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.
487 questions
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Trying to derive some properties of geometric brownian motion
I am trying to derive some properties of geometric brownian motion:
$dS_t = \mu S_t dt + \sigma S_t dW_t$
I am interested in analyzing paths that 'survive' a lower boundary $X$ i.e. always stay above $...
- 121
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Closed form for expected CRRA utility under a GBM
I was wondering if my proof for deriving a closed form solution for the expected CRRA utility of the 10-year distribution of a GBM could be checked. The proof goes as follows (I'll assume prior ...
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Estimate of GBM Return Variance
I typically see people define realized variance as the squared difference in log returns, i.e.
$$RVar = \frac{1}{T} \sum_{n=1}^N \log \left( \frac{S_{n}}{S_{n-1}} \right)^2$$
where $t_n - t_{n-1} = \...
- 23
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Complete market price and incomplete market price specification
We know that if a liquid market of an asset exists, then the standard derivative pricing theorem implies an equivalent martingale measure exists, not necessarily unique, under which the discounted ...
- 419
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ABM Crossing Times
Suppose I have a process that follows an arithmetic brownian motion
$dX_t = \sigma dW_t$
How do I calculate, within a certain interval $\Delta t$
, the expected number of times that the process will &...
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2
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Volatility in simulated paths different to monte carlo parameters
I am trying to convince myself that I have set up my monte carlo simulation correctly by looking at the results and trying to get them to agree with the model parameters. Please help me understand ...
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How is Itô's Lemma connected to Messmore's Variance Drain?
How does Itô's Lemma explain the concept of volatility drain in investment returns, and how do the associated equations illustrate this effect? I did the following considerations so far:
In financial ...
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Geometric Brownian Motion as the limit of a Binomial Tree?
Consider the price of a stock whose drift and volatility parameters are $\mu, \sigma$ respectively, over the time interval $[0, t]$. Suppose we use an $n$-stage binomial tree to model the price ...
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What are $\mu$ and $a$ in $ \mu = a + \frac{\sigma^2}{2} $
Considering GBM:
\begin{equation}
S(t_i) = S_0 \exp(a \cdot t_i + \sigma \cdot W(t_i)) = S_0 \exp\left((\mu - \frac{\sigma^2}{2}) \cdot t_i + \sigma \cdot W(t_i)\right)
\end{equation}
I am interested ...
- 129
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Identifying stochastic process from data
Suppose I am given the values of a stochastic process $S_t$ satisfying some unknown SDE from say 2000 to 2024 so I have a lot a data. How do I identify, model this stochastic process ?
First I thought ...
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Differentiating Wiener process
I have come across an expression as below
$d\left({W_t}^4\right) = 4 {W_t}^3 d\left({W_t}\right) + 6{W_t}^2 dt$
where $W_t$ is standard Wiener process.
While I understand the first part of the RHS, I ...
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Probability distribution function for boundaries on brownian motion
What would be the probability distribution function for Brownian motion with two boundaries, i.e. a stop loss and take profit. The process is a standard Brownian motion. However at values $a$ and $b$ ...
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Is there a "standard" "textbook" model for making re-financing decisions?
You have a loan with an x% interest rate. Rates fall to y%. Should you pay a fee to refinance?
Presumably not if the NPV of the saved interest is less than the fee. However, if you always refinance ...
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Sample Wiener process constrained to open (initial), high (max), low (min), close (final)
With a Brownian bridge, one can sample a Wiener process constrained to a specified initial value and a final value.
Can the same be done when the process is constrained also to have a specified ...
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Possibility of obtaining a positive mathematical expectation in a quoted currency
There is a currency pair C/USD = 1. C - currency in which I want to invest in order to make a profit in USD.
Suppose its price changes discretely: 50% - increases by 20%, 50% - decreases by 20%. This ...
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Weak stationarity of continuous ARMA process from Brockwell
I am currently working on Brockwell "Levy-driven CARMA processes" (2001) and I am stuck in the introduction.
So we have a continuous AR process (CAR(p))
\begin{align*}
X_t=e^{At}X_0+\...
- 135
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Find expected rate of return without drift based on ito process
I would like to know how to solve question (ii), I know it is a cash-or-nothing option but I have no idea how to get the expected rate of return even I use put-call parity. Could anybody guide me I ...
- 1
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Orthogonalizing brownian path
I want to improve the stability of my SDE sample (statistical properties do not change much when using a different seed). I am using a sobol brownian bridge to generate the brownian path increments dw....
