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An *Asian* option is a path-dependent
option with a payoff linked to the average value of the underlying
asset during the life (or some part of the life) of the option. They
are similar to lookback options in that there are two types of Asian
options: fixed (average price option) and floating (average strike
option). Fixed Asian options have a specified strike, while floating
Asian options have a strike equal to the average value of the underlying
asset over the life of the option.

There are four Asian option types, each with its own characteristic payoff formula:

Fixed call (average price option):

Fixed put (average price option):

Floating call (average strike option):

Floating put (average strike option):

where:

is the average price of underlying asset.

is the price of the underlying asset.

is the strike price (applicable only to fixed Asian options).

is defined using either a geometric or an arithmetic average.

The following functions support Asian options.

Function | Purpose |
---|---|

Price Asian options from a CRR binomial tree. | |

Price Asian options from an EQP binomial tree. | |

Price Asian options using an implied trinomial tree (ITT). | |

Construct an Asian option. | |

Price European or American Asian options using the Longstaff-Schwartz model. | |

Calculate prices and sensitivities of European or American Asian options using the Longstaff-Schwartz model. | |

Price European geometric Asian options using the Kemna Vorst model. | |

Calculate prices and sensitivities of European geometric Asian options using the Kemna Vorst model. | |

Price European arithmetic Asian options using the Levy model. | |

Calculate prices and sensitivities of European arithmetic Asian options using the Levy model. |

A *barrier* option is similar to a
vanilla put or call option, but its life either begins or ends when
the price of the underlying stock passes a predetermined barrier value.
There are four types of barrier options.

This option becomes effective when the price of the underlying stock passes above a barrier that is above the initial stock price. Once the barrier has knocked in, it will not knock out even if the price of the underlying instrument moves below the barrier again.

This option terminates when the price of the underlying stock passes above a barrier that is above the initial stock price. Once the barrier has knocked out, it will not knock in even if the price of the underlying instrument moves below the barrier again.

This option becomes effective when the price of the underlying stock passes below a barrier that is below the initial stock price. Once the barrier has knocked in, it will not knock out even if the price of the underlying instrument moves above the barrier again.

This option terminates when the price of the underlying stock passes below a barrier that is below the initial stock price. Once the barrier has knocked out, it will not knock in even if the price of the underlying instrument moves above the barrier again.

If a barrier option fails to exercise, the seller may pay a rebate to the buyer of the option. Knock-outs may pay a rebate when they are knocked out, and knock-ins may pay a rebate if they expire without ever knocking in.

The following functions support barrier options.

Function | Purpose |
---|---|

Price barrier options from a CRR binomial tree. | |

Price barrier options from an EQP binomial tree. | |

Price barrier options using an implied trinomial tree (ITT). | |

Construct a barrier option. |

A *basket* option is an option on
a portfolio of several underlying equity assets. Payout for a basket
option depends on the cumulative performance of the collection of
the individual assets. A basket option tends to be cheaper than the
corresponding portfolio of plain vanilla options for these reasons:

If the basket components correlate negatively, movements in the value of one component neutralize opposite movements of another component. Unless all the components correlate perfectly, the basket option is cheaper than a series of individual options on each of the assets in the basket.

A basket option minimizes transaction costs because an investor has to purchase only one option instead of several individual options.

The payoff for a basket option is as follows:

For a call:

For a put:

where:

*Si* is the price of asset *i* in
the basket.

*Wi* is the quantity of asset *i* in
the basket.

*K* is the strike price.

Financial Instruments Toolbox™ software supports Longstaff-Schwartz and Nengiu Ju models for pricing basket options. The Longstaff-Schwartz model supports both European, Bermuda, and American basket options. The Nengiu Ju model only supports European basket options. If you want to price either an American or Bermuda basket option, use the functions for the Longstaff-Schwartz model. To price a European basket option, use either the functions for the Longstaff-Schwartz model or the Nengiu Ju model.

