Questions tagged [geometric-brownian]
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46
questions
1
vote
0answers
66 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
305 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
40 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)...
1
vote
1answer
144 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
72 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
73 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 ...
1
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0answers
47 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
117 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
25 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
36 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
97 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
111 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
111 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
108 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
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0answers
37 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
64 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
79 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
42 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
50 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
286 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
60 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
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1answer
70 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
82 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
140 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
$$D(0) = S_0 \mathbb{E}...
6
votes
4answers
484 views
Find a formula for the price of a derivative paying $\max(S_T(S_T-K),0)$
Develop a formula for the price of a derivative paying
$$\max(S_T(S_T-K))$$
in the Black Scholes model.
Apparently the trick to this question is to compute the expectation under the stock measure. So,...
3
votes
2answers
240 views
Normality or Log-Normality of Regular Returns
Another old question on this site (How to simulate stock prices with a Geometric Brownian Motion?) inspired me to ask the following question: if we assume that regular returns could be normally ...
0
votes
1answer
83 views
Correctly simulating BEKK series to model asset returns
I am trying to create financial data as close as possible to that of asset returns. Using the R code I can collect some stock data and compute the return:
...
11
votes
1answer
194 views
From VG and NIG processes to GBM
I would like to find out if it is possible to reduce:
the Madan-Seneta Variance Gamma (VG) model;
the Barndorff-Nielsen Normal Inverse Gaussian (NIG) model
to the standard Black-Scholes through a ...
1
vote
3answers
177 views
Probability of a stock price using implied volatility
I have attempted to use the fact of having implied volatility, but have not been able to come up with a viable way to calculate the probability, any ideas?
Suppose that a stock $S_t$ follows a ...
0
votes
1answer
72 views
Sampling from SDE
In the case of the classic Geometric Brownian motion
$$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] $$
and ...
0
votes
0answers
63 views
Geometric brownian motion and probabilities
A stock's price movement is described by the equations $dS_t=0.02S_tdt+0.25S_tdW_t$ and $S_0=100$. An investor buys a call option on said stock with a strike price $K=95$ which expires in $T=2$ years. ...
1
vote
1answer
86 views
true or false: the risk-neutral measure is useless in this situation
Example 2 of this Wiki article on the risk-measure describes how a stock price $S_t$ that is modeled with Geometric Brownian motion with drift $\mu$
$$
dS_t = \mu S_t dt + \sigma S_t dW_t
$$
can be ...
1
vote
0answers
65 views
On Geometric Brownian motion and Itô's formula
Let $S_t$ be a geometric brownian motion such as
$$d S(t) = rS(t)dt +\sigma S(t)dW(t),$$
where $W$ is a standard Brownian motion.
With Itô's lemma and formulas $(dt)^2=dtdW_t=dW_tdt=0$ and $(dW_t)^2=...
0
votes
0answers
58 views
Convert drift and diffusion term in terms of time in the Geometric Brownian Motion framework
Assume that we have daily prices covering the period of 10 years. For calibrating the drift and diffusion parameters of the GBM model
$$S_{t+1} = S_{t}e^{[(\mu-\sigma^2/2)]\Delta t + \sigma \sqrt{\...
1
vote
1answer
86 views
Why do I get this difference when simulating geometric Brownian motion?
I tried simulating GBM using both the SDE definition and the closed form solution. The paths I get through these methods are very different. Can someone help me figure my mistake?
...
2
votes
1answer
124 views
Covariance of logarithms of geometric Brownian motion
Suppose I have a Geometric Brownian Motion process,
$$dX_t=\mu X_t dt + \sigma X_t dW_t$$
I'd like to find the covariance of $\log(X_t)$ and $\log(X_s)$ where $s<t$. We can write $\log(X_t)$ in ...
0
votes
0answers
31 views
How to mathematically calculate the probability of GBM generating difference of less than some value
I have a custom index that follows Geometric Brownian Motion (GBM) with volatility v. I started this index at 10k with 4 decimal places i.e the starting price of ...
2
votes
1answer
113 views
How To Understand the Drift of ln(S) if S Follows Geometric Brownian Motion
As we know, if an asset S follows geometric Brownian motion, under risk neutral measure, it can be expressed as $\frac{dS}{S}=rdt+\sigma dW$, by applying Ito's lemma, $d(lnS)=(r-0.5*σ^2)dt+σdW(t)$, ...
3
votes
0answers
70 views
GBM probability of hitting non constant barrier
I know there is a formula for probability of hitting a constant barrier for GBM/BM (See page 651 in Martinagle Methods in Financial Modelling).
Is there a formula for non-constant barrier?
The ...
1
vote
2answers
148 views
How to Understand Lognormal Distribution in the Following Case
I got a question and corresponding solution, but have some difficulties in understand the lognormal distribution part of it, so I really appreciate your advice:
Question: assume zero interest rate ...
0
votes
1answer
111 views
Drawing values from a lognormal distribution of a GBM
I'm looking at a GBM with parameters
$$
r=0.05 \\
\sigma=0.2 \\
K=130\\
T=0.25\\
S_0 = 100
$$
This is a process that is lognormally distributed with mean and variance given by
$
\mu = S_0e^{r T+0.5\...
3
votes
1answer
88 views
Boundaries for Call Spread
I'm reading an interview book called A Practical Guide to Quantitative Finance Interview and I have some doubts regarding part of its solution and highlighted them in bold:
Question:
What are the ...
1
vote
1answer
1k views
Simulation of Geometric Brownian Motion in R
Using R, I would like to simulate a sample path of a geometric Brownian motion using
\begin{equation*}
S(t) = S(0) \exp\left(\left(\mu - \frac{\sigma^{2}}{2}\right)t + \sigma B_{t}\right),
\end{...
4
votes
1answer
142 views
What the expectation of S^2 is from GBM? [closed]
I was at an interview and was asked to write down the SDE for GBM.
$$
dS = S\mu dt + S\sigma dX
$$
Then I was asked how I would compute the expectation of S^2. I didn't know where to start. Any ...
1
vote
2answers
228 views
What is the stock price expectation?
The Hull textbook (and accompanying technical note) says that the expected stock price $\mathbb{E}[S_T]=S_0 \exp(\mu T)$. However, the answers to a British actuarial examination (Q4 for September 2018)...
