optSensByMertonFFT
Option price and sensitivities by Merton76 model using FFT and FRFT
Syntax
Description
[
computes vanilla European option price and sensitivities by Merton76 model, using Carr-Madan
FFT and Chourdakis FRFT methods.PriceSens
,StrikeOut
] = optSensByMertonFFT(Rate
,AssetPrice
,Settle
,Maturity
,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.
[
adds optional name-value pair arguments. PriceSens
,StrikeOut
] = optSensByMertonFFT(___,Name,Value
)
Examples
Workflow for Plotting an Option Sensitivity Surface Using the Merton76 Model
Use optSensByMertonFFT
to calibrate the FFT strike grid for sensitivities, compute option sensitivities, and plot option sensitivity surfaces.
Define Option Variables and Merton76 Model Parameters
AssetPrice = 80;
Rate = 0.03;
DividendYield = 0.02;
OptSpec = 'call';
Sigma = 0.16;
MeanJ = 0.02;
JumpVol = 0.08;
JumpFreq = 2;
Compute the Option Prices for the Entire FFT (or FRFT) Strike Grid, Without Specifying "Strike"
Compute option sensitivities and also output the corresponding strikes. If the Strike
input is empty ( []
), option sensitivities will be computed on the entire FFT (or FRFT) strike grid. The FFT (or FRFT) strike grid is determined as exp(log-strike grid)
, where each column of the log-strike grid has NumFFT
points with LogStrikeStep
spacing that are roughly centered around each element of log(AssetPrice)
. The default value for NumFFT
is 2^12. In addition to the sensitivities in the first output, the optional last output contains the corresponding strikes.
Settle = datetime(2017,6,29); Maturity = datemnth(Settle, 6); Strike = []; % Strike is not specified [Delta, Kout] = optSensByMertonFFT(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ... Sigma, MeanJ, JumpVol, JumpFreq, 'DividendYield', DividendYield, 'OutSpec', "delta"); % Show the lowest and highest strike values on the FFT strike grid format [Kout(1) Kout(end)]
ans = 1×2
10138 ×
0.0000 1.8798
% Show a subset of the strikes and corresponding option sensitivities
Range = (2046:2052);
[Kout(Range) Delta(Range)]
ans = 7×2
50.4929 0.9895
58.8640 0.9801
68.6231 0.8816
80.0000 0.5283
93.2631 0.1551
108.7251 0.0241
126.7505 0.0025
Change the Number of FFT (or FRFT) Points and Compare with optSensByMertonNI
Try a different number of FFT (or FRFT) points, and compare the results with numerical integration. Unlike optSensByMertonFFT
, which uses FFT (or FRFT) techniques for fast computation across the whole range of strikes, the optSensByMertonNI
function uses direct numerical integration and it is typically slower, especially for multiple strikes. However, the values computed by optSensByMertonNI
can serve as a benchmark for adjusting the settings for optSensByMertonFFT
.
% Try a smaller number of FFT points % (e.g. for faster performance or smaller memory footprint) NumFFT = 2^10; % Smaller than the default value of 2^12 Strike = []; % Strike is not specified (will use the entire FFT strike grid) [Delta, Kout] = optSensByMertonFFT(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ... Sigma, MeanJ, JumpVol, JumpFreq, 'DividendYield', DividendYield, 'OutSpec', "delta", ... 'NumFFT', NumFFT); % Compare with numerical integration method Range = (510:516); Strike = Kout(Range); DeltaFFT = Delta(Range); DeltaNI = optSensByMertonNI(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ... Sigma, MeanJ, JumpVol, JumpFreq, 'DividendYield', DividendYield, 'OutSpec', "delta"); Error = abs(DeltaFFT-DeltaNI); table(Strike, DeltaFFT, DeltaNI, Error)
ans=7×4 table
Strike DeltaFFT DeltaNI Error
______ __________ __________ __________
12.696 0.89726 0.99002 0.092766
23.449 0.93421 0.99002 0.05581
43.312 0.94691 0.99001 0.043093
80 0.50983 0.52827 0.018446
147.76 0.004147 0.00019101 0.003956
272.93 0.001071 1.547e-09 0.001071
504.11 0.00030521 5.7578e-10 0.00030521
Make Further Adjustments to FFT (or FRFT)
If the values in the output DeltaFFT
are significantly different from those in DeltaNI
, try making adjustments to optSensByMertonFFT
settings, such as CharacteristicFcnStep
, LogStrikeStep
, NumFFT
, DampingFactor
, and so on. Note that if (LogStrikeStep
* CharacteristicFcnStep
) is 2*pi/ NumFFT
, FFT is used. Otherwise, FRFT is used.
