Questions tagged [geometric-brownian]

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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
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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 ...
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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 ...
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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 $...
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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 ...
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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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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, $ ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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{...
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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)....
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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 ...
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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
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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 ...
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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 ...
3
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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) + ...
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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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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
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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 ...
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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 ...
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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 = \...
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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
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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
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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 ...
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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 $...
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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
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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
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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}...
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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. ...
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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 ...
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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 $, ...
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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 \...
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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) ] -...
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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
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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
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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=(...
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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
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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
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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
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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 ...
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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, ...
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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
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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 $P(S_t\leq x)=1-a$ ($a:$...
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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
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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 ...
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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 ...