optSensByMertonFD
Option price and sensitivities by Merton76 model using finite differences
Syntax
Description
[
computes a vanilla European or American option price and sensitivities by the Merton76
model, using the Crank-Nicolson Adams-Bashforth (CNAB) IMEX method.PriceSens
,PriceGrid
,AssetPrices
,Times
] = optSensByMertonFD(Rate
,AssetPrice
,Settle
,ExerciseDates
,OptSpec
,Strike
,Sigma
,MeanJ
,JumpVol
,JumpFreq
)
Note
Alternatively, you can use the Vanilla
object to calculate
price or sensitivities for vanilla options. For more information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.
[
specifies options using one or more name-value pair arguments in addition to the input
arguments in the previous syntax. PriceSens
,PriceGrid
,AssetPrices
,Times
] = optSensByMertonFD(___,Name,Value
)
Examples
Compute Price and Sensitivities for an American Option Using the Merton Model
Define the option variables and Merton model parameters.
AssetPrice = 90; Strike = 100; Rate = 0.06; DividendYield = 0.1; Settle = datetime(2018,1,1); ExerciseDates = datetime(2018,4,2); Sigma = 0.40; MeanJ = -0.10; JumpVol = 0.01; JumpFreq = 1.00;
Compute the American call option price and sensitivities using the finite differences method.
OptSpec = 'Call'; [Price, Delta, Gamma, Rho, Theta, Vega] = optSensByMertonFD(Rate, AssetPrice, Settle, ExerciseDates, OptSpec, Strike,... Sigma, MeanJ, JumpVol, JumpFreq, 'DividendYield', DividendYield, 'AmericanOpt', 1,... 'OutSpec', ["Price" "Delta" "Gamma" "Rho" "Theta" "Vega"])
Price = 3.4551
Delta = 0.3211
Gamma = 0.0195
Rho = 5.6610
Theta = -11.9877
Vega = 15.5156
Input Arguments
Rate
— Continuously compounded risk-free interest rate
decimal
Continuously compounded risk-free interest rate, specified as a scalar decimal value.
Data Types: double
AssetPrice
— Current underlying asset price
numeric
Current underlying asset price, specified as a scalar numeric.
Data Types: double
Settle
— Option settlement date
datetime scalar | string scalar | date character vector
Option settlement date, specified as a scalar datetime, string, or date character vector.
To support existing code, optSensByMertonFD
also
accepts serial date numbers as inputs, but they are not recommended.
ExerciseDates
— Option exercise dates
datetime array | string array | date character vector
Option exercise dates, specified as a datetime array, string array, or date character vectors:
For a European option, use a scalar date. For a European option,
ExerciseDates
contains only one value: the option expiry date.For an American option, use a
1
-by-2
vector of dates 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-NaN
date is listed, then the option can be exercised betweenSettle
date and the single listed value inExerciseDates
.To support existing code,
optSensByMertonFD
also accepts serial date numbers as inputs, but they are not recommended.
OptSpec
— Definition of option
character vector with value of 'call'
or
'put'
| string array with value of "call"
or
"put"
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
Strike
— Option strike price value
scalar numeric
Option strike price value, specified as a scalar numeric.
Data Types: double
Sigma
— Volatility of underlying asset
scalar numeric
Volatility of the underling asset, specified as a scalar numeric.
Data Types: double
MeanJ
— Mean of the random percentage jump size
scalar decimal
Mean of the random percentage jump size (J), specified as a
scalar decimal value where log
(1+J) is normally
distributed with the mean
(log
(1+MeanJ
)-0.5*JumpVol
^2)
and the standard deviation JumpVol
.
Data Types: double
JumpVol
— Standard deviation of log
(1+J)
scalar decimal
Standard deviation of log
(1+J) where
J
is the random percentage jump size, specified as a scalar
decimal.
Data Types: double
JumpFreq
— Annual frequency of Poisson jump process
scalar numeric
Annual frequency of the Poisson jump process, specified as a scalar numeric.
Data Types: double
Name-Value Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: [Price,PriceGrid] =
optByMertonFD(Rate,AssetPrice,Settle,ExerciseDates,OptSpec,Strike,MeanJ,JumpVol,JumpFreq,'Basis',7,'OutSpec','delta')
Basis
— Day-count basis of instrument
0
(default) | numeric values: 0
,1
, 2
,
3
, 4
, 6
,
7
, 8
, 9
,
10
, 11
, 12
,
13
Day-count basis of the instrument, specified as the comma-separated pair
consisting of 'Basis'
and a scalar using a supported value:
0 = actual/actual
1 = 30/360 (SIA)
2 = actual/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 = actual/360 (ICMA)
10 = actual/365 (ICMA)
11 = 30/360E (ICMA)
12 = actual/365 (ISDA)
13 = BUS/252
For more information, see Basis.
Data Types: double
DividendYield
— Continuously compounded underlying asset yield
0
(default) | scalar numeric
Continuously compounded underlying asset yield, specified as the comma-separated
pair consisting of 'DividendYield'
and a scalar numeric.
