What is Lambda
One of the “Greeks”, lambda, is the ratio between the change in the dollar price of an option and a 1% change in the expected price volatility, also called implied volatility, of an underlying asset. Lambda tells investors how much the price of an option will change for a given change in implied volatility, even if the actual price of the underlying remains the same.
The value of Lambda is higher the later the expiration date of an option and decreases as the expiration date approaches. Just as individual options each have a lambda, an option portfolio has a net lambda which is determined by adding the lambdas of each individual position.
In the options analysis, lambda is used interchangeably with the terms vega, kappa and sigma.
Lambda FAILURE
Lambda changes in the event of significant price movements or increased volatility of the underlying asset. For example, if the price of an option increases by 10% when volatility increases by 5%, its lambda value is 2.0. The lambda is calculated by the price movement divided by the increase in volatility.
If lambda is high, the value of the option is very sensitive to small changes in volatility. If lambda is low, changes in volatility will not have much effect on the option. A positive lambda is associated with a long option and means that the option becomes more valuable as volatility increases. Conversely, a negative lambda is associated with a short option and means that the option becomes more valuable as volatility decreases.
Lambda is one of the most important Greek options. Other important options of the Greeks include:

Delta, which measures the impact of a change in the price of the underlying asset

Gamma, which measures the rate of change of the delta

Theta, which measures the impact of a change in the time remaining to expiration, also known as time decay
Lambda in action
If a share for ABC is trading at $ 40 in April and a call from MAY 45 sells for $ 2. The lambda of the option is 0.15 and the volatility is 20%.
If the underlying volatility has increased from 1% to 21%, then theoretically, the option price should go up to $ 2 + (1 x 0.15) = $ 2.15.
Alternatively, if volatility decreased from 3% to 17% instead, the option should fall to $ 2 – (3 x 0.15) = $ 1.55
Implied volatility
Implied volatility is the estimated volatility, or fluctuation, of the price of a security and is most often used when pricing options. Usually, but not always, implied volatility increases when the market is bearish or when investors expect the price of the asset to fall over time. It generally, but not always, decreases when the market is bullish or when investors believe the price will increase over time. This movement is due to the common belief that bear markets are more risky than bull markets. Implied volatility is a way to estimate future changes in the value of a security based on certain predictive factors.
As noted earlier, lambda measures the theoretical percentage change in prices for each percentage change in implied volatility. Implied volatility (IV) is calculated using an option pricing model and determines what current market prices estimate the future volatility of an underlying asset. However, the implied volatility may deviate from the realized future volatility.