optByBatesFD

Option price by Bates model using finite differences

Syntax

[Price,PriceGrid,AssetPrices,Variances,Times] = optByBatesFD(Rate,AssetPrice,Settle,ExerciseDates,OptSpec,Strike,V0,ThetaV,Kappa,SigmaV,RhoSV,MeanJ,JumpVol,JumpFreq)

[Price,PriceGrid,AssetPrices,Variances,Times] = optByBatesFD(___,Name,Value)

Description

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

[Price,PriceGrid,AssetPrices,Variances,Times] = optByBatesFD(___,Name,Value) specifies options using one or more name-value pair arguments in addition to the input arguments in the previous syntax.

Examples

Price an American Option Using the Bates Model

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 using the finite differences method.

OptSpec = 'Put';
Price = optByBatesFD(Rate, AssetPrice, Settle, ExerciseDates, OptSpec, Strike, ...
V0, ThetaV, Kappa, SigmaV, RhoSV, MeanJ, JumpVol, JumpFreq, 'AmericanOpt', 1)

Price = 11.6116

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

serial date number | date character vector | datetime | string array

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

ExerciseDates — Option exercise dates

serial date number | date character vector | datetime array | string array

Option exercise dates, specified as a serial date number, date character vector, datetime array, or string array:

• For a European option, use a scalar serial date number, date character vector, datetime object, or string scalar. 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

OptSpec — Definition of option

character vector with value of 'call' or 'put' | string with value of "call" or "put"

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

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 underling asset, specified as a scalar numeric.

Data Types: double

ThetaV — Long-term variance of underlying asset

scalar numeric

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

Data Types: double

RhoSV — Correlation between Weiner processes for underlying asset and its variance

scalar numeric

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

Data Types: double

MeanJ — Mean of random percentage jump size

scalar decimal

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

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 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: [Price,PriceGrid] = optByBatesFD(Rate,AssetPrice,Settle,ExerciseDates,OptSpec,Strike,V0,ThetaV,Kappa,SigmaV,RhoSV,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 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) | serial date number | date character vector | string array | datetime array

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

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

VarianceMax — Maximum variance for variance grid boundary

1.0 (default) | scalar numeric

Maximum variance for the 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 of variance grid for finite difference grid

200 (default) | scalar numeric

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

TimeGridSize — Number of nodes of time grid for finite difference grid

100 (default) | positive numeric scalar

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

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 — European

• 1 — American

Data Types: double

Output Arguments

Price — Option price

scalar numeric

Option price, returned as a scalar numeric.

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 the function adds exercise and ex-dividend dates 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: $$\max (St-K,0)$$

• For a put: $$\max (K-St,0)$$

where:

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

K is the strike price.

For more information, see Vanilla Option.

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:

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

where:

r is the continuous risk-free rate.

q is the continuous dividend yield.

S_t is the asset price at time t.

v_t 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 $$\ln (1+{\mu }_J)-\frac{{\delta }^2}{2}$$ and the standard deviation δ, and (1+J) has a lognormal distribution:

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

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).

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

p is the correlation between the Weiner processes W_t and $$W_t^v$$ for (-1 ≤ p ≤ 1).

μ_J is the mean of J for (μ_J > -1).

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

$${\lambda }_p$$ is the annual frequency (intensity) of Poisson process P_t 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.