# optByBatesNI

Option price by Bates model using numerical integration

## Description

example

Price = optByBatesNI(Rate,AssetPrice,Settle,Maturity,OptSpec,Strike,V0,ThetaV,Kappa,SigmaV,RhoSV,MeanJ,JumpVol,JumpFreq) computes vanilla European option price by Bates model, using numerical integration methods.

Note

Alternatively, you can use the Vanilla object to price vanilla options. For more information, see Get Started with Workflows Using Object-Based Framework for Pricing Financial Instruments.

example

Price = optByBatesNI(___,Name,Value) adds optional name-value pair arguments.

## Examples

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optByBatesNI uses numerical integration to compute option prices and then plot an option price surface.

Define Option Variables and Bates Model Parameters

AssetPrice = 80;
Rate = 0.03;
DividendYield = 0.02;
OptSpec = 'call';

V0 = 0.04;
ThetaV = 0.05;
Kappa = 1.0;
SigmaV = 0.2;
RhoSV = -0.7;
MeanJ = 0.02;
JumpVol = 0.08;
JumpFreq = 2;

Compute the Option Price for a Single Strike

Settle = datetime(2017,6,29);
Maturity = datemnth(Settle, 6);
Strike = 80;

Call = optByBatesNI(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, V0, ...
ThetaV, Kappa, SigmaV, RhoSV, MeanJ, JumpVol, JumpFreq, ...
'DividendYield', DividendYield)
Call = 5.3484

Compute the Option Prices for a Vector of Strikes

The Strike input can be a vector.

Settle = datetime(2017,6,29);
Maturity = datemnth(Settle, 6);
Strike = (76:2:84)';

Call = optByBatesNI(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, V0, ...
ThetaV, Kappa, SigmaV, RhoSV, MeanJ, JumpVol, JumpFreq, ...
'DividendYield', DividendYield)
Call = 5×1

7.5765
6.4020
5.3484
4.4173
3.6073

Compute the Option Prices 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

Call = optByBatesNI(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, V0, ...
ThetaV, Kappa, SigmaV, RhoSV, MeanJ, JumpVol, JumpFreq, ...
'DividendYield', DividendYield) % Five values in vector output
Call = 5×1

9.7516
10.3931
10.8865
11.2990
11.6491

Expand the Output for a Surface

Set the ExpandOutput name-value pair argument to "true" to expand the output into a NStrikes-by-NMaturities matrix. In this case, it is a square matrix.

Call = optByBatesNI(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, V0, ...
ThetaV, Kappa, SigmaV, RhoSV, MeanJ, JumpVol, JumpFreq, ...
'DividendYield', DividendYield, 'ExpandOutput', true) % (5 x 5) matrix output
Call = 5×5

9.7516   11.4387   12.8395   14.0588   15.1361
8.6554   10.3931   11.8344   13.0890   14.1980
7.6432    9.4149   10.8865   12.1693   13.3046
6.7153    8.5035    9.9951   11.2990   12.4553
5.8705    7.6581    9.1594   10.4771   11.6491

Compute the Option Prices 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

Call = optByBatesNI(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, V0, ...
ThetaV, Kappa, SigmaV, RhoSV, MeanJ, JumpVol, JumpFreq, ...
'DividendYield', DividendYield, 'ExpandOutput', true) % (5 x 6) matrix output
Call = 5×6

7.5765    9.7516   11.4387   12.8395   14.0588   15.1361
6.4020    8.6554   10.3931   11.8344   13.0890   14.1980
5.3484    7.6432    9.4149   10.8865   12.1693   13.3046
4.4173    6.7153    8.5035    9.9951   11.2990   12.4553
3.6073    5.8705    7.6581    9.1594   10.4771   11.6491

Compute Option Prices 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

Call = optByBatesNI(Rate, ManyAssetPrices, Settle, Maturity, OptSpec, Strike, V0, ...
ThetaV, Kappa, SigmaV, RhoSV, MeanJ, JumpVol, JumpFreq, ...
'DividendYield', DividendYield, 'ExpandOutput', true) % (5 x 4) matrix output
Call = 5×4

4.2033    6.6918    9.7516   13.2808
3.5558    5.8111    8.6554   11.9993
2.9906    5.0181    7.6432   10.7934
2.5018    4.3096    6.7153    9.6651
2.0825    3.6818    5.8705    8.6158

Plot an Option Price Surface

The Strike and Maturity inputs can be vectors. Set ExpandOutput to "true" to output the surface as a NStrikes-by-NMaturities matrix.

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)';

Call = optByBatesNI(Rate, AssetPrice, Settle, Maturity, OptSpec, Strike, V0, ...
ThetaV, Kappa, SigmaV, RhoSV, MeanJ, JumpVol, JumpFreq, ...
'DividendYield', DividendYield, 'ExpandOutput', true);

[X,Y] = meshgrid(Times,Strike);

figure;
surf(X,Y,Call);
title('Price');
xlabel('Years to Option Expiry');
ylabel('Strike');
view(-112,34);
xlim([0 Times(end)]);
zlim([0 80]);

## 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 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

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, optByBatesNI 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.

