TABLE OF CONTENTS We first introduce in 5.1 the Cox-Ross-Rubinstein based approach that prices the option using binary tree hypermodel, then we relate it to the classical Black-Scholes formula from last chapter.
In 5.2, we represent the binary tree hypermodel using a matrix which we called the CRR matrix. Then we give some examples.
We now consider in 5.1 the derivation of the classical Black-Scholes formula directly from the binary tree hypermodel. This method was pioneered by Cutland, Kopp and Willinger in [Cutland et al., 1991] (see also [Cutland, 2000], as well as [Cutland et al., 1993a], [Cutland et al., 1993b], [Cutland et al., 1995] and [Cutland et al., 1997] for further developments) and is based on the approach of Cox-Ross-Rubinstein in [Cox et al., 1979].
To price options using the Cox-Ross-Rubinstein approach, we need to work under a more strict form of no arbitrage assumption, the kind use in subsection 3.2.2, which guarantees the existence of the martingale measure between any time t and t+ ? t. This is stronger then the virtually arbitrage free in subsection 3.3.1.
Recall that m denotes the annual return rate of the underlying asset, ? its volatility and r is riskless annual interest rate.
To simplify our notation a bit, we first normalize the stock price and expiry to units ($1 and 1 year):
Then use the interest rate r ? t during the basic unit, the argument from...
