# price

## Syntax

## Description

`[`

computes the equity instrument price and related pricing information based on the pricing
object `Price`

,`PriceResult`

] = price(`inpPricer`

,`inpInstrument`

)`inpPricer`

and the instrument object
`inpInstrument`

.

`[`

adds an optional argument to specify sensitivities in addition to the required arguments
in the previous syntax.`Price`

,`PriceResult`

] = price(___,`inpSensitivity`

)

## Examples

### Use `AssetTree`

Pricer and `BlackScholes`

Model to Price `Vanilla`

Instrument

This example shows the workflow to price a `Vanilla`

instrument when you use a `BlackScholes`

model and an `AssetTree`

pricing method.

**Create Vanilla Instrument Object**

Use `fininstrument`

to create a `Vanilla`

instrument object.

VanillaOpt = fininstrument("Vanilla",'ExerciseDate',datetime(2019,5,1),'Strike',29,'OptionType',"put",'ExerciseStyle',"european",'Name',"vanilla_option")

VanillaOpt = Vanilla with properties: OptionType: "put" ExerciseStyle: "european" ExerciseDate: 01-May-2019 Strike: 29 Name: "vanilla_option"

**Create BlackScholes Model Object**

Use `finmodel`

to create a `BlackScholes`

model object.

BlackScholesModel = finmodel("BlackScholes",'Volatility',0.25)

BlackScholesModel = BlackScholes with properties: Volatility: 0.2500 Correlation: 1

**Create ratecurve Object**

Create a flat `ratecurve`

object using `ratecurve`

.

Settle = datetime(2018,1,1); Maturity = datetime(2020,1,1); Rate = 0.035; myRC = ratecurve('zero',Settle,Maturity,Rate,'Basis',1)

myRC = ratecurve with properties: Type: "zero" Compounding: -1 Basis: 1 Dates: 01-Jan-2020 Rates: 0.0350 Settle: 01-Jan-2018 InterpMethod: "linear" ShortExtrapMethod: "next" LongExtrapMethod: "previous"

**Create AssetTree Pricer Object**

Use `finpricer`

to create an `AssetTree`

pricer object for an LR equity tree and use the `ratecurve`

object for the `'DiscountCurve'`

name-value pair argument.

LRPricer = finpricer("AssetTree",'DiscountCurve',myRC,'Model',BlackScholesModel,'SpotPrice',30,'PricingMethod',"LeisenReimer",'Maturity',datetime(2019,5,1),'NumPeriods',15)

LRPricer = LRTree with properties: InversionMethod: PP1 Strike: 30 Tree: [1x1 struct] NumPeriods: 15 Model: [1x1 finmodel.BlackScholes] DiscountCurve: [1x1 ratecurve] SpotPrice: 30 DividendType: "continuous" DividendValue: 0 TreeDates: [02-Feb-2018 08:00:00 06-Mar-2018 16:00:00 08-Apr-2018 00:00:00 10-May-2018 08:00:00 11-Jun-2018 16:00:00 14-Jul-2018 00:00:00 15-Aug-2018 08:00:00 16-Sep-2018 16:00:00 ... ] (1x15 datetime)

**Price Vanilla Instrument**

Use `price`

to compute the price and sensitivities for the `Vanilla`

instrument.

`[Price, outPR] = price(LRPricer,VanillaOpt,"all")`

Price = 2.2542

outPR = priceresult with properties: Results: [1x7 table] PricerData: [1x1 struct]

outPR.Results

`ans=`*1×7 table*
Price Delta Gamma Vega Lambda Rho Theta
______ ________ ________ ______ ______ _______ ________
2.2542 -0.33628 0.044039 12.724 -4.469 -16.433 -0.76073

### Use `AssetTree`

Pricer and `BlackScholes`

Model to Price `Vanilla`

Instrument

This example shows the workflow to price a `Vanilla`

instrument when you use a `BlackScholes`

model and an `AssetTree`

pricing method for a Standard Trinomial (STT) tree.

**Create Vanilla Instrument Object**

Use `fininstrument`

to create a `Vanilla`

instrument object.

