Derivative Pricing in Complete and Incomplete Markets

Abstract

The starting point of the present paper is the Binomial Option Pricing Model. It basically assumes that there is only one possible value for the volatility of the stock price and it gives a unique arbitrage-free price of an option. This assumption is relaxed in the sense that the occurrence of a second value for the volatility is supposed to have strictly positive probability. Then it is no longer possible to find only one arbitrage-free price of the option; instead some concepts of general arbitrage pricing theory such as the Fundamental Theorem of Asset Pricing are employed to construct an interval of arbitrage-free option prices. Subsequently, a natural question to ask is under which conditions the Binomial Option Pricing Model assuming deterministic and stochastic volatility respectively agrees on arbitrage-free option prices. This paper gives a formal answer to that question by showing that the volatility used to calculate the price in the deterministic setting has to lie strictly in between the two possible volatilities used when assuming stochastic volatility. Furthermore, the Binomial Option Pricing Model is enriched to the Trinomial Option Pricing Model. In the latter model in cases of both deterministic as well as stochastic volatility an interval of arbitrage free option prices is obtained. The question of agreement on arbitrage-free prices is discussed again and a similar answer to the above situation is derived. The paper provides illustrations of the theoretical findings along examples such as European Call Options, Butterfly Spreads as well as Double Butterfly Spreads.