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# 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

example

````[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.```

example

````[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

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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

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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 a scalar numeric.

Data Types: `double`

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`

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`

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`

Option strike price value, specified as a scalar numeric.

Data Types: `double`

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

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 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: ```[Price,PriceGrid] = optByMertonFD(Rate,AssetPrice,Settle,ExerciseDates,OptSpec,Strike,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

For more information, see Basis.

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`

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 time grid for 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.

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

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 `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`.

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

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

## More About

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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.

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

`$\begin{array}{l}d{S}_{t}=\left(r-q-{\lambda }_{p}{\mu }_{j}\right){S}_{t}dt+\sigma {S}_{t}d{W}_{t}+J{S}_{t}d{P}_{t}\\ \text{prob(}d{P}_{t}=1\right)={\lambda }_{p}dt\end{array}$`

where:

r is the continuous risk-free rate.

q is the continuous dividend yield.

Wt 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 $\mathrm{ln}\left(1+{\mu }_{J}\right)-\frac{{\delta }^{2}}{2}$ 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:

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).

## 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.

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