Chapter 4: Binomial Trees

4.1 Introduction

The binomial tree approach simulates a random walk of an asset as a series of up or down movements. The up or down movements are proportional to the volatility of the asset. The value of a European option on the asset is evaluated at the expiration and this option value is propagated back through the branches of the tree to the initial-time root–node. For small time steps, the asset values of the tree replicate the log-normal distribution and thus an option price will converge to the Black–Scholes model price. An advantage of the tree approach is that an early exercise premium, for example, of an American call, can be calculated at each node. In addition, volatility is known to vary with time and this effect can be readily embedded into the tree. The time-dependent volatilities can be calibrated in a manner that is consistent with implied volatilities derived from the market listed price of options for several expirations.

4.2 Risk-Neutral Valuation

The structure of a binomial tree is such that an asset at an initial price S0 can only move up to S0u or down to S0d. Cox et al. (1979) selected to force the nodes to replicate, for example, . The probability of an up movement is p and the probability of a down movement is (1 − p). The binomial ...

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