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Ornstein-Uhlnbeck Process with Jumps

I am trying to simulate an OU Process (Vasicek version) with jumps and I would like to derive the drift and diffusion term when jumps are incorporated, which will enable me to perform monte carlo ...
wanna_be_quant's user avatar
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39 views

Diagonalization of Matrix question about Alvaro Cartea Algorithimic and High frequency trading

I'm currently reading the book "Algorithmic and High Frequency Trading" by Alvaro Cartea and 2 others. I encountered this equation and went down a rabbithole of verifying it. [some text ...
QuantDuck's user avatar
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131 views

Modeling mean-reversion for different volatility regimes

Motivation: Half-life (HL) period shows how long it takes for a mean-reverting process to return halfway to its mean after a deviation. Most commonly, an Ornstein–Uhlenbeck (OU) process is applied to ...
Sane's user avatar
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Why does Theta in an OU process relate to a mispricing of a pair's spread

I am reading this article from Alex Lipton and Marcos Lopez de Prado: A closed-form solution for optimal mean-reverting trading strategies (2020) which talks about finding optimal profit taking and ...
AlexBB's user avatar
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OLS estimation for ornstein uhlenbeck process

I am reading the following paper. In particular, in section 4 - numerical determination of OTRs, it mentions applying Ordinary Least Square on Eq(5). However, what I don't know is whether ${P_{0,0}, ...
user1769197's user avatar
1 vote
1 answer
208 views

Change of measure when the underlying dynamic is Ornstein-Uhlenbeck

Let the $r$ riskless rate to be constant. Let's consider the following underlying dynamic under the $\mathbf{P}$ “physical measure” $$dS_{t}=\mu_{t}S_{t}dt+\sigma_{t}S_{t}dW_{t}^{\mathbf{P}},$$ where $...
Kapes Mate's user avatar
3 votes
0 answers
170 views

How to calibrate an O-U process based on historical data?

Background: I have been working on my master thesis project for the last few months and gave the final presentation on the 2023-06-01. As a part of the master thesis project, I did a complete ...
Yuanlin Dong's user avatar
4 votes
1 answer
164 views

Characteristic function of Gamma-OU process

Consider the Gamma-Ornstein-Uhlenbeck process defined in the way Barndorff-Nielsen does, but consider a different long running mean $b$ which may be bigger than zero: $$dX(t) = \eta(b - X(t))dt + dZ(t)...
Tom's user avatar
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2 votes
0 answers
169 views

Bond-pricing under the Vasicek short rate model

I'm currently studying the Vasicek model of the short interest rate $$dr_t=a(\mu-r_t)dt+\sigma dW_t$$ I know how to solve this stochastic differential equation (SDE) and how to find expectation and ...
Don Abbondio's user avatar
3 votes
0 answers
328 views

Why aren't the optimal entry/exit thresholds for OU pairs trading relatively invariant to shifts in the OU mean?

The optimal entry/exit thresholds for mean reversion trading (assuming an underlying Ornstein-Uhlenbeck (OU) process) is derived in the paper "Optimal Mean Reversion Trading with Transaction ...
mpeac's user avatar
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Why do I need fancy methods to calculate half-life of mean reversion?

I am investigating ways to calculate the mean reversion half life of a mean reverting series. I am encountering things like the Ornstein – Uhlenbeck Process and various types of regression to estimate ...
Vladimir Belik's user avatar
1 vote
1 answer
445 views

American option under Ornstein-Uhlenbeck stock price

I came across with the following problem: For the Ornstein-Uhlenbeck process $(X_t, 0\leq t\leq T)$ with initial condition $X_0 = x$, find the stopping time $\tau$ that maximizes $\mathbb{E}[e^{-r\...
Aguazz's user avatar
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2 votes
0 answers
45 views

Variance of trade length for the OU-process

Based on the article by Bertram (2010), I am trying to calculate the variance of the trade length stated in equation [10] of the paper. However, the weights used has a specification that I cannot ...
RVA92's user avatar
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2 answers
309 views

Affine Structure Resolution for the Vasicek model

I would like to now how to solve the PDE of the affine structure under Vasicek.I am delineating the steps: First let's posit the OU process under a Risk Neutral Measure such as : \begin{align*} \...
lays's user avatar
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2 answers
1k views

Calibrating OU parameters using AR(1)

I have a mean reverting time series and want to find the Ornstein-Uhlenbeck (OU) parameters of it. I researched the internet and found that we can calibrate the model as a simple AR(1) process, $$\...
shananims's user avatar
2 votes
0 answers
723 views

Testing the fit of an Ornstein-Uhlenbeck process

I would like to check if a time-series follows an Ornstein-Uhlenbeck process defined by an SDE: $$dX_t - \lambda (\mu - X_t) dt = \sigma dW_t$$ where $\lambda > 0$ is the mean-reversion ...
MilTom's user avatar
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1 answer
1k views

