Mouvement Brownien Finance Pdf

Brownian motion, also known as a Wiener process, has become a cornerstone in financial modeling since its initial application by Louis Bachelier in his 1900 dissertation, "Théorie de la Spéculation." Bachelier proposed using Brownian motion to model stock price movements, laying the groundwork for much of modern quantitative finance. The core idea is that price fluctuations can be treated as random walks, where each step is independent and identically distributed (i.i.d.). This seemingly simple concept has had profound implications, shaping how we understand and model asset prices, option pricing, and risk management.

A key attribute of Brownian motion in finance is its continuous-time nature. Unlike discrete-time models, which evaluate prices at specific intervals (e.g., daily closing prices), Brownian motion allows for price changes to occur continuously. This is particularly valuable when dealing with high-frequency trading or complex derivatives where continuous monitoring is crucial. Mathematically, a standard Brownian motion, W(t), has the following properties:

• W(0) = 0 (starts at zero)
• Independent increments: For any t > s ≥ 0, W(t) - W(s) is independent of the path before time s.
• Normally distributed increments: For any t > s ≥ 0, W(t) - W(s) follows a normal distribution with mean 0 and variance t - s.
• Continuous paths: The function W(t) is continuous in t.

In financial applications, the standard Brownian motion is often modified to incorporate a drift (μ) representing the average rate of return and a volatility (σ) representing the degree of price fluctuation. This leads to the Geometric Brownian Motion (GBM), which is widely used to model stock prices. The GBM equation is often written as:

dS(t) = μS(t)dt + σS(t)dW(t)

Where S(t) is the price of the asset at time t, μ is the drift rate, σ is the volatility, and dW(t) is the increment of the standard Brownian motion. This equation states that the change in price (dS(t)) is proportional to both the current price (S(t)) and a random component driven by the Brownian motion. The drift term (μS(t)dt) represents the expected growth of the asset, while the volatility term (σS(t)dW(t)) captures the random fluctuations.

The adoption of Brownian motion and GBM has had a transformative impact on option pricing. The Black-Scholes-Merton model, developed in the 1970s, relies heavily on the assumption that asset prices follow a GBM. This model provides a theoretical framework for calculating the fair price of European-style options and has become an essential tool for options traders and risk managers.

Despite its widespread use, the Brownian motion model is not without its limitations. Real-world asset prices often exhibit characteristics that deviate from the assumptions of the model, such as:

• Fat Tails: Actual price changes often have a higher probability of extreme events (large gains or losses) than predicted by the normal distribution.
• Volatility Clustering: Periods of high volatility tend to be followed by periods of high volatility, and vice-versa. This contradicts the assumption of constant volatility in the GBM.
• Jumps: Sudden and discontinuous price jumps can occur due to unexpected news events or market shocks, which are not accounted for in standard Brownian motion.

To address these limitations, more sophisticated models have been developed, including jump-diffusion models, stochastic volatility models, and models incorporating heavier-tailed distributions. These models attempt to capture the complexities of real-world financial markets more accurately, while still building on the fundamental framework established by Brownian motion.