Options

An option is a forward contract, where an invester has to pay to purchase an option contract.

  • A call option gives the right to buy at a certain date at a certain price.
  • A put option gives the right to sell at a certain date at a certain price.
The following defintions are used in conjunction with options:
  • Exercise price or strike price, the price in the contract.
  • Expiration date or exercise date, also known as maturity is the date in the contract.
  • Exercise style: can either be European (contract may be exercised at expiration date only), American (contract may be exercised any time up to the expiration date) or Bermudan (contract may be exercised at fixed intervals).
People who buy options are called holders and those who sell options are called writers; furthermore, buyers are said to have long positions, and sellers are said to have short positions. Note that it is possible to take long and short positions in either call or put options, hence four different scenarios are possible. This is depicted in the figures below.
Call option



Figure 1: Long and short positions in call option.
In the case of a call option, expiration value (per unit of notional amount) is either:
  • zero, or
  • the difference between the value of the underlier X and the strike price ST,
whichever is greater. This may also be expressed mathematically as:
  • Payoff long position: max(ST - X, 0)
  • Payoff short position: min(X - ST, 0)
Put option



Figure 2: Long and short positions in put option.
In the case of a put option, expiration value (per unit of notional amount) is either:
  • zero, or
  • the difference between the strike price ST and the value of the underlier X,
whichever is greater. This may also be expressed mathematically as:
  • Payoff long position: max(X - ST, 0)
  • Payoff short position: min(ST - X, 0)
In addition to the payoff at expiration, one defines the intrinsic value of an option the value that an option would have, if it were about to expire. For call options, this is the difference between the underlying stock's price and the strike price. For put options, it is the difference between the strike price and the underlying stock's price. In the case of both puts and calls, if the difference between the underlying stock's price and the strike price is negative, the value is given as zero.

Note that prior to expiration, an option's market value will generally exceed this intrinsic value by an amount that is called the option's time value. An option is said to be at-the-money (ATM) if the underlier value currently equals the strike price, i.e. an ATM option would not lead to any cash flow to the holder if it were exercised immediately. Otherwise, the option is said to be in-the-money (ITM) if it has positive intrinsic value (positive cash flow to the holder of the option), or out-of-the-money (OTM) if it has zero intrinsic value (negative cash flow to the holder of the option). For example, a call is in-the-money if the underlier value is above the strike price. A put is in-the-money if the underlier value is below the strike price. As a corollary, intrinsic value in options is the in-the-money portion of the option's premium. Finally, remark that the initial cost of the option is not included in the definitions of option's value (payoff) and intrinsic value!

Examples

Examples of derivative securities are:
  • Interest Rate Caps. Definition:
    • Cap rate: threashold for a floating interest rate.
  • Bonds, see definition in Wikipedia.
  • ICONs, Index Currency Option Notes are bonds in which the amount received by the holder at maturity varies with a foreign exchange rate.
  • Range Forward contracts, flexible forward contract.

Type of traders

The following types of traders are distinguished:
  • Hedgers: reduce or eliminate risk, make an outcome more certain, not necessarily improve the outcome.
  • Speculators: take position in a market, use derivatives to get extra leverage, wish to bet on future movements of an asset.
  • Arbitrageurs: lock in a riskless profit by simultaneously entering into transactions in two or more markets, i.e. take advantage of difference in price in two different markets.

Volatilities

A volatility is a measure of a security's propensity to go up and down in price. Mathematically, this is expressed as the standard deviation from the average, in this case average performance. In general, high volatility means high unpredictability, and therefore greater risk. Numerous attempts have been made to incorporate volatility into pricing models, but the problem has always been that past volatility is not necessarily a good guide to future volatility. Generally speaking, the higher the volatility of a share, the higher the price of option/warrants on the share will be. To arrive at a volatility there are two methods employed:
  • Implied volatilities: volatilities are implied from the price of an option, since the price of an option is an indication of/based on what one expects the value of the underlier to do in the future.
  • Historic volatilities are given for a variety of time periods based on the historic values of the underlier for a given time interval. All volatilities are annualized.

Greeks

Almost every option pricer outputs five "greeks". The "greeks" denote five symbols and associated quantities, which measure how an option market value should respond to a change in some variable, such as an underlier, implied volatility, interest rate or time:
  • Delta measures first order (linear) sensitivity to an underlier.
  • Gamma measures second order (quadratic) sensitivity to an underlier.
  • Vega measures first order (linear) sensitivity to the implied volatility of an underlier.
  • Theta measures first order (linear) sensitivity to the passage of time.
  • Rho measures first order (linear) sensitivity to an applicable interest rate.
Roughly speaking, the linear sensitivity can be interpreted as change, quadratic sensitivity may be seen as rate of change (fast or slow changes). These quantities are called the Greeks because four out of the five are named after letters of the Greek alphabet. Vega is the exception. For reasons unknown, it is named after the brightest star in the constellation Lyra. At times, vega has been called kappa, but the name vega is now well established.