Function | Purpose |
---|---|

Price basket options using the Longstaff-Schwartz model. | |

Calculate price and sensitivities for basket options using the Longstaff-Schwartz model. | |

Price European basket options using the Nengjiu Ju approximation model. | |

Calculate European basket options price and sensitivity using the Nengjiu Ju approximation model. | |

Specify a basket stock structure. |

A *compound* option is basically an
option on an option; it gives the holder the right to buy or sell
another option. With a compound option, a vanilla stock option serves
as the underlying instrument. Compound options thus have two strike
prices and two exercise dates.

There are four types of compound options:

Call on a call

Put on a put

Call on a put

Put on a call

Consider the third type, a call on a put. It gives the holder the right to buy a put option. In this case, on the first exercise date, the holder of the compound option pay the first strike price and receives a put option. The put option gives the holder the right to sell the underlying asset for the second strike price on the second exercise date.

The following functions support compound options.

Function | Purpose |
---|---|

Price compound options from a CRR binomial tree. | |

Price compound options from an EQP binomial tree. | |

Price compound options using an implied trinomial tree (ITT). | |

Construct a compound option. |

A *lookback* option is
a path-dependent option based on the maximum or minimum value the
underlying asset achieves during the entire life of the option.

Financial Instruments Toolbox software supports two types of lookback options: fixed and floating. Fixed lookback options have a specified strike price, while floating lookback options have a strike price determined by the asset path. Consequently, there are a total of four lookback option types, each with its own characteristic payoff formula:

Fixed call:

Fixed put:

Floating call:

Floating put:

where:

is the maximum price of underlying stock found along the particular path followed to the node.

is the minimum price of underlying stock found along the particular path followed to the node.

is the price of the underlying stock on the node.

is the strike price (applicable only to fixed lookback options).

The following functions support lookback options.

Function | Purpose |
---|---|

Price lookback options from a CRR binomial tree. | |

Price lookback options from an EQP binomial tree. | |

Price lookback options using an implied trinomial tree (ITT). | |

Construct a lookback option based on an equity tree model. | |

Calculate prices of European lookback fixed and floating strike options using the Conze-Viswanathan and Goldman-Sosin-Gatto models. For more information, see Lookback Option. | |

Calculate prices and sensitivities of European fixed and floating strike lookback options using the Conze-Viswanathan and Goldman-Sosin-Gatto models. For more information, see Lookback Option. | |

Calculate prices of lookback fixed and floating strike options using the Longstaff-Schwartz model. For more information, see Lookback Option. | |

Calculate prices and sensitivities of lookback fixed and floating strike options using the Longstaff-Schwartz model. For more information, see Lookback Option. |

A *digital* option is
an option whose payoff is characterized as having only two potential
values: a fixed payout, when the option is in the money or a zero
payout otherwise. This is the case irrespective of how far the asset
price at maturity is above (call) or below (put) the strike.

Digital options are attractive to sellers because they guarantee a known maximum loss in the event that the option is exercised. This overcomes a fundamental problem with the vanilla options, where the potential loss is unlimited. Digital options are attractive to buyers because the option payoff is a known constant amount, and this amount can be adjusted to provide the exact quantity of protection required.

Financial Instruments Toolbox supports four types of digital options:

Cash-or-nothing option — Pays some fixed amount of cash if the option expires in the money.

Asset-or-nothing option — Pays the value of the underlying security if the option expires in the money.

Gap option — One strike decides if the option is in or out of money; another strike decides the size the size of the payoff.

Supershare — Pays out a proportion of the assets underlying a portfolio if the asset lies between a lower and an upper bound at the expiry of the option.

The following functions calculate pricing and sensitivity for digital options.

Function | Purpose |
---|---|

Calculate the price of cash-or-nothing digital options using the Black-Scholes model. | |

Calculate the price of asset-or-nothing digital options using the Black-Scholes model. | |

Calculate the price of gap digital options using the Black-Scholes model. | |

Calculate the price of supershare digital options using the Black-Scholes model. | |

Calculate the price and sensitivities of cash-or-nothing digital options using the Black-Scholes model. | |

Calculate the price and sensitivities of asset-or-nothing digital options using the Black-Scholes model. | |

Calculate the price and sensitivities of gap digital options using the Black-Scholes model. | |

Calculate the price and sensitivities of supershare digital options using the Black-Scholes model. |

A rainbow option payoff depends on the relative
price performance of two or more assets. A *rainbow* option
gives the holder the right to buy or sell the best or worst of two
securities, or options that pay the best or worst of two assets.

