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optByMertonFD

Option price by Merton76 model using finite differences

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Syntax

[Price,PriceGrid,AssetPrices,Times] = optByMertonFD(Rate,AssetPrice,Settle,ExerciseDates,OptSpec,Strike,Sigma,MeanJ,JumpVol,JumpFreq)
[Price,PriceGrid,AssetPrices,Times] = optByMertonFD(___，名称，值）

Description

example

[Price,PriceGrid,资产优势,时代] = optByMertonFD(Rate,AssetPrice,Settle,ExerciseDates,OptSpec,罢工,Sigma,MeanJ,JumpVol,JumpFreq)computes a vanilla European or American option price by the Merton76 model, using the Crank-Nicolson Adams-Bashforth (CNAB) IMEX method.

example

[Price,PriceGrid,资产优势,时代] = optByMertonFD(___,Name,Value)specifies options using one or more name-value pair arguments in addition to the input arguments in the previous syntax.

Examples

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Open Live Script

Define the option variables and Merton model parameters.

AssetPrice = 90; Strike = 100; Rate = 0.06; DividendYield = 0.1; Settle ='01 -Jan-2018'; ExerciseDates ='02-Apr-2018'; Sigma = 0.40; MeanJ = -0.10; JumpVol = 0.01; JumpFreq = 1.00;

Compute the American call option price using the finite differences method.

OptSpec ='称呼'; Price = optSensByMertonFD(Rate, AssetPrice, Settle, ExerciseDates, OptSpec, Strike,...Sigma, MeanJ, JumpVol, JumpFreq,'DividendYield', DividendYield,'AmericanOpt', 1)
Price = 3.4551

Input Arguments

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Continuously compounded risk-free interest rate, specified as a scalar decimal value.

Data Types:double

Current underlying asset price, specified as a scalar numeric.

Data Types:double

Option settlement date, specified as a scalar using a serial date number, date character vector, datetime object, or string scalar.

Data Types:double|char|datetime|string

Option exercise dates, specified as a serial date number, date character vector, datetime array, or string array:

  • For a European option, use a scalar serial date number, date character vector, datetime object, or string scalar. For a European option,ExerciseDatescontains only one value: the option expiry date.

  • For an American option, use a1-by-2vector of serial date numbers, date character vectors, datetime arrays, or string arrays to specify the exercise date boundaries. An American option can be exercised on any date between or including the pair of dates. If only one non-NaNdate is listed, then the option can be exercised betweenSettledate and the single listed value inExerciseDates.

Data Types:double|char|datetime|string

Definition of the option, specified as a scalar using a character vector or string array with a value of'call'or'put'.

Data Types:cell|string

Option strike price value, specified as a scalar numeric.

Data Types:double

Volatility of the underling asset, specified as a scalar numeric.

Data Types:double

的意思是the random percentage jump size (J），指定为标量小数值log(1+J) is normally distributed with the mean (log(1+MeanJ)-0.5*JumpVol^2）和标准偏差JumpVol.

Data Types:double

Standard deviation oflog(1+J) whereJ是随机百分比跳跃大小，指定为标量小数。

Data Types:double

Annual frequency of the Poisson jump process, specified as a scalar numeric.

Data Types:double

Name-Value Pair Arguments

Specify optional comma-separated pairs ofName,Valuearguments.Nameis the argument name andValueis the corresponding value.Namemust appear inside quotes. You can specify several name and value pair arguments in any order asNAME1，Value1，...，Namen，Valuen.

Example:[Price，PriceGrid] = OptbyMertonFD（速率，AssetPrice，Setter，Settook，Optspec，Strike，sighj，sunj，umgvol，Jumpfreq，“基础”，7）

Day-count basis of the instrument, specified as the comma-separated pair consisting of'基础'and a scalar using a supported value:

  • 0 = actual/actual

  • 1 = 30/360 (SIA)

  • 2 =实际/360

  • 3 = actual/365

  • 4 = 30/360（PSA）

  • 5 = 30/360 (ISDA)

  • 6 = 30/360 (European)

  • 7 = actual/365 (Japanese)

  • 8 = actual/actual (ICMA)

  • 9 =实际/360（ICMA）

  • 10 = actual/365 (ICMA)

  • 11 = 30/360E (ICMA)

  • 12 = actual/365 (ISDA)

  • 13 = BUS/252

For more information, see基础.