Strike = []; % Strike is not specified (will use the entire FFT or FRFT strike grid) [Delta, Kout] = optSensByMertonFFT(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ... Sigma, MeanJ, JumpVol, JumpFreq, 'DividendYield', DividendYield, 'OutSpec', "delta", ... 'NumFFT', NumFFT, 'CharacteristicFcnStep', 0.065, 'LogStrikeStep', 0.001); % Compare with numerical integration method Strike = Kout(Range); DeltaFFT = Delta(Range); DeltaNI = optSensByMertonNI(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ... Sigma, MeanJ, JumpVol, JumpFreq, 'DividendYield', DividendYield, 'OutSpec', "delta"); Error = abs(DeltaFFT-DeltaNI); table(Strike, DeltaFFT, DeltaNI, Error)
ans=7×4 table
Strike DeltaFFT DeltaNI Error
______ ________ _______ __________
79.76 0.53701 0.53701 5.6407e-12
79.84 0.5341 0.5341 5.3257e-12
79.92 0.53119 0.53119 5.0099e-12
80 0.52827 0.52827 4.6956e-12
80.08 0.52536 0.52536 4.3811e-12
80.16 0.52245 0.52245 4.0652e-12
80.24 0.51953 0.51953 3.7503e-12
% Save the final FFT (or FRFT) strike grid for future reference. For % example, it provides information about the range of Strike inputs for % which the FFT (or FRFT) operation is valid. FFTStrikeGrid = Kout; MinStrike = FFTStrikeGrid(1) % Strike cannot be less than MinStrike
MinStrike = 47.9437
MaxStrike = FFTStrikeGrid(end) % Strike cannot be greater than MaxStrike
MaxStrike = 133.3566
Compute the Option Sensitivity for a Single Strike
Once the desired FFT (or FRFT) settings are determined, use the Strike
input to specify the strikes rather than providing an empty array. If the specified strikes do not match a value on the FFT (or FRFT) strike grid, the outputs are interpolated on the specified strikes.
Settle = datetime(2017,6,29); Maturity = datemnth(Settle, 6); Strike = 80; Delta = optSensByMertonFFT(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ... Sigma, MeanJ, JumpVol, JumpFreq, 'DividendYield', DividendYield, 'OutSpec', "delta", ... 'NumFFT', NumFFT, 'CharacteristicFcnStep', 0.065, 'LogStrikeStep', 0.001)
Delta = 0.5283
Compute the Option Sensitivities for a Vector of Strikes
Use the Strike
input to specify the strikes.
Settle = datetime(2017,6,29); Maturity = datemnth(Settle, 6); Strike = (76:2:84)'; Delta = optSensByMertonFFT(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ... Sigma, MeanJ, JumpVol, JumpFreq, 'DividendYield', DividendYield, 'OutSpec', "delta", ... 'NumFFT', NumFFT, 'CharacteristicFcnStep', 0.065, 'LogStrikeStep', 0.001)
Delta = 5×1
0.6727
0.6013
0.5283
0.4565
0.3883
Compute the Option Sensitivities for a Vector of Strikes and a Vector of Dates of the Same Lengths
Use the Strike
input to specify the strikes. Also, the Maturity
input can be a vector, but it must match the length of the Strike
vector if the ExpandOutput
name-value pair argument is not set to "true"
.
