1. The Significance of the Black-Scholes Option Pricing Model
The Black-Scholes model and some of its variants have been widely used by option traders, investment banks, financial managers, insurers, and others. The expansion of derivative instruments has made international financial markets more efficient, but it has also made global markets more volatile. New technologies and the creation of new financial instruments have strengthened the interdependence between markets and market participants, not only within a single country but also across other countries and even many of them. As a result, a fluctuation or financial crisis in one market or one country is very likely to spread quickly to other countries and even to the entire world economy. China’s financial system is unsound and its capital markets are incomplete, but as reform deepens and the country moves toward internationalization, the capital markets will keep developing, the exchange system will gradually improve, and enterprises will gain more autonomy and therefore face greater risk. It is therefore essential to cultivate a financial derivatives market that hedges risk, and it is also necessary to explore the derivatives market. We have only just begun.
2. Methods of Option Pricing
Although stock options had already appeared in the United States as early as the 19th century, before 1973 these transactions were scattered across over-the-counter markets, the varieties traded were very limited, and the scale of trading was quite small. Before 1973, the only stock options traded were calls, with no puts. As a result, as late as 1968, the number of shares represented by stock option contracts traded in the United States was still only 1% of the number of shares traded on the New York Stock Exchange. On April 26, 1937, the world’s first centralized options market β the Chicago Board Options Exchange β was officially established.
3. The Binomial Tree Pricing Method
The binomial tree option pricing model was first proposed by Cox, Ross, and Rubinstein (1979), based on the no-arbitrage principle and the risk-neutral principle. This model has many advantages: it is a simple model, easy to program, and applicable to option pricing with large and complex data. It can analyze option pricing from multiple angles, and if extended to multiple periods, the binomial model becomes a powerful method for valuing options whose future cash flows depend on the market prices of other assets.
The binomial option pricing model assumes that stock prices fluctuate in only two directions, up and down, and that the probability and magnitude of each up (or down) movement remain constant throughout the entire horizon under consideration. The model divides the life of the option under consideration into several stages, simulates all possible development paths of the underlying stock over its entire life based on the historical volatility of the stock price, and for each node on each path calculates the exercise payoff of the warrant and the warrant price computed by the discounting method. For American warrants, since they can be exercised early, the theoretical price of the warrant at each node should be the larger of the exercise payoff and the warrant price computed by discounting.
4. Monte Carlo Simulation
The theoretical basis of the Monte Carlo method for option pricing is the risk-neutral pricing principle, its theoretical foundation is probability theory and mathematical statistics, and its essence is to predict the average return of an option by simulating the price path of the underlying asset and thereby obtain an estimate of the option price. In general, the Monte Carlo simulation method for option pricing consists of the following steps (using a European call option as an example):
- Under the risk-neutral measure, simulate one random path of the underlying asset’s price from the initial moment to the expiration date.
- Calculate the option’s payoff at expiration along this path, and discount the payoff at the risk-free rate.
- Repeat the first two steps to obtain a large sample of discounted option payoffs.
- Take the sample mean to obtain the Monte Carlo simulated value of the option price.
5. The Black-Scholes Option Pricing Model
The first complete option pricing model was created by Myron Scholes and Fischer Black and made public in 1973. The Black-Scholes Option Pricing Model, which they created and developed, laid the foundation for the rational pricing of various derivative financial instruments priced by market price changes in emerging derivative financial markets, including stocks, bonds, currencies, and commodities.
5.1 Black-Scholes Model Assumptions
- The security price follows geometric Brownian motion; the return within each small interval follows a normal distribution, and any two different intervals are independent of each other. During the life of the option, the variance of the security price and the risk-free rate are constant.
- Short selling is allowed, and the proceeds may be used freely.
- There are no transaction costs or taxes, and all securities are perfectly divisible.
- During the life of the derivative security, the underlying security pays no cash income.
- There are no risk-free arbitrage opportunities.
- Security trading is continuous, and price changes are continuous as well.
- During the life of the derivative security, the risk-free rate is constant.
- European option. A European option can be exercised only on the expiration date.
5.2 Black-Scholes Model Formula
\begin{equation}
\frac{\partial \mathrm C}{ \partial \mathrm t } + \frac{1}{2}\sigma^{2} \mathrm S^{2} \frac{\partial^{2} \mathrm C}{\partial \mathrm C^2}+ \mathrm r \mathrm S \frac{\partial \mathrm C}{\partial \mathrm S}\ =\mathrm r \mathrm C \label{eq:1}
\end{equation}
The price of a European option can be calculated as
\[C(S,t)= N(d_1)S - N(d_2)Ke^{-rt}\]
\begin{equation}
\mathrm d_1= \frac{1}{\sigma \sqrt{\mathrm t}} \left[\ln{\left(\frac{S}{K}\right)} + t\left(r + \frac{\sigma^2}{2} \right) \right]
\end{equation}
\begin{equation}
\mathrm d_2= \frac{1}{\sigma \sqrt{\mathrm t}} \left[\ln{\left(\frac{S}{K}\right)} + t\left(r - \frac{\sigma^2}{2} \right) \right]
\end{equation}
\begin{equation}
N(x)=\frac{1}{\sqrt{2\pi}} \int_{-\infty}^{x} \mathrm e^{-\frac{1}{2}z^2} dz
\end{equation}
C = the initial fair price of the option;
S = the current stock price
K = the strike price of the option
r = the risk-free rate (a number between 0 and 1)
\(\sigma\) = the stock’s volatility of return (a number between 0 and 1)
t = the time to option expiration (years)
N = the cumulative normal distribution function
5.3 Black-Scholes Option Calculation Example
The current stock price is 42 yuan, the strike price of a European option is 40 yuan, the time is 6 months from now, the risk-free rate is 10% per year, and the stock’s volatility is 20% per year. That is, S=42, K=40, r=0.10, \(\sigma\)=0.20, t=0.5.
Substituting into the formula:
\begin{equation}
\mathrm d_1= \frac{1}{0.2 \sqrt{\mathrm 0.5}} \left[\ln{\left(\frac{42}{40}\right)} + 0.5\left(0.10 + \frac{0.2^2}{2} \right) \right]=0.7693
\end{equation}
\begin{equation}
\mathrm d_2= \frac{1}{0.2 \sqrt{\mathrm 0.5}} \left[\ln{\left(\frac{42}{40}\right)} + 0.5\left(0.10 - \frac{0.2^2}{2} \right) \right]=0.6278
\end{equation}
If the option is a call option, its value C is:
\[ C_1= N(0.7693) \times 42 - N(0.6278)\times 40 \times e^{-0.1 \times 0.5} = 4.76 \]
If the option is a put option, its value C is:
\[ C_2= N(-0.6278)\times40\times e^{-0.1\times 0.5}-N(-0.7693)\times 42 = 0.81 \]
