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# optstocksensbylr

Determine option prices or sensitivities using Leisen-Reimer binomial tree model

## Syntax

``PriceSens = optstockbylr(LRTree,OptSpec,Strike,Settle,ExerciseDates)``
``PriceSens = optstockbylr(___,Name,Value)``

## Description

example

````PriceSens = optstockbylr(LRTree,OptSpec,Strike,Settle,ExerciseDates)` calculates option prices or sensitivities using a Leisen-Reimer binomial tree model.```

example

````PriceSens = optstockbylr(___,Name,Value)` adds optional name-value pair arguments for `AmericanOpt` and `OutSpec`.```

## Examples

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This example shows how to compute option prices and sensitivities using a Leisen-Reimer binomial tree model. Consider European call and put options with an exercise price of \$100 that expire on December 1, 2010. The underlying stock is trading at \$100 on June 1, 2010 and has a volatility of 30% per annum. The annualized continuously compounded risk-free rate is 7% per annum. Using this data, compute the price, delta and gamma of the options using the Leisen-Reimer model with a tree of 25 time steps and the `PP2` method.

```AssetPrice = 100; Strike = 100; ValuationDate = 'June-1-2010'; Maturity = 'December-1-2010'; % define StockSpec Sigma = 0.3; StockSpec = stockspec(Sigma, AssetPrice); % define RateSpec Rates = 0.07; Settle = ValuationDate; Basis = 1; Compounding = -1; RateSpec = intenvset('ValuationDate', ValuationDate, 'StartDates', Settle, ... 'EndDates', Maturity, 'Rates', Rates, 'Compounding', Compounding, 'Basis', Basis); % build the Leisen-Reimer (LR) tree with 25 time steps LRTimeSpec = lrtimespec(ValuationDate, Maturity, 25); % use the PP2 method LRMethod = 'PP2'; TreeLR = lrtree(StockSpec, RateSpec, LRTimeSpec, Strike, 'method', LRMethod); % compute prices and sensitivities using the LR model: OptSpec = {'call'; 'put'}; OutSpec = {'Price', 'Delta', 'Gamma'}; [Price, Delta, Gamma] = optstocksensbylr(TreeLR, OptSpec, Strike, Settle, ... Maturity, 'OutSpec', OutSpec)```
```Price = 10.1332 6.6937 ```
```Delta = 0.6056 -0.3944 ```
```Gamma = 0.0185 0.0185 ```

## Input Arguments

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Stock tree structure, specified by `lrtree`.

Data Types: `struct`

Definition of the option as `'call'` or `'put'`, specified as a `NINST`-by-`1` cell array of character vectors with values `'call'` or `'put'`.

Data Types: `char` | `cell`

Option strike price value, specified with nonnegative integer:

• For a European option, use a `NINST`-by-`1` vector of strike prices.

• For a Bermuda option, use a `NINST`-by-`NSTRIKES` vector of strike prices. Each row is the schedule for one option. If an option has fewer than `NSTRIKES` exercise opportunities, the end of the row is padded with `NaN`s.

• For an American option, use a `NINST`-by-`1` vector of strike prices.

Data Types: `double`

Option exercise dates, specified as a vector of date character vectors or serial date numbers where each row is the schedule for one option and the last element of each row must be the same as the maturity of the tree.

• For a European option, use a `NINST`-by-`1` vector of dates. For a European option, there is only one `ExerciseDate` on the option expiry date.

• For a Bermuda option, use a `NINST`-by-`NSTRIKEDATES` vector of dates.

• For an American option, use a `NINST`-by-`1` vector of exercise dates. For the American type, the option can be exercised on any tree data between the `ValuationDate` and tree maturity.

Data Types: `double` | `char`

### 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 single quotes (`' '`). You can specify several name and value pair arguments in any order as `Name1,Value1,...,NameN,ValueN`.

Example: ```[Price,Delta,Gamma] = optstocksensbylr(LRTree,OptSpec,Strike,Settle,ExerciseDates,'OutSpec',OutSpec)```

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Option type, specified as a `NINST`-by-`1` vector of flags with values:

• `0` — European or Bermuda

• `1` — American

Data Types: `double`

Define outputs specifying `NOUT`- by-`1` or `1`-by-`NOUT` cell array of character vectors with possible values of `'Price'`, `'Delta'`, `'Gamma'`, `'Vega'`, `'Lambda'`, `'Rho'`, `'Theta'`, and `'All'`.

`OutSpec = {'All'}` specifies that the output should be `Delta`, `Gamma`, `Vega`, `Lambda`, `Rho`, `Theta`, and `Price`, in that order. This is the same as specifying `OutSpec` to include each sensitivity:

Example: ```OutSpec = {'delta','gamma','vega','lambda','rho','theta','price'}```

Data Types: `char` | `cell`

## Output Arguments

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Expected future prices or sensitivities values, returned as a `NINST`-by-`1` vector.

Data Types: `double`

## References

[1] Leisen D.P., M. Reimer. “Binomial Models for Option Valuation – Examining and Improving Convergence.” Applied Mathematical Finance. Number 3, 1996, pp. 319–346.