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, optByBatesNI 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.

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

Option strike price value, specified as a NINST-by-1, NRows-by-1, NRows-by-NColumns vector of strike prices.

For more information on the proper dimensions for Strike, see the name-value pair argument ExpandOutput.

Data Types: double

Initial variance of the underling asset, specified as a scalar numeric value.

Data Types: double

Long-term variance of the underling asset, specified as a scalar numeric value.

Data Types: double

Mean revision speed for the underling asset, specified as a scalar numeric value.

Data Types: double

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

Data Types: double

Correlation between the Weiner processes for the underlying asset and its variance, specified as a scalar numeric value.

Data Types: double

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

Standard deviation of log(1+J) where J is the random percentage jump size, specified as a scalar decimal value.

Data Types: double

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: Price = optByBatesNI(Rate,AssetPrice,Settle,Maturity,OptSpec,Strike,V0,ThetaV,Kappa,SigmaV,RhoSV,MeanJ,JumpVol,JumpFreq,'Basis',7)

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

Data Types: double

Continuously compounded underlying asset yield, specified as the comma-separated pair consisting of 'DividendYield' and a scalar numeric value.

Data Types: double

Volatility risk premium, specified as the comma-separated pair consisting of 'VolRiskPremium' and a scalar numeric value.

Data Types: double

Flag indicating "Little Heston Trap" formulation by Albrecher et al, specified as the comma-separated pair consisting of 'LittleTrap' and a logical:

• true — Use the Albrecher et al formulation.

• false — Use the original Heston formation.

Data Types: logical

Absolute error tolerance for numerical integration, specified as the comma-separated pair consisting of 'AbsTol' and a scalar numeric value.

Data Types: double

Relative error tolerance for numerical integration, specified as the comma-separated pair consisting of 'RelTol' and a scalar numeric value.

Data Types: double

Numerical integration range used to approximate the continuous integral over [0 Inf], specified as the comma-separated pair consisting of 'IntegrationRange' and a 1-by-2 vector representing [LowerLimit UpperLimit].

Data Types: double

Framework for computing option prices and sensitivities using numerical integration of models, specified as the comma-separated pair consisting of 'Framework' and a scalar string or character vector with the following values:

• "heston1993" or 'heston1993' — Method used in Heston (1993)

• "lewis2001" or 'lewis2001' — Method used in Lewis (2001)

Data Types: char | string

Flag to expand the outputs, specified as the comma-separated pair consisting of 'ExpandOutput' and a logical:

• true — If true, the outputs are NRows-by- NColumns matrices. NRows is the number of strikes for each column and it is determined by the Strike input. For example, Strike can be a NRows-by-1 vector, or a NRows-by-NColumns matrix. NColumns is determined by the sizes of AssetPrice, Settle, Maturity, and OptSpec, which must all be either scalar or NColumns-by-1 vectors.

• false — If false, the outputs are NINST-by-1 vectors. Also, the inputs Strike, AssetPrice, Settle, Maturity, and OptSpec must all be either scalar or NINST-by-1 vectors.

Data Types: logical

## Output Arguments

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Option prices, returned as a NINST-by-1, or NRows-by-NColumns, depending on ExpandOutput.

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### 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: $\mathrm{max}\left(St-K,0\right)$

• For a put: $\mathrm{max}\left(K-St,0\right)$

where:

St is the price of the underlying asset at time t.

K is the strike price.

### Bates Stochastic Volatility Jump Diffusion Model

The Bates model (Bates (1996)) is an extension of the Heston model, where, in addition to stochastic volatility, the jump diffusion parameters similar to Merton (1976) were also added to model sudden asset price movements.

The stochastic differential equation is:

$\begin{array}{l}d{S}_{t}=\left(r-q-{\lambda }_{p}{\mu }_{J}\right){S}_{t}dt+\sqrt{{v}_{t}}{S}_{t}d{W}_{t}+J{S}_{t}d{P}_{t}\\ d{v}_{t}=\kappa \left(\theta -{v}_{t}\right)dt+{\sigma }_{v}\sqrt{{v}_{t}}d{W}_{t}\\ \text{E}\left[d{W}_{t}d{W}_{t}^{v}\right]=pdt\\ \text{prob(}d{P}_{t}=1\right)={\lambda }_{p}dt\end{array}$

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.

J is the random percentage jump size conditional on the jump occurring, where ln(1+J) is normally distributed with mean $\mathrm{ln}\left(1+{\mu }_{J}\right)-\frac{{\delta }^{2}}{2}$ and the standard deviation δ, and (1+J) has a lognormal distribution:

$\frac{1}{\left(1+J\right)\delta \sqrt{2\pi }}\mathrm{exp}\left\{{\frac{-\left[\mathrm{ln}\left(1+J\right)-\left(\mathrm{ln}\left(1+{\mu }_{J}\right)-\frac{{\delta }^{2}}{2}\right]}{2{\delta }^{2}}}^{2}\right\}$

v0 is the initial variance of the asset price at t = 0 (v0> 0).

