Chapter 11 - Asset pricing in dynamic (B, S)-markets

Abstract

This chapter investigates the discrete (B, S)-model for the dynamic pricing of derivative securities in financial markets. In general, the dynamics of the current amount hold in the fairly risky asset B is modeled by the stochastic process B and the current amount of the risky asset S by the stochastic process S. The fundamental theorem of asset pricing connects the important concepts of arbitrage (economics) and martingale measures (probability). This plays a fundamental role in the analysis of financial markets. This is particularly true for (B, S)-market model with dynamic probabilities. Under some fairly mild assumption of having nonnegative semi-martingale structure for B and S, it is possible to show that both B and S must be governed by the linear stochastic difference equations. The aim is at deriving pricing and hedging formulas for the (B, S)-market governed with dynamic probabilities for the interest rates p instead of the commonly studied binomial model with static probabilities for the distribution of p. To have a good grasp of it, enough study should be done on existence and equivalence of martingale measures, the fundamental theorem of dynamic asset pricing with dynamic multinomial distributions, the completeness of the dynamic (B, S)-market, the fair price, and optimal hedging strategies. One of the most important problems in financial markets is to calculate the fair premium c at time 0 which assures to purchase an asset with a derivative security at later time. The theories and practices discussed in this chapter attempt to solve this problem.