Converting Hourly Ornstein–Uhlenbeck Models to Daily Simulations

# How to transform Ornstein-Uhlenbeck parameters from hourly to daily?


# How to transform Ornstein-Uhlenbeck parameters from hourly to daily?












I get the parameters (long-term mean, volatility, mean-reversion speed, correlation) of two correlated Ornstein-Uhlenbeck processes via a likelihood estimation from hourly data. If I want to transform these to use them to create a daily - instead of hourly - simulation (tree or Monte Carlo), what do I have to do? Thanks in advance.

## Answer by simmy (score 1)

https://quant.stackexchange.com/a/25981

You can aggregate your starting hourly data to obtain daily data and re-estimate the parameters, then simulate. Alternatvely, with your parameters already obtained, you can simulate hourly data and make a post-simulation aggregation to have daily data.

## Answer by M. Jeunesse (score 1)

https://quant.stackexchange.com/a/25985

Let $X^h$ be your hourly process

Let $X^d$ be your daily process

Let $\delta$ be one day

you have

$$X^d_t=\frac{1}{\delta}\int_{t-\delta}^{t}X^h_s ds$$

$$dX^h_t = a(b-X^h_t)dt + \sigma dB_t$$

$$\Delta X^d_t := X^d_{t+\delta}-X^d_t =\frac{1}{\delta}\int_{t-\delta}^t\left(X^h_{u+\delta}-X^h_{u}\right)du$$

so it is a gaussian random variable by knowns results on OU.

You can express it and compute $Cov(\Delta X^{d}_{k\delta},\Delta X^d_{j\delta})$

You will then be able to conclude.

### Details

by known results :

$$X^h_{t+\delta}-X^h_t=(b-X_{t})(1-e^{-a\delta})+\int_{t}^{t+\delta}e^{a(u-t)}dB_u$$

so:

$$\begin{split} X^d_{t+\delta}-X^d_t &= (b-X^d_t)(1-e^{-2a\delta})+\int_{t-\delta}^{t}\frac{1}{\delta}\int_{u}^{u+\delta}e^{a(s-u)}dB_s du \\ & = (b-X^d_t)(1-e^{-2a\delta})+\int_{t}^{t+\delta}\frac{1}{\delta}\int_{u-\delta}^{u}e^{a(s-u+\delta)}dB_s du \\ \end{split} $$