Settle = datetime(2017,6,29); Maturity = datemnth(Settle, [12 18 24 30 36]); % Five maturities Strike = [76 78 80 82 84]'; % Five strikes Delta = optSensByMertonFFT(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ... Sigma, MeanJ, JumpVol, JumpFreq, 'DividendYield', DividendYield, 'OutSpec', "delta", ... 'NumFFT', NumFFT, 'CharacteristicFcnStep', 0.065, ... 'LogStrikeStep', 0.001) % Five values in vector output
Delta = 5×1
0.6419
0.5907
0.5565
0.5311
0.5110
Expand the Outputs for a Surface
Set the ExpandOutput
name-value pair argument to "true"
to expand the outputs into NStrikes
-by-NMaturities
matrices. In this case, they are square matrices.
[Delta, Kout] = optSensByMertonFFT(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ... Sigma, MeanJ, JumpVol, JumpFreq, 'DividendYield', DividendYield, 'OutSpec', "delta", ... 'NumFFT', NumFFT, 'CharacteristicFcnStep', 0.065, ... 'LogStrikeStep', 0.001, 'ExpandOutput', true) % (5 x 5) matrix output
Delta = 5×5
0.6419 0.6305 0.6245 0.6204 0.6173
0.5922 0.5907 0.5905 0.5905 0.5905
0.5422 0.5507 0.5565 0.5607 0.5637
0.4927 0.5112 0.5229 0.5311 0.5372
0.4447 0.4725 0.4898 0.5020 0.5110
Kout = 5×5
76 76 76 76 76
78 78 78 78 78
80 80 80 80 80
82 82 82 82 82
84 84 84 84 84
Compute the Option Sensitivities for a Vector of Strikes and a Vector of Dates of Different Lengths
When ExpandOutput
is "true"
, NStrikes
do not have to match NMaturities
. That is, the output NStrikes
-by-NMaturities
matrix can be rectangular.
Settle = datetime(2017,6,29); Maturity = datemnth(Settle, 12*(0.5:0.5:3)'); % Six maturities Strike = (76:2:84)'; % Five strikes Delta = optSensByMertonFFT(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ... Sigma, MeanJ, JumpVol, JumpFreq, 'DividendYield', DividendYield, 'OutSpec', "delta", ... 'NumFFT', NumFFT, 'CharacteristicFcnStep', 0.065, ... 'LogStrikeStep', 0.001, 'ExpandOutput', true) % (5 x 6) matrix output
Delta = 5×6
0.6727 0.6419 0.6305 0.6245 0.6204 0.6173
0.6013 0.5922 0.5907 0.5905 0.5905 0.5905
0.5283 0.5422 0.5507 0.5565 0.5607 0.5637
0.4565 0.4927 0.5112 0.5229 0.5311 0.5372
0.3883 0.4447 0.4725 0.4898 0.5020 0.5110
Compute the Option Sensitivities for a Vector of Strikes and a Vector of Asset Prices
When ExpandOutput
is "true"
, the output can also be a NStrikes
-by-NAssetPrices
rectangular matrix by accepting a vector of asset prices.
Settle = datetime(2017,6,29); Maturity = datemnth(Settle, 12); % Single maturity ManyAssetPrices = [70 75 80 85]; % Four asset prices Strike = (76:2:84)'; % Five strikes Delta = optSensByMertonFFT(Rate, ManyAssetPrices, Settle, Maturity, OptSpec, Strike, ... Sigma, MeanJ, JumpVol, JumpFreq, 'DividendYield', DividendYield, 'OutSpec', "delta", ... 'NumFFT', NumFFT, 'CharacteristicFcnStep', 0.065, ... 'LogStrikeStep', 0.001, 'ExpandOutput', true) % (5 x 4) matrix output
Delta = 5×4
0.3796 0.5157 0.6419 0.7472
0.3315 0.4637 0.5922 0.7043
0.2874 0.4137 0.5422 0.6592
0.2474 0.3664 0.4927 0.6128
0.2117 0.3224 0.4447 0.5657
Plot Option Sensitivity Surfaces
Use the Strike
input to specify the strikes. Increase the value for NumFFT
to support a wider range of strikes. Also, the Maturity
input can be a vector. Set ExpandOutput
to "true"
to output the surfaces as NStrikes
-by-NMaturities
matrices.
