# Generalised Geometric Brownian Motion: Theory and Applications to Option Pricing

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## Abstract

**:**

## 1. Introduction

## 2. Background

#### 2.1. Standard GBM

#### 2.2. Black–Scholes Formula

#### 2.3. Subdiffusive GBM

## 3. Generalised GBM

#### 3.1. Subordination Approach

**Remark**

**1.**

**Remark**

**2.**

**Remark**

**3.**

#### 3.2. Generalised BS Formula

#### 3.3. Calculation of Moments

#### 3.4. Exponentially Truncated Subdiffusive GBM

#### 3.5. Combined Standard and Subdiffusive GBM

#### 3.6. Mix of Subdiffusive GBMs

## 4. Numerical Experiments

## 5. Empirical Example

**Moneyness in TSLA:**Let us now turn our attention to Figure 4, where we use TSLA data gathered on 1st March 2018 on options which expire on 16th March 2018 to examine the dependence of the sGBM, the mix of GBM-sGBM and the mix of sGBM models on $\alpha $ in predicting the option price. We discover that, in general, the TSLA asset price dynamics are best described as a sGBM, and the minimal error occurs around $\alpha =0.2$. For large $\alpha $, the mix of sGBM becomes the best performer, because it inherently includes a process with a lower subdiffusion and, thus, it has a close resemblance to the sGBM process with low $\alpha $ than the other models. For every $\alpha $, the mix of GBM-sGBM has the worst performance since it includes a term with a normal diffusion.

**Maturity in Apple (AAPL):**Next, we use AAPL data that were gathered for at-the-money options on 28th February 2018 and examine how the maturity T affects the performance of the same models in predicting the option price. We focus solely on data for at-the-money options in order to remove the potential bias in the prediction error that arises from the potential moneyness effect, which as we saw in the above paragraph might arise (i.e., we discovered that the error rate with respect to $\alpha $ (${\alpha}_{1}$) depends on the moneyness of the option).

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

GBM | Geometric Brownian motion |

sGBM | Subdiffusive geometric Brownian motion |

gGBM | Generalised geometric Brownian motion |

BS | Black-Scholes |

CTRW | Continuous time random walk |

MSD | Mean squared displacement |

ML | Mittag-Leffler |

ATM | at-the-money |

RMSE | Root mean squared error |

TSLA | Tesla |

AAPL | Apple |

## Appendix A. Solution of the Fokker-Planck Equation for Standard GBM

## Appendix B. Derivation of the Fokker-Planck Equation for gGBM from CTRW Theory

## Appendix C. General Results for nth Moment

## Appendix D. Fox H-Function

## References

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**Figure 1.**Generalised geometric Brownian motion (gGBM) properties. (

**a**) An example for simulated individual trajectories of gGBM for different memory kernels: standard GBM (blue solid line), subdiffusive GBM (sGBM) (red dashed line), a mix of standard GBM and sGBM (yellow dotted line), and a mix of sGBM (violet dot-dashed line). (

**b**) Numerical estimation for the first moment in GBM, sGBM, a mix of standard GBM and sGBM and a mix of sGBM as a function of time. (

**c**) Same as (

**b**), only for the second moment. (

**d**) Empirical PDF for the logarithmic return at $t=1$ year estimated from 1000 realisations of gGBM. (

**a**–

**d**) In the simulations, $\mu =0.03$ and ${\sigma}^{2}=0.02$. Moreover, for the sGBM case we set $\alpha =0.8$, for the mix GBM-sGBM case, we set $\alpha =0.8$ and ${w}_{1}={w}_{2}=0.5$, and for the mix of sGBM case ${\alpha}_{1}=0.8$, ${\alpha}_{2}=0.6$, and ${w}_{1}={w}_{2}=0.5$.

**Figure 2.**gGBM skewness and excess kurtosis. (

**a**) Skewness for the distribution of logarithmic return for sGBM, a mix of standard GBM and sGBM and a mix of sGBM at $t=1$ year as a function of $\alpha $ (for sGBM and the mix of GBM-sGBM) or ${\alpha}_{1}$ (for the mix of sGBMs). (

**b**) Same as (

**a**), only for the excess kurtosis. (

**a**,

**b**) The PDF for the logarithmic return at $t=1$ year is estimated from 1000 realisations of gGBM. In the simulations, $\mu =0.03$ and ${\sigma}^{2}=0.02$. Moreover, for the mix GBM-sGBM case, we set ${w}_{1}={w}_{2}=0.5$, and for the mix of sGBM case ${\alpha}_{2}=0.8$ and ${w}_{1}={w}_{2}=0.5$.

