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

  • options_a7a9870d411fa7be56753e7bbd48731b3acb06b9.png is the price of the stock
  • options_94840bddfb4d81597649c069eef4b1be11ab3b54.png the price of a derivative as a function of time and stock price
  • options_2ad0edbaf4ef69bf2aacb0db37e3bb57beb00477.png the price of a European call option
  • options_ab864c2d5cf63c253ace3bb559a62863544b0d41.png the price of a European put option
  • options_d80b633218f04f7a25f6a9dd7e4617a38e88f263.png the strike price of the option
  • options_b4bc85fbe6b1085924f1ba268e01893b52ddad89.png the annualized risk-free interest rate, continuously compounded
  • options_8700f80c4895c66c1fb785c5b0cf5c4b6f532c75.png the drift rate of options_a7a9870d411fa7be56753e7bbd48731b3acb06b9.png, annualized
  • options_f2eebbd62e41be9d5a2457b4bd291dd096896884.png the std. dev. of stock's returns
  • options_eb5d809ed7c492fae7d4927a6fc9a5e22f9b3831.png a time in years with options_b3731d37cd447bd6c31809075a6be43b3d0b04ec.png being the expiry
  • $Π the value of the portfolio
  • options_59917483c3aefd6a10b7a0e0daeb37959e241530.png denotes the /std. normal cumulative distribution function, and options_f09da1a2700e107affe1baf536200343464e5c00.png the std. normal density:

    options_77c325d7725daa22026835c0fdde4779138db6c5.png

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
  • 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:

options_568d2c2ecf069c9ff07292b6b38aaf2bba7a8d4c.png

  • 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