Settle = datetime(2017,6,29); Maturity = datemnth(Settle, 12*[1/12 0.25 (0.5:0.5:3)]'); Times = yearfrac(Settle, Maturity); Strike = (2:2:200)'; % Increase 'NumFFT' to support a wider range of strikes NumFFT = 2^13; [Delta, Gamma, Rho, Theta, Vega] = optSensByMertonFFT(... Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, ... Sigma, MeanJ, JumpVol, JumpFreq, 'DividendYield', DividendYield, ... 'NumFFT', NumFFT, 'CharacteristicFcnStep', 0.065, 'LogStrikeStep', 0.001, ... 'OutSpec', ["delta", "gamma", "rho", "theta", "vega"], ... 'ExpandOutput', true); [X,Y] = meshgrid(Times,Strike); figure; surf(X,Y,Delta); title('Delta'); xlabel('Years to Option Expiry'); ylabel('Strike'); view(-112,34); xlim([0 Times(end)]);
figure; surf(X,Y,Gamma) title('Gamma') xlabel('Years to Option Expiry') ylabel('Strike') view(-112,34); xlim([0 Times(end)]);
figure; surf(X,Y,Rho) title('Rho') xlabel('Years to Option Expiry') ylabel('Strike') view(-112,34); xlim([0 Times(end)]);
figure; surf(X,Y,Theta) title('Theta') xlabel('Years to Option Expiry') ylabel('Strike') view(-112,34); xlim([0 Times(end)]);
figure; surf(X,Y,Vega) title('Vega') xlabel('Years to Option Expiry') ylabel('Strike') view(-112,34); xlim([0 Times(end)]);
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 numeric value using a scalar or a
NINST
-by-1
or
NColumns
-by-1
vector.
For more information on the proper dimensions for AssetPrice
,
see the name-value pair argument ExpandOutput
.
Data Types: double
Settle
— Option settlement date
datetime array | string array | date character vector
Option settlement date, specified as a
NINST
-by-1
or
NColumns
-by-1
vector using a datetime array,
string array, or date character vectors. The Settle
date must be
before the Maturity
date.
To support existing code, optSensByMertonFFT
also
accepts serial date numbers as inputs, but they are not recommended.
For more information on the proper dimensions for Settle
, see
the name-value pair argument ExpandOutput
.
Maturity
— Option maturity date
datetime array | string array | date character vector
Option maturity date, specified as a
NINST
-by-1
or
NColumns
-by-1
vector using a datetime array,
string array, or date character vectors.
To support existing code, optSensByMertonFFT
also
accepts serial date numbers as inputs, but they are not recommended.
For more information on the proper dimensions for Maturity
, see
the name-value pair argument ExpandOutput
.
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
NINST
-by-1
or
NColumns
-by-1
vector using a cell array of
character vectors or string arrays with values 'call'
or
'put'
.
For more information on the proper dimensions for OptSpec
, see
the name-value pair argument ExpandOutput
.
Data Types: cell
| string
Strike
— Option strike price value
numeric
Option strike price value, specified as a
NINST
-by-1
,
NRows
-by-1
,
NRows
-by-NColumns
vector of strike
prices.
If this input is an empty array ([]
), option prices are
computed on the entire FFT (or FRFT) strike grid, which is determined as
exp(log-strike grid)
. Each column of the log-strike grid
has'NumFFT'
points with 'LogStrikeStep'
spacing
that are roughly centered around each element of log(AssetPrice)
.