**Figure 3.**gGBM at-the-money (ATM) implied volatility and volatility skew. (

**a**) ATM implied volatility for sGBM, a mix of standard GBM and sGBM and a mix of sGBM for an option with $K={x}_{0}=1$, $T=0.083$ years (one month) as a function of $\alpha $ (for sGBM and the mix of GBM-sGBM) or ${\alpha}_{1}$ (for the mix of sGBMs). (

**b**) Same as (

**a**), only for the ATM volatility skew. (

**a**,

**b**) We assume that $r=0.02$. Moreover, for the mix GBM-sGBM case, we set ${w}_{1}={w}_{2}=0.5$, and for the mix of sGBM case ${\alpha}_{2}=0.8$ and ${w}_{1}={w}_{2}=0.5$.

**Figure 4.**Moneyness in Tesla (TSLA). Root mean squared error (RMSE) of the predicted option price as a function of $\alpha $ (${\alpha}_{1}$) for sGBM, a mix of GBM-sGBM and a mix of sGBM. The inset plot gives the difference between the predicted TSLA option price ${C}_{g}$ and its real value C as a function of the strike price of the option for various choices of $\alpha $. The data is taken on 1st March 2020 and describe the value of TSLA options that expire on 16th March 2020. For the mix GBM-sGBM case we set ${w}_{1}={w}_{2}=0.5$, and for the mix of sGBM case ${\alpha}_{2}=0.8$ and ${w}_{1}={w}_{2}=0.5$.

**Figure 5.**Maturity in at-the-money AAPL options. Root mean squared error (RMSE) of the prediction of the AAPL at-the-money option price with data taken on 28th February 2018 as a function of $\alpha $ for various maturity periods T (measured in years). (

**a**) The gGBM model is sGBM. (

**b**) The gGBM model is a mix of GBM-sGBM. (

**c**) The gGBM model is a mix of sGBM. For the mix GBM-sGBM case we set ${w}_{1}={w}_{2}=0.5$, and for the mix of sGBM case ${\alpha}_{2}=0.8$ and ${w}_{1}={w}_{2}=0.5$.

**Table 1.**Minimum prediction error and optimal $\alpha $ (${\alpha}_{1}$) for ATM AAPL options. For the mix GBM-sGBM case we set ${w}_{1}={w}_{2}=0.5$, and for the mix of sGBM case ${\alpha}_{2}=0.8$ and ${w}_{1}={w}_{2}=0.5$.

Maturity | sGBM | GBM-sGBM | mix of sGBM | |||
---|---|---|---|---|---|---|

(in Years) | $min\mathit{\alpha}$ | RMSE | $min\mathit{\alpha}$ | RMSE | $min{\mathit{\alpha}}_{1}$ | RMSE |

0.006 | 0.89 | 0.28 | 0.84 | 0.28 | 0.88 | 0.31 |

0.025 | 0.96 | 0.39 | 0.93 | 0.39 | 0.93 | 0.48 |

0.044 | 0.97 | 0.46 | 0.93 | 0.46 | 0.97 | 0.6 |

0.063 | 0.99 | 0.52 | 0.98 | 0.52 | 0.98 | 0.71 |

0.101 | 1.00 | 0.72 | 0.99 | 0.72 | 0.99 | 1.00 |

0.140 | 1.00 | 0.61 | 0.99 | 0.60 | 0.99 | 0.91 |

0.888 | 0.33 | 4.52 | 0.35 | 6.38 | 0.38 | 5.47 |

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**MDPI and ACS Style**

Stojkoski, V.; Sandev, T.; Basnarkov, L.; Kocarev, L.; Metzler, R.
Generalised Geometric Brownian Motion: Theory and Applications to Option Pricing. *Entropy* **2020**, *22*, 1432.
https://doi.org/10.3390/e22121432

**AMA Style**

Stojkoski V, Sandev T, Basnarkov L, Kocarev L, Metzler R.
Generalised Geometric Brownian Motion: Theory and Applications to Option Pricing. *Entropy*. 2020; 22(12):1432.
https://doi.org/10.3390/e22121432

**Chicago/Turabian Style**

Stojkoski, Viktor, Trifce Sandev, Lasko Basnarkov, Ljupco Kocarev, and Ralf Metzler.
2020. "Generalised Geometric Brownian Motion: Theory and Applications to Option Pricing" *Entropy* 22, no. 12: 1432.
https://doi.org/10.3390/e22121432