Documentation

# optSensByBatesFD

Option price and sensitivities by Bates model using finite differences

## Description

example

[PriceSens,PriceGrid,AssetPrices,Variances,Times] = optSensByBatesFD(Rate,AssetPrice,Settle,ExerciseDates,OptSpec,Strike,V0,ThetaV,Kappa,SigmaV,RhoSV,MeanJ,JumpVol,JumpFreq) computes a vanilla European or American option price and sensitivities by the Bates model, using the alternating direction implicit (ADI) method.

example

[PriceSens,PriceGrid,AssetPrices,Variances,Times] = optSensByBatesFD(___,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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Define the option variables and Bates model parameters.

AssetPrice = 90;
Strike = 100;
Rate = 0.03;
Settle = '01-Jan-2018';
ExerciseDates = '02-Jul-2018';

V0 = 0.04;
ThetaV = 0.04;
Kappa = 2;
SigmaV = 0.25;
RhoSV = -0.5;
JumpVol = 0.4;
MeanJ = exp(-0.5+JumpVol.^2/2)-1;
JumpFreq = 0.2;

Compute the American put option price and sensitivities using the finite differences method.

OptSpec = 'Put';
[Price, Delta, Gamma, Rho, Theta, Vega, VegaLT] = optSensByBatesFD(Rate, AssetPrice, Settle, ExerciseDates,...
OptSpec, Strike, V0, ThetaV, Kappa, SigmaV, RhoSV, MeanJ, JumpVol, JumpFreq, 'AmericanOpt', 1,...
'OutSpec', ["Price" "Delta" "Gamma" "Rho" "Theta" "Vega" "VegaLT"])
Price = 11.6116
Delta = -0.6578
Gamma = 0.0366
Rho = -20.1643
Theta = -4.8425
Vega = 14.3216
VegaLT = 8.1849

## Input Arguments

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

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 object or string scalar:

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

• For an American option, use a 1-by-2 vector 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-NaN date is listed, then the option can be exercised between Settle date and the single listed value in ExerciseDates.

Data Types: double | char | datetime | string

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

Data Types: cell | string

Option strike price value, specified as a scalar numeric.

Data Types: double

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

Data Types: double

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

Data Types: double

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

Data Types: double

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

Data Types: double

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

Data Types: double

Mean of the random percentage jump size (J), specified as a scalar decimal 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

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

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 of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: [PriceSens,PriceGrid] = optSensByBatesFD(Rate,AssetPrice,Settle,ExerciseDates,OptSpec,Strike,V0,ThetaV,Kappa,SigmaV,RhoSV,MeanJ,JumpVol,JumpFreq,'Basis',7,'OutSpec','delta')

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

Data Types: double

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

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

Ex-dividend dates, specified as the comma-separated pair consisting of 'ExDividendDates' and an NDIV-by-1 vector 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

Maximum variance for the variance grid boundary, specified as the comma-separated pair consisting of 'VarianceMax' as a scalar numeric.

Data Types: double

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

Number of nodes of the variance grid for the finite difference grid, specified as the comma-separated pair consisting of 'VarianceGridSize' 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 — American

Data Types: double

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 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

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Option price or sensitivities, returned as a numeric. The name-value pair argument OutSpec determines the types and order of the outputs.

Grid containing prices calculated by the finite difference method, returned as a two-dimensional grid with size AssetGridSizeTimeGridSize. The number of columns is not necessarily equal to the TimeGridSize because the function adds exercise and ex-dividend dates to the time grid. PriceGrid(:, :, end) contains the price for t = 0.

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

Variances corresponding to the second dimension of PriceGrid, returned as a vector.

Times corresponding to the third dimension of PriceGrid, returned as a vector.

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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 [1] extends the Heston model by including stochastic volatility and (similar to Merton) jump diffusion parameters in the modeling of 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:

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

where:

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).

## References

[1] Bates, D. S. "Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in Deutsche Mark Options." The Review of Financial Studies. Vol. 9, Number 1, 1996.