Questions tagged [geometric-brownian]
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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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How can we simulate daily return based on multi-factor model?
This is an interesting question for simulation. The question is a bit lengthy but I'm trying my best to make it super clear here.
Now I have some multi-factor model, say some US barra risk model from ...
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High/Low range under GBM - Analytic solution?
Does anybody know of an analytical solution to the expected high / low range for an asset that follows a GBM process over sampling frequency dt? I have ran numerical simulations and find that the ...
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Reinvesting the dividends of a dividend paying stock
Suppose we have a dividend paying stock that has the following dynamics:
$$dS_t=S_t((\mu-q)dt+\sigma dW_t)$$
With a continuous dividend yield $q$. What is the portfolio $Y_t$ that results out of ...
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Price a contingent claim with payoff $(S_{1T}-S_{2T})^+$ at time $T$
Two stocks are modelled as follows:
$$dS_{1t}=S_{1t}(\mu_1dt+\sigma_{11}dW_{1t}+\sigma_{12}dW_{2t})$$
$$dS_{2t}=S_{2t}(\mu_2dt+\sigma_{21}dW_{1t}+\sigma_{22}dW_{2t})$$
with $dW_{1t}dW_{2t}=\rho dt$....
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1
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Solving the SDE for GBM [closed]
Let's assume that we have the following stochastic differential equation:
$dX_t = \mu X_t dt + \sigma X_tdW_t$
and that we have to prove that this is its solution:
$X_t = X_0 \exp\left(\left(\mu -{\...
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2
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258
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Why do we adjust the drift in the geometric brownian motion
I am building a monte carlo based on the GMB, and I am having a hard time understanding why we subtract 1/2 variance from the drift. If I have a drift of 12% and a volatility of 50%, that would give ...
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1
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Geometric Brownian motion and semi-martingality
I am trying to learn by myself stochastic calculus, and I have a question regarding semi-martingale stochastic process and geometric Brownian motion (GBM).
We know that a stochastic process $S_t$ is ...
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42
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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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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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Dynamics of independent Geometric Brownian Motions under risk-neutral measure Q
Suppose I have two Geometric Brownian motions and a bank account:
$$dB_t=rB_tdt$$
$$
dS=S(\alpha dt + \sigma dW_t)
$$
$$
dY = Y(\beta dt + \delta dV_t)
$$
Where $dW_t$ and $dV_t$ are independent ...
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Distribution of discrete Geometric average and Stock Price
If we have $$S_t = S_0 e^{(r-\frac{1}{2} \sigma ^2) +\sigma W_t}$$ and a discrete geometric average of stock prices $$G_n = (\prod_{i=1}^{n} S_{t_i})^{\frac{1}{n}} $$ where the monitoring points are ...
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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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Bessel Correction and Geometric Brownian Motion
Does it make sense to use bessel's correction for standard deviation and variance when fitting the drift and volatility parameters of geometric brownian motion to historical return data for a security....
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Floating Strike Geometric Averaged Asian Option Pricing
How can I use the risk neutral evaluation to price an asian option with floating strike using the continuous geometric average? I have tried searching for ways to do it but have found almost nothing.
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On the operational process of fractional and delay Brownian motions (FGBM/GDBM) governing respective market scenarios
I have some knowledge about the fabrication of a stochastic differential equation (SDE) governing asset price ($S(t)$) dynamics (This answer helped me up to some extend).
For instance, I am little bit ...
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References for path-dependent GBMs or continuous time analog of discrete time filters
Consider a path-dependent GBM model for a stock price:
$$dS_t = \mu(t, S_.)S_tdt + \sigma(t, S_.) S_t dB_t,$$
where $\mu, \sigma : [0,\infty)\times C_{[0,\infty)}\to \mathbb{R}$ are previsible path-...
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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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how to estimate Geometric Brownian Motion parameters on long timeseries [closed]
I'm working on a 50-years financial timeseries and I would like to simulate GBM paths from it.
The first thing I'm supposed to do is to estimate the drift $\mu$ and the volatility $\sigma$ parameters.
...
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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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315
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Cholesky decomposition reduces volatility of simulated Wiener Process / Brownian Motions
I am trying to simulate $n$ correlated geometric brownian motions (GBM) given a specified correlation matrix $\Sigma$ by following this procedure which uses Cholesky decomposition.
However, when I ...
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Dynamics of FX rate
I've see a couple of places where a FX rate, denoted $X$, such as EURUSD (quoted as "the number of USD needed to buy 1 EUR") is modeled with a diffusion process / Geometric Brownian Motion ...
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What are common ways to realistically simulate the stock market using historical market data?
I am currently using the FinRL library to try to automate Trading using Reinforcement Learning. However, I wanted to understand how FinRL simulates the stock market using historical data. I read here ...
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What is the Kurtosis of Returns in Geometric Brownian Motion?
Suppose that $dS_t=S_t(\mu\mathop{dt}+\sigma\mathop{dW_t})$ which has solution
$$S_t=S_0\exp\left(t\left(\mu+\frac{\sigma^2}{2}\right)+\sigma W_t\right),$$
such that $W_t$ is a Wiener process, $\mu$ ...
