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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Construction of Ito Integral - simple integrands [closed]

I am trying to understand the application of the below concept: Question 1 how can [W(t1) - W(t0)] = [W(t1) - W(0)] =[W(t1) - 0] = gain? In this calculation the purchase price is not taken into ...
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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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Milstein Discretization of Heston Model

Given the following representation of the Heston Model: $$d\left(\begin{array}{l}S_{t} \\ V_{t}\end{array}\right)=\left(\begin{array}{c}\mu S_{t} \\ \nu-\varrho V_{t}\end{array}\right) d t+\left(\...
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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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Correlated Geometric Brownian Motion - Drift rate for different stocks from different countries

I am valuing a structured product where the payout function depends on the paths of two assets. The key in my valuation is to use Monte Carlo simulations of a payout function tied to a geometric ...
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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

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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Antithetic method in GaussianMultiPathGenerator quantlib

I am trying to generate MS simulation paths as well as antithetic paths, however when I try to use the Pathgenerator.antithetic(), it gives me the exact same result of the Pathgenerator.next(). here's ...
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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 generate normalized factor scores for beta exposure

I'm working on building a time series momentum model (TSMOM) based on price alone for currency pairs. I'm implementing a paper that produces a buy/sell signal based on geometric brownian motion and a ...
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4 votes
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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 answers
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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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Infill lower frequency data: Brownian Bridge

Given monthly returns data, I would like to infill those to get daily returns. Roughly estimates imply that annual volatility is about 1.5x of SPY. One option that came up in my initial research was ...
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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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How to calculate the expectation of complicated Brownian motions?

I'm not very familiar with the Brownian motions stuff and I wish to receive your help. My question is that I have an objective equation: theta is variable that I want to maximize the objective ...
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5 votes
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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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Geometric brownian motion small timesteps high volatility

I'm trying to generate some sample geometric brownian motion paths for an asset which is traded 24/7 without interruption and is highly volatile (upwards to 150% implied volatility on options markets)....
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2 votes
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Equivalent local martingale measure vs. equvalent martingale measure in a Brownian setup

Assume you have the standard financial market built up of a Brownian motion. I have seen some books say that an equivalent local martingale measure imples no arbitrage, and some say that an equivalent ...
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Generate a Fractional Gaussian Noise

I am trying to simulate a Fractional Gaussian Noise using Fast Fourrier algorithm.However,I couldn't even if I could retrieve my original covariance matrix such : $E\left[ X(t)X(s) \right] = \frac{1}{...
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Second variation of a Brownian motion under jump-diffusion process

I am trying to solve exercise 15.3 from the book The concepts and practice of mathematical finance where it is asked Suppose the $\log S_t$ follows a Brownian motion over the period $[0, 1]$ except ...
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Ito's lemma $f(t,W_t^2)$

Let $f$ be a function of $t$ and $W_t^2$. a)Find a function $f$ such that $f(t,W_t^2)$ is a $F_{t^-}$ martingale, with $F$ the Brownian filtration. b)Use Ito's lemma to show that $f(t,W_t^2)$ is a ...
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Sampling change in the driving brownian motion of a CIR process

I have volatility driven by a CIR process: $$\mathrm{d}v_t = \kappa (\bar{v}-v_t)\mathrm{d}t + \omega \sqrt{v_t}\mathrm{d}W_v\text{.}\tag{1}$$ I am working with several (complicated) approximations of ...
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Finding the PDE and replicating strategy of a european contigent claim [duplicate]

Suppose that we have the Black and Scholes model where the interest rate and the volatility are time varying: $dB(t)=r(t)B(t)dt$ and $dS(t)=S(t)b(t)dt+S(t)\sigma(t)dW(t), S(0)=s>0$ where $r,b,\...
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2 votes
1 answer
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Solving an SDE using Ito's Lemma

Suppose that $Z(t)=e^{-\int_0^t \theta'(s)dW(s)-\frac{1}{2}\int_0^t ||\theta(s)||^2ds}$ with $\theta()=\sigma^{-1}()[b()-r()]$, $\sigma()>0$ and invertable and $W()$ a Wiener process There is also ...
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3 votes
1 answer
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Brownian Bridge general case

The SDE for the Brownian bridge is the following: $dY_t=\frac{b-Y(t)}{1-t}dt+dW(t)$ with solution: $Y(t)=Y(0)(1-t)+bt+(1-t)\int_0^t \dfrac{dW(s)}{1-s}$ Can someone help me on proving that $$\lim_{t\...
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8 votes
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On a time integral of Brownian motion up to the hitting time

Just come up with a 'simple' and interesting problem that I've been struggling to deal with for some time. Consider a filtered probability space $(\Omega, \mathcal{F}, \{\mathcal{F}_t\}_{t\in[0,T]},\...
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3 votes
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MGF of Generalised Itô Integral

The following derivation produces a moment closure problem - I would appreciate any insight. It may seem trivial at first glance, but the key aspect is the integrand dependence on $t$. Consider $W_t$ ...
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Distribution and Analytical solution of a GBM with stochastic interest rate?

We model the exchange rate $S_t$ with a geometric Brownian motion and the USD and EUR interest rates $r_u$ and $r_e$ each according to the Vasicek model. Under the domestic equivalent martingale ...
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Drift Term in Black-Scholes Model Martingale

How would I prove that a Black-Scholes Model is not a Martingale if it has drift. In many cases it is just stated as a fact (without proof). For instance if Im looking at: $$dS_{t} = \mu S_{t} + \...
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Sum of discretely sampled BM

If an underlying follows lognormal GM with no drift $dS_t = \sigma S_t dW_t $ and $A_N = \Sigma_{i=1}^{N} S_{t_i}$. How to compute variance of $A_N$?
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1 vote
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Deriving Law of Motion by Ito's Lemma

I've been trying to derive the law of motion for the stochastic process above using Ito's Lemma, given Geometric Brownian Motion with it's law of motion shown below: I've managed to take the partial ...
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4 votes
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Conditional expectation of integral of brownian motion

I am trying to calculate $$\mathbb{E}\biggl[\biggl(\int_s^t W_u du\biggl)^2 \biggl|W_s=x, W_t=y\biggl] $$ where $W$ is a Standard Brownian Motion and $s\leq u \leq t$. Any help or tips would be ...
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Monte carlo simulations giving biased output [closed]

I wrote code to simulate the stock price using geometric brownian motion. My code is as follows: ...
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3 votes
1 answer
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EMM for Bachelier model

The stock price is assumed to evolve as $S_{t}=S_{0}+\mu t+\sigma B_{t}$, where $S_{0}>0, \mu>0$ and the process $B_{t}$ is Brownian motion. The saving account is assumed to be $\beta_{t}=e^{r t}...
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4 votes
1 answer
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Covariance of two Brownian Motions

During revision, I came across the following question in a past paper: Suppose $(B_t, t\geq0)$ is a standard Brownian motion. Compute for $0<s<t$ the covariance $$cov(tB_{3t}-B_{2t}+5, B_s-1).$$ ...
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