STOCK-PRICE MODELS AND OPTION PRICING Dong Myung CHUNG Department of Mathematics, The Catholic University of Korea , Bucheon, 420-743, KOREA
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ABSTRACT Financial mathematics may be regarded as the branch of the applied mathematics which are concerned with the financial markets. Generally, financial mathematics derives, and extends the mathematical models to describe the dynamics of stock prices in the financial market and uses stochastic calculus to obtain the fair price of the derivative of the stock. In terms of practice, financial mathematics also overlaps heavily with the field of financial engineering and computational finance. Arguably, all three are largely synonymous, although the latter two focus on application, while the former focuses on modelling of stocks and option pricing. In this tutorial, we will go through three concepts which might gain you insights into the theory of stock price models and the theory of option pricing. We will first discuss some of stochastic processes which are suitable for an adequate description of the dynamics of underlying assets such as bonds and stocks. We will next discuss the concepts of arbitrage-free pricing and risk neutral valuation, which are indeed the fundamental concepts of the theory of option pricing. The following is the outline to be discussed in this tutorial: I. Stock-price models : 1. Diffusion models: log returns can be modelled by a Wiener process - Black- Scholes model 2. Pure jump models: log returns can be modelled by a Levy process - Levy models such as VG model and CGMY model II. Asset pricing models 1. Absolute pricing modes - Equilibrium pricing models - CAPM, 2. Relative pricing models - Arbitrage-free pricing - Black-Scholes models III. Risk-neutral valuation 1. Method of binomial trees - Risk-neutral measure 2. Martingale measure approach
REFERENCES 1.. Mikosch, T. Elementary Stochastic Calculus with Finance in View, World Scientific, Singapore, 1998. 2. Shreve, S. E. Stochastic Calculus for Finance I: The Binomial Asset Pricing Model, Springer, New York, 2005. 3. Schoutens, W. Levy processes in Finance: Pricing Financial Derivatives, John Wiley, West Sussex, 2003.
