Black-Scholes option pricing model, Black-Shoals option pricing model.

1997 10 June10, the 29th Nobel Prize in Economics was awarded to two American scholars, robert merton, a professor at Harvard Business School, and Myron Scholes, a professor at Stanford University. The Black Scholes option pricing model they founded and developed laid the foundation for the reasonable pricing of various derivative financial instruments in emerging derivative financial markets, including stocks, bonds, currencies and commodities.

In the early 1970s, Scholes and his colleague, the late mathematician fischer black, developed a complex option pricing formula. At the same time, Merton also found the same formula and many other useful conclusions about options. As a result, these two papers were published in different journals almost simultaneously. Therefore, Black-Scholes pricing model can also be called Black-Scholes-Merton pricing model. Merton expanded the connotation of the original model and applied it to many other forms of financial transactions. The Swedish Academy of Royal Sciences praised their research achievements in option pricing as the most outstanding contribution to economic science in the next 25 years.

[Editor] B-S option pricing model (hereinafter referred to as B-S model) and its assumptions.

[Editor] (1)B-S model has seven important assumptions.

1, the stock price behavior obeys the lognormal distribution model;

2. Risk-free interest rate and financial asset return variables are unchanged within the validity period of the option;

3. There is no friction in the market, that is, there is no tax and transaction cost, and all securities are completely separable;

4. Financial assets have no dividends and other income during the validity period of the option (giving up this assumption);

5. This option is a European option, that is, it cannot be implemented before the option expires.

6. There is no risk-free arbitrage opportunity;

7. The securities trading is continuous;

8. Investors can borrow money at a risk-free interest rate.

[Editor] (B) The B-S pricing formula that won the Nobel Prize in Economics

C = S * N(d 1)？ Le? rTN(d2)

These include:

C- initial reasonable price of option

L- option delivery price

S- the current price of the financial assets traded.

T- period of validity of option

R- risk-free interest rate h of continuous compound interest.

σ2- annualized variance

N()- cumulative probability distribution function of normally distributed variables, here are two points to explain:

First of all, the risk-free interest rate in this model must be in the form of continuous compound interest. A simple or discontinuous risk-free interest rate (set as r0) is usually compounded once a year, while R requires the interest rate to be compounded continuously. R0 must be converted into r to be substituted in the above formula. The conversion relationship between them is: r = ln( 1+r0) or r0=Er- 1. For example, r0=0.06, then r=ln( 1+0.06)=0853, that is, 100 invests 583% in continuous compound interest, and the next year gets 106. This result is consistent with the answer directly calculated by r0=0.06.

The second is the relative expression of the option validity period T, that is, the ratio of the option validity days to 365 days a year. If the option is valid for 100 days, then.

[Editor] Derivation and Application of B-S Pricing Model

(I) Derivation of B-S model The derivation of B-S model begins with call options. For call options, the maturity value is:

E[G] = E[max(St？ l，O)]

Where e [g] is the expected maturity value of the call option.

ST- Market value of financial assets traded at maturity.

L- option delivery (exercise) price

There are two possible expiration situations:

1, if St > L, option execution takes effect within the price, max(St? l，O) = St？ L

2. if ST

Max (St.? l，O) = 0

Therefore:

Where: p: (ST > L) probability e [st | st >; L]: establish (St > l) discount E[G] according to the expected value of ST, and get the initial reasonable price of the option;

C = Pe？ rT(E[St | St & gt； l】？ L) such option pricing is transformed into determining p and e [ST | ST >; L].

First, define income. Consistent with the interest rate, the return is the logarithmic value of the ratio of the market price (st) to the current price (S) on the delivery date of financial assets options, that is, the return = lnSt/S = ln(St/L). Assuming that the return of 1 obeys lognormal distribution, that is, ln (ST/L) ~ so e [ln (ST/S) = μ t, ST/S ~ can prove that the relative price expectation is greater than eμt, that is, E[St/S] = eμt+σ2T2 = eeT, so μ t =

Secondly, find (probability p of St), that is, the probability that the return is greater than (LS). It is known that normal distribution has the following properties: PR06 [ξ >; x] = 1？ N(x？ μ σ), where:

ζ: a random variable with normal distribution

X: key value

Expected value of μ: zeta

Sigma: the standard deviation of Zeta

Therefore: p = pr06 [ST >1] = pr06 [lnst/s] > lnLS = :LN？ lnLS？ (r？ σ2)TσTnc4 is composed of symmetry: 1? N(d) = N(？ d)P = NlnSL + (r？ σ2)TσTarS .

