READING 55: BINOMIAL TREES

Binomial Model

        The value of a European option can be calculated using a binomial tree, as the probability-weighted expected value of the option at maturity discounted at the risk-free rate.
         Given the volatility of the underlying stock and the length of the steps in the binomial tree, the size of the up- and down-move factors are calculated as:

         The risk-neutral probabilities of up and down moves are calculated as:

         π_d = probability of a down move = 1 − π_u
         where:
         r = annual continuously compounded risk-free rate
         The value of the comparable European put option can be calculated using put-call parity, which is put = call − stock + Xe^–rT.
         The value of an American option reflects the early exercise features. An American option will be exercised at the end of the first period if the intrinsic value is greater than the discounted value of the expected option payoff at the end of the second period.

Assessing Volatility

        The higher the standard deviation, the greater the dispersion between stock prices in up and down states. Therefore volatility, as measured here by standard deviation, can be captured by evaluating stock prices at each time period considered in the tree.

Risk-Neutral Valuation

        The sizes of the upward and downward movements are defined as functions of the volatility and the length of the "steps" in the binomial model.

Increasing the Number of Time Periods

        As the period covered by a binomial model is divided into arbitrarily small, discrete time periods, the model results converge to those of the continuous-time model.

Modifying the Binomial Model

        The binomial option pricing model can be altered to value a stock that pays a continuous dividend yield, q.

         π_d = 1 − π_u
         Options on stock indices are valued in a similar fashion to stocks with dividends.
         For options on currencies, upward probability in the binomial model is altered by replacing e^rt with e^{(rDC − rFC)t} such that:

         The binomial model can also incorporate the unique characteristics of options on futures.

Using Delta to Develop a Replicating Portfolio

        The delta of a stock option, Δ, tells us how many units of the stock to hold per call option to be shorted in order to make a hedge work. Delta is computed as the ratio of the change in the stock option to the change in the underlying stock price.