θ is the long-term variance level for (θ > 0).

κ is the mean reversion speed for (κ > 0).

σv is the volatility of variance for (σv > 0).

p is the correlation between the Weiner processes Wt and ${W}_{t}^{v}$ for (-1 ≤ p ≤ 1).

μJ is the mean of J for (μJ > -1).

δ is the standard deviation of ln(1+J) for (δ ≥ 0).

${\lambda }_{p}$ is the annual frequency (intensity) of Poisson process Pt for (${\lambda }_{p}$ ≥ 0).

The characteristic function ${f}_{Bate{s}_{j}\left(\varphi \right)}$ for j = 1 (asset price mean measure) and j =2 (risk-neutral measure) is:

where

ϕ is the characteristic function variable.

ƛVolRisk is the volatility risk premium.

τ is the time to maturity for (τ = T - t).

i is the unit imaginary number for (i2= -1).

The definitions for Cj and Dj under “The Little Heston Trap” by Albrecher et al. (2007) are:

$\begin{array}{l}{C}_{j}=\left(r-q\right)i\varphi \tau +\frac{\kappa \theta }{{\sigma }_{v}{}^{2}}\left[\left({b}_{j}-p{\sigma }_{v}i\varphi -{d}_{j}\right)\tau -2\mathrm{ln}\left(\frac{1-{\epsilon }_{j}{e}^{-{d}_{j}\tau }}{1-{\epsilon }_{j}}\right)\right]\\ Dj=\frac{{b}_{j}-p{\sigma }_{v}i\varphi -{d}_{j}}{{\sigma }_{v}^{2}}\left(\frac{1-{e}^{-{d}_{j}\tau }}{1-{\epsilon }_{j}{e}^{-{d}_{j}\tau }}\right)\\ {\epsilon }_{j}=\frac{{b}_{j}-p{\sigma }_{v}i\varphi -{d}_{j}}{{b}_{j}-p{\sigma }_{v}i\varphi +{d}_{j}}\end{array}$

### Numerical Integration Method Under Heston (1993) Framework

Numerical integration is used to evaluate the continuous integral for the inverse Fourier transform.

The numerical integration method under the Heston (1993) framework is based on the following expressions:

$\begin{array}{l}Call\left(K\right)={S}_{t}{e}^{-q\tau }{P}_{1}-K{e}^{-r\tau }{P}_{2}\\ Put\left(K\right)=Call\left(K\right)+K{e}^{-r\tau }-{S}_{t}{e}^{-q\tau }\\ {P}_{j}=\frac{1}{2}+\frac{1}{\pi }\underset{0}{\overset{\infty }{\int }}\mathrm{Re}\left[\frac{{e}^{-i\varphi \mathrm{ln}\left(K\right)}{f}_{j}\left(\varphi \right)}{i\varphi }\right]d\varphi \end{array}$

where

r is the continuous risk-free rate.

q is the continuous dividend yield.

St is the asset price at time t.

K is the strike.

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.

fj(ϕ) is the characteristic function for Pj(j = 1,2).

P1 is the probability of St > K under the asset price measure for the model.

P2 is the probability of St > K under the risk-neutral measure for the model.

Where j = 1,2 so that f1(ϕ) and f2(ϕ) are the characteristic functions for probabilities P1 and P2, respectively.

This framework is chosen with the default value “Heston1993” for the Framework name-value pair argument.

### Numerical Integration Method Under Lewis (2001) Framework

Numerical integration is used to evaluate the continuous integral for the inverse Fourier transform.

The numerical integration method under the Lewis (2001) framework is based on the following expressions:

$\begin{array}{l}Call\left(k\right)={S}_{t}{e}^{-q\tau }-\frac{\sqrt{K}{e}^{-\tau t}}{\pi }\underset{0}{\overset{\infty }{\int }}\mathrm{Re}\left[{K}^{-iu}{f}_{2}\left(\varphi =\left(u-\frac{i}{2}\right)\right)\frac{1}{{u}^{2}+\frac{1}{4}}\right]du\\ Put\left(K\right)=Call\left(K\right)=K{e}^{-\tau t}-{S}_{t}{e}^{-q\tau }\end{array}$

where

r is the continuous risk-free rate.

q is the continuous dividend yield.

St is the asset price at time t.

K is the strike.

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.

u is the characteristic function variable for integration, where $\varphi =\left(u-\frac{i}{2}\right)$.

f2(ϕ) is the characteristic function for P2.

P2 is the probability of St > K under the risk-neutral measure for the model.

This framework is chosen with the value “Lewis2001” for the Framework name-value pair argument.

## 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] 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. No. 2. 1993.

[3] Lewis, A. L. “A Simple Option Formula for General Jump-Diffusion and Other Exponential Levy Processes.” Envision Financial Systems and OptionCity.net, 2001.

## Version History

Introduced in R2018a

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