Suppose that c1 and p1 are the prices of a European average...

Suppose that $c_1$ and $p_1$ are the prices of a European average price call and a European average price put with strike price $K$ and maturity $T, c_2$ and $p_2$ are the prices of a European average strike call and European average strike put with maturity $T$, and $c_3$ and $p_3$ are the prices of a regular European call and a regular European put with strike price $K$ and maturity $T$. Show that $c_1+c_2-c_3=p_1+p_2-p_3$.

Key Concepts

European Options

European options are financial derivatives that can only be exercised at their expiration date. Their pricing models, such as Black-Scholes, rely on the behavior of the underlying asset up to a fixed maturity, making them fundamental in understanding more complex option structures. They serve as a benchmark in the valuation and comparison of exotic options.

Asian Options

Asian options, often referred to as average options, have payoffs based on the average price of the underlying asset over a predetermined period rather than its price at a single point in time. This group includes both average price options, where the average is used in calculating the payoff, and average strike options, where the averaged value determines the strike price. Their structure helps reduce volatility effects and offers more stable outcomes over the option’s life.

Regular European Options

Regular European options are standard call and put options with fixed strike prices and maturity dates. Their valuation is directly tied to the difference between the asset’s price at expiration and the preset strike price. They provide a reference point for understanding how additional features, such as averaging in exotic options, alter option pricing.

Put-Call Parity

Put-call parity is a fundamental relationship in option pricing that links the prices of European calls and puts with identical strike prices and expiration dates. It is based on no-arbitrage arguments, ensuring that any combination of long and short positions in these derivatives will not result in a riskless profit. In extended contexts like options with averaging features, put-call parity principles still apply, confirming the consistency and balance in pricing different option structures.

No-Arbitrage Principle

The no-arbitrage principle is a core concept in financial economics asserting that in efficient markets it is impossible to earn a risk-free profit through trading strategies. This principle underlies all modern derivative pricing, including the put-call parity relationships, by ensuring that the option prices reflect inherent pricing consistency. It guarantees that various combinations of option contracts, even with additional features such as averaging, align to prevent arbitrage opportunities.

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