Stock Price Models with Stochastic Volatility

Archil Gulisashvili
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Archil Gulisashvili: Ohio University

Chapter Chapter 2 in Analytically Tractable Stochastic Stock Price Models, 2012, pp 37-65 from Springer

Abstract: Abstract Stock price models with stochastic volatility have been developed in the last decades to improve the performance of the celebrated Black-Scholes model. The volatility of the stock in such a model is described by a nonnegative stochastic process. For instance, in the Hull-White model, a geometric Brownian motion plays the role of stochastic volatility, in the Stein-Stein model, the volatility is represented by an Ornstein-Uhlenbeck process, or by the absolute value of this process, while in the Heston model, the volatility process is the square root of a CIR-process. Chapter 2 focuses on stochastic volatility models. In addition, it presents Girsanov’s theorem, risk-neutral measures, and market prices of risk. It is explained in Chap. 2 how to use Girsanov’s theorem to find risk-neutral measures in uncorrelated stochastic volatility models, and how to overcome complications, arising in the case of a non-zero correlation between the stock price and the volatility. The chapter presents results of C. Sin, concerning risk-neutral measures in the correlated Hull-White model. Sin’s results show that the existence of such measures is determined by the possibility of explosions in finite time for solutions of certain auxiliary stochastic differential equations.

Keywords: Stock Price; Stochastic Differential Equation; Stochastic Volatility; Standard Brownian Motion; Geometric Brownian Motion (search for similar items in EconPapers)
Date: 2012
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprfcp:978-3-642-31214-4_2

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DOI: 10.1007/978-3-642-31214-4_2

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