Calibrating the Ornstein-Uhlenbeck process with an additional parameter

Firstly I find the spread between two cointegrated time-series $Y_t$ and $Z_t$ by finding the best slope parameter $\beta$ in the equation $spread_t = Y_t - \beta Z_t$ (via Cointegrated Dickey-Fuller ...
MilTom's user avatar
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1 vote
1 answer
562 views

Simulating exponential Vasicek/Ornstein-Uhlenbeck

I am trying to simulate commodity prices using the exponential Vasicek/Ornstein-Uhlenbeck model from Schwartz 1997 p. 926 Equation (1). I am using the closed form solution from Vega 2018 p. 5 Equation ...
Tharmis's user avatar
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4 votes
1 answer
522 views

Ornstein–Uhlenbeck process – integration by parts

While deriving the solution for the stochastic differential equation that models the Ornstein–Uhlenbeck process, Paul Wilmott (Paul Wilmott on Quantitative Finance, chapter 4, page 87) performs the ...
Vinícius Lopes Simões's user avatar
3 votes
1 answer
242 views

Digital call under Ornstein-Uhlenbeck dynamics

I am trying to price a digital option with payoff $\mathbb{I}_{S_T>K}$, where $S_t$ follows the Ornstein-Uhlenbeck dynamics $\mathrm{d}S_t=rS_t\mathrm{d}t+\sigma\mathrm{d}W^{\mathbb{Q}}_t$ in the ...
user107224's user avatar
0 votes
1 answer
313 views

Pairs trading by transforming two cointegrated series into a mean-reverting process?

I am slightly confused about the following. Let us assume I have two cointegrated time-series. I would like to model their 'cointegration' by a mean-reverting Ornstein-Uhlenbeck process since if they ...
MilTom's user avatar
  • 165
1 vote
1 answer
314 views

Estimating Ornstein-Uhlenbeck process drift

What is the easiest way to obtain a drift parameter of O-U process given I have $\mu$? Is it ok to linearize the O-U process like so: $P_{t} = \mu + \phi(P_{t-1}-\mu)+\xi_t$ Form vectors from historic ...
spacemonkey's user avatar
3 votes
1 answer
498 views

Simulating Bid-Ask Spreads

I would like to examine the impact of the volatility on the transaction costs (Bid-ask spread). In my case I would like to examine this for power prices. However, I don't have access to actual order ...
Question Anxiety's user avatar
0 votes
1 answer
250 views

Vasicek model - Bond price and volatility

Why does the bond price under the Vasicek model increase as the rate volatility increases? What is the intuition behind this?
actuarialboi9's user avatar
1 vote
2 answers
3k 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: ...
Shile Wen's user avatar
  • 111
1 vote
1 answer
665 views

Is Ornstein–Uhlenbeck process the continuous-time correspondence of AR(1) process?

I see the AR(1) process (with $|\alpha| < 1$) can be written in the following way: $$x_{t+1} = \alpha x_t + \epsilon_t$$ $$\Delta x_t = - (1 - \alpha) x_t + \epsilon_t$$ which looks quite like the ...
DiveIntoML's user avatar
2 votes
0 answers
297 views

Develop an option pricing equation by Ornstein Uhlenbeck process

I know that Black-Scholes equation is based that the Equity price has a Geometrical Brownian movement. Can I develop from the same principles( now with transaction cost) that Black Scholes is ...
Hernan's user avatar
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4 votes
1 answer
679 views

Solution to a Geometric Ornstein Uhlenbeck Process $dX_t = \kappa(\theta - X_t)dt + \sigma X_t dW_t$

I've been searching for the solution to the modified Ornstein-Uhlenbeck process \begin{equation*} dX_t = \kappa(\theta - X_t)dt + \sigma X_t dW_t \end{equation*} but it surprisingly hard to find. The ...
Freelunch's user avatar
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5 votes
2 answers
1k views

Modelling EUR/USD rate with Ornstein-Uhlenbeck model

I have a data set of daily EUR/USD rate for time period 2000-2018. My goal is to model future behaviour of this financial time series using Ornstein-Uhlenbeck model: $$d X_t = \alpha (\theta - X_t) ...
Metod Jazbec's user avatar
4 votes
1 answer
769 views

What is a stochastic processes which reasonably captures commodity price dynamics?

What stochastic processes (and corresponding probability distributions) empirically capture spot/forward commodity prices and forward term structures? Background I want to use discounted cash flow ...
David Addison's user avatar
6 votes
0 answers
3k views

How to trade the Ornstein-Uhlenbeck process?

My question comes from this paper, which is a short version of Avellaneda's paper The picture bellow provides a summary of the equations. Do I understand correctly that in order to trade OU process I ...
Artem Korol's user avatar