Asset prices Brownian motion

For the piuposes of employing option pricing models, the d3mamic behaviour of asset prices is usually described as a function of what is known as a Wiener process, which is also known as Brownian motion. The noise or volatility component is described by an adapted Brownian or Wiener process, and involves introducing a random increment to the standard random process. This is described next. 

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, ... 

It is possible to generalise Ito s formula in order to produce a multi-dimensional formula, which can then be used to construct a model to price interest-rate derivatives or other asset-class options where there is more than one variable. To do this, we generahse the formula to apply to situations where the d5mamic function/() is dependent on more than one Ito process, each expressed as a standard Brownian motion. 

The uncertainty in asset price dynamics is described as having two sources, both represented by independent standard Brownian motions. These are denoted... 

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 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. 

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