The Binomial Pricing Model
The binomial model is an options pricing model which was developed by William Sharpe in 1978. Today, one finds a large variety of pricing models which differ according to their hypotheses or the underlying instruments upon which they are based (stock options, currency options, options on interest rates).
The Black & Scholes Model
The Black & Scholes model was published in 1973 by Fisher Black and Myron Scholes. It is one of the most popular options pricing models. It is noted for its relative simplicity and its fast mode of calculation: unlike the binomial model, it does not rely on calculation by iteration.
The intention of this section is to introduce you to the basic premises upon which this pricing model rests. A complete coverage of this topic is material for an advanced course
The Black-Scholes model is used to calculate a theoretical call price (ignoring dividends paid during the life of the option) using the five key determinants of an option's price: stock price, strike price, volatility, time to expiration, and short-term (risk free) interest rate.
The original formula for calculating the theoretical option price (OP) is as follows:
Where:
The variables are:
S = stock price
X = strike price
t = time remaining until expiration, expressed as a percent of a year
r = current continuously compounded risk-free interest rate
v = annual volatility of stock price (the standard deviation of the short-term returns over one year).
ln = natural logarithm
N(x) = standard normal cumulative distribution function
e = the exponential function
Lognormal distribution: The model is based on a lognormal distribution of stock prices, as opposed to a normal, or bell-shaped, distribution. The lognormal distribution allows for a stock price distribution of between zero and infinity (ie no negative prices) and has an upward bias (representing the fact that a stock price can only drop 100 per cent but can rise by more than 100 per cent).
Risk-neutral valuation: The expected rate of return of the stock (ie the expected rate of growth of the underlying asset which equals the risk free rate plus a risk premium) is not one of the variables in the Black-Scholes model (or any other model for option valuation). The important implication is that the price of an option is completely independent of the expected growth of the underlying asset. Thus, while any two investors may strongly disagree on the rate of return they expect on a stock they will, given agreement to the assumptions of volatility and the risk free rate, always agree on the fair price of the option on that underlying asset.
The key concept underlying the valuation of all derivatives -- the fact that price of an option is independent of the risk preferences of investors -- is called risk-neutral valuation . It means that all derivatives can be valued by assuming that the return from their underlying assets is the risk free rate.
Limitation: Dividends are ignored in the basic Black-Scholes formula, but there are a number of widely used adaptations to the original formula, which I use in my models, which enable it to handle both discrete and continuous dividends accurately.
However, despite these adaptations the Black-Scholes model has one major limitation: it cannot be used to accurately price options with an American-style exercise as it only calculates the option price at one point in time -- at expiration. It does not consider the steps along the way where there could be the possibility of early exercise of an American option.
As all exchange traded equity options have American-style exercise (ie they can be exercised at any time as opposed to European options which can only be exercised at expiration) this is a significant limitation.
The exception to this is an American call on a non-dividend paying asset. In this case the call is always worth the same as its European equivalent as there is never any advantage in exercising early.
Advantage: The main advantage of the Black-Scholes model is speed -- it lets you calculate a very large number of option prices in a very short time. Since, high accuracy is not critical for American option pricing (eg when animating a chart to show the effects of time decay) using Black-Scholes is a good option. But, the option of using the binomial model is also advisable for the relatively few pricing and profitability numbers where accuracy may be important and speed is irrelevant. You can experiment with the Black-Scholes model using on-line options pricing calculator.
The Binomial Model
The binomial model breaks down the time to expiration into potentially a very large number of time intervals, or steps. A tree of stock prices is initially produced working forward from the present to expiration. At each step it is assumed that the stock price will move up or down by an amount calculated using volatility and time to expiration. This produces a binomial distribution, or recombining tree, of underlying stock prices. The tree represents all the possible paths that the stock price could take during the life of the option.
At the end of the tree -- ie at expiration of the option -- all the terminal option prices for each of the final possible stock prices are known as they simply equal their intrinsic values.
Next the option prices at each step of the tree are calculated working back from expiration to the present. The option prices at each step are used to derive the option prices at the next step of the tree using risk neutral valuation based on the probabilities of the stock prices moving up or down, the risk free rate and the time interval of each step. Any adjustments to stock prices (at an ex-dividend date) or option prices (as a result of early exercise of American options) are worked into the calculations at the required point in time. At the top of the tree you are left with one option price.
Advantage: The big advantage the binomial model has over the Black-Scholes model is that it can be used to accurately price American options. This is because, with the binomial model it's possible to check at every point in an option's life (ie at every step of the binomial tree) for the possibility of early exercise (eg where, due to eg a dividend, or a put being deeply in the money the option price at that point is less than the its intrinsic value).
Where an early exercise point is found it is assumed that the option holder would elect to exercise and the option price can be adjusted to equal the intrinsic value at that point. This then flows into the calculations higher up the tree and so on.
Limitation: As mentioned before the main disadvantage of the binomial model is its relatively slow speed. It's great for half a dozen calculations at a time but even with today's fastest PCs it's not a practical solution for the calculation of thousands of prices in a few seconds which is what's required for the production of the animated charts in my strategy evaluation modelFor Stock advice: Saturday watch on Market Outlook
