The Greeks are used as risk measures that represent how sensitive the price of derivatives are to change. This is useful as risks can be treated in isolation and thus allows for tuning in a portfolio to reach a desired level of risk. The values are called 'the Greeks' as they are denoted by Greek letters. Each will be presented in turn as an introduction:

## Black-Scholes Formula and Python Implementation

The Black-Scholes model was first introduced by Fischer Black and Myron Scholes in 1973 in the paper "The Pricing of Options and Corporate Liabilities". Since being published, the model has become a widely used tool by investors and is still regarded as one of the best ways to determine fair prices of options.

## Implied Volatility Calculations with Python

Implied volatility $\sigma_{imp}$ is the volatility value $\sigma$ that makes the Black-Scholes value of the option equal to the traded price of the option.

Recall that in the Black-Scholes model, the volatility parameter $\sigma$ is the only parameter that can't be directly observed. All other parameters can be determined through market data (in the case of the risk-free rate $r$ and dividend yield $q$ and when the option is quoted. This being the case, the volatility parameter is the result of a numerical optimization technique given the Black-Scholes model.

## Put-Call Parity of Vanilla European Options and Python Implementation

In the paper, it is stated the premium of a call option implies a certain fair price for the corresponding put option (same asset, strike price and expiration date). The Put-Call Parity is used to validate option pricing models as any pricing model that produces option prices which violate the parity should be considered flawed.

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