optSensByHestonFD
Option price and sensitivities by Heston model using finite differences
Syntax
Description
[
computes a vanilla European or American option price and sensitivities by the Heston model,
using the alternating direction implicit (ADI) method.PriceSens
,PriceGrid
,AssetPrices
,Variances
,Times
] = optByHestonFD(Rate
,AssetPrice
,Settle
,ExerciseDates
,OptSpec
,Strike
,V0
,ThetaV
,Kappa
,SigmaV
,RhoSV
)
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
,Variances
,Times
] = optByHestonFD(___,Name,Value
)
Examples
Compute an American Option Price and Sensitivities Using the Heston Model
Define the option variables and Heston model parameters.
AssetPrice = 10; Strike = 10; Rate = 0.1; Settle = datetime(2017,1,1); ExerciseDates = datetime(2017,4,2); V0 = 0.0625; ThetaV = 0.16; Kappa = 5.0; SigmaV = 0.9; RhoSV = 0.1;
Compute the American put option price and sensitivities.
OptSpec = 'Put'; [Price,Delta,Gamma,Rho,Theta,Vega,VegaLT] = optSensByHestonFD(Rate, AssetPrice, Settle, ExerciseDates, ... OptSpec, Strike, V0, ThetaV, Kappa, SigmaV, RhoSV, 'AmericanOpt', 1, ... 'OutSpec', ["Price" "Delta" "Gamma" "Rho" "Theta" "Vega" "VegaLT"])
Price = 0.5188
Delta = -0.4472
Gamma = 0.2822
Rho = -0.9234
Theta = -1.1614
Vega = 0.8998
VegaLT = 1.0921
Input Arguments
Rate
— Continuously compounded risk-free interest rate
scalar decimal
Continuously compounded risk-free interest rate, specified as a scalar decimal.
Data Types: double
AssetPrice
— Current underlying asset price
scalar 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, optSensByHestonFD
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, there is only one
ExerciseDates
value and this is the option expiry date.For an American option, use a
1
-by-2
vector of exercise date boundaries. The option can be exercised on any tree date between or including the pair of dates on that row. If only one non-NaN
date is listed, the option can be exercised between theSettle
date and the single listedExerciseDate
.
To support existing code, optSensByHestonFD
also
accepts serial date numbers as inputs, but they are not recommended.
OptSpec
— Definition of option
cell array of character vector with values 'call'
or
'put'
| string array with values "call"
or
"put"
Definition of the option, specified as a scalar using a cell array of character
vectors or string arrays with values '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
V0
— Initial variance of underlying asset
scalar numeric
Initial variance of the underlying asset, specified as a scalar numeric.
Data Types: double
ThetaV
— Long-term variance of underlying asset
scalar numeric
Long-term variance of the underlying asset, specified as a scalar numeric.
Data Types: double
Kappa
— Mean revision speed for the variance of underlying asset
scalar numeric
Mean revision speed for the variance of the underlying asset, specified as a scalar numeric.
Data Types: double
SigmaV
— Volatility of the variance of underlying asset
scalar numeric
Volatility of the variance of the underlying asset, specified as a scalar numeric.
Data Types: double
RhoSV
— Correlation between Wiener processes for underlying asset and its variance
scalar numeric
Correlation between the Wiener processes for the underlying asset and its variance, 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: [PriceSens,PriceGrid,AssetPrices,Variances,Times] =
optSensByHestonFD(Rate,AssetPrice,Settle,ExerciseDates,OptSpec,Strike,V0,ThetaV,Kappa,SigmaV,RhoSV,'Basis',7)
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 a
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 a
NDIV
-by-1
vector using a datetime array,
string array, or date character vectors.
To support existing code, optSensByHestonFD
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
Maximum price for price grid boundary, specified as the comma-separated pair
consisting of 'AssetPriceMax'
and a positive scalar.
Data Types: single
| double
VarianceMax
— Maximum variance to use for variance grid boundary
1.0
(default) | scalar numeric
Maximum variance to use for variance grid boundary, specified as the
comma-separated pair consisting of 'VarianceMax'
as a scalar
numeric.