For more information on the proper dimensions for Strike
, see
the name-value pair argument ExpandOutput
.
Data Types: double
Sigma
— Volatility of underlying asset
numeric
Volatility of the underling asset, specified as a scalar numeric value.
Data Types: double
MeanJ
— Mean of the random percentage jump size
decimal
Mean of the random percentage jump size (J), specified as a
scalar decimal value where log
(1+J) is normally
distributed with mean
(log
(1+MeanJ
)-0.5*JumpVol
^2)
and the standard deviation JumpVol
.
Data Types: double
JumpVol
— Standard deviation of log
(1+J)
decimal
Standard deviation of log
(1+J) where
J
is the random percentage jump size, specified as a scalar
decimal value.
Data Types: double
JumpFreq
— Annual frequency of Poisson jump process
numeric
Annual frequency of Poisson jump process, specified as a scalar numeric value.
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,StrikeOut] =
optSensByMertonFFT(Rate,AssetPrice,Settle,Maturity,OptSpec,Strike,Sigma,MeanJ,JumpVol,JumpFreq,'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 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) | numeric
Continuously compounded underlying asset yield, specified as the comma-separated
pair consisting of 'DividendYield'
and a scalar numeric
value.
Data Types: double
OutSpec
— Define outputs
["price"]
(default) | string array with values "price"
, "delta"
,
"gamma"
, "vega"
, "rho"
, and
"theta"
| cell array of character vectors 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.
Note
"vega"
is the sensitivity with respect the initial
volatility sqrt(V0
).
Example: OutSpec =
["price","delta","gamma","vega","rho","theta"]
Data Types: string
| cell
NumFFT
— Number of grid points in the characteristic function variable
4096
(default) | numeric
Number of grid points in the characteristic function variable and in each column
of the log-strike grid, specified as the comma-separated pair consisting of
'NumFFT'
and a scalar numeric value.
Data Types: double
CharacteristicFcnStep
— Characteristic function variable grid spacing
0.01
(default) | numeric
Characteristic function variable grid spacing, specified as the comma-separated
pair consisting of 'CharacteristicFcnStep'
and a scalar numeric
value.
Data Types: double
LogStrikeStep
— Log-strike grid spacing
2*pi/NumFFT/CharacteristicFcnStep
(default) | numeric
Log-strike grid spacing, specified as the comma-separated pair consisting of
'LogStrikeStep'
and a scalar numeric value.
Note
If
(LogStrikeStep
*CharacteristicFcnStep
) is
2*pi
/NumFFT
, FFT is used. Otherwise,
FRFT is used.
Data Types: double
DampingFactor
— Damping factor for Carr-Madan formulation
1.5
(default) | numeric
Damping factor for Carr-Madan formulation, specified as the comma-separated pair
consisting of 'DampingFactor'
and a scalar numeric value.
Data Types: double
Quadrature
— Type of quadrature
"simpson"
(default) | character vector with values:'simpson'
or
'trapezoidal'
| string array with values: "simpson"
or
"trapezoidal"
Type of quadrature, specified as the comma-separated pair consisting of
'Quadrature'
and a single character vector or string array with a
value of 'simpson'
or 'trapezoidal'
.
Data Types: char
| string
ExpandOutput
— Flag to expand the outputs
false
(outputs are
NINST
-by-1
vectors) (default) | logical with value of true
or false
Flag to expand the outputs, specified as the comma-separated pair consisting of
'ExpandOutput'
and a logical:
true
— Iftrue
, the outputs areNRows
-by-NColumns
matrices.NRows
is the number of strikes for each column and it is determined by theStrike
input. For example,Strike
can be aNRows
-by-1
vector, or aNRows
-by-NColumns
matrix. IfStrike
is empty,NRows
is equal toNumFFT
.NColumns
is determined by the sizes ofAssetPrice
,Settle
,Maturity
, andOptSpec
, which must all be either scalar orNColumns
-by-1
vectors.false
— Iffalse
, the outputs areNINST
-by-1
vectors. Also, the inputsStrike
,AssetPrice
,Settle
,Maturity
, andOptSpec
must all be either scalar orNINST
-by-1
vectors.