Third, find the established St > expected value of St under L. Since E[St | St > L] is in the range of L to∞ of normal distribution,

e[St | St]& gt； = SerTN(d 1)N(d2)

These include:

Finally, p, e [ST | ST] >: L] is substituted (C = Pe? rT(E[St | St & gt； l】？ L))B-S pricing model: C = SN(d 1)? Le? rTN(d2)

(B) Derivation of put option pricing formula

B-S model is the pricing formula of call option. According to the parity theory of put option, the pricing model of effective option can be deduced. According to the theory of call-put parity, the combination of buying a stock and a put option on the stock is equivalent to buying a call option under the same conditions and issuing a risk-free discount bond at the option delivery price, which is as follows:

S + Pe(S，T，L) = Ce(S，T，L) + L( 1 + r)？ T

Transferred items:

Pe(S，T，L) = Ce(S，T，L) + L( 1 + r)？ t？ s，

Substituting into B-S model, we get:

This is the initial price pricing model of put options.

(C) B-S model application examples

Assuming that the current price of a stock in the market is 164, the risk-free continuous compound interest γ is 0.052 1, and the market variance σ2 is 0.084 1, then the initial reasonable price of the option with the strike price l of 165 and the validity period t of 0.0959 is calculated as follows:

① find d 1:

=0.0328

② Find d2:

③ Look up the standard normal distribution function table and get: n (0.03) = 0.5120n (-0.06) = 0.471.

(4) c:

c = 164×0.5 120- 165×e-0.052 1×0.0959×0.476 1 = 5.803

So the reasonable price of this option is 5.803 in theory. If the actual market price of the option is 5.75, it means that the option is undervalued. It is profitable to buy call options without transaction costs.

[Editor] Develop B-S model, share out bonus.

The B-S model only solves the option pricing problem of non-dividend stocks, and Merton developed the B-S model and applied it to dividend stock options.

(1) has a known discontinuous dividend. Suppose a stock pays a known dividend Dt at a certain time T (that is, the ex-dividend date) within the validity period of the option, just remove the present value of the dividend from the current price S of the stock and substitute the adjusted stock value S' into the B-S model: S' = S? Dte？ RT. If there are other incomes within the validity period, they shall be deducted one by one according to the provisions of this Law. Thus, the B-S model is modified into a new formula:

(2) The existence of continuous dividend refers to the continuous dividend of stocks at a known dividend rate (set to δ). If the annual dividend yield δ of a company's stock is 0.04, the present value of the stock is 164, and the expected dividend for that year is 164×004= 6.56. It is worth noting that this bonus is paid in four quarters, that is, every quarter164; In fact, it grew naturally with the continuous reinvestment of the smallest unit of the dollar, accumulating to 6.56 a year. Because the stock price fluctuates constantly throughout the year, the actual dividend is also changing, but the dividend rate is fixed. Therefore, this model does not require the dividend to be known or fixed, it only requires the dividend to be fixed according to the payment ratio of the stock price.

The present value of dividends here is: s (1-e-Δ t), so S'=S? E-Δ t, replace s with S', and get the option pricing formula of continuous dividend payment: C=S? E-δT？ N(D 1)-L？ E-γT？ North D2

[Edit] The influence of B-S mode

Since the B-S model 1973 was first published in Journalofpo Litical Economy, the traders of Chicago Board Options Exchange immediately realized its importance and quickly compiled the B-S model into a computer and applied it to the newly opened Chicago Board Options Exchange. With the development of computer and communication technology, the application scope of this formula is expanding. Nowadays, the model and some variants have been widely used by option traders, investment banks, financial managers, insurance companies and so on. The expansion of derivatives makes the international financial market more efficient, but it also makes the global market more volatile. The creation of new technologies and new financial instruments has strengthened the interdependence between the market and market participants, not only within one country, but also involving other countries and even many countries. Therefore, the fluctuation or financial crisis of a market or a country is very likely to be quickly transmitted to other countries and even the whole world economy. China's financial system is not perfect and its capital market is not perfect. However, with the deepening of reform and moving closer to internationalization, the capital market will continue to develop, the exchange system will be improved day by day, and enterprises will have more autonomy and face greater risks. Therefore, it is necessary to cultivate financial derivatives to avoid risks, and it is also necessary to develop derivative markets. We just started.

