optSensByMertonFD

Option price and sensitivities by Merton76 model using finite differences

Syntax

[PriceSens,PriceGrid,AssetPrices,Times] = optSensByMertonFD(Rate,AssetPrice,Settle,ExerciseDates,OptSpec,Strike,Sigma,MeanJ,JumpVol,JumpFreq)

[PriceSens,PriceGrid,AssetPrices,Times] = optSensByMertonFD(___,Name,Value)

Description

[PriceSens,PriceGrid,AssetPrices,Times] = optSensByMertonFD(Rate,AssetPrice,Settle,ExerciseDates,OptSpec,Strike,Sigma,MeanJ,JumpVol,JumpFreq) computes a vanilla European or American option price and sensitivities by the Merton76 model, using the Crank-Nicolson Adams-Bashforth (CNAB) IMEX method.

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

Examples

Compute Price and Sensitivities for an American Option Using the Merton Model

Define the option variables and Merton model parameters.

AssetPrice = 90;
Strike = 100;
Rate = 0.06;
DividendYield = 0.1;
Settle = '01-Jan-2018';
ExerciseDates = '02-Apr-2018';

Sigma = 0.40;
MeanJ = -0.10;
JumpVol = 0.01;
JumpFreq = 1.00;

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

OptSpec = 'Call';

[Price, Delta, Gamma, Rho, Theta, Vega] = optSensByMertonFD(Rate, AssetPrice, Settle, ExerciseDates, OptSpec, Strike,...
Sigma, MeanJ, JumpVol, JumpFreq, 'DividendYield', DividendYield, 'AmericanOpt', 1,...
'OutSpec', ["Price" "Delta" "Gamma" "Rho" "Theta" "Vega"])

Price = 3.4551

Delta = 0.3211

Gamma = 0.0195

Rho = 5.6610

Theta = -11.9877

Vega = 15.5156

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 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 array, or string array.

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 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

OptSpec — Definition of option

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

Definition of the option, specified as a scalar using a character vector or string array 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

Sigma — Volatility of underlying asset

scalar numeric

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

Data Types: double

MeanJ — Mean of the random percentage jump size

scalar decimal

Mean of the random percentage jump size (J), specified as a scalar decimal value 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 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: [Price,PriceGrid] = optByMertonFD(Rate,AssetPrice,Settle,ExerciseDates,OptSpec,Strike,MeanJ,JumpVol,JumpFreq,'Basis',7,'OutSpec','delta')

'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

'AssetGridSize' — Size of asset grid for the 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

'TimeGridSize' — Number of nodes of time grid for the 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 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

'OutSpec' — Define outputs

['price'] (default) | cell array of character vectors with values 'price', 'delta', 'gamma', 'vega', 'rho', and 'theta' | string array 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.

Example: OutSpec = ['price','delta','gamma','vega','rho','theta']

Data Types: string | cell

Output Arguments

PriceSens — Option price or sensitivities

numeric

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

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 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.

Times — Times

vector

Times corresponding to the second 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.

Merton Jump Diffusion Model

The Merton jump diffusion model [2] extends the Black-Scholes model by using the Poisson process to include jump diffusion parameters in the modeling of sudden asset price movements (both up and down).

The stochastic differential equation is

$$\begin{array}{l}d{S}_t=(r-q-{\lambda }_p{\mu }_j)S_{t}dt+\sigma S_{t}d{W}_t+J{S}_{t}d{P}_t\\ \text{prob(}d{P}_t=1)={\lambda }_{p}dt\end{array}$$

where:

r is the continuous risk-free rate.

q is the continuous dividend yield.

W_t is the Weiner process.

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:

μ_J is the mean of J for (μ_J > -1).

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

ƛ_p is the annual frequency (intensity) of Poisson process P_tfor (ƛ_p ≥ 0).

σ is the volatility of the asset price for (σ > 0).

References

[1] Cont, R., and E. Voltchkova. “A Finite Difference Scheme for Option Pricing in Jump Diffusion and Exponential Lévy Models.” SIAM Journal on Numerical Analysis. Vol. 43, Number 4, 2005, pp. 1596-1626.

[2] Merton, R. "Option Pricing When Underlying Stock Returns Are Discontinuous." The Journal of Financial Economics. Vol 3. 1976, pp. 125-144.