Questions tagged [brownian-motion]
In mathematics, Brownian motion is described by the Wiener process; a continuous-time stochastic process named in honor of Norbert Wiener.
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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 ...
3
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1answer
56 views
General Dynamics of a Tradable Asset under the Risk Neutral Measure
Is it true that every tradable asset must have a log-normal dynamics under the risk neutral measure where the drift term is the short rate $r$? I.e., is it true that if $X$ is a tradable asset then
$$\...
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Expectation of number of hits by a brownian motion
If we denote $\tau_i$ the sequence of stopping times defined by:
$\tau_i = \inf(t>\tau_{i-1} : |B(t)-B(\tau_{i-1})| > a)$, $\tau_0=0$.
If we denote N the number of stopping times below T.
What ...
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0answers
62 views
Theoretical Expected Maximum Drawdown vs Empirical Maximum Drawdown
I have been looking at the approach for calculating the expected maximum drawdown of a Brownian Motion [1] and the corresponding function maxddStats in the fBasics package in R [2].
I do not ...
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160 views
+50
Geometric Brownian Motion and Energy-Efficiency Investments
Suppose the payoff $X$ on an investment follows a Geometric Brownian Motion:
$$
dX/X = \mu dt + \sigma dz\ ,
$$
for $dz$ an increment of a Wiener process. I wish to compute the expected present value ...
2
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0answers
64 views
law of absolute of max of brownian motion
What is the law of $\max\left(|B_t|\right)$ for $t$ in $[0,T]$ and $B_t$ is a Brownian motion?
Any references for properties of this process?
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0answers
67 views
Brownian function and Clark's formula
I was reading a paper (link) from Richard Bass about Brownian functionals, and came across the following passage :
Let $X_t$ be a Brownian motion, $F_t$ its filtration and $g$ a real-valued function. ...
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62 views
Alternatives to Lognormality for negative Prices
If I would want to use a different type of distributions (i.e. to allow for negative prices) f.e. a beta distribution how would I have to start to proceed to apply it f.e. to a SDE of the type of a ...
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26 views
Modelling Power Prices
Since electricity prices involve strong seasonality, jump components as well as negative prices and can not be modelled by the GBM, what models/distributions exist, which would allow for modelling ...
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1answer
74 views
Infinitesimal generator - Is it obtained from a stochastic process or It can construct the process
We can see here that the generator is an operator which can be determined for a stochastic process. But, in the answers and comments here we can see that the brownian motion on sphere can be ...
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4answers
418 views
Ito Integral of functions of Brownian motion
How does one show that:
$$ \mathbb{E}\left[ \int f(W_s)dWs \right] = 0 $$
For all $f()$ that are powers of $W(s)$?? I assume that one would have to go via the definition of Ito integral and express ...
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40 views
Simulating correlated stock paths to calculate VaR
So I wanted to generate a Monte Carlo simulation for two correlated assets to derive then the VaR as a quantile of the generated distributions. My code is the following, where the input parameters are ...
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56 views
Instantaneous correlation in the 2 factor Hull White model
I'm trying to understand which parameter controls the instantaneous correlation in the 2 F HW model. As in, correlation b/w 2 rates observed at the same time. My thinking is as follows:
$$Rate(1)=P(t,...
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59 views
Summary of Pricing Options of Log-Normal Claims Using Black's Formula
Cross posted from here.
Let $B$ be a $Q$-Brownian motion and $X^{s,x}$ given by
$$dX_t = X_t(\mu_t dt + \sigma_t dB_t),\quad X_s = x$$
for $\mu, \sigma$ deterministic. Let $\mu_{s,t}=\int_s^t \mu_u du$...
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116 views
How to fix my Ornstein-Uhlenbeck parameter MLE in Python?
I am trying to fit time-series data into an Ornstein-Uhlenbeck process.
Here is my code so far:
...
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4answers
413 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,...
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4answers
224 views
Price of Call Option with or without jumps
Suppose two assets in the Black Scholes world have the same volatility, but different drifts and that one has downward jumps at random times. How does this affect the option prices?
I would have ...
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How to price a down-and-out leveraged barrier call option using Brownian motion?
I am trying to price a type of leveraged down-and-out (LDAO) barrier call option, using geometric Brownian motion.
My python script is below. I am not sure how to correctly model the increasing ...
3
votes
1answer
187 views
Simulate stock prices with Geometric Brownian Motion motion with mu and signa based on 'normal' or continuous compounding?
