Questions tagged [geometric-brownian]
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75
questions
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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, $ ...
0
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1answer
52 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
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0answers
53 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
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1answer
42 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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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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30 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
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1answer
79 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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53 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
94 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
75 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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93 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
84 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
104 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:
$$\frac12 \sigma^2 V^2 F_{vv}(V,t) + \mu V F_{v}(V,t) - r F(V,t) + ...
1
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1answer
172 views
What does it mean to "compute" an Itô integral?
I'm reading Shreve's Stochastic Calculus for Finance II. On page 191, Exercise 4.6, we are given the problem
Exercise 4.6. Let $S(t)=S(0)\exp\Big \{\sigma W(t)+(\alpha-\frac{1}{2}\sigma^2)t\Big\}$ be ...
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315 views
Probability Distribution at each Simulation Period using Geometric Brownian Motion
I am using the equation $S_t = S_0e^{(\mu-\frac{\sigma^2}{2})t+\sigma\epsilon\sqrt{t}} $ to simulate a financial metric at each $t$, where $t=1$ and $T=5$. Stated in plain English, I am trying to ...
4
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1answer
107 views
Default intensity in Black-Cox model
Consider the model by Black and Cox (Journal of Finance, 1976).
The default intensity function is defined in the usual way: $$h(t) \equiv - \frac{\partial \log P[\tau > t| \mathcal{F}_t]}{\partial ...
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0answers
32 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
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1answer
157 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 = \...
1
vote
1answer
149 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
72 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
92 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 $...
0
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0answers
35 views
VAR Monte Carlo GBM vs Selecting Normal Dist Returns
I am running a VaR calculation and have seen two ways of doing it in several places online.
One simply assumes normal distribution of returns and selects n number of returns from the normal ...
1
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0answers
129 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
136 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
308 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
841 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
88 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
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1answer
54 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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0answers
58 views
Probability of Hitting time of Brownian motion
Let $B =\{ B(t); t \ge 0\}$ be Brownian motion. What is the probability that $B$
hits state one and then state minus one before time one?
My take: Let $T_x = \inf \{ t\ge 0 : B(t) = x\}$, the first ...
1
vote
2answers
264 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
338 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
108 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
275 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
253 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
459 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
74 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
486 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
45 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
votes
1answer
257 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
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1answer
121 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
211 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
174 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 ...
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0answers
142 views
Source on multivariate correlated geometric Brownian motion returns, not prices
Can anyone provide a source that formulates how to generate multivariate geometric Brownian motion returns using the Cholesky method with target correlation matrix, instead of correlated GBM prices?
...
2
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0answers
75 views
Theoretical Expected Maximum Drawdown vs Empirical Maximum Drawdown
I have been looking at the approach for calculating the expected maximum drawdown of a Brownian Motion [1] and the corresponding function maxddStats in the fBasics package in R [2].
I do not ...
1
vote
1answer
140 views
Transition density of geometric Brownian motion with time-dependent drift and volatility
Can you provide a reference to the transition density of the scalar geometric Brownian Motion with time-dependent drift and volatility, i.e. the scalar process $X = (X_t)_{t\geq 0}$ defined by the SDE
...
0
votes
1answer
80 views
Can a down-and-out barrier call option be priced using the Black & Scholes formula or should it be approximated?
I am trying to price of a Down-and-Out Barrier call option with leverage. When the price of the underlying asset hits a certain barrier (B), the option becomes worthless. The issuer of these options ...
2
votes
2answers
349 views
Market price of risk on two assets
Under the assumptions of the Black--Scholes model, I read that the market price of risk of two assets $S_1$ and $S_2$ are the same, if they both follow Geometric Brownian motion driven by the same ...
