A stock price is currently 40. Over each of the next two...

A stock price is currently $$$40$$. Over each of the next two 3-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 12% per annum with continuous compounding. (a) What is the value of a 6 month European put option with a strike price of $$42$$? (b) What is the value of a 6-month American put option with a strike price of $$42$$?

Key Concepts

Intrinsic Value and Early Exercise for American Options

The concept of intrinsic value refers to the immediate exercise payoff of an option. For American options, which allow early exercise, the possibility of taking immediate advantage of intrinsic value creates a complex decision process where the value of exercising the option early is compared with the discounted expected payoff of holding the option, leading to the determination of optimal exercise strategies.

Continuous Compounding and Discounting

Continuous compounding is a method used to calculate the future and present values of cash flows by assuming that interest accrues continuously. When discounting expected option payoffs, the continuous compounding formula, which uses the exponential function, provides a more precise measurement of the time value of money over short periods, such as the 3-month intervals in this model.

European vs. American Options

European and American options differ primarily in their exercise rights; European options can only be exercised at expiration, while American options can be exercised at any time before expiration. This distinction affects the pricing, particularly for puts, where early exercise might be optimal under certain conditions, making American options generally more valuable than their European counterparts.

Risk-Neutral Valuation

Risk-neutral valuation is a method used in financial mathematics where all investors are assumed to be indifferent to risk, allowing the expected return on all securities to be set equal to the risk-free rate. In this approach, future payoffs are weighted by risk-neutral probabilities and discounted at the risk-free rate to compute their present value, simplifying the option pricing process.

Binomial Option Pricing Model

The binomial option pricing model is a discrete-time framework used to evaluate options by simulating possible future price paths of the underlying asset over multiple time steps. In each period, the asset price can move up or down by specified factors, and the model involves working backwards from the option’s terminal payoff to its present value by discounting expected values under risk-neutral probabilities.

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