Geometric Brownian motion (GBM) model basically suggests whether the distribution of asset returns is normal or lognormal. However, many empirical studies have revealed that return distributions are usually not normal. These studies, time and again, discover evidence of non-normality, such as heavy tails and excess kurtosis.

This work was aimed at analysing the GBM with a sequential Monte Carlo (SMC) technique based on

The SMC or particle filter based on the

The particle filter based on the

Results show that the GBM model based on student’s

The proposed model which is heavier tailed than the normal does not only provide an approximate solution to non-normal estimation problem.

The GBM model based on student’s

Most of the models utilised in the description of financial time series are written in terms of a continuous time diffusion _{t} that satisfies the stochastic differential equation (SDE):

where _{t} ~ _{t} and _{t} denote the volatility and drift function, respectively. This class of parametric models has been extensively used to portray the dynamics of financial variables, including stock prices, interest rates and exchange rates. A stochastic process _{t} is said to follow a geometric Brownian motion (GBM) if it satisfies the above SDE.

The GBM is one of the most popular stochastic processes and undoubtedly an effective instrument in modelling and predicting the random changes in stock prices that evolve over time. It is essentially useful for index price study because the process in question assumes that percentage changes are independent and identically distributed over equal and non-overlapping time length (Luenberger

In this article, we extended our investigations by introducing a GBM model based on the

A number of studies have been conducted in the area of GBM as a model for stock prices. Some scholars have tried extending, and hence improving, the standard GBM model. Duplantier (

Thao (

Geometric Brownian motion has been extensively used as a model for stock prices, commodity prices, growth in demand for products and services, and real options analysis (Benninga & Tolkowsky

Works on modelling return distributions of financial assets also exist. The most used are the normal, the lognormal and the non-Gaussian stable distributions. Other types of distributions, such as the student’s

Hsu, Miller and Wichern (

Theodossiou (

Geometric Brownian motion is the stochastic process used in the Black–Scholes methodology to model the evolution of prices in time. As in a typical structural model, let us consider a firm with its value of the asset _{t} following a GBM:

where _{t} ~ _{t}_{t} controls the ‘trend’ of this trajectory and the term _{t}_{t} controls the random noise in the trajectory. Nevertheless, one of the foremost challenges in applying this model to financial market data is the fact that the underlying asset value process is unobservable.

Applying the Ito’s formula (see Lamberton & Lapeyre

The stochastic process, as characterised by

Taking the exponential of both sides and inserting the initial condition _{0}, we obtain the solution. The analytical solution of this GBM is given by:

This SDE is principally significant in the modelling of many asset classes.

where

The parameters

As Brigo et al. (

The likelihood function is denoted as:

where _{θ}

Let _{θ}

The likelihood function is maximised to get the optimal estimators

As stated earlier, the Wiener process _{t} is assumed to follow a normal distribution with mean 0 and variance

let

The expression

From the above, the specific log-likelihood function of the GBM is found by replacing the equivalent Weiner process such that:

The natural logarithm of the likelihood function is differentiated in terms of

where

Determining

Asset return distributions are frequently presumed to follow either a normal or a lognormal distribution. It can also follow GBM based on the Gaussian process. However, many empirical studies have shown that return distributions are usually not normal. They often find evidence of non-normality, such as heavy tails, excess kurtosis and finite moments. One class of fat-tailed distributions with the potential to give a better approximation to the distribution of stock returns is the

An extension of the version of the GBM model, wherein it is assumed that the random noise process, _{t}, is a student’s _{t}, effects a change in the equation:

The distribution of the error term for this specification according to Shimada and Tsukuda (

The SMC, also known as particle filter algorithm (Gordon, Salmond & Smith

Assuming that we have at time

Kitagawa and Sato (

For

Repeat the following steps for

For

For

For

Generate

This Monte Carlo filter returns

In this section, with known model parameters, we apply the particle filter algorithm based on t-distribution to update the information about the underlying asset value process recursively from the observed times series of stock prices.

With known parameters Θ = {_{t}; _{t};

The algorithm for the filtering is an extension of Godsill, Doucet and West’s (_{t},| _{t}) for each

Given

Generate

Generate a random number

Compute

Compute

Generate

Resample from

As averred by Lawrence, Klimberg and Lawrence (_{t}; and forecast value at time period _{t}. The mean absolute percentage error (MAPE) seems to be the most widely used to evaluate the forecasting method that considers the effect of the magnitude of the actual values. It is a measure of prediction accuracy of a forecasting method in statistics. It usually expresses accuracy as a percentage and is defined as follows:

The difference between _{t} and _{t} is divided by the actual value _{t} again. The absolute value in this calculation is summed for every forecasted point in time and divided by the number of fitted points

We apply the above-described methodology to model the stock prices of five firms of the NSE – each from five different sectors, namely the banking sector (Guaranteed Trust Bank [GTB]), oil and gas sector (Oando), construction sector (Julius Berger [Jberger]), health care sector (GlaxoSmith) and industrial goods sector (Chemical & Allied Product [CAP]) over the period 02 January 2010 to 31 December 2014.