Data Types: double
AssetGridSize
— Size of asset grid for 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
VarianceGridSize
— Number of nodes for variance grid for finite difference grid
200
(default) | scalar numeric
Number of nodes for the variance grid for finite difference grid, specified as the
comma-separated pair consisting of 'VarianceGridSize'
and a scalar
numeric.
Data Types: double
TimeGridSize
— Number of nodes of time grid for 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 values [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'
, 'theta'
, and
'vegalt'
| string array with values "price"
, "delta"
,
"gamma"
, "vega"
, "rho"
,
"theta"
, and "vegalt"
Define outputs, specified as the comma-separated pair consisting of
'OutSpec'
and an NOUT
- by-1
or a 1
-by-NOUT
string array or cell array of
character vectors with the supported values.
Note
'vega'
is the sensitivity with respect to the initial
volatility sqrt(V0
). In contrast, 'vegalt'
is the sensitivity with respect to the long-term volatility
sqrt(ThetaV
).
Example: OutSpec =
{'price','delta','gamma','vega','rho','theta','vegalt'}
Data Types: string
| cell
Output Arguments
PriceSens
— Option price and sensitivities
scalar numeric
Option price and sensitivities, returned as a scalar numeric.
OutSpec
determines the types and order of the outputs.
PriceGrid
— Grid containing prices calculated by the finite difference method
numeric
Grid containing prices calculated by the finite difference method, returned as a
three-dimensional grid with size AssetGridSize
⨉
VarianceGridSize
⨉ TimeGridSize
. The depth
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.
Variances
— Variances
vector
Variances corresponding to the second dimension of PriceGrid
,
returned as a vector.
Times
— Times
vector
Times corresponding to the third 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.
Heston Stochastic Volatility Model
The Heston model is an extension of the Black-Scholes model, where the volatility (square root of variance) is no longer assumed to be constant, and the variance now follows a stochastic (CIR) process. This allows modeling the implied volatility smiles observed in the market.
The stochastic differential equation is:
where
r is the continuous risk-free rate.
q is the continuous dividend yield.
St is the asset price at time t.
vt is the asset price variance at time t
v0 is the initial variance of the asset price at t = 0 for (v0 > 0).
θ is the long-term variance level for (θ > 0).
κ is the mean reversion speed for the variance for (κ > 0).
σv is the volatility of the variance for (σv > 0).
p is the correlation between the Wiener processes Wt and Wvt for (-1 ≤ p ≤ 1).
References
[1] Heston, S. L. “A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options.” The Review of Financial Studies. Vol 6, Number 2, 1993.
Version History
Introduced in R2018bR2022b: Serial date numbers not recommended
Although optSensByHestonFD
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
optstockbyfd
| optstocksensbyfd
| optByHestonFD
| optByLocalVolFD
| optSensByLocalVolFD
| optByBatesFD
| optSensByBatesFD
| optByMertonFD
| optSensByMertonFD
| Vanilla
MATLAB Command
You clicked a link that corresponds to this MATLAB command:
Run the command by entering it in the MATLAB Command Window. Web browsers do not support MATLAB commands.
Select a Web Site
Choose a web site to get translated content where available and see local events and offers. Based on your location, we recommend that you select: .
You can also select a web site from the following list:
How to Get Best Site Performance
Select the China site (in Chinese or English) for best site performance. Other MathWorks country sites are not optimized for visits from your location.
Americas
- América Latina (Español)
- Canada (English)
- United States (English)
Europe
- Belgium (English)
- Denmark (English)
- Deutschland (Deutsch)
- España (Español)
- Finland (English)
- France (Français)
- Ireland (English)
- Italia (Italiano)
- Luxembourg (English)
- Netherlands (English)
- Norway (English)
- Österreich (Deutsch)
- Portugal (English)
- Sweden (English)
- Switzerland
- United Kingdom (English)