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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): $$\mathrm{max}(0,{S}_{av}-X)$$

Fixed put (average price option): $$\mathrm{max}(0,X-{S}_{av})$$

Floating call (average strike option): $$\mathrm{max}(0,S-{S}_{av})$$

Floating put (average strike option): $$\mathrm{max}(0,{S}_{av}-S)$$

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). | |

Price Asian options using a standard trinomial tree (STT). | |

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). | |

Price barrier options using a standard trinomial tree (STT). | |

Price barrier option using finite difference method. | |

Calculate barrier option price and sensitivities using finite difference method. | |

Price a European barrier option using Black-Scholes option pricing model. | |

Calculate price and sensitivities for a European barrier option using Black-Scholes option pricing model. | |

Price a barrier option using Longstaff-Schwartz model. | |

Calculate price and sensitivities for a barrier option using Longstaff-Schwartz model. | |

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: $$\mathrm{max}({\displaystyle \sum Wi\ast Si-K;0)}$$

For a put: $$\mathrm{max}({\displaystyle \sum K-Wi\ast Si;0)}$$

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

### 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 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). | |

Price compound options using a standard trinomial tree (STT). | |

Construct a compound option. |

A *convertible bond* is a financial
instrument that combines equity and debt features. It is a bond with
the embedded option to turn it into a fixed number of shares. The
holder of a convertible bond has the right, but not the obligation,
to exchange the convertible security for a predetermined number of
equity shares at a preset price. The debt component is derived from
the coupon payments and the principal. The equity component is provided
by the conversion feature.

Convertible bonds have several defining features:

Coupon — The coupon in convertible bonds are typically lower than coupons in vanilla bonds since investors are willing to take the lower coupon for the opportunity to participate in the company’s stock via the conversion.

Maturity — Most convertible bonds are issued with long-stated maturities. Short-term maturity convertible bonds usually do not have call or put provisions.

Conversion ratio — Conversion ratio is the number of shares that the holder of the convertible bond will receive from exercising the call option of the convertible bond:

`Conversion ratio = par value convertible bond/conversion price of equity`

For example, a conversion ratio of 25 means a bond can be exchanged for 25 shares of stock. This also implies a conversion price of $40 (1000/25). This, $40, would be the price at which the owner would buy the shares. This can be expressed as a ratio or as the conversion price and is specified in the contract along with other provisions.

Option type:

Callable Convertible: a convertible bond that is callable by the issuer. The issuer of the bond forces conversion, removing the advantage that conversion is at the discretion of the bondholder. Upon call, the bondholder can either convert the bond or redeem at the call price. This option enables the issuer to control the price of the convertible bond and if necessary refinance the debt with a new cheaper one.

Puttable Convertible: a convertible bond with a put feature that allows the bondholder to sell back the bond at a premium on a specific date. This option protects the holder against rising interest rates by reducing the year to maturity.

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

Price convertible bonds using a CRR binomial tree with the Tsiveriotis and Fernandes model. | |

Price convertible bonds using an EQP binomial tree with the Tsiveriotis and Fernandes model. | |

Price convertible bonds using an implied trinomial tree with the Tsiveriotis and Fernandes model. | |

Price convertible bonds using a standard trinomial tree with the Tsiveriotis and Fernandes model. | |

Construct a |

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. So, there are a total of four lookback option types, each with its own characteristic payoff formula:

Fixed call: $$\mathrm{max}(0,{S}_{\mathrm{max}}-X)$$

Fixed put: $$\mathrm{max}(0,X-{S}_{\mathrm{min}})$$

Floating call: $$\mathrm{max}(0,S-{S}_{\mathrm{min}})$$

Floating put: $$\mathrm{max}(0,{S}_{\mathrm{max}}-S)$$

where:

$${S}_{\mathrm{max}}$$ 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). | |

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 when 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: $$\mathrm{max}(St-K,0)$$

For a put: $$\mathrm{max}(K-St,0)$$

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 American options prices using the Barone-Adesi-Whaley option pricing model. | |

Calculate the American options prices and sensitivities using the Barone-Adesi-Whaley option pricing model. | |

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 vanilla option prices using finite difference method. | |

Calculate vanilla option prices and sensitivities using finite difference method. | |

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 an STT tree. | |

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

Calculate the price and sensitivities 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:

$$\mathrm{max}(X1-X2-K,0)$$

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. |

A *forward option* is a non-standardized
contract between two parties to buy or to sell an asset at a specified
future time at a price agreed upon today. The buyer of a forward option
contract has the right to hold a particular forward position at a
specific price any time before the option expires. The forward option
seller holds the opposite forward position when the buyer exercises
the option. A call option is the right to enter into a long forward
position and a put option is the right to enter into a short forward
position. A closely related contract is a futures contract. A forward
is like a futures in that it specifies the exchange of goods for a
specified price at a specified future date. The table below displays
some of the characteristics of forward and futures contracts.

