Asset prices dynamics

The implications of this new model class are in contrast to most term structure models discussed in the literature, which assume that the bond markets are complete and fixed income derivatives are redundant securities. Collin-Dufresne and Goldstein [ 18] and Heiddari and Wu [36] show in an empirical work, using data of swap rates and caps/floors that there is evidence for one additional state variable that drives the volatility of the forward rates. Following from that empirical findings, they conclude that the bond market do not span all risks driving the term structure. This framework is rather similar to the affine models of equity derivatives, where the volatility of the underlying asset price dynamics is driven by a subordinated stochastic volatility process (see e.g. Heston [38], Stein and Stein [71] and Schobel and Zhu [69]). 

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

In other words, assuming a complete market stochastic volatility model implies that the short rate is modeled directly, while the traded asset (bond) has to be derived. Therefore, only the direct modeling of the bond price dynamics, together with stochastic volatility leads to an incomplete market model analog to the stochastic volatility models of equity markets". ... 

Duffle D (1996) Dynamic Asset Pricing Theory. Princeton University Press,... 

Models that seek to value options or describe a yield curve also describe the dynamics of asset price changes. The same process is said to apply to changes in share prices, bond prices, interest rates and exchange rates. The process by which prices and interest rates evolve over time is known as a stochastic process, and this is a fundamental concept in finance theory. Essentially, a stochastic process is a time series of random variables. Generally, the random variables in a stochastic process are related in a non-random manner, and so therefore we can capture them in a probability density function. A good introduction is given in Neftci (1996), and following his approach we very briefly summarise the main features here. 

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, ... 

A filtration is a family F = F,), f e T of variables F,CF which is increasing in level in the sense that F C F, whenever s,t T,sdynamic information structure, and Ft represents the information available to the investor at time t. The behaviour of the asset price is seen by the increase in filtration, which implies that more and more data are assimilated over time, and historical data is incorporated into the current price, rather than disregarded or forgotten. A filtration F = F is said to be augmented if F, is augmented for each time t. This means that only Fq is augmented. A stochastic process W is described as being adapted to the filtration F if for each fixed t G T, the random variable X ... 

All valuation models must capture a process describing the dynamics of the asset price. This was discussed at the start of the chapter and is a central tenet of derivative valuation models. Under the Black-Scholes model for example, the price dynamics of a risk-bearing asset St under the risk-neutral probability function Q are given by... 

In the following chapter, we tie in the work on dynamics of asset prices to option valuation models. 

From oiu understanding of derivatives, we know that option pricing models such as Black-Scholes assume that asset price returns follow a lognormal distribution. The dynamics of interest rates and the term structure is the subject of... 

A Markov process is one where the path is dependent on the present state of the process only, so that all historical data, including the path taken to arrive at the present state, is irrelevant. So in a Markov process, all data up to the present is contained in the present state. The dynamics of asset prices are frequently assumed to follow a Markov process, and in fact it represents a semi-strong form efficient market. It is written... 

The assumption of a constant interest rate. This is possibly the model s most unrealistic assumption. Not only are rates dynamic, but those at the short end of the yield curve often move in the opposite direction from asset prices, particularly the prices of bonds and bond options. 

Zhao, W., YS. Zheng. 2000. Optimal Dynamic Pricing for Perishable Assets with Nonhomogeneous Demand. Management Science 46(3), 375-388. 

W. Zhao and Y. S. Zheng. Optimal dynamic pricing for perishable assets with nonhomogeneous demand. Management Science, 46(3) 375-388, 2000. 

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