Understanding Equity Exotic Options :: Equity Derivatives (Financial Derivatives Toolbox™)

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Understanding Equity Exotic Options

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Introduction Asian Option Barrier Option Basket Option Compound Option Lookback Option Digital Option Rainbow Option Bermuda Put and Call Schedule

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Bermuda Put and Call Schedule

Introduction

Financial Derivatives Toolbox software supports eight types of equity exotic options. Support for all of these equity exotic option types additionally includes American and European puts and calls.

Asian Option

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 options types, each with its own characteristic payoff formula:

Fixed call:
Fixed put:
Floating call:
Floating put:

where:

is the average 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 Asian options).

is defined using either a geometric or an arithmetic average.

The following functions support Asian options

Function	Purpose
`asianbycrr`	Price Asian option from a CRR binomial tree.
`asianbyeqp`	Price Asian option from an EQP binomial tree.
`asianbyitt`	Price Asian options using an implied trinomial tree (ITT).
`instasian`	Construct an Asian option.

Barrier Option

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.

Up Knock-In

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.

Up Knock-Out

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.

Down Knock-In

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.

Down Knock-Out

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.

Rebates

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
`barrierbycrr`	Price barrier option from a CRR binomial tree.
`barrierbyeqp`	Price barrier option from an EQP binomial tree.
`barrierbyitt`	Price barrier options using an implied trinomial tree (ITT).
`instbarrier`	Construct a barrier option.

Basket Option

A basket option is an option on a portfolio of several underlying 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.

The following functions support basket options.

Function	Purpose
`basketbyls`	Price basket options using the Longstaff-Schwartz model.
`basketsensbyls`	Calculate price and sensitivities for basket options using the Longstaff-Schwartz model.
`basketbyju`	Price European basket options using the Nengjiu Ju approximation model.
`basketsensbyju`	Calculate European basket options price and sensitivity using the Nengjiu Ju approximation model.
`basketstockspec`	Specify basket stock structure.

Compound Option

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

Note The payoff formulas for compound options are too complex for this discussion. If you are interested in the details, consult the paper by Mark Rubinstein entitled "Double Trouble," published in Risk 5 (1991).

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 is allowed to pay the first strike price and receive 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
`compoundbycrr`	Price compound option from a CRR binomial tree.
`compoundbyeqp`	Price compound option from an EQP binomial tree.
`compoundbyitt`	Price compound options using an implied trinomial tree (ITT).
`instcompound`	Construct a compound option.

Lookback Option

A l ookback 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 Derivatives 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 compound options

Function	Purpose
`lookbackbycrr`	Price lookback option from a CRR binomial tree.
`lookbackbyeqp`	Price lookback option from an EQP binomial tree.
`lookbackbyitt`	Price lookback options using an implied trinomial tree (ITT).
`instlookback`	Construct a lookback option.

Digital 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 Derivatives Toolbox software 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.

Financial Derivatives Toolbox supports the following functions to calculate pricing and sensitivity for digital options.

Function	Purpose
`cashbybls`	Calculate the price of cash-or-nothing digital options using the Black-Scholes model.
`assetbybls`	Calculate the price of asset-or-nothing digital options using the Black-Scholes model.
`gapbybls`	Calculate the price of gap digital options using the Black-Scholes model
`supersharebybls`	Calculate the price of supershare digital options using the Black-Scholes model
`cashsensbybls`	Calculate the price and sensitivities of cash-or-nothing digital options using the Black-Scholes model.
`assetsensbybls`	Calculate the price and sensitivities of asset-or-nothing digital options using the Black-Scholes model.
`gapsensbybls`	Calculate the price and sensitivities of gap digital options using the Black-Scholes model.
`supersharesensbybls`	Calculate the price and sensitivities of supershare digital options using the Black-Scholes model.

Rainbow Option

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 Derivatives Toolbox software 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.

Financial Derivatives Toolbox supports the following Rainbow options for speculating/hedging on two equity assets.

Function	Purpose
`minassetbystulz`	Calculate the European rainbow option price on minimum of two risky assets using the Stulz option pricing model.
`minassetsensbystulz`	Calculate the European rainbow option prices and sensitivities on minimum of two risky assets using the Stulz pricing model.
`maxassetbystulz`	Calculate the European rainbow option price on maximum of two risky assets using the Stulz option pricing model.
`maxassetsensbystulz`	Calculate the European rainbow option prices and sensitivities on maximum of two risky assets using the Stulz pricing model.

Bermuda Put and Call Schedule

A Bermuda option is somewhat like a hybrid of American and European options. It can be exercised on predetermined dates only, usually once a month. In Financial Derivatives Toolbox software, the relevant information for a Bermuda option is indicated in two input matrices:

Strike — Contains the strike price values for the option.
ExcerciseDates — Contains the schedule when the option can be exercised.

Understanding Equity Trees

Computing Prices and Sensitivities for Equity Derivatives Using Trees

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