Forwards | Futures |
---|---|

Customized contracts | Standardized contracts |

Over the counter traded | Exchange traded |

Exposed to default risk | Clearing house reduces default risk |

Mostly used for hedging | Mostly used by hedgers and speculators |

Settlement at the end of contract (no Margin required) | Daily changes are settled day by day (Margin required) |

Delivery usually takes place | Delivery usually never happens |

The payoff for a forward option, where the value of a forward
position at maturity depends on the relationship between the delivery
price (*K*) and the underlying price (*S** _{T}*)
at that time, is:

For a long position: $${f}_{T}={S}_{T}-K$$

For a short position: $${f}_{T}=K-{S}_{T}$$

The following functions support pricing a forwards option.

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

Price options on forwards using the Black option pricing model. | |

Determine option prices and sensitivities on forwards using the Black pricing model. |

A *future option* is a standardized contract
between two parties to buy or sell a specified asset of standardized
quantity and quality for a price agreed upon today (the futures price)
with delivery and payment occurring at a specified future date, the
delivery date. The contracts are negotiated at a futures exchange,
which acts as an intermediary between the two parties. The party agreeing
to buy the underlying asset in the future, the "buyer" of the contract,
is said to be "long", and the party agreeing to sell the asset in
the future, the "seller" of the contract, is said to be "short."

Forwards | Futures |
---|---|

Customized contracts | Standardized contracts |

Over the counter traded | Exchange traded |

Exposed to default risk | Clearing house reduces default risk |

Mostly used for hedging | Mostly used by hedgers and speculators |

Settlement at the end of contract (no Margin required) | Daily changes are settled day by day (Margin required) |

Delivery usually takes place | Delivery usually never happens |

A futures contract is the delivery of item *J* at
time *T* and:

There exists in the market a quoted price $$F(t,T)$$, which is known as the futures price at time

*t*for delivery of*J*at time*T*.The price of entering a futures contract is equal to zero.

During any time interval [

*t*,*s*], the holder receives the amount $$F(s,T)-F(t,T)$$ (this reflects instantaneous marking to market).At time

*T*, the holder pays $$F(T,T)$$ and is entitled to receive*J*. Note that $$F(T,T)$$ should be the spot price of*J*at time*T*.

The following functions support pricing a futures option.

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

Price options on futures using the Black option pricing model. | |

Determine option prices and sensitivities on futures using the Black pricing model. |

`asianbycrr`

| `asianbyeqp`

| `asianbyitt`

| `asianbykv`

| `asianbylevy`

| `asianbyls`

| `asiansensbykv`

| `asiansensbylevy`

| `asiansensbyls`

| `assetbybls`

| `assetsensbybls`

| `barrierbycrr`

| `barrierbyeqp`

| `barrierbyitt`

| `basketbyju`

| `basketbyls`

| `basketsensbyju`

| `basketsensbyls`

| `basketstockspec`

| `basketstockspec`

| `cashbybls`

| `cashsensbybls`

| `chooserbybls`

| `compoundbycrr`

| `compoundbyeqp`

| `compoundbyitt`

| `crrprice`

| `crrsens`

| `crrtimespec`

| `crrtree`

| `eqpprice`

| `eqpsens`

| `eqptimespec`

| `eqptree`

| `gapbybls`

| `gapsensbybls`

| `impvbybjs`

| `impvbyblk`

| `impvbybls`

| `impvbyrgw`

| `instasian`

| `instbarrier`

| `instcompound`

| `instlookback`

| `instoptstock`

| `ittprice`

| `ittsens`

| `itttimespec`

| `itttree`

| `lookbackbycrr`

| `lookbackbycvgsg`

| `lookbackbyeqp`

| `lookbackbyitt`

| `lookbackbyls`

| `lookbackbyls`

| `lookbacksensbycvgsg`

| `lookbacksensbyls`

| `lookbacksensbyls`

| `lrtimespec`

| `lrtree`

| `maxassetbystulz`

| `maxassetsensbystulz`

| `minassetbystulz`

| `minassetsensbystulz`

| `optpricebysim`

| `optstockbybjs`

| `optstockbyblk`

| `optstockbybls`

| `optstockbycrr`

| `optstockbyeqp`

| `optstockbyitt`

| `optstockbylr`

| `optstockbyls`

| `optstockbyrgw`

| `optstocksensbybjs`

| `optstocksensbyblk`

| `optstocksensbybls`

| `optstocksensbylr`

| `optstocksensbyls`

| `optstocksensbyrgw`

| `spreadbybjs`

| `spreadbykirk`

| `spreadbyls`

| `spreadsensbybjs`

| `spreadsensbykirk`

| `spreadsensbyls`

| `stockspec`

| `supersharebybls`

| `supersharesensbybls`

| `treepath`

| `trintreepath`

- Understanding Equity Trees
- Pricing Equity Derivatives Using Trees
- Creating Instruments or Properties
- Graphical Representation of Equity Derivative Trees
- Compute Option Prices on a Forward
- Compute Forward Option Prices and Delta Sensitivities
- Compute the Option Price on a Future
- Pricing European Call Options Using Different Equity Models
- Pricing Asian Options
- Equity Derivatives Using Closed-Form Solutions
- Pricing Using the Bjerksund-Stensland Model

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