Questions tagged [geometric-brownian]
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80
questions
3
votes
1answer
93 views
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 ...
0
votes
1answer
75 views
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 ...
1
vote
2answers
203 views
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 ...
1
vote
0answers
32 views
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 $...
0
votes
1answer
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 ...
0
votes
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 ...
0
votes
0answers
89 views
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, $ ...
1
vote
1answer
99 views
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 ...
4
votes
0answers
55 views
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 ...
0
votes
1answer
60 views
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 ...
0
votes
0answers
15 views
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 ...
0
votes
0answers
31 views
Valuation of security when reaching hitting time under GBM
I'm trying to find a formula to value the following security:
Where equation (2) is given by
I already have that :
I looking for formula to value this security in the real-world.
Can someone please ...
0
votes
1answer
86 views
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 ...
0
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0answers
63 views
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{...
0
votes
1answer
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)....
0
votes
0answers
78 views
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 ...
1
vote
0answers
102 views
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:
...
2
votes
1answer
85 views
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 ...
1
vote
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 ...
3
votes
1answer
64 views
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:
$$\frac12 \sigma^2 V^2 F_{vv}(V,t) + \mu V F_{v}(V,t) - r F(V,t) + ...
1
vote
1answer
185 views
What does it mean to "compute" an Itô integral?
I'm reading Shreve's Stochastic Calculus for Finance II. On page 191, Exercise 4.6, we are given the problem
Exercise 4.6. Let $S(t)=S(0)\exp\Big \{\sigma W(t)+(\alpha-\frac{1}{2}\sigma^2)t\Big\}$ be ...
0
votes
0answers
358 views
Probability Distribution at each Simulation Period using Geometric Brownian Motion
I am using the equation $S_t = S_0e^{(\mu-\frac{\sigma^2}{2})t+\sigma\epsilon\sqrt{t}} $ to simulate a financial metric at each $t$, where $t=1$ and $T=5$. Stated in plain English, I am trying to ...
4
votes
1answer
122 views
Default intensity in Black-Cox model
Consider the model by Black and Cox (Journal of Finance, 1976).
The default intensity function is defined in the usual way: $$h(t) \equiv - \frac{\partial \log P[\tau > t| \mathcal{F}_t]}{\partial ...
0
votes
0answers
34 views
Do simulated values for IV need to be linked to the simulated series of underlying prices when used together in a Monte Carlo Simulation?
I've been using thousands of simulated stock price series generated with mean and standard deviation of daily returns and Geometric Brownian Motion, and then running these simulated price series ...
0
votes
1answer
239 views
Solution to geometric Brownian motion with time dependent volatility and drift?
I am able to compute the general solution of a standard geometric Brownian motion, but I'm struggling to find the general solution for a GBM where volatility and mean depend on time, $$\text{d}S_t = \...
0
votes
0answers
41 views
generating synthetic asset prices
I would like to use geometric brownian motion (gbm) in order to generate artificial asset prices. I know that gbm has constant volatility, therefore I somehow converted it to stochastic in a very ...
1
vote
1answer
170 views
Simulation of Geometric Brownian Motion
I generate 10000 random binomial paths for a stock whose price is from S(0) = 10 out to S(t) where t = 1 year. Assume geometric Brownian motion for the stock price with a drift of 15% per year and a ...
2
votes
1answer
79 views
stock price path simulation using GBM, is it possible to run the same simulation over and over again?
When I simulate a stock's price path using geometric brownian motion I am sometimes able to get a pretty good forecast that fits the real values very well. But if I run the simulation again, the ...
1
vote
1answer
103 views
What is the difference between log volatility and simple volatility in a GBM? [closed]
What difference do they make? Why do many people seem to find more accurate simulations with log volatility?
standard volatility in GBM is defined as $\sigma = \frac{1}{N}\sum_{i=1}^N(x_i-\mu)$ where $...
1
vote
0answers
175 views
Calibrate Geometric Brownian Motion of trading volume time series
Let's say I'm modeling the trading volume of a stock price (e.g. Apple Inc.) to follow a Geometric Brownian Motion and I want to estimate the parameters (i.e. drift and volatility) using historical ...
1
vote
1answer
140 views
Simulating correlated Geometric Brownian Motion with lag
I know that it is possible to simulate two correlated GBM in e.g. Matlab (Generating Correlated Asset Paths in MATLAB) based on cholesky decomposition. However, they take as input the correlation ...
9
votes
1answer
328 views
Change of numéraire for two risky assets without bank account (Margrabe’s formula?)
I am considering two risky assets following the usual correlated GBM given by
$$\frac{\mathrm{d}S^{(i)}_t}{S^{(i)}_t}=\mu_i\mathrm{d}t+\sigma_i\mathrm{d}W^{(i)}_t,\quad i\in\{1,2\}$$
with
$$\mathrm{d}...
1
vote
1answer
1k views
Geometric Brownian Motion simulation in Python: strange results
I am trying to simulate Geometric Brownian Motion in Python, however the results that I get seem very strange and in my opinion they can't be correct. My goal is to simulate each day of 1 year. ...
1
vote
1answer
89 views
Copula analytic formula for $max(S_T^1 - K, 0) 1_{\{L<S_T^2<U\}}$
Consider the payoff function
$$ V_T = max(S_T^1 - K, 0) 1_{\{L<S_T^2<U\}} = (S_T^1 - K)1_{\{S_T^1 > K\}}1_{\{L<S_T^2<U\}}$$
where $S_T^1$ and $S_T^2$ are two GBM distributed stocks with ...
