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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Ito Process: How to calculate expected return?
On page 300 of Hull's Options, Futures and Other Derivatives
It is tempting to suggest that a stock price follows a generalized Wiener process; that is, that it has a constant expected drift rate and ...
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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
$$
z_1(s) = \exp \left[ - \int_t^s b(y(r),r) dr \right]
$...
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How do your solve for trader's optimal demand in market similar to Kyle's model?
Suppose that $(\Omega,\mathcal{F},\mathbb{P})$ is a standard probability space and $Z_t=(Z_t^1,Z_t^2)$ is a two dimensional Brownian motion with the filtration $\mathcal{F}^Z_{t}$ and $Z_t^1$, $Z_t^2$ ...
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Maximum likelihood estimation of system of correlated SDEs
I have the following system of SDEs (which you can think of as 3 different stocks)
$$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 ...
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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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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
$$E\left[\int_t^\infty e^{-r(s-t)} V_s ds | V_t \right]...
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Mean of diffusion term not zero using NORMINV? [closed]
maybe this is a question considered too basic for all of you but im new so please excuse:
I wanted to buid a simulation in excel using the usual suspect(STANDNORMINV(RAND()) and i tried to calculate ...
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Conditional Expectation of Integral of Squared Brownian Motion - PDE Approach
I am looking to compute the following using Ito's formula.
$$u(t,\beta_t) = \mathbb{E}(\int_t^T\beta_s^2ds|\beta_t)$$
Knowing the properties of brownian motion, it is rather easy to show that the ...
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Integral of Function of Brownian Motion w.r.t Time (Context: Computing Quadratic Variation)
I am looking to compute the quadratic variation of $$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$$...
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Quantile function for fractional Brownian motion (fBm)
If anyone could help me to understand if it is possible calculate the quantile function for fBm?
I’ve checked several papers([1],[2],[3]), and although several works stated that it is centralised ...
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Construction of Itos integral
I am trying to understand the below:
Question 1 how can [W(t1) - W(t0)] = [W(t1) - W(0)] =[W(t1) - 0] =some positive number be a profit or loss?
In this calculation the purchase price is not taken ...
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Simulating Iterated Brownian Motions
I was going through an interesting article (https://arxiv.org/pdf/1112.3776.pdf) while I was trying to read about subordinated processes. I wanted to simulate subordinated processes (in R or python) ...
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How did Bachelier characterize the Brownian motion?
The model for a stock price
$$
dS_t=\mu dt + \sigma dB_t
$$
where $B_t$ is a Brownian motion on $(\Omega, \mathcal{F},P)$, is commonly attributed to the work that Bachelier has carried out in his PhD ...
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Hitting time of Brownian motion with drift using Feynman-Kac
I was studying this question from "A Practical Guide to Quantitative Finance Interviews" and was having some trouble understanding one solution. Please advise if misunderstood anything or if ...
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Does the Lévy characterization imply that the price process of any asset is a Brownian motion?
While studying Brownian motion applied to mathematical finance, I came across these lecture notes by prof Steve Lalley. In the prologue, he gives this explanation for the occurrence of Brownian motion ...
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NFT Floor Price
I'm interested in modeling NFT Floor Price. Specifically, I'm trying to answer the question:
Given current bid-ask info on an NFT collection, what is the probability distribution of the lowest ask ...
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Implication of unique risk neutral measure
I'm reading Shreve Stochastic Calculus II, theorem 5.4.9 (Second fundamental theorem of asset pricing),
This is the part that confuses me :
suppose there is only one risk-neutral measure. This ...
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Dividend Dynamics under Q Measure / Using Girsanov Theorem with Covariance
I want to find the value of a dividend stream. I can do it under the P-measure, but now I would also like to do it under the Q-measure but cant figure out how to derive the dynamics of the dividend ...
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Integral of brownian motion wrt. time over [t;T]
From the post Integral of Brownian motion w.r.t. time we have an argument for
$$\int_0^t W_sds \sim N\left(0,\frac{1}{3}t^3\right).$$
However, how does this generalise for the interval $[t;T]$? I.e. ...
