Given that options are a crucial instrument both in the securities market and in strategic transactions, I will touch on a more complex but more realistic model for their valuation – the binomial method.
In previous notes, I discussed two simple methods for option valuation (the portfolio replication and the risk-free world assumption methods). The binomial method means making these simple methods more realistic.
While the simple models had only one period and one change, in the binomial model, we assume that changes occur over multiple steps.
Now, I will discuss the binomial variant of the risk-free world assumption model. The asset replication method is conceptually similar.
Below in the photo, one-step, two-step, and multi-step variants (discrete distribution) are shown. The logic is simple. Suppose we want to value a 6-month option; the model can be built on the assumption that the price will change once, twice, or several times over 6 months. The range of price changes at each step is determined based on historical statistics.
Formulas that allow us to determine the range of deviations based on historical volatility are as follows:
[ 1 + \text{upside change} = u = e^{σ√h} ]
[ 1 + \text{downside change} = d = \frac{1}{u} ]
[ e = \text{base for natural logarithms} = 2.718 ] (exponential constant)
[ σ = \text{STDV of stock returns} ] (standard deviation of the stock price)
[ h = \text{interval as a fraction of a year} ] (time interval as a fraction of the year, since the standard deviation is annual)
Several important points:
- In the binomial model, the price changes at each step by the same percentage parameters derived from the formulas above.
- Accordingly, the probabilities of increase and decrease in the risk-free world also remain unchanged (see the simple model in the previous note).
- The branches of the binomial tree provide figures reflecting the stock price, but we need to convert these figures into the option value. For example, for a Call option, where the stock price exceeds the strike price, the option value will be equal to the corresponding positive difference, and where the stock price is below the strike price, the option value will be zero.
The binomial model resembles a decision tree, with the expected outcome at the end of each branch. However, in our case, we do not choose a single branch; instead, we calculate the weighted average of all branches to obtain the fair value of the option.
P.S.
There is an even more realistic model for option valuation, where the number of steps goes to infinity, and the distribution becomes continuous – the so-called Black-Scholes model (to be discussed in the next note). It works well in practice but cannot be applied to all options, which is why the binomial model is also used in practice.
Source:
“Principles of Corporate Finance” – by F. Allen, R. A. Brealey, & S. Myers