blsgamma

Black-Scholes sensitivity to underlying delta change

Syntax

Gamma = blsgamma(Price,Strike,Rate,Time,Volatility)

Gamma = blsgamma(___,Yield)

Description

Gamma = blsgamma(Price,Strike,Rate,Time,Volatility) returns gamma, the sensitivity of delta to change in the underlying asset price. blsgamma uses normpdf, the 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 gamma values for a Vanilla, Barrier, Touch, DoubleTouch, or Binary instrument using a BlackScholes model.

Note

blsgamma 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.

Gamma = blsgamma(___,Yield) adds an optional argument for Yield.

Examples

Find Gamma for a Change in the Underlying Asset Price

This example shows how to find the gamma, the sensitivity of delta to a change in the underlying asset price.

Gamma = blsgamma(50, 50, 0.12, 0.25, 0.3, 0)

Gamma = 
0.0512

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

Gamma — Delta to change in underlying security price

numeric

Delta to change in underlying security price, returned as a numeric value.

More About

Gamma

A gamma sensitivity measures the rate of change of an option's delta in response to a change in the price of the underlying asset.

In other words, while delta tells you how much the price of an option might move, gamma tells you how fast the option's delta itself will change as the price of the underlying asset moves. For instance, if an option has a gamma of 0.10, its delta is expected to change by 0.10 for every $1 move in the underlying asset's price.

References

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

Version History

Introduced in R2006a