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

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

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

Examples

Compute Option Prices and Sensitivities Using a Leisen-Reimer Binomial Tree Model

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 = 2×1

   10.1332
    6.6937

Delta = 2×1

    0.6056
   -0.3944

Gamma = 2×1

    0.0185
    0.0185

Input Arguments

LRTree — Stock tree structure

structure

Stock tree structure, specified by lrtree.

Data Types: struct

OptSpec — Definition of option

cell array of character vectors with values 'call' or 'put'

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

Strike — Option strike price values

vector of nonnegative integers

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 NaNs.

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

Data Types: double

Settle — Settlement or trade date

serial date number | date character vector

Settlement or trade date, specified as an NINST-by-1 matrix using serial date numbers or date character vectors.

Data Types: double | char

ExerciseDates — Option exercise dates

serial date number | date character vector

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 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)

'AmericanOpt' — Option type

0 European or Bermuda (default) | values: [0,1]

Option type, specified as the comma-separated pair consisting of 'AmericanOpt' and a NINST-by-1 vector of flags with values:

• 0 — European or Bermuda

• 1 — American

Data Types: double

'OutSpec' — Define outputs

{'Price'} (default) | cell array of character vectors with values: 'Price', 'Delta', 'Gamma', 'Vega', 'Lambda', 'Rho', 'Theta', and 'All'

Define outputs, specified as the comma-separated pair consisting of 'OutSpec' and a 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

PriceSens — Expected future prices or sensitivities values

vector

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

Data Types: double

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.

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.