Derivatives Analytics with Python: Data Analysis, Models, Simulation, Calibration and Hedging

CHAPTER 5 Complete Market Models

5.1 Introduction

Ever since the publication of the seminal works by Black, Scholes and Merton (BSM) in 1973 (cf. Black and Scholes (1973) and Merton (1973)), the BSM model—which is a continuous market model—and associated option pricing formulas have been considered a benchmark for option pricing. Benchmark in the sense that they provide closed-form solutions in a simple but still somehow realistic setting. The original and famous formula is derived in the papers on the basis of two different arguments. The first one in Black and Scholes (1973) is an equilibrium argument saying that a risk-less portfolio should yield the risk-less interest rate in equilibrium. The second, and rather widely applicable, one from Merton (1973) is that the value of a (European) option should equal the value of a portfolio that, in combination with an appropriate trading strategy, perfectly replicates the payoff at maturity. It is essentially the key argument of the general arbitrage pricing theory presented in Chapter 4.

Several years later, in 1979, Cox, Ross and Rubinstein presented (cf. Cox et al. (1979)) their binomial option pricing model. This model assumes in principle a BSM economy but in discrete time with discrete state space. Whereas the BSM model necessitates advanced mathematics and the handling of partial differential equations (PDE), the CRR analysis relies on fundamental probabilistic concepts only. Their representation of uncertainty by binomial ...