Questions tagged [geometric-brownian]
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63
questions
3
votes
1answer
58 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:
$$\frac12 \sigma^2 V^2 F_{vv}(V,t) + \mu V F_{v}(V,t) - r F(V,t) + ...
1
vote
1answer
149 views
What does it mean to “compute” an Itô integral?
I'm reading Shreve's Stochastic Calculus for Finance II. On page 191, Exercise 4.6, we are given the problem
Exercise 4.6. Let $S(t)=S(0)\exp\Big \{\sigma W(t)+(\alpha-\frac{1}{2}\sigma^2)t\Big\}$ be ...
0
votes
0answers
66 views
Probability Distribution at each Simulation Period using Geometric Brownian Motion
I am using the equation $S_t = S_0e^{(\mu-\frac{\sigma^2}{2})t+\sigma\epsilon\sqrt{t}} $ to simulate a financial metric at each $t$, where $t=1$ and $T=5$. Stated in plain English, I am trying to ...
4
votes
1answer
90 views
Default intensity in Black-Cox model
Consider the model by Black and Cox (Journal of Finance, 1976).
The default intensity function is defined in the usual way: $$h(t) \equiv - \frac{\partial \log P[\tau > t| \mathcal{F}_t]}{\partial ...
0
votes
0answers
30 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
74 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, $$\text{d}S_t = \...
1
vote
1answer
121 views
Simulation of Geometric Brownian Motion
I generate 10000 random binomial paths for a stock whose price is from S(0) = 10 out to S(t) where t = 1 year. Assume geometric Brownian motion for the stock price with a drift of 15% per year and a ...
2
votes
1answer
64 views
stock price path simulation using GBM, is it possible to run the same simulation over and over again?
When I simulate a stock's price path using geometric brownian motion I am sometimes able to get a pretty good forecast that fits the real values very well. But if I run the simulation again, the ...
1
vote
1answer
87 views
What is the difference between log volatility and simple volatility in a GBM? [closed]
What difference do they make? Why do many people seem to find more accurate simulations with log volatility?
standard volatility in GBM is defined as $\sigma = \frac{1}{N}\sum_{i=1}^N(x_i-\mu)$ where $...
0
votes
0answers
28 views
VAR Monte Carlo GBM vs Selecting Normal Dist Returns
I am running a VaR calculation and have seen two ways of doing it in several places online.
One simply assumes normal distribution of returns and selects n number of returns from the normal ...
1
vote
0answers
71 views
Calibrate Geometric Brownian Motion of trading volume time series
Let's say I'm modeling the trading volume of a stock price (e.g. Apple Inc.) to follow a Geometric Brownian Motion and I want to estimate the parameters (i.e. drift and volatility) using historical ...
1
vote
1answer
73 views
Simulating correlated Geometric Brownian Motion with lag
I know that it is possible to simulate two correlated GBM in e.g. Matlab (Generating Correlated Asset Paths in MATLAB) based on cholesky decomposition. However, they take as input the correlation ...
9
votes
0answers
227 views
Change of numéraire for two risky assets without bank account (Margrabe’s formula?)
I am considering two risky assets following the usual correlated GBM given by
$$\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
497 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
81 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
52 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
vote
0answers
45 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 \...
0
votes
0answers
54 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
217 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
331 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
83 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)...
2
votes
1answer
202 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
156 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
277 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
65 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) ...
1
vote
1answer
147 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
34 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
43 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
176 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
117 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
174 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
157 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 ...
1
vote
0answers
110 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
votes
0answers
69 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
119 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
0answers
43 views
Simulating correlated stock paths to calculate VaR
So I wanted to generate a Monte Carlo simulation for two correlated assets to derive then the VaR as a quantile of the generated distributions. My code is the following, where the input parameters are ...
0
votes
1answer
68 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
326 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 ...
0
votes
1answer
44 views
Applicability of the Ito's lemma [duplicate]
Ito's lemma is used to find the stochastic process of the function of a ...
0
votes
0answers
63 views
Arbitrage free pricing of option to trade stocks
Consider Black-Scholes model with constant interest rate r and stocks with prices $S_t^A$ and $S_t^B$ that satisfy the SDE's $dS_t^A = S_t^A(\mu^A dt + \sigma^A dB_t)$ and $dS_t^B = S_t^B(\mu^B dt + \...
0
votes
1answer
71 views
What is the meaning that Geometric Brownian motion is leptokurtic? [closed]
Does this have any relation to the symmetry of the normal distribution?
2
votes
0answers
132 views
What are some alternatives to Geometric Brownian motion that can be used in the Black-Scholes? [closed]
I hear that there are many extensions to the black scholes model to make it more realistic, however, GBM does not account for volatile swings. Is there any sort of alternative approach to use instead?
1
vote
2answers
153 views
Compute the price of a derivative which pays $\log(S_T)S_T$ in the Black Scholes world
Compute the price of a derivative which has pays $\log(S_T)S_T$, you can assume that the Black Scholes model is valid.
Using the stock measure we can write the expectation as
$$D(0) = S_0 \mathbb{E}...
6
votes
4answers
603 views
Find a formula for the price of a derivative paying $\max(S_T(S_T-K),0)$
Develop a formula for the price of a derivative paying
$$\max(S_T(S_T-K))$$
in the Black Scholes model.
Apparently the trick to this question is to compute the expectation under the stock measure. So,...
3
votes
2answers
337 views
Normality or Log-Normality of Regular Returns
Another old question on this site (How to simulate stock prices with a Geometric Brownian Motion?) inspired me to ask the following question: if we assume that regular returns could be normally ...
0
votes
1answer
85 views
Correctly simulating BEKK series to model asset returns
I am trying to create financial data as close as possible to that of asset returns. Using the R code I can collect some stock data and compute the return:
...
11
votes
1answer
240 views
From VG and NIG processes to GBM
I would like to find out if it is possible to reduce:
the Madan-Seneta Variance Gamma (VG) model;
the Barndorff-Nielsen Normal Inverse Gaussian (NIG) model
to the standard Black-Scholes through a ...
1
vote
3answers
333 views
Probability of a stock price using implied volatility
I have attempted to use the fact of having implied volatility, but have not been able to come up with a viable way to calculate the probability, any ideas?
Suppose that a stock $S_t$ follows a ...
0
votes
1answer
103 views
Sampling from SDE
In the case of the classic Geometric Brownian motion
$$dS_t = \mu S_t dt + \sigma S_tdW_t$$
we solve it as
$$ S_t = S_0 \exp\left[ \left(\mu - \frac{\sigma^2}{2}\right)t + \sigma dW_t\right] $$
and ...
0
votes
0answers
66 views
Geometric brownian motion and probabilities
A stock's price movement is described by the equations $dS_t=0.02S_tdt+0.25S_tdW_t$ and $S_0=100$. An investor buys a call option on said stock with a strike price $K=95$ which expires in $T=2$ years. ...
