Jump diffusion geometric Brownian motion with the re-invested dividend yielding asset and transaction cost

Introduction

The Black-Scholes model, developed by Fischer Black and Myron Scholes in 1973 [1] , is a fundamental tool in financial mathematics for pricing options. This model revolutionized the options market by providing a systematic method for valuing European-style options. The Black-Scholes framework consists of two primary assets: a riskless bond with a constant interest rate and a stock whose price follows a geometric Brownian motion (GBM). The stock price dynamics are given by:

Where S(t) represents the stock price at time t, S 0 is the initial stock price, µ is the drift term (constant rate of return), σ is the volatility, and B t is the standard Brownian motion [2] . The standard Black While these assumptions simplify the mathematical formulation, they do not fully capture the complexities of real-world markets. Various modifications, such as stochastic volatility models and jump diffusion models, have been developed to address these limitations.

A crucial component of the Black-Scholes model is the geometric Brownian motion (GBM), which governs stock price evolution. The differential equation for GBM is given by: (2) where W(t) is the Wiener process representing the random fluctuations in stock price dynamics. The equation consists of two components: 1. Deterministic Growth (µS(t)dt)-Models the expected return of the stock. 2. Random Fluctuations (σS(t) dW(t))-Captures stochastic variations due to market randomness. Despite its utility, empirical studies suggest that stock prices may exhibit sudden jumps due to unexpected economic events, financial announcements, or external shocks. The geometric Brownian motion fails to account for these discontinuous movements, leading to the development of more advanced models, such as the jump diffusion model introduced by Merton. This model extends the standard GBM by incorporating discrete jumps in asset prices, allowing for a more accurate representation of real-world financial markets [3] . Our current model, A Jump Diffusion Geometric Brownian Motion with the Re-Invested Dividend Yielding Asset and Transaction Cost, expands upon these advancements by introducing an additional layer of realism to asset price modeling. Unlike traditional jump diffusion models, our approach integrates: 1. Re-Invested Dividend Yields-Captures the compounding effect of reinvesting dividends, influencing asset growth dynamics. 2. Transaction Costs-Incorporates the friction of trading costs, making the model more applicable to real-world financial markets. 3. Jump Diffusion Process-Enhances price modeling by allowing for sudden, stochastic price jumps in asset valuation.

This extended framework provides a more comprehensive tool for pricing and risk assessment by accommodating factors that traditional models overlook. It not only refines derivative pricing but also improves investment decision-making in markets where dividends and trading costs play a significant role [4] . In conclusion, while the Black-Scholes model remains a cornerstone of financial mathematics, its assumptions limit its applicability in real-world markets. The incorporation of stochastic volatility, jump processes, dividend reinvestments, and transaction costs offers a more robust framework for modeling asset price dynamics and improving derivative pricing strategies.

Methodology

In this section, we recall some key basic concepts, methodologies and the result that are fundamental in the study:

Stochastic Process

A stochastic process can be denoted as X = [X(t); t ∈ (0; α)] which is a collection of random variables such that for each t in the index set (0; α) which specifies the time interval over which the process is defined, X(t) is a random variable that represents the state of the process at time t. A discrete time stochastic is the one where the value of the variable can only change at a certain fixed points in time. On the other hand, in a continuous time stochastic process, change can take any value within a certain range [3] .

Markov stochastic process

A Markov stochastic process, or simply a Markov process, is a specific type of stochastic process where the future movements of the process depend solely on its current state.

Wiener process (Brownian Motion)

The Wiener process W(t), also known as Brownian motion, is a continuous-time stochastic process that has wide applications in various fields, including finance, physics, and statistics. The Wiener process, denoted as W(t) for t ≥ 0, is characterized by the following properties: 1. Normal increments: The increments of the Wiener process, W(t) -W (s), where 0 ≤ s < t are normally distributed.

Specifically, W(t) -W (s) ∽N(0, t -s) meaning that the increment has a mean of 0 and a variance equal to the time deference t -s. 2. Independent increments: The increments of the Wiener process are independent of each other. This property states that for disjoint time intervals, the increments are statistically independent. For example, W (t1) -W (s1) and W (t2) -W (s2) are independent when t1 < s1 < t2 < s2. This independence property allows for the modeling and analysis of complex systems involving random processes.

