Finance and Stochastics
Опубликовано на портале: 22-06-2006Martin Bibby, Michael Sorensen Finance and Stochastics. 1996. Vol. 1. P. 25-41.
In the present paper we consider a model for stock prices which is a generalization of the model behind the Black-Scholes formula for pricing European call options. We model the log-price as a deterministic linear trend plus a diffusion process with drift zero and with a diffusion coefficient (volatility) which depends in a particular way on the instantaneous stock price. It is shown that the model possesses a number of properties encountered in empirical studies of stock prices. In particular the distribution of the adjusted log-price is hyperbolic rather than normal. The model is rather successfully fitted to two different stock price data sets. Finally, the question of option pricing based on our model is discussed and comparison to the Black-Scholes formula is made. The paper also introduces a simple general way of constructing a zero-drift diffusion with a given marginal distribution, by which other models that are potentially useful in mathematical finance can be developed.
Опубликовано на портале: 12-12-2002Helyette Geman, Marc Yor, Dilip B. Madan Finance and Stochastics. 2002. Vol. 6. No. 1. P. 63-90.
Stochastic volatility and jumps are viewed as arising from Brownian subordination given here by an independent purely discontinuous process and we inquire into the relation between the realized variance or quadratic variation of the process and the time change. The class of models considered encompasses a wide range of models employed in practical financial modeling. It is shown that in general the time change cannot be recovered from the composite process and we obtain its conditional distribution in a variety of cases. The implications of our results for working with stochastic volatility models in general is also described. We solve the recovery problem, i.e. the identification the conditional law for a variety of cases, the simplest solution being for the gamma time change when this conditional law is that of the first hitting time process of Brownian motion with drift attaining the level of the variation of the time changed process. We also introduce and solve in certain cases the problem of stochastic scaling. A stochastic scalar is a subordinator that recovers the law of a given subordinator when evaluated at an independent and time scaled copy of the given subordinator. These results are of importance in comparing price quality delivered by alternate exchanges.