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Option pricing is the process of determining the premium, or price, of an option contract. Options are primarily priced based on several variables such as the current stock price (S), the strike price (K), time to expiration (T), volatility of the underlying stock (\( \sigma \)), and the risk-free interest rate (r).
One key model used for pricing options is the Black-Scholes Model, which provides a theoretical estimate for the price of European-style options. However, when the price is not available from the model, it can also be deduced through put-call parity.
Put-call parity is a pivotal concept helping interrelate the prices of puts and calls, ensuring no arbitrage opportunities exist. When dealing with options where current pricing details aren't fully disclosed, put-call parity gives us a close approximation by using related market elements in its equation. This assures a fair comparison between market conditions and theoretical models.

Put-call parity is a fundamental principle in options pricing that defines a relationship between the prices of European call and put options with the same strike price and expiration. The principle is given by the equation \( C + Ke^{-rT} = P + S \), where \( C \) is the call option value, \( P \) is the put option value, \( K \) is the strike price, and \( S \) is the current stock price.
This parity ensures that no arbitrage opportunity can exist between the options, as the relationship maintains a balance between their respective prices.
When determining the price of an unknown option, this equation can be rearranged and used, assuming certain variables when necessary, to provide a necessary estimate of the option's value. This systematic approach ensures traders can leverage the parity to make informed decisions even when complete market data isn't fully available.

The Black-Scholes Model is widely used for pricing European call and put options. Introduced by Fischer Black and Myron Scholes in 1973, it allows investors to determine theoretical prices based on several variables.
The model incorporates factors like the spot price of the underlying asset, the option's strike price, the time until expiration, risk-free interest rate, and market volatility. The extensive use of this model is due to its ability to provide a theoretically fair valuation of options under the assumption of normal market conditions.
The basic formula for a call option is:

• \( C = S_0N(d_1) - Ke^{-rT}N(d_2) \)
• where \( N(d) \) is the cumulative distribution function of the standard normal distribution.

This is a groundbreaking model as it offers an analytical solution, unlike other methods which require extensive calculations or simulations.

The delta of a call option is a measure of how much the price of the option is expected to change when the price of the underlying stock changes. Essentially, delta is a ratio that compares the change in the price of the underlying asset to the corresponding change in the price of the option.
Mathematically, it is represented as \( \Delta = e^{-\delta T} * N(d1) \). This formula accounts for variables such as time decay and the distribution's cumulative function. For call options specifically, delta values range between 0 and 1. A delta of 0.5, for example, suggests that if the stock price increases by \(1, the option price is likely to increase by \)0.50.
Understanding delta is crucial for creating delta-hedged portfolios, where the goal is to neutralize market movement risk by buying or selling shares equivalent to the option's delta. This calculated adjustment helps investors maintain a balanced risk profile against volatility.