A one-step binomial model is best described within a two-state world where the price of a stock will either go up once or down once, and the change will occur one step ahead at the end of the holding period.
THE REPLICATING PORTFOLIO
The replicating portfolio is a concept that holds that the outlay for a bankruptcy-free stock position should be the same as the outlay for a long call position with the same payoff.
SYNTHETIC CALL REPLICATION
A combination of the hedge ratio, the stock price, & the present value of the borrowings can be used to price the call option: call price = hedge ratio x [stock price — PV(borrowing)]
The one-step binomial model can also be expressed in terms of probabilities and call prices. 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:
U = size of the up-move factor = eσ√t
D = size of the down-move factor = e-σ√t= 1/eσ√t =1/U
σ = annual volatility of the underlying asset’s returns
t = the length of the step in the binomial model
The risk-neutral probabilities of upward and downward movements are then calculated as follows:
πu= probability of an up move =(eπ– D)/ (U—D)
πd = probability of a down move = 1 — πu
where: r = continuously compounded annual risk-free rate
We can calculate the value of an option on the stock by:
In the two-period and multi-period models, the tree is expanded to provide for a greater number of potential outcomes.
A high standard deviation will result in a large difference between the stock price in an up state, SU, and the stock price in a down state, SD. If the standard deviation were zero, the binomial tree would collapse into a straight line and SU would equal SD.
Therefore volatility, as measured here by standard deviation, can be captured by evaluating stock prices at each time period considered in the tree.
The binomial option pricing model can be altered to value a stock that pays a continuous dividend yield, q.
πu = (e(r—q)t—D)/ (U-D)
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 ert with e(rdc-rfc)t such that:
πu= (erDC-rFC)t – D)/(U-D)
The binomial model can also incorporate the unique characteristics of options on futures.
Valuing American options with a binomial model requires the consideration of the ability of the holder to exercise early. In the case of a two-step model, that means determining whether early exercise is optimal at the end of the first period. If the payoff from early exercise (the intrinsic value of the option) is greater than the option’s value (the present value of the expected payoff at the end of the second period), then it is optimal to exercise early.
If we shorten the length of the intervals in a binomial model, there are more intervals over the same time period, more branches to consider, and more possible ending values. If we continue to shrink the length of intervals in the model until they are what mathematicians call “arbitrarily small,” we approach continuous time as the limiting case of the binomial model.
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