optstocksensbybjs

Determine American option prices or sensitivities using Bjerksund-Stensland 2002 option pricing model

Syntax

PriceSens = optstocksensbybjs(RateSpec,StockSpec,Settle,Maturity,OptSpec,Strike)

PriceSens = optstocksensbybjs(___,Name,Value)

Description

PriceSens = optstocksensbybjs(RateSpec,StockSpec,Settle,Maturity,OptSpec,Strike) computes American option prices or sensitivities using the Bjerksund-Stensland 2002 option pricing model.

Note

optstocksensbybjs computes prices of American options with continuous dividend yield using the Bjerksund-Stensland option pricing model. All sensitivities are evaluated by computing a discrete approximation of the partial derivative. This means that the option is revalued with a fractional change for each relevant parameter, and the change in the option value divided by the increment, is the approximated sensitivity value.

PriceSens = optstocksensbybjs(___,Name,Value) adds an optional name-value pair argument for OutSpec.

Examples

Compute American Option Prices and Sensitivities Using the Bjerksund-Stensland 2002 Option Pricing Model

This example shows how to compute American option prices and sensitivities using the Bjerksund-Stensland 2002 option pricing model. Consider four American put options with an exercise price of $100. The options expire on October 1, 2008. Assume the underlying stock pays a continuous dividend yield of 4% and has a volatility of 40% per annum. The annualized continuously compounded risk-free rate is 8% per annum. Using this data, calculate the delta, gamma, and price of the American put options, assuming the following current prices of the stock on July 1, 2008: $90, $100, $110 and $120.

Settle = 'July-1-2008';
Maturity = 'October-1-2008';
Strike = 100;
AssetPrice = [90;100;110;120];
Rate = 0.08;
Sigma = 0.40;
DivYield = 0.04;

% define the RateSpec and StockSpec
StockSpec = stockspec(Sigma, AssetPrice, {'continuous'}, DivYield);

RateSpec = intenvset('ValuationDate', Settle, 'StartDates', Settle,...
'EndDates', Maturity, 'Rates', Rate, 'Compounding', -1);

% define the option type
OptSpec = {'put'};

OutSpec = {'Delta', 'Gamma', 'Price'};

[Delta, Gamma, Price] = optstocksensbybjs(RateSpec, StockSpec, Settle, Maturity,...
OptSpec, Strike, 'OutSpec', OutSpec)

Delta = 4×1

   -0.6572
   -0.4434
   -0.2660
   -0.1442

Gamma = 4×1

    0.0217
    0.0202
    0.0150
    0.0095

Price = 4×1

   12.9467
    7.4571
    3.9539
    1.9495

Input Arguments

RateSpec — Interest-rate term structure

structure

Interest-rate term structure (annualized and continuously compounded), specified by the RateSpec obtained from intenvset. For information on the interest-rate specification, see intenvset.

Data Types: struct

StockSpec — Stock specification for underlying asset

structure

Stock specification for the underlying asset. For information on the stock specification, see stockspec.

stockspec handles several types of underlying assets. For example, for physical commodities the price is StockSpec.Asset, the volatility is StockSpec.Sigma, and the convenience yield is StockSpec.DividendAmounts.

Data Types: struct

Settle — Settlement or trade date

serial date number | date character vector

Settlement or trade date, specified as serial date number or date character vector using a NINST-by-1 vector.

Data Types: double | char

Maturity — Maturity date for option

serial date number | date character vector

Maturity date for option, specified as serial date number or date character vector using a NINST-by-1 vector.

Data Types: double | char

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 value

nonnegative vector

Option strike price value, specified as a nonnegative NINST-by-1 vector.

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: [Delta,Gamma,Price] = optstocksensbybjs(RateSpec,StockSpec,Settle,Maturity,OptSpec,Strike,'OutSpec',OutSpec)

'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] Bjerksund, P. and G. Stensland. “Closed-Form Approximation of American Options.” Scandinavian Journal of Management. Vol. 9, 1993, Suppl., pp. S88–S99.

[2] Bjerksund, P. and G. Stensland. “Closed Form Valuation of American Options.” Discussion paper 2002 (https://www.scribd.com/doc/215619796/Closed-form-Valuation-of-American-Options-by-Bjerksund-and-Stensland#scribd)