blstheta

Black-Scholes sensitivity to time-until-maturity change

Syntax

[CallTheta,PutTheta] = blstheta(Price,Strike,Rate,Time,Volatility)

[CallTheta,PutTheta] = blstheta(___,Yield)

Description

[CallTheta,PutTheta] = blstheta(Price,Strike,Rate,Time,Volatility) returns the call option theta CallTheta, and the put option theta PutTheta.

Theta is the sensitivity in option value with respect to time and is measured in years. CallTheta or PutTheta can be divided by 365 to get Theta per calendar day or by 252 to get Theta by trading day.

blstheta uses normcdf, the normal cumulative distribution function, and normpdf, the normal probability density function, in the Statistics and Machine Learning Toolbox™.

In addition, you can use the Financial Instruments Toolbox™ object framework with the BlackScholes (Financial Instruments Toolbox) pricer object to obtain price and theta values for a Vanilla, Barrier, Touch, DoubleTouch, or Binary instrument using a BlackScholes model.

Note

blstheta can handle other types of underlies like Futures and Currencies. When pricing Futures (Black model), enter the input argument Yield as:

Yield = Rate

When pricing currencies (Garman-Kohlhagen model), enter the input argument Yield as:

Yield = ForeignRate

where ForeignRate is the continuously compounded, annualized risk-free interest rate in the foreign country.

[CallTheta,PutTheta] = blstheta(___,Yield) adds an optional argument for Yield.

Examples

Compute the Black-Scholes Sensitivity to Time-Until-Maturity Change (Theta)

This example shows how to compute theta, the sensitivity in option value with respect to time.

[CallTheta, PutTheta] = blstheta(50, 50, 0.12, 0.25, 0.3, 0)

CallTheta = 
-8.9630

PutTheta = 
-3.1404

Input Arguments

Price — Current price of underlying asset

numeric

Current price of the underlying asset, specified as a numeric value.

Data Types: double

Strike — Exercise price of option

numeric

Exercise price of the option, specified as a numeric value.

Data Types: double

Rate — Annualized, continuously compounded risk-free rate of return over life of option

positive decimal

Annualized, continuously compounded risk-free rate of return over the life of the option, specified as a positive decimal value.

Data Types: double

Time — Time (in years) to expiration of the option

numeric

Time (in years) to expiration of the option, specified as a numeric value.

Data Types: double

Volatility — Annualized asset price volatility

positive decimal

Annualized asset price volatility (annualized standard deviation of the continuously compounded asset return), specified as a positive decimal value.

Data Types: double

Yield — Annualized, continuously compounded yield of the underlying asset over life of option

0 (default) | decimal

(Optional) Annualized, continuously compounded yield of the underlying asset over the life of the option, specified as a decimal value. For example, for options written on stock indices, Yield could represent the dividend yield. For currency options, Yield could be the foreign risk-free interest rate.

Data Types: double

Output Arguments

CallTheta — Call option theta

numeric

Call option theta, returned as a numeric value.

PutTheta — Put option theta

numeric

Put option theta, returned as a numeric value.

More About

Theta

A theta sensitivity measures the rate at which the price of an option decreases as time passes, all else being equal.

Theta is essentially a quantification of time decay, which is a key concept in options pricing. Theta provides an estimate of the dollar amount that an option's price would decrease each day, assuming no movement in the price of the underlying asset and no change in volatility. For example, if a call option has a theta of -0.05, it means that the option's price is expected to decrease by $0.05 per day, holding all other factors constant.

References

[1] Hull, John C. Options, Futures, and Other Derivatives. 5th edition, Prentice Hall, 2003.

Version History

Introduced in R2006a