Continuous and Discrete-Time Option Pricing and Interval Market Model
In this chapter, we introduce the continuous-time interval market model and derive the game-theoretic problem whose solution gives the pricing function as a function of the option type (the terminal payment), maturity, and current underlying stock’s marke
- PDF / 208,992 Bytes
- 15 Pages / 439.36 x 666.15 pts Page_size
- 53 Downloads / 238 Views
Continuous and Discrete-Time Option Pricing and Interval Market Model
7.1 Introduction Recently, there has been a large body of literature on robust control optimization, with applications to various fields including mathematical finance. See, e.g., [27, 41, 42]. Most aim to exploit the power of modern computer tools to solve complex problems whose stochastic formulation is essentially out of reach. Instances of such works will appear in the later parts of this volume. In contrast, we concentrate in this part on the simplest problems – one underlying asset and the less sophisticated interval model – and strive to push the analytic investigation as far as possible. In some sense, being very pretentious, this may be seen as a counterpart, in the robust control option pricing literature, of the Black–Scholes theory in the stochastic literature, the representation theorem playing the role of the Black–Scholes closedform pricing formula. The role of probabilities in finance is further discussed in other parts of the book. But special mention must be made of the outstanding book by Shafer and Vovk [136]. They start from the same analysis of hedging as we do in terms of a game against “nature,” very much in the spirit of the present book. But where we conclude that we can do without probability theory, they claim that this is probability theory. Or rather they claim that this can be an alternative to Kolmogorov’s measuretheoretical foundation of probability theory, and they proceed to recover many results, such as asymptotic and ergodic theorems, from this new viewpoint. Our seller’s price is their “upper expectation” – of which they claim, as do we implicitly, that it is the price likely to be found on the market. Yet, they are more interested in recovering classical probabilities and elaborating on the classic Black–Scholes theory than in providing alternative models and tackling the problems of transaction costs and discrete trading that we consider. But the relationship of that theory to our model deserves further consideration.
P. Bernhard et al., The Interval Market Model in Mathematical Finance, Static & Dynamic Game Theory: Foundations & Applications, DOI 10.1007/978-0-8176-8388-7 7, © Springer Science+Business Media New York 2013
91
92
7 Continuous and Discrete-Time Option Pricing and Interval Market Model
7.1.1 A New Theory of Option Pricing? In a series of papers [34, 38–40, 70, 143], we introduced a robust control approach to option pricing and, more specifically, to the design of a hedging portfolio and management strategy using the interval model for the market and a robust control approach to hedging. An overview of the theory available at that time appeared in [35], and the most complete account to date is in [142]. But no complete account is available in print yet. This theory arose from a failed attempt to extend to the discrete trading case the probability-free robust control approach of Chap. 2 with no assumption on the sequence of market prices of the underlying stock. It soon became clear that
Data Loading...