VanillaOpt = fininstrument("Vanilla",'ExerciseDate',datetime(2019,5,1),'Strike',29,'OptionType',"put",'ExerciseStyle',"european",'Name',"vanilla_option")

VanillaOpt = Vanilla with properties: OptionType: "put" ExerciseStyle: "european" ExerciseDate: 01-May-2019 Strike: 29 Name: "vanilla_option"

**Create BlackScholes Model Object**

Use `finmodel`

to create a `BlackScholes`

model object.

BlackScholesModel = finmodel("BlackScholes",'Volatility',0.25)

BlackScholesModel = BlackScholes with properties: Volatility: 0.2500 Correlation: 1

**Create ratecurve Object**

Create a flat `ratecurve`

object using `ratecurve`

.

Settle = datetime(2018,1,1); Maturity = datetime(2020,1,1); Rate = 0.035; myRC = ratecurve('zero',Settle,Maturity,Rate,'Basis',1)

myRC = ratecurve with properties: Type: "zero" Compounding: -1 Basis: 1 Dates: 01-Jan-2020 Rates: 0.0350 Settle: 01-Jan-2018 InterpMethod: "linear" ShortExtrapMethod: "next" LongExtrapMethod: "previous"

**Create AssetTree Pricer Object**

Use `finpricer`

to create an `AssetTree`

pricer object for an Standard Trinomial equity tree and use the `ratecurve`

object for the `'DiscountCurve'`

name-value pair argument.

STTPricer = finpricer("AssetTree",'DiscountCurve',myRC,'Model',BlackScholesModel,'SpotPrice',30,'PricingMethod',"StandardTrinomial",'Maturity',datetime(2019,5,1),'NumPeriods',15)

STTPricer = STTree with properties: Tree: [1x1 struct] NumPeriods: 15 Model: [1x1 finmodel.BlackScholes] DiscountCurve: [1x1 ratecurve] SpotPrice: 30 DividendType: "continuous" DividendValue: 0 TreeDates: [02-Feb-2018 08:00:00 06-Mar-2018 16:00:00 08-Apr-2018 00:00:00 10-May-2018 08:00:00 11-Jun-2018 16:00:00 14-Jul-2018 00:00:00 15-Aug-2018 08:00:00 16-Sep-2018 16:00:00 ... ] (1x15 datetime)

**Price Vanilla Instrument**

Use `price`

to compute the price and sensitivities for the `Vanilla`

instrument.

`[Price, outPR] = price(STTPricer,VanillaOpt,"all")`

Price = 2.2826

outPR = priceresult with properties: Results: [1x7 table] PricerData: [1x1 struct]

outPR.Results

`ans=`*1×7 table*
Price Delta Gamma Vega Lambda Rho Theta
______ _______ ________ _____ _______ _______ ________
2.2826 -0.2592 0.030949 12.51 -3.8981 -16.516 -0.73845

## Input Arguments

`inpInstrument`

— Instrument object

`Vanilla`

object | `Barrier`

object | `Asian`

object | `Lookback`

object

Instrument object, specified as a scalar or vector of previously created instrument
objects. Create the instrument objects using `fininstrument`

. The following
instrument objects are supported:

**Data Types: **`object`

`inpSensitivity`

— List of sensitivities to compute

`[ ]`

(default) | string array with values `"Price"`

, `"Delta"`

,
`"Gamma"`

, `"Vega"`

, `"Theta"`

,
`"Rho"`

, `"Lambda"`

, and
`"All"`

| cell array of character vectors with values `'Price'`

,
`'Delta'`

, `'Gamma'`

, `'Vega'`

,
`'Theta'`

, `'Rho'`

, `'Lambda'`

, and
`'All'`

(Optional) List of sensitivities to compute, specified as an
`NOUT`

-by-`1`

or a
`1`

-by-`NOUT`

cell array of character vectors or
string array with possible values of `'Price'`

,
`'Delta'`

, `'Gamma'`

, `'Vega'`

,
`'Theta'`

, `'Rho'`

, `'Lambda'`

, and
`'All'`

.