Note
If you enter a value for DividendYield
, then set
DividendAmounts
and ExDividendDates
=
[ ]
or do not enter them. If you enter values for
DividendAmounts
and ExDividendDates
,
then set DividendYield
= 0
.
Data Types: double
DividendAmounts
— Cash dividend amounts
[ ]
(default) | vector
Cash dividend amounts, specified as the comma-separated pair consisting of
'DividendAmounts'
and an
NDIV
-by-1
vector.
Note
Each dividend amount must have a corresponding ex-dividend date. If you enter
values for DividendAmounts
and
ExDividendDates
, then set
DividendYield
= 0
.
Data Types: double
ExDividendDates
— Ex-dividend dates
[]
(default) | datetime array | string array | date character vector
Ex-dividend dates, specified as the comma-separated pair consisting of
'ExDividendDates'
and an
NDIV
-by-1
vector using a datetime array,
string array, or date character vectors.
To support existing code, optSensByMertonFD
also
accepts serial date numbers as inputs, but they are not recommended.
AssetPriceMax
— Maximum price for price grid boundary
if unspecified, value is calculated based on asset price
distribution at maturity (default) | positive scalar numeric
Maximum price for the price grid boundary, specified as the comma-separated pair
consisting of 'AssetPriceMax'
and a positive scalar numeric.
Data Types: double
AssetGridSize
— Size of asset grid for the finite difference grid
400
(default) | scalar numeric
Size of the asset grid for finite difference grid, specified as the
comma-separated pair consisting of 'AssetGridSize'
and a scalar
numeric.
Data Types: double
TimeGridSize
— Number of nodes of time grid for the finite difference grid
100
(default) | positive numeric scalar
Number of nodes of the time grid for finite difference grid, specified as the
comma-separated pair consisting of 'TimeGridSize'
and a positive
numeric scalar.
Data Types: double
AmericanOpt
— Option type
0
(European) (default) | scalar with value of [0,1]
Option type, specified as the comma-separated pair consisting of
'AmericanOpt'
and a scalar flag with one of these values:
0
— European1
— American
Data Types: double
OutSpec
— Define outputs
['price']
(default) | cell array of character vectors with values 'price'
,
'delta'
, 'gamma'
, 'vega'
,
'rho'
, and 'theta'
| string array with values "price"
, "delta"
,
"gamma"
, "vega"
, "rho"
, and
"theta"
Define outputs, specified as the comma-separated pair consisting of
'OutSpec'
and a NOUT
- by-1
or a 1
-by-NOUT
string array or cell array of
character vectors with supported values.
Example: OutSpec =
['price','delta','gamma','vega','rho','theta']
Data Types: string
| cell
Output Arguments
PriceSens
— Option price or sensitivities
numeric
Option price or sensitivities, returned as a numeric. The name-value pair argument
OutSpec
determines the types and order of the outputs.
PriceGrid
— Grid containing prices calculated by the finite difference method
grid
Grid containing prices calculated by the finite difference method, returned as a
two-dimensional grid with size AssetGridSize
⨉
TimeGridSize
. The number of columns is not necessarily equal to
the TimeGridSize
because exercise and ex-dividend dates are added
to the time grid. PriceGrid(:, :, end)
contains the price for
t = 0
.
AssetPrices
— Prices of the asset
vector
Prices of the asset corresponding to the first dimension of
PriceGrid
, returned as a vector.
Times
— Times
vector
Times corresponding to the second dimension of PriceGrid
,
returned as a vector.
More About
Vanilla Option
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.
For more information, see Vanilla Option.
Merton Jump Diffusion Model
The Merton jump diffusion model [2] extends the Black-Scholes model by using the Poisson process to include jump diffusion parameters in the modeling of sudden asset price movements (both up and down).
The stochastic differential equation is
where:
r is the continuous risk-free rate.
q is the continuous dividend yield.
Wt is the Wiener process.
J is the random percentage jump size conditional on the jump
occurring, where ln
(1+J) is normally distributed with
mean and the standard deviation δ, and (1+J) has a lognormal distribution:
where:
μJ is the mean of J for (μJ > -1).
δ is the standard deviation of
ln
(1+J) for (δ≥ 0).
ƛp is the annual frequency (intensity) of Poisson process Ptfor (ƛp ≥ 0).
σ is the volatility of the asset price for (σ > 0).
References
[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.
Version History
Introduced in R2019aR2022b: Serial date numbers not recommended
Although optSensByMertonFD
supports serial date numbers,
datetime
values are recommended instead. The
datetime
data type provides flexible date and time
formats, storage out to nanosecond precision, and properties to account for time
zones and daylight saving time.
To convert serial date numbers or text to datetime
values, use the datetime
function. For example:
t = datetime(738427.656845093,"ConvertFrom","datenum"); y = year(t)
y = 2021
There are no plans to remove support for serial date number inputs.
See Also
optByLocalVolFD
| optSensByLocalVolFD
| optByHestonFD
| optSensByHestonFD
| optByBatesFD
| optSensByBatesFD
| optByMertonFD
| Vanilla
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