A stock price is currently 50. It is known that at the end...

A stock price is currently $$\$ 50$$. It is known that at the end of 6 months it will be either $$\$ 60$$ or $$\$ 42$$. The risk-free rate of interest with continuous compounding is $12 \%$ per annum. Calculate the value of a 6-month European call option on the stock with an exercise price of $$\$ 48$$. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers.

Key Concepts

Replicating Portfolio

A replicating portfolio is constructed by combining positions in the underlying asset and risk-free bond in such a way that its payoff exactly mimics the option's payoff. This concept underlines the idea that if two portfolios have identical future payoffs, they must be priced equally today, reinforcing the no-arbitrage condition.

Binomial Option Pricing Model

This model represents the evolution of an asset's price over a single time period by assuming two possible outcomes, typically 'up' and 'down' moves. It forms the basic framework for calculating derivative prices, where the option’s value is determined based on these possible future states and the probabilities of each occurring.

Risk-Neutral Valuation

Risk-neutral valuation involves adjusting real-world probabilities to a 'risk-neutral' framework, where all assets are assumed to grow at the risk-free rate. In this method, the expected payoff of an option is computed under these probabilities and then discounted at the risk-free rate to yield its current value, ensuring a consistent and arbitrage-free price.

No-Arbitrage Principle

The no-arbitrage principle asserts that it should not be possible to secure a risk-free profit with zero net investment. In option pricing, this concept ensures that the price derived from replicating portfolios or risk-neutral methods is consistent, as any discrepancy would allow traders to exploit price differences and earn arbitrage profits.

Discounting with Continuous Compounding

Discounting with continuous compounding involves converting future cash flows to their present value using an exponential decay factor based on the risk-free rate. This method is crucial for pricing derivatives in a continuous-time framework, and it is especially relevant when interest rates are quoted on a continuous compounding basis.

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