Tracing Market Vibrations: The Mathematical Bridge Between Wiener Process and Black-Scholes Model
In financial mathematics, the Black-Scholes Model has gained significant attention for its analytical approach to pricing European-style options. This model is closely tied to the Geometric Brownian Motion (GBM), a well-established method for modeling financial asset prices. But before we delve into GBM, it’s essential to understand its foundation: the Wiener Process.
The Wiener Process is a fundamental concept in stochastic calculus, and its extension into GBM provides the mathematical basis for the Black-Scholes Model. Let’s discuss through this progression, from the Wiener Process to GBM, and then to the Black-Scholes Model, highlighting the importance of each step in understanding option pricing.
The Wiener Process:
The Wiener Process (\( W_t \)) is defined by four properties: independence, stationarity, normality, and continuity. It models the randomness in the evolution of a variable over continuous time.
Geometric Brownian Motion:
The Geometric Brownian Motion (GBM) is a common model used to depict the trajectory of stock prices over time in a continuous domain. The stochastic differential equation (SDE) for GBM is given by:
\[ dS_t = \mu S_t dt + \sigma S_t dW_t \]Let’s break down each term:
- \( S_t \): This is the stock price at time \( t \).
- \( \mu \): Known as the drift coefficient, it represents the expected return of the stock, and captures the systematic, deterministic trends affecting the stock price.
- \( \sigma \): This is the volatility coefficient, reflecting the magnitude of the stock’s price fluctuations.
- \( dW_t \): This term represents a Wiener process or Brownian motion, embodying the random, stochastic fluctuations in the stock price. It’s a standard normal random variable over the infinitesimally small interval \( dt \).
The entire equation models the dynamics of a stock price over time, combining deterministic trends (drift term) and stochastic shocks (diffusion term).
Black-Scholes Partial Differential Equation (PDE):
The Black-Scholes Model (BSM) is built on the foundation laid by the GBM. It aims to determine the theoretical value of European-style options. The core of the Black-Scholes Model is the Black-Scholes PDE given by:
\[ \frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} – rV = 0 \]
Dissecting the PDE:
- \( V \): This is the option price, which depends on the stock price \( S \) and time \( t \).
- \( \frac{\partial V}{\partial t} \): This term represents the rate of change of option price with respect to time, often referred to as the option’s Theta (\( \Theta \)).
- \( \frac{\partial^2 V}{\partial S^2} \): This term, known as Gamma (\( \Gamma \)), reflects how the Delta (\( \Delta \), rate of change of option price with respect to stock price) changes with a change in stock price.
- \( \frac{\partial V}{\partial S} \): This is the Delta of the option.
- \( r \): This is the risk-free interest rate, representing the time value of money.
- \( \sigma \): As before, this is the volatility of the stock.
The Black-Scholes PDE encapsulates how the option price evolves over time, considering the stock price movements, time decay, and volatility.
From GBM SDE to Black-Scholes PDE: Derivation of the Black-Scholes Formula
The starting point is the stochastic differential equation for GBM:
$$dS_t = \mu S_t dt + \sigma S_t dW_t$$
We have already discussed on it earlier.
Itô's Lemma:
To transition from GBM to the Black-Scholes formula, apply Itô’s Lemma to the function \(f(t, S_t)\) where \(f\) is twice differentiable. Itô’s Lemma yields the following expression:
$$df(t, S_t) = \left( \frac{\partial f}{\partial t} + \mu S_t \frac{\partial f}{\partial S_t} + \frac{1}{2} \sigma^2 S_t^2 \frac{\partial^2 f}{\partial S_t^2} \right) dt + \sigma S_t \frac{\partial f}{\partial S_t} dW_t$$
Black-Scholes Partial Differential Equation (PDE):
Construct a portfolio consisting of one option and a certain number of shares of the stock to eliminate the risk. Let \(V(t, S_t)\) be the price of the option, and let \(-\Delta\) be the number of shares held. The value of the portfolio is:
$$\Pi(t) = V(t, S_t) – \Delta S_t$$
Differentiate \(\Pi(t)\) with respect to \(t\) using Itô’s Lemma to obtain a riskless portfolio (i.e., one whose value does not depend on \(dW_t\)). By choosing \(\Delta\) such that the term in front of \(dW_t\) vanishes, obtain an expression for \(\Delta\) and eliminate \(dW_t\) from the expression for \(d\Pi\). This yields the Black-Scholes PDE:
$$\frac{\partial V}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} – rV = 0$$
Solving the Black-Scholes PDE:
Solve the Black-Scholes PDE with the boundary condition that \(V(T, S_T)\) equals the payoff of the option at expiry. This leads to the Black-Scholes formula for a European call option:
$$C(S_t, t) = S_t N(d_1) – X e^{-r(T-t)} N(d_2)$$
and for a European put option:
$$P(S_t, t) = X e^{-r(T-t)} N(-d_2) – S_t N(-d_1)$$
where
$$d_1 = \frac{1}{\sigma \sqrt{T-t}} \left( \ln \left( \frac{S_t}{X} \right) + \left( r + \frac{\sigma^2}{2} \right)(T-t) \right)$$
$$d_2 = d_1 – \sigma \sqrt{T-t}$$
The Black-Scholes formula provides a theoretical estimate of the prices of European-style options. This formula assumes that stock prices follow a Geometric Brownian Motion, which is consistent with the continuous compounding and continuous trading assumptions inherent in the Black-Scholes model. Through this derivation, we see how the Black-Scholes formula emerges naturally from the dynamics of the stock price as modeled by the GBM, under the risk-neutral measure.
Conclusion:
The transition from the Wiener Process to the Black-Scholes formula is a compelling illustration of the application of stochastic calculus in financial mathematics. The transition from GBM to the Black-Scholes Model demonstrates a profound blend of stochastic calculus and financial modeling. The GBM serves as a bedrock for deriving the Black-Scholes PDE, which in turn leads to the Black-Scholes formula, a cornerstone in the field of financial derivatives pricing.