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# BEATING RIVALS

Our model reproduces
**key features of financial assets and their markets**, outperforming both classical as well as
cutting-edge stochastic and time series models.

We judge how realistic synthetic series are by measuring some of their “stylized facts”, which existing models fail to reproduce.

## Myriad is going to be compared against the three following models:

### 01

#### GBM

(Geometric Brownian Motion)

GBM applies **drift and volatility factors** to an underlying Wiener process. The stochastic
differential equation for GBM is given by,

Where:

- dSt is the change in the asset price, S, at time t;
- μ is the percentage drift expected per annum;
- dt represents time;
- σ is the daily volatility expected in the asset price;
- Wt is a Wiener process (Brownian Motion).

### 02

#### MERTON JUMP DIFUSSION

GBM does not exhibit price jumps as found in real price series, because the underlying process is continuous. To
address this limitation, Robert Merton proposed
**adding a jump process to an underlying GBM** which follows a compound Poisson distribution.

Where:

- Nt is the compound Poisson process with rate λ and Yi is log-normally distributed

### 03

#### SARIMAX

(Seasonal Autoregressive Integrated Moving Average with eXogenous components)

This model is an extension of ARIMA that explicitly supports univariate time series data with a seasonal component and adds exogenous regressors to the forecast.

The SARIMAX model is **the most recent state-space model for time series analysis**, but it is still
unable to capture the complexity of financial time series.

## REAL AND SYNTHETIC SERIES COMPARISON

B

FI

FI

Eq

Eq

Real

B

FI

FI

Eq

Eq

GBM

B

FI

FI

Eq

Eq

Myriad

1.0

0.8

0.4

0.0

-0.4

-0.8

-1.0

#### CORRELATION

Correlation between assets (in this case monthly returns) are crucial characteristics of price series.

GBM doesn't capture the correlation between series.

B = Balanced

FI = Fixed Income

Eq = Equity

#### DENSITY

The probability densities of returns present high kurtosis, skewness and very heavy tails.

GBM assumes a normal distribution of returns, while this is not the case in real markets.

#### AUTOCORRELATION

Absolute returns of real series present a degree of autocorrelation between consecutive days, decreasing slowly with time. This is sometimes interpreted as a sign of long-range dependencies present in the markets.

GBM can’t reproduce the autocorrelation present in real markets because of the normality assumption.

#### VOLATILITY CLUSTERING

Real financial assets undergo different states of volatility, which tend to cluster at certain times.

GBM can’t reproduce the volatility clustering present in real markets.