Data Types:double

连续复合的基本资产产量，指定为逗号分隔对'DividendYield'and a scalar numeric.

Note

If you enter a value forratidendyield, then setDividendAmountsandExdividendEddates=[ ]or do not enter them. If you enter values forDividendAmountsandExdividendEddates, then setratidendyield=0.

Data Types:double

Cash dividend amounts, specified as the comma-separated pair consisting of“DividendAmounts”and anNDIV-by-1vector.

Note

Each dividend amount must have a corresponding ex-dividend date. If you enter values forDividendAmountsandExdividendEddates, then setratidendyield=0.

Data Types:double

Ex-dividend dates, specified as the comma-separated pair consisting of“ exdividendeddates”and anNDIV-by-1vector of serial date numbers, date character vectors, string arrays, or datetime arrays.

Data Types:double|char|string|datetime

Maximum price for the price grid boundary, specified as the comma-separated pair consisting of'AssetPriceMax'and a positive scalar numeric.

Data Types:double

Size of the asset grid for the finite difference grid, specified as the comma-separated pair consisting of'AssetGridSize'and a scalar numeric.

Data Types:double

Number of nodes of the time grid for the finite difference grid, specified as the comma-separated pair consisting of'TimeGridSize'and a positive numeric scalar.

Data Types:double

Option type, specified as the comma-separated pair consisting of'AmericanOpt'and a scalar flag with one of these values:

  • 0— European

  • 1- 美国人

Data Types:double

Output Arguments

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Option price, returned as a scalar numeric.

Grid containing prices calculated by the finite difference method, returned as a two-dimensional grid with sizeAssetGridSize⨉TimeGridSize. The number of columns is not necessarily equal to theTimeGridSizebecause exercise and ex-dividend dates are added to the time grid.PriceGrid(:, :, end)包含价格t=0.

Prices of the asset corresponding to the first dimension ofPriceGrid, returned as a vector.

时代corresponding to the second dimension ofPriceGrid, returned as a vector.

More About

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Vanilla Option

Avanilla option是一个类别的选项,包括只有莫st 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: max ( S t − K , 0 )

  • For a put: max ( K − S t , 0 )

where:

Stis the price of the underlying asset at timet.

K是罢工价格。

For more information, seeVanilla Option.

Merton Jump Diffusion Model

默顿跳跃扩散模型[2]通过使用泊松过程来扩展黑色 - choles模型，以在突然资产价格移动的建模中包括跳跃扩散参数（上下）。

The stochastic differential equation is

d S t = ( r − q − λ p μ j ) S t d t + σ S t d W t + J S t d P t prob( d P t = 1 ) = λ p d t

where:

ris the continuous risk-free rate.

qis the continuous dividend yield.

Wtis the Weiner process.

Jis the random percentage jump size conditional on the jump occurring, whereln(1+J）正态分布 ln ( 1 + μ J ) − δ 2 2 and the standard deviation δ, and (1+J) has a lognormal distribution:

1 ( 1 + J ) δ 2 π exp { − [ ln ( 1 + J ) − ( ln ( 1 + μ J ) − δ 2 2 ] 2 δ 2 2 }

where:

μJis the mean ofJfor (μJ> -1）。

δis the standard deviation ofln(1+J) for (δ≥ 0).

ƛpis the annual frequency (intensity) of Poisson processPtfor (ƛp≥ 0).

σis the volatility of the asset price for (σ> 0).

参考

[1] Cont, R., and E. Voltchkova. “A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Lévy Models.”SIAM Journal on Numerical Analysis.Vol. 43, Number 4, 2005, pp. 1596-1626.

[2] Merton, R. "Option Pricing When Underlying Stock Returns Are Discontinuous."The Journal of Financial Economics.Vol 3. 1976, pp. 125-144.

See Also

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