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Option pricing boundary condition
I am currently working on this paper "https://arxiv.org/abs/2305.02523" about travel time options and I am stuck at Theorem 14 page 20. The proof is similar to Theorem 7.5.1, "...
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Pricing PDE of Asian option by Shreve
I am currently working on "Stochastic Calculus for finance II, continuous time model" from Shreve. In chapter 7.5 Theo 7.5.1 he derives a pricing PDE with boundary conditions for an Asian ...
- 135
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Volatility of a stochastic Process given by an SDE
I am currently working on this thesis: http://arks.princeton.edu/ark:/88435/dsp01vd66w212h and i am stuck on page 199. There we have a portfolio $P=\alpha F+\beta G $ with $\alpha +\beta =1$ and ...
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Moments of the integral of the exponential of Brownian motion/Normal random variable
I'm studying arithmetic Asian options and there is integral of the following form: $$X_T=\int_0^T e^{\sigma W_t+\left(r-\frac{\sigma^2}{2}\right)t}dt,$$
where $W_t$ is a Brownian motion/Wiener process....
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The conditionnal law of a brownian motion
Please, I have a question about the conditionnal law of a brownian motion.
Here is the statement:
We have $\mathcal{B}_{h}$ the $\sigma$-field generated by the $\left(S_{t_{k}}, k=0, \ldots, N\right)$...
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91
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Deriving probability of hitting stop loss given annual return and Sharpe
Suppose I have a strategy with a mean return and defined Sharpe. Given a preset stop loss, I want to calculate the probability of the stop being hit.
In the example below I use the following ...
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79
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Change of numeraire : quotient
Let's consider $X_1(t)$ a geometric brownian motion (with variable volatility) and $X_2(t)$ a Brownian bridge :
$dX_1(t) = \mu X_1(t) dt + \sigma_1(t) X_1(t) dW(t)$
$dX_2(t) = \frac{b - X_2(t)}{T - t} ...
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Brownian Motion as a Limit of Simpler Models
Let $Δ$ be a small increment of time, and consider a process such that every $Δ$ time units the value of the process either increases by the amount $σ \cdot sqrt(Δ)$ with probability $p$ or decreases ...
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modelling time series using semi-martingale process
During this week lecture my professor said that the semimartingale( brownian motion contamined by noise) is a model in reduced form because we do not specify the dynamic which leads to price ...
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Return distribution with stop loss
Assume there is a investment with payoff going like a brownian motion, i.e. $dS=\mu dt+\sigma dW$, for simplicity, setting $\mu= \sigma=1$. At $t=1$, the payoff distribution is $P=Normal(1,1)$. If we ...
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How to apply CLT on scaled symmetric random walk--Shreve unclear
"Theorem 3.2.1 (Central limit)" in the book "Stochastic Calculus for Finance II Continuous-Time Models" by Steven Shreve says:
Theorem. Fix $t\geq0$. As $n\to \infty$, the ...
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Integrated Brownian motion
I occasionally see a post here: Integral of brownian motion wrt. time over [t;T].
This post has the conclusion that $\int_t^T W_s ds = \int_t^T (T-s)dB_s$.
However, here is my derivation which is ...
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Pricing equation with two correlated states
Consider the following asset pricing setting for a perpetual defaultable coupon bond with price $P(V,c)$, where $V$ is the value of the underlying asset and $c$ is a poisson payment that occurs with ...
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Quadratic Variation Of Mixed Brownian Motion and Poisson Process
I am trying to solve this problem where we're asked to compute the quadratic variation of a process.
I assume that it is necessary to apply Ito's formula but not sure how to get the right solution.
...
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multivariate geometric brownian motion equivalent martingale measure
Suppose $W$ is a $\mathbb{P}$-Brownian motion and the process $S$ follows a geometric $\mathbb{P}$-Brownian motion model with respect to $W$. $S$ is given by
\begin{equation}
dS(t) = S(t)\big((\mu - ...
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Confusion about the formula for gain process in a financial market
In this wikipedia page, we consider the following financial market
The formulas for the stocks are given here
And the gain process of a portfolio $\pi$ is defined such that
From what I understand, ...
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Equivalent definition of brownian motion
I'm having a question about this characterization of Brownian Motion :
Theorem :
If a process : $\big( X_t \big)_{t\geq 0}$ satisfies these conditions,
$\big( X_t \big)_{t\geq 0}$ is a Gaussian ...
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If the price of a stock follows a Geometric Brownian motion, then does stock return depends on past stock returns? [closed]
Got this question from my homework. I think if past returns are keep raising then current return should also be positive, but the answer is it's not related to past returns, why?
