Stochastic Modeling

Markov Chain

Markov Chains in Stock Market

The Random Walk of Stock Market

Getting Equilibrium Matrix in Markov Chain

Weiner Process

Simulating Geometric Brownian Motion

Essentials of Markovian Models

GBM is a stochastic process commonly used to model stock prices. It is a continuous-time stochastic process where the logarithm of the stochastic process follows a Brownian motion.

Here is the formula for Geometric Brownian Equation –

$$ S(t) = S(0) \times e^{(\mu – \frac{\sigma^2}{2})t + \sigma W(t)} $$

Where:

- \( S(t) \) is the stock price at time \( t \).
- \( S(0) \) is the initial stock price.
- \( \mu \) is the drift rate (in this case, the annualized drift of 0.2).
- \( \sigma \) is the volatility (in this case, the annualized volatility of 0.3).
- \( W(t) \) is a Wiener process or Brownian motion.

This equation elegantly encapsulates both the deterministic (drift) and stochastic (diffusion) aspects of asset price dynamics, forming a bedrock for various financial modeling and derivative pricing endeavors.

The term \(\sigma W(t)\) in the GBM formula introduces randomness into the stock price evolution, and it’s driven by the Wiener process \(W(t)\).

$$ dS_t = \mu S_t dt + \sigma S_t dW_t $$

The equation is fundamental to finance, and it models the dynamics of an asset price over time under the geometric Brownian motion (GBM) framework. Let’s dissect this equation to better understand its components and implications:

The formula \( S(t) = S(0) \times e^{(\mu – \frac{\sigma^2}{2})t + \sigma W(t)} \) for Geometric Brownian Motion (GBM) is derived from the stochastic differential equation (SDE) \( dS_t = \mu S_t dt + \sigma S_t dW_t \). They are two representations of the same stochastic process.

The left-hand side \(dS_t\) represents the incremental change in the asset price \(S\) at time \(t\).

The first term on the right-hand side, \(\mu S_t dt\), is known as the drift term. It represents the deterministic trend in the asset price, often reflecting the expected return over a small time interval \(dt\).

- \(\mu\): Known as the drift coefficient, it is analogous to the expected return of the asset. In a way, it aims to capture the general trend of the asset price over time.
- \(S_t\): The asset price at time \(t\).
- \(dt\): A small increment in time.

The second term on the right-hand side, \(\sigma S_t dW_t\), is known as the diffusion term. It captures the stochastic (random) movements in the asset price.

- \(\sigma\): Known as the volatility coefficient, it quantifies the extent of the asset’s price fluctuations.
- \(dW_t\): Represents a Wiener Process increment over the small time interval \(dt\), which embodies the random shock to the asset price.

The Wiener Process, denoted by \(dW_t\), is a fundamental stochastic process that represents a standard Brownian motion. It captures the random component of the asset price movement over the time interval \(dt\).

This stochastic differential equation models asset price dynamics under Geometric Brownian Motion (GBM), which assumes that returns are normally distributed and the logarithm of asset prices forms a Brownian motion with drift. It’s a crucial assumption underlying many financial models, including the Black-Scholes Model for options pricing.

Here’s a concise walkthrough of how the formula for GBM comes about from the SDE:

**1. Stochastic Differential Equation (SDE):**

The SDE

$$ dS_t = \mu S_t dt + \sigma S_t dW_t $$

is the continuous-time model for the dynamics of an asset price \( S_t \).

**2. Integration:**

To get the formula for \( S(t) \), we need to solve this SDE. The method of solving involves Itô’s Lemma, a fundamental result in stochastic calculus.

**3. Itô’s Lemma Application:**

Applying Itô’s Lemma, we can find a differential equation for \( \ln(S_t) \). The procedure leads to the differential equation

$$ d(\ln(S_t)) = (\mu – \frac{\sigma^2}{2}) dt + \sigma dW_t $$.

**4. Integration of the Result:**

Integrating this result over the interval \([0, t]\), we obtain:

$$ \ln(S_t) – \ln(S_0) = (\mu – \frac{\sigma^2}{2}) t + \sigma W(t) $$.

**5. Exponential Transformation:**

Taking the exponential of both sides to eliminate the natural logarithm, we get:

$$ S(t) = S(0) \times e^{(\mu – \frac{\sigma^2}{2})t + \sigma W(t)} $$.

This formula for \( S(t) \) represents the asset price at time \( t \) in terms of the initial asset price \( S(0) \), the drift rate \( \mu \), the asset volatility \( \sigma \), and a standard Brownian motion \( W(t) \) up to time \( t \).

It provides a closed-form solution for the asset price under the assumptions of Geometric Brownian Motion, which is pivotal for various applications in finance, including options pricing via the Black-Scholes Model.

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