Modelling of stock price changes: A real analysis approach

Abstract. In this paper a real analysis approach to stock price modelling is considered. A stock price and its return are defined in a duality to each other provided there exist suitable limits along a sequence of nested partitions of a time interval, mimicking sum and product integrals. It extends the class of stochastic processes susceptible to theoretical analysis. Also, it is shown that extended classical calculus is applicable to market analysis whenever the local 2–variation of sample functions of the return is zero, or is determined by jumps if the process is discontinuous. In particular, an extended Riemann-Stieltjes integral is used in that case to prove several properties of trading strategies.

[1]  Jordan Stoyanov,et al.  ESSENTIALS OF STOCHASTIC FINANCE , 2000 .

[2]  A Closer Look at the Implications of the Stable Paretian Hypotheses , 1975 .

[3]  R. S. Clarkson.  An Actuarial Theory of Option Pricing , 1997, British Actuarial Journal.

[4]  Ernst Eberlein,et al.  On Modeling Questions In Security Valuation , 1992 .

[5]  Sergio M. Focardi,et al.  Modeling the Market: New Theories and Techniques , 1997 .

[6]  L. C. Young,et al.  An inequality of the Hölder type, connected with Stieltjes integration , 1936 .

[7]  J. Harrison,et al.  Continuous Price Processes in Frictionless Markets Have Infinite Variation , 1984 .

[8]  E. Wong,et al.  On the Convergence of Ordinary Integrals to Stochastic Integrals , 1965 .

[9]  C. Doléans-Dade,et al.  Quelques applications de la formule de changement de variables pour les semimartingales , 1970 .

[10]  Walter Willinger,et al.  Dynamic spanning without probabilities , 1994 .

[11]  J. Harrison,et al.  Martingales and stochastic integrals in the theory of continuous trading , 1981 .

[12]  M. Lévy.  Le Mouvement Brownien Plan , 1940 .

[13]  Hans Föllmer,et al.  Calcul d'ito sans probabilites , 1981 .

[14]  Philip Protter,et al.  FROM DISCRETE- TO CONTINUOUS-TIME FINANCE: WEAK CONVERGENCE OF THE FINANCIAL GAIN PROCESS' , 1992 .

[15]  Gian-Carlo Rota,et al.  Discrete thoughts: Essays on mathematics, science, and philosophy , 1986 .

[16]  Donna Salopek,et al.  Tolerance to Arbitrage , 1998 .

[17]  D. Freedman.  Brownian motion and diffusion , 1971 .

[18]  Financial Economics — an Investment Actuary's Viewpoint , 1996, British Actuarial Journal.

[19]  T. Andersen.  THE ECONOMETRICS OF FINANCIAL MARKETS , 1998, Econometric Theory.

[20]  S. Pliska.  Introduction to Mathematical Finance: Discrete Time Models , 1997 .

[21]  R. Dudley,et al.  Product integrals, young integrals and p-variation , 1999 .

[22]  Freddy Delbaen,et al.  No-arbitrage, change of measure and conditional Esscher transforms , 1996 .