Data Types: logical
Output Arguments
PriceSens
— Option prices or sensitivities
numeric
Option prices or sensitivities, returned as a
NINST
-by-1
, or
NRows
-by-NColumns
, depending on
ExpandOutput
. The name-value pair argument
OutSpec
determines the types and order of the outputs.
StrikeOut
— Strikes corresponding to Price
numeric
Strikes corresponding to Price
, returned as a
NINST
-by-1
, or
NRows
-by-NColumns
, depending on
ExpandOutput
.
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 (Merton (1976)) is a different extension of the Black-Scholes model, where sudden asset price movements (both up and down) are modeled by adding the jump diffusion parameters with the Poisson process.
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:
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).
The characteristic function for j = 1 (asset prices measure) and j = 2 (risk-neutral measure) is:
where
ϕ is the characteristic function variable.
τ is the time to maturity (τ = T- t).
i is the unit imaginary number ( i2 = -1).
Carr-Madan Formulation
The Carr and Madan (1999) formulation is a popular modified implementation of Heston (1993) framework.
Rather than computing the probabilities P1 and P2 as intermediate steps, Carr and Madan developed an alternative expression so that taking its inverse Fourier transform gives the option price itself directly.
where
r is the continuous risk-free rate.
q is the continuous dividend yield.
St is the asset price at time t.
τ is time to maturity (τ = T-t).
Call(K) is the call price at strike K.
Put(K) is the put price at strike K.
i is a unit imaginary number (i2= -1).
ϕ is the characteristic function variable.
α is the damping factor.
u is the characteristic function variable for integration, where ϕ = (u - (α+1)i).
f2(ϕ) is the characteristic function for P2.
P2 is the probability of St > K under the risk-neutral measure for the model.
To apply FFT or FRFT to this formulation, the characteristic function variable for
integration, u, is discretized into
NumFFT
(N) point with the step size
CharacteristicFcnStep
(Δu), and the log-strike
k is discretized into N points with the step size
LogStrikeStep
(Δk).
The discretized characteristic function variable for integration, uj(for j = 1,2,3,…,N), has a minimum value of 0 and a maximum value of (N-1) (Δu), and it approximates the continuous integration range from 0 to infinity.
The discretized log-strike grid,
kn(for n =
1, 2, 3, N) is approximately centered around
ln
(St),
with a minimum value of
and a maximum value of
Where the minimum allowable strike is
and the maximum allowable strike is
As a result of the discretization, the expression for the call option becomes
where
Δu is the step size of discretized characteristic function variable for integration.
Δk is the step size of discretized log-strike.
N is the number of FFT or FRFT points.
wj is the weights for quadrature used for approximating the integral.
FFT is used to evaluate the above expression if Δk and Δu are subject to the following constraint:
otherwise, the functions use the FRFT method described in Chourdakis (2005).
References
[1] Bates, D. S. “Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options.” The Review of Financial Studies. Vol 9. No. 1. 1996.
[2] Carr, P., and D.B. Madan. “Option Valuation Using the Fast Fourier Transform.” Journal of Computational Finance. Vol 2. No. 4. 1999.
[3] Cont, R. and P. Tankov. Financial Modeling with Jump Processes. Chapman & Hall/CRC Press, 2004.
[4] Chourdakis, K. “Option Pricing Using Fractional FFT.” Journal of Computational Finance. 2005.
[5] Merton, R. “Option Pricing When Underlying Stock Returns are Discontinuous.” Journal of Financial Economics. Vol 3. 1976.
Version History
Introduced in R2018aR2022b: Serial date numbers not recommended
Although optSensByMertonFFT
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.
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