Option Pricing Models
[Black Scholes]
[Binomial Tree]
[Monte Carlo ]
The Nobel Prize Winning Models! The idea of Black-Scholes approach
is as follows: Assuming stock price movement is a Wiener process, it is derived an
differential equations on the relations of option price and the underlying stock price.
A formula of the option price can be derived by solving the differential equation.
Applying Black-Scholes Analysis to different instruments, we create different models:
The models are based on the following strict assumptions:
- The option is the European-type, that is, exercisable only on expiration.
- There are no taxes, no transaction costs and no stock-borrowing costs.
- The price change of the securities follows a log-normal distribution.
- There are no restrictions on short sales of the underlying securities.
- Trading in the market is continuous.
- There are no arbitrages in the market.
- The risk-free interest rate is constant during the term of the option.
- No dividends are paid on the underlying security during the term of the option.
Nevertheless, the Black-Scholes models are the most widely used models for pricing options.
Binomial valuation models are intuitive and useful models proposed first by Cox, Ross and Rubinstein.
The models assume that stock price either moves up or down by a proportional amount in each short interval of time.
Binomial valuation models are also heavily used in bond analysis. Although we assume options are
European-style here for comparison purpose, the models can be easily extended to American-style options.
Applying binomial valuation to different instruments, we create different models:
Monte Carlo Simulation models are statistical models based on large number of random trials.
It can be used for European-style options but can provide great efficiency if several variables
are to be estimated.
Applying Monte Carlo simulation to different instruments, we create different models: