# Options

## Table of Contents

## Notation

**call option**: buyer / owner of the option has the right to*buy***put option**: buyer / owner of the option has the right to*sell***strike price**or**exercise price**: the pixed price at which the owner of the option can buy (in*call*), or sell (in*put*) the underlying security or commodity*call*(*put*) is**in-the-money**if the strike price is*below*(*above*) the market price*call*(*put*) is**out-of-the-money**if the strike price is*above*(*below*) the market price

## Option-styles

The **style** or **family** of an option is the class into which the option falls, usually defined by the dates on which the option may be *exercised*.

The vast majority of options are either *European* or *American* (style) options.

The key differences between *American* and *European* options relates to when the options can be *exercised*:

**European option**may be exercised only at the*expiration date*of the option, i.e. at a single pre-defined point in time.**American option**may be exercised at*any time before*the expiration date.

## Pricing

### Black-Scholes

- Model for dynamics of financial market containing derivative investment instruments
**Key idea**: hedge the option by buying and selling the underlying asset in the just the right away, thus eliminate risk

#### Notation

- is the
*price of the stock* - the price of a derivative as a function of time and stock price
- the price of a European call option
- the price of a European put option
- the
*strike price*of the option - the
*annualized risk-free interest rate*, continuously compounded - the
*drift rate*of , annualized - the std. dev. of stock's returns
- a time in years with being the
*expiry* - $Π the value of the portfolio
denotes the /std. normal cumulative distribution function, and the std. normal density:

#### Assumptions

Assumptions on the *underlying assets*:

*Riskeless rate*: rate of return on the riskless asset is*constant*and thus called the risk-free interest rate*Random walk*: instantaneous log return of the stock price is infinitesimal random walk*with drift*; it's a*geometric Brownian motion*, and we will assume its drift and volatility are constant- If
*time-varying*, one can deduce a modified Black-Scholes formula, as long as the volatility is not random

- If
- Stock does not pay dividends

Assumptions on the *market*:

- No arbitrage
- Possible to borrow and lend any amount, even fractional, of cash at a riskless rate
- Possible ot buy and sell any amount, even fractional, of the stock (includes short-selling)
- Above transactions do not incur any fees or costs (i.e. frictionless market)

The **Black-Scholes equation** is a partial differential equation, which describes the price of the option over time:

- Gives a theoretical estimate of the price of European-style options, and shows that it has a
*unique*price regardless of the risk of the security and its expected return