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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133 views

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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3answers
242 views

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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60 views

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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1answer
90 views

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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50 views

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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48 views

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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1answer
93 views

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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1answer
122 views

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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33 views

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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1answer
181 views

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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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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2answers
119 views

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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59 views

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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1answer
146 views

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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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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113 views

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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1answer
71 views

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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36 views

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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1answer
98 views

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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191 views

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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2answers
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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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58 views

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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1answer
123 views

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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1answer
167 views

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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232 views

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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102 views

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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57 views

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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2answers
155 views

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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1answer
100 views

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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1answer
107 views

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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1answer
261 views

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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1answer
87 views

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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1answer
94 views

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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1answer
445 views

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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1answer
182 views

How is the formula of Quadratic Variation of Brownian Motion derived? [closed]

This is a follow up on this question on quant SE: The question mentions for a Brownian motion : $X_t = X_0 + \int_0^t\mu ds + \int_0^t\sigma dW_t $ , the quadratic variation is calculated as $dX_t ...
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290 views

Covariance between integral of brownian motion and brownian motion

Let $$ I = \int_0^1W_tdt, $$ where $W_t$ is a Brownian motion. From Integral of Brownian motion w.r.t. time we have that $$ \mathbb{E}[I]=0, $$ by Fubini's theorem. And that $$ \mathbb{V}\text{ar}[I] =...
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1answer
119 views

Calculating futures price

Consider a world as follows: $$\frac{dB}{B} = r_tdt$$ $$\frac{dS}{S} = r_tdt - 0.05dW_1 + 0.5dW_2$$ $$dr_t = 0.2 dW_1$$ where $r_0=0$. The Wiener processes $W_1$ and $W_2$ are independent. The price ...
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95 views

Fractional Brownian Motion's Covariance Proof

Let's have the non independent Brownian motion such : $B_{H}(r)=\frac{1}{A(H)} \int_{R}\left[\left\{(r-s)_{+}\right\}^{H-1 / 2}-\left\{(-s)_{+}\right\}^{H-1 / 2}\right] \mathrm{d} B(s), \quad r \in R$ ...
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1answer
138 views

Integral of Brownian motion w.r.t. time and integral not starting at zero

I'm new to stochastic calculus and try to calculate (1) mean and (2) variance of $$\int_s^t W_u du$$ where $W_u$ is a Brownian motion. I already found this helpful answer, where it was shown that $\...
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1answer
99 views

Hermite polynomials as martingales [closed]

Let $\left\{W_{t}: t \geq 0\right\}$ be a standard B.M. on the filtered probability space $\left(\Omega, \mathcal{F},\left\{\mathcal{F}_{t}\right\}_{t \geq 0}, \mathbb{P}\right)$. Define the Hermite ...
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1answer
95 views

Mutual variation of Brownian motions

Let $\{W^1\}_{t\geq0}$ and $\{W^2\}_{t\geq0}$ be two Brownian motions with correlation coefficient $\rho \in [0, 1]$, i.e., $\mathbb{E}[(W^1(t)-W^1(s))(W^2(t)-W^2(s))]=\rho(t-s)$ for all $t,s \geq 0$. ...
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1answer
185 views

What does it mean to "compute" an Itô integral?

I'm reading Shreve's Stochastic Calculus for Finance II. On page 191, Exercise 4.6, we are given the problem Exercise 4.6. Let $S(t)=S(0)\exp\Big \{\sigma W(t)+(\alpha-\frac{1}{2}\sigma^2)t\Big\}$ be ...
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1answer
202 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)=...
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2answers
262 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 ...
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1answer
151 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 ...
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1answer
105 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-...
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1answer
90 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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