Continuity of paths:

The Wiener process has almost surely continuous paths. This means that the sample paths of the process are continuous functions of time with probability 1. The continuous paths property distinguishes the Wiener process from other stochastic processes that exhibit jumps or discontinuities. i.e W (t) is a continuous function of t.

where s and t are functions of time [3] .

Generalised Wiener Process

The Generalized Wiener process has continuous sample paths, like the standard Wiener process, making it suitable for modeling continuous phenomena. However, its long-range dependence property allows it to capture complex behaviors and trends that standard Brownian motion cannot represent. A generalised Wiener process defines the dynamics of X in terms of a deterministic component adt and a stochastic component bdW as;

where a represents the mean rate or drift of the variable X and b represents the variance rate or diffusivity of X. It quantifies the magnitude of the random fluctuations or noise added to the process, dt is the infinitesimal time interval (often referred to as an "infinitesimal change in time") and dW represents the Wiener process or Brownian motion, which introduces randomness or variability to the path followed by X. By incorporating both the deterministic and stochastic components, the generalized Wiener process allows for modeling variables that exhibit both expected trends and random fluctuations. In a small interval ∆t, the change in the value of X denoted as ∆X is given by;

Ito's Lemma

Ito's Lemma is a powerful tool in stochastic calculus and is widely used in the derivation of various financial models, such as the Black-Scholes option pricing model. It allows for the analysis and manipulation of functions of stochastic processes, enabling the calculation of derivatives and the development of stochastic differential equations that describe the dynamics of complex systems involving randomness. By applying Ito's Lemma, one can analyze and model the behavior of financial derivatives, asset prices, and other stochastic processes, taking into account the impact of random fluctuations and incorporating uncertainty into mathematical models. This is the formula used for solving stochastic differential equations. Suppose that the value X follows an Ito's Process, then dX = a(X; t)dt + b(X; t)dW, (9

where dW is a wiener process, a and b are functions of X and t. The variable X has a drift rate of a and variance of b2. Ito's Lemma shows that the function G(X; t) twice differentiable in X and once in t, is also an an Ito's process given by;

Geometric Brownian motion

A geometric Brownian motion (GBM) is a continuous-time stochastic process that is commonly used to model the price dynamics of financial assets such as stocks, currencies, and commodities. It is an extension of the standard Brownian motion (also known as the Wiener process) that incorporates exponential growth and volatility. A specific type of Ito's process is the geometric brownian motion of the form;

where a(X, t) = aX and b(X, t) = bX [3] in the equation above, the geometric Brownian motion has been applied in stock pricing and is given as;

where St represents the value of the asset at time t, µ is the average rate of return or drift of the asset per unit time and σ is the standard deviation of the asset's returns or volatility per unit time and dW(t) is a Wiener process increment, representing a random shock or noise in the system. The term µStdt represents the deterministic growth component, which accounts for the expected return or trend in the stock price while the term σStdW(t) represents the random or stochastic component which is scaled by the stock price St and the volatility σ. The equation is expressed as;

The solution to this stochastic differential equation is given by the following exponential process:

where S0 is the initial value of the asset at time t = 0, and W (t) represents the Wiener process.

Results

To study and derive the price of a jump diffusion Geometric Brownian motion with the re-invested dividend and transaction cost, Geometric Brownian motion of the stock price, Geometric Brownian motion with jump process, Geometric Brownian motion with dividend yielding asset with transaction cost was considered.

Derivation of Stock price following Geometric Brownian motion

Ito's Lemma is a fundamental tool in stochastic calculus used to describe the dynamics of a stochastic process like stock price following Geometric Brownian motion. Ito's lemma is given by the following formula;

References

1. Black F, Scholes N. The pricing of options and corporate liabilities. Journal of Political Economy. 1973;81(3):637-659.
2. Merton RC. Option pricing when underlying stock returns are discontinuous. Journal of Financial Economics. 1976;3(12):125-144.
3. Opondo MO, Oduor DB, Odundo F. Jump diffusion logistic Brownian motion with dividend yielding asset. International Journal of Mathematics and Applications. 2021;9(4):25-34.
4. Oduor DB. A combination of dividend and jump diffusion process on Heston model in deriving Black-Scholes equation. International Journal of Applied Mathematics and Statistics. 2022;7(1a):770-779.
5. Li Y, Mao X, Song Y, Jian T. Optimal investment and proportional reinsurance strategy under the mean-reverting OrnsteinUhlenbeck process and net profit condition. AIMS Journals. 2010.