Asset prices stochastic processes

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]). 

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 price processes of shares and bonds, as well as interest rate processes, are stochastic processes. That is, they exhibit a random change over time. For the purposes of modelling, the change in asset prices is divided into two components. These are the drift of the process, which is a deterministic element, also called the mean, and the random component known as the noise, also called the volatility of the process. 

The standard Wiener process is a close approximation of the behaviour of asset prices but does not account for some specific aspects of market behaviour. In the first instance, the prices of financial assets do not start at zero, and their price increments have positive mean. The variance of asset price moves is also not always unity. Therefore, the standard Wiener process is replaced by the generalised Wiener process, which describes a variable that may start at something other than zero, and also has incremental changes that have a mean other than zero as well as variances that are not unity. The mean and variance are still constant in a generalised process, which is the same as the standard process, and a different description must be used to describe processes that have variances that differ over time these are known as stochastic integrals (Figure 2.3). 

We noted at the start of the chapter that the price of an option is a function of the price of the underlying stock and its behaviour over the life of the option. Therefore, this option price is determined by the variables that describe the process followed by the asset price over a continuous period of time. The behaviour of asset prices follows a stochastic process, and so option pricing models must capture the behaviour of stochastic variables behind the movement of asset prices. To accurately describe financial market processes, a financial model will depend on more than one variable. Generally, a model is constructed where a function is itself a function of more than one variable. Ito s lemma, the principal instrument in continuous time finance theory, is used to differentiate such functions. This was developed by a mathematician, Ito (1951). Here we simply state the theorem, as a proof and derivation are outside the scope of the book. Interested readers may wish to consult Briys et al. (1998) and Hull (1997) for a background on Ito s lemma we also recommend Neftci (1996). Basic background on Ito s lemma is given in Appendices B and C. 

The standard stochastic differential equation for the process of an asset price S, is given in the form... 

A variable (such as an asset price) may be assumed to have a lognormal distribution if the natural logarithm of the variable is normally distributed. So, if an asset price S follows a stochastic process described by... 

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 ... 

From a practical point of view, we can safely assume that the majority of stochastic processes representing prices of traded financial assets are adapted to the filtration F and that the short rate process r = r(r) >o is a predictable process, meaning that r(t) is Ff i measurable. This implies that B t) is also Ff i measurable and this condition is automatically satisfied for continuous or left-continuous processes. 

The approach is similar to the historical simulation method, except that it creates the hypothetical changes in prices by random draws from a stochastic process. It consists of simulating various outcomes of a state variable (or more than one in case of multifactor models), whose distribution has to be assumed, and pricing the portfolio with each of the results. A state variable is the factor underlying the price of the asset that we want to estimate. It could be specified as a macroeconomic variable, the short-term interest rate or the stock price, depending on the economic problem. 

Pricing an option therefore requires knowing the value of both />, the probability that the option will expire in the money, and E[Sr 5t- > A] — X, its expected payoff should this happen. In calculating/, the probability function is modeled. This requires assuming that asset prices follow a stochastic process. 

Note that although a key assumption of the model is that interest rates are constant, in the case of hond options, it is applied to an asset price that is essentially an interest rate assumed to follow a stochastic process. 

All this suggests that asset-price behavior is more accurately described by nonstandard price processes, such as the jump diffusion model or a stochastic volatility, than by a model assuming constant volatility. For more-detailed discussion of the volatility smile and its implications, interested readers may consult the works listed in the References section. 

---