
votes
0answers
69 views
Theoretical Expected Maximum Drawdown vs Empirical Maximum Drawdown
I have been looking at the approach for calculating the expected maximum drawdown of a Brownian Motion [1] and the corresponding function maxddStats in the fBasics package in R [2].
I do not ...
3
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0answers
181 views
Geometric Brownian Motion and Energy-Efficiency Investments
Suppose the payoff $X$ on an investment follows a Geometric Brownian Motion:
$$
dX/X = \mu dt + \sigma dz\ ,
$$
for $dz$ an increment of a Wiener process. I wish to compute the expected present value ...
2
votes
0answers
116 views
law of absolute of max of brownian motion
What is the law of $\max\left(|B_t|\right)$ for $t$ in $[0,T]$ and $B_t$ is a Brownian motion?
Any references for properties of this process?
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0answers
70 views
Brownian function and Clark's formula
I was reading a paper (link) from Richard Bass about Brownian functionals, and came across the following passage :
Let $X_t$ be a Brownian motion, $F_t$ its filtration and $g$ a real-valued function. ...
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0answers
70 views
Alternatives to Lognormality for negative Prices
If I would want to use a different type of distributions (i.e. to allow for negative prices) f.e. a beta distribution how would I have to start to proceed to apply it f.e. to a SDE of the type of a ...
0
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1answer
68 views
Modelling Power Prices
Since electricity prices involve strong seasonality, jump components as well as negative prices and can not be modelled by the GBM, what models/distributions exist, which would allow for modelling ...
0
votes
1answer
82 views
Infinitesimal generator - Is it obtained from a stochastic process or It can construct the process
We can see here that the generator is an operator which can be determined for a stochastic process. But, in the answers and comments here we can see that the brownian motion on sphere can be ...
2
votes
4answers
763 views
Ito Integral of functions of Brownian motion
How does one show that:
$$ \mathbb{E}\left[ \int f(W_s)dWs \right] = 0 $$
For all $f()$ that are powers of $W(s)$?? I assume that one would have to go via the definition of Ito integral and express ...
0
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0answers
43 views
Simulating correlated stock paths to calculate VaR
So I wanted to generate a Monte Carlo simulation for two correlated assets to derive then the VaR as a quantile of the generated distributions. My code is the following, where the input parameters are ...
0
votes
1answer
115 views
Instantaneous correlation in the 2 factor Hull White model
I'm trying to understand which parameter controls the instantaneous correlation in the 2 F HW model. As in, correlation b/w 2 rates observed at the same time. My thinking is as follows:
$$Rate(1)=P(t,...
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
875 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
586 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
316 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 ...
