Geometric Brownian motion

The above discussion is used to derive a model of the behaviom of asset prices sometimes referred to as geometric Brownian motion. The dynamics of the asset price X are represented by the ltd process shown in Equation (2.18), where there is a drift rate of a and a variance rate of b X, ... 

Geometric Brownian motion is a commonly used model, which assumes that changes in asset prices are uncorrelated over time and that small movements in prices can be described by... 

The Monte Carlo method, however, is prone to model risk. If the stochastic process chosen for the underlying variable is unrealistic, so will be the estimate of VaR. This is why the choice of the underlying model is particularly important. The geometric Brownian motion model described above adequately describes the behavior of some financial variables, but certainly not that of short-term fixed-income securities. In the Brownian motion, shocks on prices are never reversed. This does not represent the price process for default-free bonds, which must converge to their face value at expiration. 

The markets assume that the state variables evolve through a geometric Brownian motion, or Weiner process. It is therefore possible to model their evolution using a stochastic differential equation. The market also assumes that the cash flow stream of assets such as bonds and equities is a function of the state variables. 

The zero-coupon bond yield has thus far been described as a stochastic process following a geometric Brownian motion that drifts with no discernible trend. This description is incomplete. It implies that the yield will either rise or fall continuously to infinity, which is clearly not true in practice. To be more realistic, the model needs to include a term capturing the fact that interest rates move up and down in a cycle. The short rate s expected direction of change is the second parameter in an interest rate model. This is denoted in some texts by a letter such as a or i , in others by /i. The short-rate process can therefore be described as function (4.3). 

Because this process is a geometric Brownian motion, it has two important properties. First, the drift rate is equal to the expected change in... 

The behavior of underlying asset prices follows a geometric Brownian motion, or Weiner process, with a variance rate proportional to the square root of the price. This is stated formally in (8.11). 

The Black model refers to the underlying assets or commodity s spot price, S t). This is defined as the price at time t payable for immediate delivery, which, in practice, means delivery up to two days forward. The spot price is assumed to follow a geometric Brownian motion. The theoretical price, F t,T), of a futures contract on the underlying asset is the price agreed at time t for delivery of the asset at time T and payable on delivery. When t= T, the futures price equals the spot price. As explained in chapter 12, futures contracts are cash settled every day through a clearing mechanism, while forward contracts involve neither daily marking to market nor daily cash settlement. 

The volatility figure used in a B-S computation is constant and derived mathematically, assuming that asset prices move according to a geometric Brownian motion. In reality, however, asset prices that are either very high or very low do not move in this way. Rather, as a price rises, its volatility increases, and as it falls, its variability decreases. As a result, the B-S model tends to undervalue out-of-the-money options and overvalue those that are deeply in the money. 

Park, C. Padgett, W. Accelerated degradation models for failure based on geometric brownian motion and... 

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