I tried to ask ...
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Feynman-Kac formula: Ito's lemma for exponentiated integrals $e^{-\int b dr}$
Consider the stochastic process
$$
dy = f(y,s)ds + g(y,s)dw
$$
where, $w$ is Brownian motion.
Now consider the following exponentiated integral
$$
z_1(s) = \exp \left[ - \int_t^s b(y(r),r) dr \right]
$...
2
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How do your solve for trader's optimal demand in market similar to Kyle's model?
Suppose that $(\Omega,\mathcal{F},\mathbb{P})$ is a standard probability space and $Z_t=(Z_t^1,Z_t^2)$ is a two dimensional Brownian motion with the filtration $\mathcal{F}^Z_{t}$ and $Z_t^1$, $Z_t^2$ ...
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Maximum likelihood estimation of system of correlated SDEs
I have the following system of SDEs (which you can think of as 3 different stocks)
$$dX_t^1 = \mu_t X_t^1 dt + \sigma_t X_t^1 dW_t^1$$
$$dX_t^2 = \mu_2 dt + \sigma_2 dW_t^2$$
$$dX_t^3 = \mu_3 dt + \...
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Estimating the knockout probability of a discretely observed autocall note
For simplicity, let's suppose the underlier follows a Geometric Brownian Motion $S_t\sim\text{GBM}(\mu, \sigma), t\ge 0$ with $S_0=1$. A discretely-observed binary autocall note is a derivative ...
- 913
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On first and last zeros before t in a Brownian Motion
Suppose we have the following random variables, given a fixed $t$ we define the last zero before $t$ and the first zero after $t$:
\begin{align*}
\alpha_t &= \sup\left\{ s\leq t: B(s) = 0 \...
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How to derive this HJB equation?
I'm reading the paper by J.Gatheral and A.Schied (2012) - "Optimal Trade Execution under
Geometric Brownian Motion in the Almgren and Chriss Framework".
On page 6, the authors provide a ...
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How to simulate from instantaneously correlated Brownian motions?
Say I have obtained a distribution for different forward rates F_k such that:
$$
dF_k (t) = \sigma (t) * F_k (t) * dW_k(t)
$$
with
$$ dW_k(t) * dW_l(t) = \rho_{k,l} (t) dt.
$$
From this I want to ...
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Standard Brownian Motion and Exponential Martingale calculation [closed]
Let $W(t)$ be a standard brownian motion and let $Z(t) = \exp (\lambda W(t) - \frac{1}{2}\lambda^2 t).$ In Xinfeng Zhou's Green Book in the section on Brownian Motion (p.130) he writes as part of the ...
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Expectation of Bt^4 given BS [closed]
What is the expectation of Bt^4 and Bt^3 given Bs? Given t>s.
I understand that the expectation of Bt given Bs is Bs and that the expectation of Bt^2 given Bs is something like Bs - s + t.
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508
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4th Order Brownian Motion Martingale [closed]
I understand the first order MG of brownian motion is Bt.. the second order is Bt^2 - t and the third order is bt^3 - 3tBt. How can I find the fourth and beyond order of a Brownian Motion Martingale?
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Discounted expectation of generic $\mathbb{C}^2$ function
Consider a standard geometric Brownian motion $V_t$ with drift $\mu<r$ and standard deviation $1$.
It holds that the discounted expectation is
$$E\left[\int_t^\infty e^{-r(s-t)} V_s ds | V_t \right]...
- 327
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0
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37
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Mean of diffusion term not zero using NORMINV? [closed]
maybe this is a question considered too basic for all of you but im new so please excuse:
I wanted to buid a simulation in excel using the usual suspect(STANDNORMINV(RAND()) and i tried to calculate ...
- 11
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303
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Conditional Expectation of Integral of Squared Brownian Motion - PDE Approach
I am looking to compute the following using Ito's formula.
$$u(t,\beta_t) = \mathbb{E}(\int_t^T\beta_s^2ds|\beta_t)$$
Knowing the properties of brownian motion, it is rather easy to show that the ...
3
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1
answer
167
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Integral of Function of Brownian Motion w.r.t Time (Context: Computing Quadratic Variation)
I am looking to compute the quadratic variation of $$S_t = S_0e^{\sigma B_t}$$ where $B_t$ is Brownian Motion. Applying Itô's lemma, I having the following
$$(dS_t)^2 = S_0^2\sigma^2e^{2\sigma B_t}dt$$...