### 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

### Formula

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