0
votes
1answer
56 views
Volatility of a function of an asset
Suppose that $ G $ is a function of the underlying asset $ S $, which follows a geometric Brownian motion. Suppose that $ \sigma_{S} $ and $ \sigma_{G} $ are the volatilities of $ S $ and $ G $, ...
1
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0answers
48 views
Moments of a SDE: a detail on the information set
Very basic questions. Let $(z_t)_{t \geq 0}$ be a standard Brownian motion and let
$$dS_t = \mu S_t dt + \sigma S_t dz_t.$$
When we write $E\left( S_t \right)$, do we mean $E\left( S_t \big| F_0 \...
1
vote
2answers
345 views
Integral of the square of Brownian motion using definition of variance
Let $B = \{ B(t); t \ge 0\}$ and let $Z = \{ Z(t); t \ge 0 \}$ where $$Z(t) = \int_0^t B^2(s) ds.$$ How do we find $E[Z(t)]$ and $E[Z^2 (t)]$ in order to get the variance $Var [Z^2(t)] = E[Z^2 (t) ] -...
2
votes
2answers
340 views
Proving $\mathbb{P}(S_t<0|S_0=s_0)=0$ for Geometric BM
I am trying to prove that for the geometric Brownian motion of a stock $\textrm{d}S_t=\mu S_t\textrm{d}t+\sigma S_t\textrm{d}B_t$ with strictly positive constants $\mu$ and $\sigma$ and and $S_0=s_0&...
1
vote
1answer
128 views
Discrete Dividend GBM process
I'm trying to derive the risk neutral process for a stock with both continuous and discrete dividends. In particular, suppose the forward level process at time, $t$ is given by $F(S_t, t, T) = e^{(r-y)...
3
votes
1answer
287 views
Pricing an Option with payoff $\left(1-\frac{K}{S_t}\right)^{+}$
Let $S_t=S_0 \exp\left\{rt+0.5\sigma^2t+\sigma W_t\right\}$ be the usual GBM model for a Stock price under the money-market numeraire.
Suppose we want to price an option with payoff at maturity: $C_T=(...
1
vote
2answers
282 views
Bond price distribution if yield assumed log-normal
Suppose we assume that yields on a zero-coupon bond that matures at time $T$ follow a log-normal process of the type $y(t,T)=y(t_0,T)e^{-0.5\sigma^2t+\sigma W_t}$ under the T-forward measure.
Then, I ...
2
votes
0answers
592 views
Simulating correlated Geometric Brownian Motion in Python
I want to simulate two correlated Geometric Brownian Motion processes in Python. I found an implementation from Matlab (https://www.goddardconsulting.ca/matlab-monte-carlo-assetpaths-corr.html) and ...
2
votes
0answers
82 views
Why does higher volatility for ATM Call Option lead to a lower risk-neutral probability of expiring ITM?
This is a follow-up question on the discussion in the thread here, from which I borrow the graph below depicting $N(d_2)$ (i.e. the risk neutral probability of a Call option expiring in the money) ...
3
votes
3answers
532 views
Probability of an Option maturing In-the-money vs. Volatility
How will the probability of an option ending up in the money change if the volatility of the underlying stock increases?
Intuitively, I think the answer to this is that if volatility goes up the ...
0
votes
0answers
48 views
Simulating two correlated time series using GBM [duplicate]
My situation is the following: I have two time series TS1 and TS2, whereas TS1 is a stock price. According to literature, TS2 is positively correlated to TS1. Furthermore, since TS1 is a stock price, ...
1
vote
0answers
47 views
Is there a relation between the so-called volatility drag and the sigma term in Black-Scholes' model? [duplicate]
The closed-form solution of Black Scholes Dynamics
$dS_t=S_t(\mu dt +\sigma dW_t$) is
$$S_t=S_0 e^{(\mu -\sigma ^2/2) t+\sigma dW_t}.$$
The $-\sigma^2/2$ term is quite similar to the volatility drag ...
2
votes
1answer
291 views
VaR and Expected Shortfall for Geometric Brownian Motion
Given that $dS_t=\mu S_tdt+\sigma S_tdW_t$ ,a risk free rate r and defining Value at Risk and Expected Shortfall as
$VaR_{t,a}=S_0e^{rt}-x$ where $x$ is the amount such that $P(S_t\leq x)=1-a$ ($a:$...
0
votes
1answer
124 views
Why am I struggling to replicate the Black-Scholes price of an option stochastically?
I am currently trying to replicate the Black-Scholes price of a call option using stochastic simulations of the price moves of the underlying. My code is as follows:
...
1
vote
1answer
230 views
Monte Carlo simulations of correlated stocks by Geometric Brownian motion
I am trying to simulate using a Geometric Brownian Motion process three autocorrelated stocks.
In particular, I need to simulate three different matrices with 1000 scenarios each using a Monte Carlo ...
1
vote
2answers
186 views
Simulating artificial asset prices: Random walk vs Brownian motion?
How well can each simulate the real-life behavior of stock prices, and what considerations or (dis-)advantages must we be aware of when deciding to use each:
Random walk with drift
Random walk ...