Rainbow options are popular because of the lower premium cost of the structure relative to the purchase of two separate options. The lower cost reflects the fact that the payoff is generally lower than the payoff of the two separate options.

Financial Instruments Toolbox supports two types of rainbow options:

Minimum of two assets — The option holder has the right to buy(sell) one of two risky assets, whichever one is worth less.

Maximum of two assets — The option holder has the right to buy(sell) one of two risky assets, whichever one is worth more.

The following rainbow options speculate/hedge on two equity assets.

Function | Purpose |
---|---|

Calculate the European rainbow option price on minimum of two risky assets using the Stulz option pricing model. | |

Calculate the European rainbow option prices and sensitivities on minimum of two risky assets using the Stulz pricing model. | |

Calculate the European rainbow option price on maximum of two risky assets using the Stulz option pricing model. | |

Calculate the European rainbow option prices and sensitivities on maximum of two risky assets using the Stulz pricing model. |

A *vanilla option* is a category
of options that includes only the most standard components. A vanilla
option has an expiration date and straightforward strike price. American-style
options and European-style options are both categorized as vanilla
options.

The payoff for a vanilla option is as follows:

For a call:

For a put:

where:

*St* is the price of the underlying asset at
time *t*.

*K* is the strike price.

The following functions support specifying or pricing a vanilla option.

Function | Purpose |
---|---|

Calculate the price of a European, Bermuda, or American stock option using a CRR tree. | |

Calculate the price of a European, Bermuda, or American stock option using an EQP tree. | |

Calculate the price of a European, Bermuda, or American stock option using an ITT tree. | |

Calculate the price of a European, Bermuda, or American stock option using the Leisen-Reimer (LR) binomial tree model. | |

Price options using the Black-Scholes option pricing model. | |

Calculate option prices and sensitivities using the Black-Scholes option pricing model. | |

Calculate American call option prices using the Roll-Geske-Whaley option pricing model. | |

Calculate American call option prices and sensitivities using the Roll-Geske-Whaley option pricing model. | |

Price American options using the Bjerksund-Stensland 2002 option pricing model. | |

Calculate American option prices and sensitivities using the Bjerksund-Stensland 2002 option pricing model. | |

Price vanilla options using the Longstaff-Schwartz model. | |

Calculate vanilla option prices and sensitivities using the Longstaff-Schwartz model. | |

Specify a European or Bermuda option. |

A Bermuda option resembles a hybrid of American and European options. You exercise it on predetermined dates only, usually monthly. In Financial Instruments Toolbox software, you indicate the relevant information for a Bermuda option in two input matrices:

`Strike`— Contains the strike price values for the option.`ExerciseDates`— Contains the schedule when you can exercise the option.

A *spread option* is an option written
on the difference of two underlying assets. For example, a European
call on the difference of two assets *X1* and *X2* would
have the following pay off at maturity:

where:

*K* is the strike price.

The following functions support spread options.

Function | Purpose |
---|---|

Price European spread options using the Kirk pricing model. | |

Calculate European spread option prices and sensitivities using the Kirk pricing model. | |

Price European spread options using the Bjerksund-Stensland pricing model. | |

Calculate European spread option prices and sensitivities using the Bjerksund-Stensland pricing model. | |

Price European or American spread options using the Alternate Direction Implicit (ADI) finite difference method. | |

Calculate price and sensitivities of European or American spread spread options using the Alternate Direction Implicit (ADI) finite difference method. | |

Price European or American spread options using Monte Carlo simulations. | |

Calculate price and sensitivities for European or American spread options using Monte Carlo simulations. |

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