The model assumes that:
First, the process of stock price generation is a geometric random walk process, and the stock price obeys binomial distribution. Like Boucher-Shaw model, in bopm model, stock price fluctuations are independent of each other and have the same distribution, but this distribution is binomial, not lognormal. In other words, the validity period of the option is divided into n equal intervals, and at the end of each interval, the stock price will rise or fall to a certain extent, so:
(attached {Figure})
Let snj represent the stock price after the nth interval, during which it is assumed that the stock price rises j times and falls (n-j) times, then:
(attached {Figure})
Second, risk-neutral economy. Because of the existence of continuous trading opportunities, the price of options has nothing to do with investors' risk preference. It is equal to a certain value because deviating from this value will create arbitrage opportunities, and market forces will bring it back to its original level. Suppose the current stock price is s[0] and the buyer's option expires after a certain time interval, then the stock price will either rise to s[ 1 1] or fall to s[ 10], that is:
(attached {Figure})
According to the assumption of risk neutrality, any asset should have the same expected rate of return, otherwise arbitrage will occur. That is to say, the future value of risk-free bonds, stocks and buyer's options at this time satisfies the following relationship:
(attached {Figure})
In the above formula, q represents the probability of the stock price rising, so the price of the option is equivalent to the discounted value of its expected price. The above analysis can be further extended to the determination of buyer's option price in n interval. First, we need to calculate the expected value of the buyer's option price. Assuming that in n intervals, the call option is still an impairment option before the stock price rises by k times, and its intrinsic value is still 0, but it has intrinsic value between k times and n times, then:
(attached {Figure})
({Figure}) The previous analysis did not consider the existence of dividends. Suppose a stock will pay a certain dividend at T, the dividend factor is F, and the ex-dividend date is the same as the interest payment date, then the stock price will fall by fs[t] on the ex-dividend date.
(attached {Figure})
For American options, we need to consider the situation of early execution:
If it is executed in advance at t, its price is equal to the intrinsic value; If not, you can get the corresponding price according to the previous derivation. The final price of point T should be the maximum value under the conditions of early execution and no early execution. Namely:
({Figure}) According to the parity relation of European options, the option price of the seller can be directly derived from the option of the buyer, while the American option cannot. Using the above method to deduce the American buyer's option price, we can also get:
(attached {Figure})
This is the pricing formula of American seller option. From the derivation of the above bopm model, we can see its main features:
1. The variables that affect the option price mainly include the market price of the basic commodity (S), the option agreement price (X), the risk-free interest rate (R), the stock price fluctuation factor (U, D), the dividend factor (F) and the number of ex-dividends. In fact, U and D describe the deviation of stock price, so compared with Boucher-Shaw model, the main factors considered by bopm are basically the same as the former, but due to the increased discussion on dividends, it has more advantages in the pricing of dividend options and American options.
2. According to the characteristics of binomial distribution, as long as U, D and P are properly defined in the bopm model, the option pricing problem under jump condition can be answered. This is beyond the scope of Boucher-Shaw model. At the same time, when n reaches a certain scale, the binomial distribution tends to normal distribution. As long as U, D and P are selected correctly, bopm model will approach Boucher-Shaw model.
Like Boucher-Shaw model, binomial distribution pricing model has also been extended to the option pricing of foreign exchange, interest rate and futures, which has been highly valued by theoretical and industrial circles.
Third, the evaluation of western option pricing theory
Western option pricing theory represented by Black-Scholes model and bopm model is gradually enriched and matured with the expansion and development of option trading, especially on-site option trading. These theories are basically based on the practice of option trading, which directly serves this practice and has certain scientific value and reference significance.
First of all, the model summarizes the factors that affect the option price as basic commodity price, agreed price, option validity period, deviation of basic commodity price, risk-free interest rate and dividend, and thinks that option price is a function of these factors, namely:
C or p=(s, x, t, σ, γ, d)
On this basis, the calculation formula of option price is obtained, which is very operable. For example, in Boucher-Shaw model, S, X and T can be obtained directly, and γ can also be obtained through the yield of government bonds with the same maturity. Therefore, using this model to evaluate only needs the corresponding σ value, that is, the price deviation of basic commodities. In practice, the σ value can be obtained by analyzing the historical price, or assuming that the market price of the exercised option is an equilibrium price and substituting it into the corresponding variable (called implied volatility at this time). So that the operation is more convenient. At the same time, this generalization is based on the inherent characteristics of options and is the result of consideration in a unified capital market. Its analysis touches on the essence of option price, trying to reveal what option price should be, rather than what it may be, which is a big step forward from the early econometric pricing model.
Secondly, the model has strong practicability and has a certain guiding role in option trading. Boucher-Shaw model and binomial distribution model have been compiled into computer software, which has become an effective tool for investors to analyze the option market. The financial sector has also compiled a ready-made option price calculation table based on the model, which is convenient to use, clear at a glance and convenient for investors. As Robert Haier said in the book Trading and Investment of Bond Options: "The (Boucher-Shaw) model has been proved to be very accurate under the premise of meeting the basic assumptions and has become a standard tool in option trading." Specifically, the application of these models in practice is mainly reflected in two aspects: 1. Guide the transaction. With the help of the model, investors can find over-priced or under-priced options in the market, buy over-priced options and sell over-priced options, and make profits from them. At the same time, according to its evaluation, we can formulate corresponding options trading strategies. In addition, some useful parameters can be obtained from the model, such as delta value, which reflects the change of option price caused by the change of basic commodity price by one unit, and is a very useful indicator for adjusting option position hedging. In addition, there is γ value (a sensitive index to measure the change of δ value); Q value (the sensitivity or elasticity of the option price to time change under the premise that the basic commodity price remains unchanged), value (the change of the option price caused by the change of interest rate by one percentage point), etc. These parameters have important reference value for portfolio management and option strategy adjustment. 2. Study market behavior. This pricing model can be used to examine the efficiency of the market, and it is also of certain significance to deepen the study of the option market.