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American option under Ornstein-Uhlenbeck stock price
I came across with the following problem:
For the Ornstein-Uhlenbeck process $(X_t, 0\leq t\leq T)$ with initial
condition $X_0 = x$, find the stopping time $\tau$ that maximizes
$\mathbb{E}[e^{-r\...
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Value of trading strategy
A trading strategy is defined as follows: starting capital $v_0 = 5$ and 1 risky asset holdings $\varphi_t = 3W_t^2-3t$ where $W$ is a Wiener process.
The problem is to find the probability of the ...
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115
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Probability the stock price (following geometric Brownian motion) hits the upper boundary U before there is a retracement from the max by amount R?
I am looking for the probability that the stock price/Geometric Brownian Motion hits the upper boundary U, before there is a retracement (from the maximum price) that exceeds amount R. In other words,...
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130
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Modelling the instantaneous funding spread as a log-normal process
Let us consider a stochastic market model with a fixed short (risk-free) rate $r\in\mathbb{R}$. A trader can obtain unsecured funding at a rate $f_t:=r+s_t$ where $s_t$ is its stochastic funding ...
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Estimating volatility of a geometric Brownian motion at different sample rates
I have troubles estimating volatility (= standard deviation of log returns) when the data is re-sampled at different sample frequencies.
Problem
I have generated a time series data using a geometric ...
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Understanding the expected value of the average
I've been looking into Asian Options pricing. Part of the process is about looking for the expected value of the average of a time series undergoing e.g. geometric brownian motion.
I came across this ...
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Why would exchange rates follow a geometric brownian motion?
I'm reading Shreve's Stochastic Calculus for Finance.
On page 382, he begins talking about exchange rates:
Finally, there is an exchange rate $Q(t)$, which gives units of domestic currency per unit ...
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From the perspective of a company, when is the right time to start paying dividends?
I am trying to understand geometric Brownian motion as it relates to the present discounted value of future dividend payments.
I am supposing that a company has a revenue stream $f(t)$. This is just $...
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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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105
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Performance of dollar cost averaging
If we're investing money into a stock $S$ at a continuous rate, $C$, what is the probability distribution of the amount we have invested?
For example, modelling a stock as GBM without contributions, $ ...
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How to simulate correlated stock prices (not returns)
Suppose we have two stocks following GBMs. Drift and volatility are calculated based on historical data. Furthermore the stocks are assumed to be correlated (i.e. they move together, if stock 1 goes ...
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How to compute this current value using no arbitrage condition?
Suppose $X_t$ is a geometric Brownian motion with drift $\mu$ and volatility $\sigma$. $X_0$ is known. You have a machine that produces something worth $X_t$ at random times $t$ generated by a Poisson ...
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170
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GBM drift when simulating correlation betwenn GBM with Cholesky Decomposition
I am currently trying to simulate correlated GBM paths and I found the Cholesky Composition for it. From my understanding, the Cholesky Decomposition can be used to create correlated random variables ...
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Interpretation deferment rate
Geometric Brownian motion related to house prices can be written as
$dH_t=(r-g)H_tdt+\sigma H_tdW_t$.
How should I interpret $g$?
The literature calls this a deferment rate. But I don't understand how ...
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Distribution of Geometric Brownian Motion drawdowns from realizations of multivariate Normal and Laplace distributions
I am trying to simulate the distribution of Geometric Brownian Motion drawdowns from samples of multivariate Normal and Laplace distributions under the same covariance structure. Drawdowns are defined ...
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Monte Carlo Simulation of GBM Process has a Very High Variance - Explanation Needed as to why?
I use Geometric Brownian Motion (GMB) to simulate a share price from March 24, 2020 to March 24 as follow:
\begin{equation}
S_t=S_{t-1}exp((rf-0.6\sigma^2)*(2)+\sigma*sqrt(2)*\mathcal{N}(0,1))
\end{...
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386
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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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Finding the optimal strategy of a stock trading game
Assume there is a stock, its price at time $t$ is $S(t)$. Its price is changing according to a geometric brownian motion, that is $dS=S(\mu dt+\sigma dW)$. You start with $\\\$1$ and you are allowed ...
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Equivalence of Call Option on $S_T$ and Put Option on $\frac{1}{S_T}$ in FX Markets
Part 1: I am trying to price an option in the FX world. It naturally pays in the domestic currency, but in this case the payout currency must be the foreign currency. For example, consider the payoff:
...
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How to compute the Present Value of this path-dependent option?
I have an option whose payoff depends on its value at two times $T_1$ and $T_2$ as follows.
$$V(t) = \mathbb{E}^{Q}[\mathbb{1}_{S(T_1)>B} (S(T_2)-K)^+)],$$
where the stock price follows the GBM ...
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181
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Deriving Law of Motion by Ito's Lemma
I've been trying to derive the law of motion for the stochastic process above using Ito's Lemma, given Geometric Brownian Motion with it's law of motion shown below:
I've managed to take the partial ...
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106
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Cauchy-Euler ODE with indicator function in coefficient
Consider the following Cauchy-Euler ODE, which is in particular the asset pricing equation for a (perpetual coupon defaultable) bond:
$$\frac12 \sigma^2 V^2 F_{vv}(V,t) + \mu V F_{v}(V,t) - r F(V,t) + ...
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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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