A stock price is currently 40. It is known that at the end...

A stock price is currently $$\$ 40$$. It is known that at the end of 3 months it will be either $$\$ 45$$ or $$\$ 35$$. The risk-free rate of interest with quarterly compounding is $8 \%$ per annum. Calculate the value of a 3-month European put option on the stock with an exercise price of $$\$ 40$$. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers.

Key Concepts

Binomial Option Pricing Model

The binomial option pricing model is a discrete-time framework used to price options by modeling the underlying asset's possible price movements over a series of time steps. By considering possible up and down movements of the asset's price, it creates a replicating portfolio that mimics the option's payoff. This method is especially useful for options with a short time horizon or nonstandard conditions, as it provides a clear structure to incorporate different possible outcomes in the valuation process.

Risk-Neutral Valuation

Risk-neutral valuation is a fundamental concept in financial derivatives pricing, where all investors are assumed to be indifferent to risk. Under this approach, the actual probabilities of outcomes are replaced with risk-neutral probabilities that ensure that the discounted expected payoff of a derivative equals its current price. This method simplifies computation by allowing the use of the risk-free rate for discounting, regardless of the investor's risk preferences.

No-Arbitrage Principle

The no-arbitrage principle asserts that in an efficient market, there should be no possibility of obtaining a risk-free profit through arbitrage. This principle underlies many pricing models and ensures that the prices of financial instruments, including options, are consistent with the costs of creating replicating portfolios. It is used in option pricing to derive the conditions under which the replicating portfolio exactly mimics the option's payoff, thereby ensuring no arbitrage opportunities exist.

European Options

European options are a type of financial derivative that can only be exercised at the expiration date, as opposed to American options which can be exercised at any time before expiration. This distinction simplifies the pricing of European options since the valuation does not need to account for early exercise possibilities, making them suitable candidates for models like the binomial option pricing model with terminal payoff considerations.

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