Option pricing model
Option is a financial derivative instrument, which gives the holder the right to buy or sell the underlying assets at a certain price at a certain time in the future, but not the obligation. Option pricing model is a mathematical formula to calculate the price of buying or selling options under specific assumptions.
historical background
In the late 1970s and early 1980s, the option market began to develop rapidly, but there was no theoretical framework to explain option pricing. In 1973, Fisher Blake, Robert Merton and Meren Scholes proposed the Black Scholes option pricing model, also known as the BS model. This model has proved to be a revolutionary achievement, which has made remarkable progress in the research of option market.
Important parameters of options
Options are priced according to two key parameters: exercise price and maturity time.
The exercise price is the purchase or sale price of the underlying asset agreed in the contract. When the option expires, if the underlying asset price is higher than the exercise price, the buyer (call option) will gain; If the price is lower than the exercise price, the buyer's income is zero, and the seller (put option) will receive a premium. The expiration time is the date when the option contract disappears. Options with longer maturities are more valuable because they give more opportunities for changes in underlying asset prices. BS model
The BS model is a mathematical formula for calculating the price of European options. It assumes the following five conditions: no arbitrage opportunities, all securities in the market are fully transferable, no transaction costs, the underlying asset price follows the geometric Brownian motion (i.e. the stock price of Brownian motion), and investors can fully short stocks. The BS model models the option price based on stochastic differential equation. Its price formula includes the following parameters: option type (call or put), underlying asset price, exercise price, residual maturity, risk-free interest rate and volatility of underlying assets. The formula for calculating the option price is:
C = S × N(d1) ？ K × e^(？rT) × N(d2)
Where, C is the price of European call options, S is the price of underlying assets, K is the exercise price, r is the risk-free interest rate, T is the residual maturity, and N (d1) and N (d2) are standard normal distribution functions.
The advantage of BS model is that it is simple to use and fast in calculation, but it has some defects. For example, the BS model cannot cope with changes in market volatility, that is, the occurrence of the Black Swan event. In addition, the BS model assumes that the underlying asset price follows the geometric Brownian motion. However, the stock market often has asymmetric distribution, soaring and falling and other abnormal events, which are different from the geometric Brownian motion.
Other option pricing models
In addition to BS model, there are many other option pricing models, such as diffusion jump model, Local Volatility model, stochastic volatility model, etc. Each model is based on different continuous time distribution functions, stock price behavior and calculation methods to apply to specific types of options and market situations.
conclusion
Option pricing model is the basis for evaluation, trading and arbitrage of derivative financial instruments. BS model is one of the most widely used option pricing models, but we should pay attention to its limitations. In practice, the reasonable choice of models depends on the required performance and the accuracy of assumptions.