The data series is transformed into daily log returns series so that we can obtain stationary series. Descriptive statistical summary is obtained to view the data for the daily stock prices and returns of all the indices.

^{-6}, which is less than 0.01.

Descriptive statistical summary for the daily stock prices.

Index | Prices |
Returns |
||||||||
---|---|---|---|---|---|---|---|---|---|---|

Mean | Standard deviation | Skewness | Kurtosis | Jarque-Bera | Mean | Standard deviation | Skewness | Kurtosis | Jarque-Bera | |

Guaranteed Trust Bank (GTB) | 40.3754 | 18.4573 | 0.6718 | 3.3487 | 194.387 |
0.0030950 | 0.0278 | −2.4739 | 29.4686 | 684.195 |

Oando | 70.9741 | 47.4833 | 1.0297 | 4.8445 | 202.394 |
−0.00062810 | 0.0352 | −1.3034 | 54.4961 | 312.573 |

Julius Berger (JBerger) | 46.8880 | 25.5413 | 0.9931 | 4.2569 | 207.281 |
0.0041886 | 0.0458 | −1.5541 | 42.6513 | 497.263 |

GlaxoSmith | 76.9873 | 48.2532 | 1.1542 | 4.2342 | 213.237 |
0.0070399 | 0.0300 | −0.1911 | 30.2326 | 513.240 |

Julius Berger (CAP) | 49.3441 | 27.3775 | 0.2783 | 3.6845 | 195.142 |
0.000875734 | 0.0367 | −3.6781 | 20.7944 | 795.142 |

GTB, Guaranteed Trust Bank; JBerger, Julius Berger; CAP, Chemical & Allied Product.

Guaranteed Trust Bank (GTB) (a) stock pricing over time, (b) returns over time.

Oando (a) stock pricing over time, (b) returns over time.

JBerger (a) stock pricing over time, (b) returns over time.

GlaxoSmith (a) stock pricing over time, (b) returns over time.

Chemical and Allied Product (CAP) (a) stock pricing over time, (b) returns over time.

The stock prices of each of the five firms of the NSE for the year 2010–2014 were used to derive the drift and volatility.

Drift and volatility values of stock prices.

Index | Drift ( |
Volatility ( |
---|---|---|

GTB | 0.0720 | 0.2816 |

Oando | 0.0485 | 0.2794 |

JBerger | 0.0514 | 0.2723 |

GlaxoSmith | 0.0354 | 0.2837 |

CAP | 0.0624 | 0.2808 |

GTB, Guaranteed Trust Bank; JBerger, Julius Berger; CAP, Chemical & Allied Product.

These two parameters (drift and volatility) were then used to create the geometric Brownian path for both the GBM normal and student’s

Estimated parameters of the geometric Brownian motion normal and student’s

Index | GBM normal |
GBM student’s |
||||||
---|---|---|---|---|---|---|---|---|

Log- likelihood | AIC | Log-likelihood | AIC | |||||

GTB | 0.45 | 0.37 | −2797 | 4359 | 0.34 | 1.09 | −2740 | 4248 |

Oando | 0.12 | 0.33 | −2341 | 4686 | 0.30 | 1.27 | −2331 | 4568 |

JBerger | 0.10 | 0.40 | −2149 | 4302 | 0.37 | 1.03 | −2135 | 4176 |

GlaxoSmith | 0.23 | 0.36 | −2344 | 4255 | 0.41 | 1.04 | −1234 | 4234 |

CAP | 0.34 | 0.44 | −2783 | 4684 | 0.32 | 1.23 | −1345 | 4221 |

CAP, Chemical & Allied Product, JBerger, Julius Berger; AIC, Akaike information criterion; GBM, geometric Brownian motion; GTB, Guaranteed Trust Bank.

The log-likelihood for the GBM student’s

The GBM normal and student’s

Evaluation statistic-distribution comparison of techniques based on the normal and student’s

Models | Mean absolute percentage deviation |
---|---|

GBM normal | 0.0967 = 9.67% |

GBM student’s |
0.0652 = 6.52% |

GBM, Geometric Brownian motion.

Graphically, for a single run, the estimation results obtained from running these two models are shown in

Brownian path for Guaranteed Trust Bank (GTB).

Brownian path for Oando.

Brownian path for JBerger.

Brownian path for GlaxoSmith.

Brownian path for Chemical & Allied Product.

Geometric Brownian motion model usually assumes that the distribution of asset returns is either normal or lognormal. Previous approaches to the estimation of GBM model have revealed that return distributions are usually not normal. In this article, a GBM model based on the

The descriptive statistics for the five indices of the NSE are given in

In evaluating the proposed GBM level of precision, the model parameters are estimated. A particle filter technique based on student’s

The plot of volatility estimation for each of the five firms’ stock prices is shown in

This work presented an extension of the random noise process, _{t}, in the GBM model from normal to student’s

The authors declare that they have no financial or personal relationships that may have inappropriately influenced them in writing this manuscript.

B.E.N. was the project leader and was responsible for most of the theoretical and experimental work done. B.E.N. contributed by introducing an extension of the random noise process in the GBM model from normal to student"s