[Editor ]B-S model test? Criticism and development

Since the B-S model came out, it has been widely concerned and praised, and some scholars have also conducted in-depth tests on its accuracy. But at the same time, many economists have expressed different views on the problems existing in the model, and extended it from the perspective of perfecting and developing the B-S model.

From 65438 to 0977, Calais, an American scholar, tested the Boucher-Shaw model for the first time by using the equity data listed on the Chicago Board Options Exchange. Since then, many scholars have made useful explorations in this field. One of the most influential representatives is Tripi? Chiras? Mannasz()？ Macbeth and Merville, etc. Generally speaking, these tests have gained the following general views:

1. model is satisfactory for the valuation of flat options, especially for those options that have a remaining validity of more than two months and do not pay dividends.

2. For the options with high value or impairment, the valuation of the model has a big deviation, which will overestimate the impairment options and underestimate the appreciation options.

3. There is a big error in option valuation near the maturity date.

4. When the deviation is too high or too low, the call option with low deviation will be underestimated and the call option with high deviation will be overestimated. But on the whole, Boucher-Shaw model is relatively accurate, and it is a pricing model with strong practical value.

The inspection of Boucher-Shaw model focuses on the analysis of actual statistical data and the evaluation of its performance. However, there are also some studies that start with theoretical analysis and put forward the problems existing in Boucher-Shaw model, which are reflected in the discussion on the rationality of the model assumptions. Many scholars believe that the assumption of this model is too strict, which affects its reliability, specifically in the following aspects:

First, the hypothesis of stock price distribution. One of the core assumptions of Boucher-Shaw model is that stock price fluctuation satisfies geometric Wiener process, so the distribution of stock price is lognormal, that is to say, stock price is continuous. Merton? Cox? Robinstein and ross once pointed out that stock price changes include not only lognormal distribution, but also jumps caused by major events, and it is not comprehensive to ignore the latter. They use binomial distribution instead of lognormal distribution to construct the corresponding option pricing model.

Secondly, the assumption of continuous trading. Theoretically, investors can constantly adjust their positions between options and stocks to obtain a risk-free portfolio. But in practice, this adjustment is bound to be restricted by many factors: 1. It is often difficult for investors to borrow or lend money at the same risk-free interest rate; 2. The separability of stock is limited by specific environment; 3. Frequent adjustments will inevitably increase transaction costs. Therefore, discontinuous transactions often occur in reality. At this time, investors' risk preference will inevitably affect the price of options, which is not considered in Boucher-Shaw model.

Thirdly, assuming that the deviation of the stock price remains unchanged is also inconsistent with the actual situation. Blackburn's later research shows that as the stock price rises, its variance generally decreases, rather than being independent of the stock price level. Some scholars (including Blackburn) tried to extend Boucher-Shaw model to solve the problem of dispersion of change, but so far no satisfactory progress has been made.

In addition, regardless of the existence of transaction costs and deposits, it is also inconsistent with reality. The assumption that the underlying stock of the option does not pay dividends further limits the wide application of the model. Many scholars believe that the time and quantity of dividend distribution will have a substantial impact on the option price, which can not be ignored. Some of them have made appropriate adjustments to the model to reflect the impact of dividends. Specifically, if it is a European call option, the adjustment method is to replace the original stock price with the present value of the stock price minus dividend (D), while other input variables remain unchanged, so it can be substituted into Boucher-Shaw model. If it is an American buyer's option, the situation is a little more complicated. The first step is to adjust according to the above method, and then get the price without executing in advance. The second step is to estimate the price of the option if it is executed immediately before the ex-dividend date, replace the actual stock price with the adjusted stock price, and replace the validity period with the time from the ex-dividend date? The exercise price (x-d) after dividend adjustment replaces the actual exercise price, and variables such as risk-free interest rate and stock price deviation can be substituted into the model. The third step is to select the larger value of the option in the above two cases as the equilibrium price of the option. It should be pointed out that this adjustment is very difficult under the complicated dividend situation.