impvbybjs

Determine implied volatility using Bjerksund-Stensland 2002 option pricing model

Syntax

Volatility = impvbybjs(RateSpec,StockSpec,Settle,Maturity,OptSpec,Strike,OptPrice)

Volatility = impvbybjs(___,Name,Value)

Description

Volatility = impvbybjs(RateSpec,StockSpec,Settle,Maturity,OptSpec,Strike,OptPrice) computes implied volatility using the Bjerksund-Stensland 2002 pricing model.

Note

impvbybjs computes implied volatility of American options with continuous dividend yield using the Bjerksund-Stensland option pricing model.

Volatility = impvbybjs(___,Name,Value) adds optional name-value pair arguments.

Examples

Compute the Implied Volatility Using the Bjerksund-Stensland 2002 Option Pricing Model

This example shows how to compute implied volatility using the Bjerksund-Stensland 2002 option pricing model. Consider three American call options with exercise prices of $100 that expire on July 1, 2008. The underlying stock is trading at $100 on January 1, 2008 and pays a continuous dividend yield of 10%. The annualized continuously compounded risk-free rate is 10% per annum and the option prices are $4.063, $6.77 and $9.46. Using this data, calculate the implied volatility of the stock using the Bjerksund-Stensland 2002 option pricing model.

AssetPrice = 100;
Settle = datetime(2008,1,1);
Maturity = datetime(2008,6,1);
Strike = 100;
DivAmount = 0.1;
Rate = 0.1;

% define the RateSpec and StockSpec
RateSpec = intenvset('ValuationDate', Settle, 'StartDates', Settle,...
'EndDates', Maturity, 'Rates', Rate, 'Compounding', -1, 'Basis', 1);

StockSpec = stockspec(NaN, AssetPrice, {'continuous'}, DivAmount);

OptSpec = {'call'};
OptionPrice = [4.063;6.77;9.46];

ImpVol =  impvbybjs(RateSpec, StockSpec, Settle, Maturity, OptSpec,...
Strike, OptionPrice)

ImpVol = 3×1

    0.1633
    0.2723
    0.3810

The implied volatility is 15% for the first call, and 25% and 35% for the second and third call options.

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 date

datetime array | string array | date character vector

Settlement date, specified as a NINST-by-1 vector using a datetime array, string array, or date character vectors.

To support existing code, impvbybjs also accepts serial date numbers as inputs, but they are not recommended.

Maturity — Maturity date

datetime array | string array | date character vector

Maturity date for the American option, specified as a NINST-by-1 vector using a datetime array, string array, or date character vectors.

To support existing code, impvbybjs also accepts serial date numbers as inputs, but they are not recommended.

OptSpec — Definition of option

character vector with values 'call' or 'put'

Definition of the option from which the implied volatility is derived, specified as a NINST-by-1 cell array of character vectors with a value of 'call' or 'put'.

Data Types: char | cell

Strike — Option strike price value

nonnegative scalar | nonnegative vector

Option strike price value, specified as a nonnegative scalar or NINST-by-1 vector of strike price values. Each row is the schedule for one option.

Data Types: double

OptPrice — American option price

nonnegative scalar | nonnegative vector

American option prices from which the implied volatility of the underlying asset is derived, specified as a nonnegative scalar or NINST-by-1 vector.

Data Types: double

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN, where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter.

Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: Volatility = impvbybjs(RateSpec,StockSpec,Settle,Maturity,OptSpec,Strike,OptPrice,'Limit',[0.2 20],'Tolerance',1e-5)

Limit — Lower and upper bound of implied volatility search interval

[0.1 10] (10% to 1000% per annum) (default) | positive vector

Lower and upper bound of implied volatility search interval, specified as the comma-separated pair consisting of 'Limit' and a 1-by-2 positive vector.

Data Types: double

Tolerance — Implied volatility search termination tolerance

1e-6 (default) | positive scalar

Implied volatility search termination tolerance, specified as the comma-separated pair consisting of 'Tolerance' and a positive scalar.

Data Types: double

Output Arguments

Volatility — Expected implied volatility values

vector

Expected implied volatility values, returned as a NINST-by-1 vector. If no solution can be found, a NaN is returned.

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.

Version History

Introduced in R2008b

R2022b: Serial date numbers not recommended

Although impvbybjs supports serial date numbers, datetime values are recommended instead. The datetime data type provides flexible date and time formats, storage out to nanosecond precision, and properties to account for time zones and daylight saving time.

To convert serial date numbers or text to datetime values, use the datetime function. For example:

t = datetime(738427.656845093,"ConvertFrom","datenum");
y = year(t)

y =

        2021

There are no plans to remove support for serial date number inputs.