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Integration of exponential raised with Brownian Motion wrt the Brownian Motion
I have to derive several things for my thesis, however, I have the following expression:
$$
\int^{t}_{0} \exp\{\sigma W_{t}\}.dW_{t}
$$
Does anyone know what the solution for this is?
Kind regards.
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Why does the diffusion term remain the same when we change pricing measure?
Consider some Itô process $dS(t)=\mu(t)dt+\sigma(t)dW^{\mathbb P}_{t}$ under the measure $\mathbb P$, where $W^{\mathbb P}$ is a $\mathbb P$-Brownian motion
In plenty of interest rate examples, I have ...
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Expectation of functions with Brownian Motion embedded
Trying to solve a problem set with:
Let $W_t$ be a Brownian Motion and $X_t = e^{izW_t}$ where $z$ is real, $i = \sqrt{-1}$.
I need to find $\mathbb{E}\left(X_t\right)$... I am a bit stuck.
I have ...
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In the Black-Scholes model with stochastic interest rates, what are the 3 assets used to compute measures?
Suppose I have a model with 2 primary assets, a stock $S$ and a short rate.
The stock will be driven by a Brownian motion $W_1$. The short rate will be random and will be driven by a Brownian motion $...
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Integral of brownian increments
I'm stuck at a problem and I'm not sure on how to proceed. My question is how would one go about and integrate the following
$$\sigma\int_{t}^{T}\mathrm{e}^{a\cdot u}\cdot (W_{u}-W_{t})du.$$
I've been ...
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GBM - How to make make annualized dividend reflected in one quarter
I want to simulate the price path of a stock for one quarter using geometric Brownian motion. The stock has a continuous dividend yield of 5% based on the annual dividend yield. However, historically ...
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Drift rate in Geometric Brownian Motion
I have two questions regarding the drift term in the geometric Brownian motion that I cannot find any clear answers to online.
When would we use risk-free rate as drift and when would we use the ...
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Brownian Bridge from timestep 1 to timestep @ expiration, proper mathematical way to generate
When I was learning finance, we didn't cover the subject of Brownian Bridges. So I am trying to learn the proper way of generating paths when you have an arithmetic Asian option which has an ...
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Lemma (maybe) to imply the sign of the sensitivity to correlation
Can anybody please help me to understaind if this result is true ?
Let $\pi=\mathbb{E}\left(f(X_{T})g(Y_{T})\right)$
where $f$ and $g$ are increasing functions.
Hence, $\pi$ is increasing with respect ...
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Solving SDE using integration factor and Ito's lemma [closed]
I don't understand how to define such integration factor in order to solve SDE, for example, as was shown in Solving $dX_{t} = \mu X_{t} dt + \sigma dW_{t}$ and Solving Stochastic Differential ...
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How can there be Brownian motions under different measures?
According to the definition, for a Brownian motion it holds that
$W_0 = 0$,
and
$W_t - W_s \in N(0, t-s), \quad t > s$.
This implies that $W_t \in N(0, t)$, for all $t \geq 0$. Hence, the ...
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How to integrate Itô integral w.r.t time?
Let $W_t$ be a Brownian motion.
How to calculate the following integral
$$
I:=\int_0^t\left( \int_0^u(u-s)dW_s\right) du?
$$
My attempt so far is:
First note that
$$
\int_0 ^u (u-s)dW_s = \int_0^u ...
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2
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Python - Problem of random numbers in MC simulation
I am interested in estimating the price of a European Call Option using the Montecarlo simulation, to get a good approximation of the analytical Black Scholes formula, so a very simple task. ...
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standard/brownian market with different brownian motion
Consider for simplicity the following brownian market:
$$dS^0_t= r S^0_tdt$$
$$dS^1_t= S^1_t(r dt + dW^1_t + dW^2_t) $$
where the filtration is generated by $W^1,W^2$
Consider now $W_t:= \frac{1}{2}(W^...
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Covariance of (fractional) Brownian motions with different Hurst parameters
I'd like to calculate the covariance function for fractional Brownian motions
$$
E_t \left[ dW^H(t) dW^{H'}(t) \right]
$$
but where the Hurst parameters are not equal: $H \neq H'$.
My first idea would ...
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