I have written a simple script for modelling stock prices using Geometric Brownian Motion. The time series I am downloading are daily adjusted closing prices. My aim is to be able to change the ...
4
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0answers
72 views
mixing fractional Brownian motions
Given two Brownian motions $W_t^1, W_t^2$, we can have them correlated by
$$W_t^1 = \rho W_t^2+\sqrt{1-\rho^2}Z_t$$
where $W_t^{2}$ and $Z_t$ are independent of each other.
My question then: is there ...
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1answer
82 views
Convert option inputs to standard Brownian motion
I want to know the probability that the strike price of an option is touched.
My input values are:
P = price
S = strike
v = vol
t = time to expiration
According ...
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1answer
70 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 ...
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29 views
How to expand lognormal approximation of Brownian motion
How can we expand this sum? $\sum_{i=1}^n (e^{rt_i-\frac{1}{2}\sigma^2t_i+\sigma w_{t_i}})^2$ where: $w_{t_i}$ is a standard Brownian motion.
If we let $m_t=e^{-\frac{1}{2}\sigma^2t_i+\sigma w_{t_i}}$...
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1answer
68 views
Ito's lemma for a Forward
I'm trying to understand the derivation of Ito's process with respect to a Forward $F$ on a stock $S$ that pays a constant dividend yield, say $y$. Stock follows brownian motion $\\$
$dS_{t} = S_{t}(\...
1
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1answer
82 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 ...
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0answers
31 views
How to find the derivative for a multi-factor geometric brownian motion model
Does anyone know how to find the derivative for a multi-factor geometric brownian motion model $ \frac { dS_{i}}{S_{i}} $. I have seen solutions for the standard GBM model however I suspect that the ...
1
vote
1answer
65 views
Properties of integrated GBM
(I asked this question on MSE but I think it might have more success here)
Good day,
I was going over some exercises and I stumbled upon a question that, for its solution, requires me to find/...
2
votes
1answer
81 views
Brownian motion and Stochastic Integration
I have two questions relating stochastic integration which perhaps could be answered together.
First question:
First of all, I don't really understand why we can't use Riemann-Stieltjes integration ...
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0answers
22 views
Minimal bounds to enclose most sample paths of a GBM (Geometric Brownian Motion)
For a (generalized) Brownian motion $Y = F(t,W)$, starting at $InitialValue$ and running for a total of $T$ time, if I want to "enclose" (in a visual way) "most" of the possible sample paths, I could ...
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1answer
58 views
Sample path simulation using two random variables
I was wondering if there is a way of generating a sample path of a Geometric Brownian Motion using two independent standard normal random variables instead of just one.
The exact scheme that uses ...
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44 views
Compo/Quanto Adjustment & Multivariate Ito
Related to the issue that I have raised here, I am facing another question. As the rule here is 1 question / 1 post, I take the opportunity to ask it below:
By exploring StackExchange, I noticed the ...
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1answer
124 views
Conditional distribution of $X_t = \int_0^t W_s \mathrm{d}s$
What is the conditional distribution of $$X_t = \int_0^t W_s \mathrm{d}s$$with respect to $W_t = x$?
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1answer
62 views
Exact solution stock price with Vasicek interest rate model
Define two correlated stock price- and interest rate (Vasicek) processes, governed by the Wiener processes $W^{S}(t)$ and $W^{r}(t)$
$$dS(t)=r(t)S(t)dt+\sigma S(t)dW^{S}(t)$$
$$dr(t)=\kappa(\theta-r(...
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43 views
Simulate correlated Brownian motions conditioned on future state(s)
Consider a model defined by 2 geometric Brownian motions
$$dY_{1}(t) = \sigma_{2} Y_{1}(t)dW_{1}(t)$$
$$dY_{2}(t) = \sigma_{2} Y_{2}(t)dW_{2}(t)$$
with $Y_{1}(0) = y_{1}$, $Y_{2}=y_{2}$ and $dW_{1}(...
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1answer
121 views
Difference between $W_t$ and $X_t= \sqrt{t}Z$
$W_t$ is a brownian motion and $X_t= \sqrt{t}Z$, where: $Z\sim N(0,1)$.
How to show that for a bounded continuous $f$ process, $$U_t = \int_0^t (f(W_s))ds$$ and $$V_t = \int_0^t (f(X_s))ds$$ have the ...
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Distribution of the random variable $X (t) = ā« s*B_sds$ [duplicate]
What is the distribution of the random variable $$X (t) = ā« s*B_sds ?$$ The integral is taken over [0,t].
$$B_s$$ is a brownian motion.
