4. Evaluation of real options using the Black-Scholz model
4.1. The binomial model and the Black-Scholz model
Connection between the binomial model and the Black-Scholz model
Considering the application of binomial estimation of real options, we proceeded from the fact that the number of links in the decision tree is discrete and known to us.
Indeed, logic requires that the links of a tree be arranged in such a way as to correspond to the frequency of taking "fateful" solutions for the company or project. That is, the nodes of the tree should be simultaneously those points in time in which strategic decisions are taken on the reduction, development, switching of business, etc.
And what if the business requires constant monitoring and the situation can change at any moment?
We construct a multi-tiered binary tree, and then we will increase the number of links in it, reducing the time intervals between its nodes. As the length of each time interval approaches zero more and more, the discrete binomial model turns into the Black-Scholz model, reflecting a continuous process in time.
Thus, the main difference between the models is that one is discrete and assumes the existence of a previously known finite number of intervals (links) of the binary tree, and the other is continuous and is based on the fact that the number of links in the tree is infinitely large, and the length of each interval are respectively infinitely small.
The formal record of the Black-Scholz model, derived to evaluate the premium on the European CALL option, is as follows:
d2 = d (- al/7 C0 is the current price of the CALL option;
50 - the current price of the underlying asset. It is assumed that the asset does not bring current income (dividend, coupon); X is the exercise price of the option; e is the base of the natural logarithm (e = 2.718); r - rate of risk-free yield, calculated by the method of continuous interest,
where r j - the annual rate of risk-free yield, the share of sd; T, t - time before execution of the option CALL; In - the sign of the natural logarithm;
о - standard deviation of the price of the underlying asset for the year, unit share;
N (d) is the cumulative function of the normal distribution. The table of its values is presented in Appendix 2.
The formula is derived from the risk-neutral approach and assumes that the option is European, and the current asset is not accrued for the underlying asset. The option's settlement price depends on the probability that it will be winning by the time it is executed. The probability in the formula is taken into account with the help of the factors N (& iquest; 0. The logarithmically normal distribution is taken as the probabilistic model of the price of the basic asset.
However, if the task is to encapsulate the US real option, then, as mentioned above, the model can also be applied for its conservative estimate (the price of a European option is the lower limit for the price of an American option with the same release conditions).
4.2. Evaluation of real CALL options using the Black-Scholz model
Let's illustrate the use of the Black-Scholz model for evaluating real options CALL. With this purpose, we consider three practical problems in which such estimates are usually required:
• Estimation of growth prospects (option for replicating the experience or increasing production capacity);
• valuation of intangible assets, such as rights, patents, licenses, etc .;
• Evaluation of business in general and M & A projects (so-called acquisitions projects).
Option for future development
When analyzing the prospects for future development, the value of an option is usually added to the value of a business or project, determined by the traditional OS-technology. As the exercise price options are used capital investments in development (expansion, replication of experience). The current value of the underlying asset (5) is the cash flow estimate that is generated by the business today (quite often it is smaller than the execution chain). The time (7) in the Black-Scholz model with respect to real options is the time during which it is possible to decide on expanding the business.
As risk-free rates, in practice, the so-called pseudo-riskless profitability levels are used, which represent a risk-free yield (most often US Treasury bills), increased by the size of the premium, depending on the credit rating of the country where the assessment takes place (Table 6.4. 1).
TABLE 6.4.1. Risk-free rate with country risk premium (pseudo-risk-based yield levels)
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