### Black Scholes

This model is a partial differential equation that can be used to provide a theoertical estimate for the following:

• European Call option with no dividend

• European Put option with no dividend

### History

This model was developed by Fischer Black, Robert Merton and Myron Scholes in 1973.
The approach used is to hedge the option by buying and selling the exact amount of underlying asset
This type of hedge is called delta hedging.
By continuously adjusting the proportions of stock and options in a portfolio, the investor can create a riskless hedge portfolio.

### Assumptions

• The price of the underlying asset follows a geometric brownian motion

• The options are european and can only be execised at expiry

• No dividends are paid during the life of the option

• No commissions

• The risk-free interest rate and volatility of the underlying are known and constant

• Returns on the underlying are normally distributed

Options on equities
The binomial distribution provides a framework for calculating the expected option value
For European options you can use Monte Carlo or Numerical Integration

Formula for calculating the value of a European call option on a non-dividend paying stock using the five following factors:
1) Stock Price (S)
2) Strike Price (K)
3) Time to Expiration (t)
4) Volatility (o)
5) Interest Rate (r)

#### Black Scholes Extended

Options on equities with continuous dividends
options on commodities
options on foreign exchange
options on bonds