Suppose that c1, c2, and c3 are the prices of European call...

Suppose that $c_1, c_2$, and $c_3$ are the prices of European call options with strike prices $K_1$, $K_2$, and $K_3$, respectively, where $K_3>K_2>K_1$ and $K_3-K_2=K_2-K_1$. All options have the same maturity. Show that $$ c_2 \leqslant 0.5\left(c_1+c_3\right) $$ (Hint: Consider a portfolio that is long one option with strike price $K_1$, long one option with strike price $K_3$, and short two options with strike price $K_2$.)

Suppose that $c_1, c_2$, and $c_3$ are the prices of European call options with strike prices $K_1$, $K_2$, and $K_3$, respectively, where $K_3>K_2>K_1$ and $K_3-K_2=K_2-K_1$. All options have the same maturity. Show that
$$
c_2 \leqslant 0.5\left(c_1+c_3\right)
$$
(Hint: Consider a portfolio that is long one option with strike price $K_1$, long one option with strike price $K_3$, and short two options with strike price $K_2$.)

Key Concepts

European Call Options

A European call option gives the holder the right, but not the obligation, to buy an underlying asset at a specified strike price at a predetermined future date. Understanding European call options is essential because their pricing, payoffs, and no-arbitrage properties form the building blocks for more complex option strategies and pricing inequalities.

Arbitrage Pricing Theory

Arbitrage pricing theory relies on the idea that if a portfolio is constructed so that its payoff is nonnegative in every scenario, then its initial cost must also be nonnegative to prevent risk-free profits. This principle is key when proving pricing inequalities, as it allows us to compare the cost of portfolios with similar payoffs.

Price Convexity

Price convexity in the context of option pricing refers to the property that the price of a European call option is a convex function of its strike price. This convexity implies that the option price at an intermediate strike is always less than or equal to a weighted average of option prices at surrounding strikes, a property that can be exploited to derive inequalities such as the one stated in the question.

Option Spread Strategies

Option spread strategies involve combining multiple options with different strike prices to construct portfolios with specific payoff patterns. In this question, a spread is created by being long calls with lower and higher strikes and short calls with an intermediate strike. Analyzing such spreads helps in understanding risk management, arbitrage opportunities, and the convexity of option prices.

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