`inpSensitivity = {'All'}`

or ```
inpSensitivity =
["All"]
```

specifies that the output is `'Delta'`

,
`'Gamma'`

, `'Vega'`

, `'Theta'`

,
`'Rho'`

, `'Lambda'`

, and `'Price'`

.
Using this syntax is the same as specifying `inpSensitivity`

to include
each sensitivity.

inpInstrument | Supported Sensitivities |
---|---|

`Asian` | `{'delta','gamma','vega','theta','rho','lambda','price'}` |

`Barrier` | `{'delta','gamma','vega','theta','rho','lambda','price'}` |

`Lookback` | `{'delta','gamma','vega','theta','rho','lambda','price'}` |

`Vanilla` | `{'delta','gamma','vega','theta','rho','lambda','price'}` |

**Note**

Sensitivities are calculated based on yield shifts of 1 basis point, where the ShiftValue = 1/10000. All sensitivities are returned as dollar sensitivities. To find the per-dollar sensitivities, divide the sensitivities by their respective instrument price.

**Example: **```
inpSensitivity =
{'delta','gamma','vega','price'}
```

**Data Types: **`string`

| `cell`

## Output Arguments

`Price`

— Instrument price

numeric

Instrument price, returned as a numeric.

`PriceResult`

— Price result

`PriceResult`

object

Price result, returned as a `PriceResult`

object. The object has
the following fields:

`PriceResult.Results`

— Table of results that includes sensitivities (if you specify`inpSensitivity`

)`PriceResult.PricerData`

— Structure for pricer data that depends on the instrument that is being priced`Asian`

and`Lookback`

have an empty (`[]`

)`PricerData`

field because the pricing functions for these instruments cannot unambiguously assign a price to any node but the root node.`Vanilla`

and`Barrier`

have the following shared fields for`PriceResult.PricerData.PriceTree`

:`PTree`

contains the clean prices.`ExTree`

contains the exercise indicator arrays. Each element of the cell array is an array where`1`

indicates that an option is exercised and`0`

indicates that an option is not exercised.`dObs`

contains the date of each level of the tree.`tObs`

contains the observation times.`Probs`

contains the probability arrays. Each element of the cell array contains the up, middle, and down transition probabilities for each node of the level.

## More About

### Delta

A *delta* sensitivity measures the rate at which
the price of an option is expected to change relative to a $1 change in the price of the
underlying asset.

Delta is not a static measure; it changes as the price of the underlying asset changes (a concept known as gamma sensitivity), and as time passes. Options that are near the money or have longer until expiration are more sensitive to changes in delta.

### Gamma

A *gamma* sensitivity measures the rate of change
of an option's delta in response to a change in the price of the underlying asset.

In other words, while delta tells you how much the price of an option might move, gamma tells you how fast the option's delta itself will change as the price of the underlying asset moves. This is important because this helps you understand the convexity of an option's value in relation to the underlying asset's price.

### Vega

A *vega* sensitivity measures the sensitivity of
an option's price to changes in the volatility of the underlying asset.

Vega represents the amount by which the price of an option would be expected to change for a 1% change in the implied volatility of the underlying asset. Vega is expressed as the amount of money per underlying share that the option's value will gain or lose as volatility rises or falls.

### Theta

A *theta* sensitivity measures the rate at which
the price of an option decreases as time passes, all else being equal.

Theta is essentially a quantification of time decay, which is a key concept in options pricing. Theta provides an estimate of the dollar amount that an option's price would decrease each day, assuming no movement in the price of the underlying asset and no change in volatility.

### Rho

A *rho* sensitivity measures the rate at which the
price of an option is expected to change in response to a change in the risk-free interest
rate.

Rho is expressed as the amount of money an option's price would gain or lose for a one percentage point (1%) change in the risk-free interest rate.

### Lambda

A *lambda* sensitivity measures the percentage
change in an option's price for a 1% change in the price of the underlying asset.

Lambda is a measure of leverage, indicating how much more sensitive an option is to price movements in the underlying asset compared to owning the asset outright.

## Version History

**Introduced in R2021a**

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