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2answers
190 views
Why is $S(t) = e^{\alpha + \beta t + \sigma W(t)}$ used as a model for prices?
Why is the Geometric Brownian Motion defined as $S(t) = e^{\alpha + \beta t + \sigma W(t)}$ used as a model for stock prices? $S(t)$ has a lognormal distribution which is right skewed. Another problem ...
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1answer
77 views
Intuition behind prices modeled by Geometric Brownian Motion
Suppose that we model a price $P_t$ to evolve per
$$\frac{dP_t}{P_t}=\mu dt+\sigma dW_t$$
for $\mu\in\mathbb{R}$ and $\sigma>0$. The unique strong solution to this diffusion is
$$P_t=P_0e^{(\mu-\...
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1answer
144 views
Integration of a deterministic function w.r.t. a Brownian motion
Help me solve this problem:
Let $W_t$ be a Brownian motion and suppose
$X_t = \int_{0}^{t}\delta _{s}dW_{s}$
where $\delta _{s}$ is a deterministic function. Then show that $X_t$ is a Gaussian ...
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1answer
27 views
Find Arithmetic Brownian Motion's transition density
Consider the following stochastic differential equation, an Arithmetic Brownian Motion: šš(š”) = š šš” + š šš(š”) . Find its solution, integrating from t to T, then find its transition density. ...
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1answer
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Prove that $d\hat{W}_t = dW_t - \frac{1}{N_t} \cdot dN_t\cdot dW_t$ gives a Brownian motion under forward measure
Let $N_t$ be a numeraire and $(W_t)$ be the standard Brownian motion under the risk-neutral probability measure $P$.
Recall that forward measure $\hat{P}$ is defined as the Radon-Nikodym derivative:
$...
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1answer
110 views
Calculation of a process's drift
Let $X_t:=e^{W_t}$ where $W_t$ follows the Wiener process. Calculate the drift.
The answer is given as $X_t/2$. My attempt at a solution (which I'm afraid is poor from a mathematical standpoint):
I ...
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57 views
Two- (multi) dimensional geometric Brownian Motion
I am trying to calculate the value of a Basket Option with two stocks and the following information:
S1 = 100, S2 = 120, r = 0.06 L = Volatilitymatrix = ((0.3, 0.1), (0.0, 0.2)), weight of Stock 1 = 1/...
4
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1answer
105 views
Advantages of pathwise calculus over stochastic calculus in continuous self-financing trading models
I am new to stochastic calculus but the statement below confuses me:
Beside the issue of the impossible consensus on a probability measure,
the representation of the gain from trading lacks a ...
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1answer
369 views
What is a Brownian motion “under the risk-neutral” measure?
I understand that the risk-neutral measure is one under which the discounted price (acc. to the risk-free rate) of any asset is a martingale. But we also see notation like $\mathbb{W}^Q_t$ to denote a ...
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Are the increments of a stochastic process driven by fractional Brownian motion independent?
I'm studying the following equation
$$\tag1
dX_t = \mu X_t dt + \sigma X_t dB^H_t
$$
where $B^H$ is the fractional Brownian motion (fBm) of Hurst parameter $H\in(0,1)$, that is a continuous ...
3
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1answer
179 views
Solving Stochastic Differential Equation for Geometric Brownian Motion with time-dependent drift
Given the stochastic differential equation:
$$dZ_t = -Z_t \theta_t dB_t, \quad Z_0 = 1.$$
for an adapted process $\theta_t$ and Brownian motion $B_t$, how exactly do I apply ItƓ's Lemma to obtain:
...
4
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2answers
176 views
Solution to SDE being Evolution of Price Process
I am trying to explain the concept of a solution to SDE being the model for the evolution of a price process. How would you do this to someone who doesn't have a financial engineering background?
...
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1answer
77 views
Showing BM $W(s)$ is independent of $W(t)-W(s)$ [closed]
Consider $0\leq s<t$ where $t,s$ represent time index.
I want to show a Brownian motion $W(s)$ is independent of $W(t)-W(s)$.
Specifically, show that $E[W(s)(W(t)-W(s))]=0$
Proof:
Writing $W(s)$...
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0answers
42 views
Risk-neutral Simple Return Moment Log-return Moment
I am trying to find a way to link Risk-neutral moment of simple return to risk-neutral moment of log-returns.
Specifically, by making the same standard assumptions of the Black-Scholes model with the ...
