A 1-month European put option on a non-dividend-paying stock...

A 1-month European put option on a non-dividend-paying stock is currently selling for $$\$ 2.50$$. The stock price is $$\$ 47$$, the strike price is $$\$ 50$$, and the risk-free interest rate is $6 \%$ per annum. What opportunities are there for an arbitrageur?

Key Concepts

Arbitrage Strategy Construction

When an option is mispriced relative to its theoretical bounds, an arbitrage strategy can be constructed. This typically involves buying the undervalued option and simultaneously taking offsetting positions in the underlying asset or risk-free securities to lock in a risk-free profit as prices converge to arbitrage-free levels.

Discounting in Option Pricing

When pricing options, future cash flows — such as the strike price paid at expiration — must be discounted back to their present values using the risk-free rate. This adjustment accounts for the time value of money and is critical in deriving correct pricing bounds for options.

No-Arbitrage Principle

This is the fundamental finance concept that securities must be priced so that there is no opportunity to earn a risk?free profit by trading them. In options pricing, it implies that mispricings—where the option is priced below or above established theoretical bounds—create opportunities for arbitrageurs to construct trading strategies that lock in a profit without risk.

Lower Bound for European Put Prices

For a European put option, the no-arbitrage lower bound is defined by the formula P ? Ke^(–rT) – S, where K is the strike price, r is the risk-free rate, T is the time to expiration, and S is the underlying asset’s price. This bound ensures that the option's price does not fall below a level that would make immediate profit possible by constructing a replicating portfolio.

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