A stock price is currently 100. Over each of the next two...

A stock price is currently $$\$ 100$$. Over each of the next two 6-month periods it is expected to go up by $10 \%$ or down by $10 \%$. The risk-free interest rate is $8 \%$ per annum with continuous compounding. What is the value of a 1-year European call option with a strike price of $$\$ 100$$ ?

Key Concepts

European Call Option

A European call option is a financial derivative that gives the holder the right, but not the obligation, to purchase an underlying asset at a predetermined strike price on a specific expiration date. This instrument is commonly used in risk management and speculative strategies, and its valuation depends solely on the payoff at maturity, not on the path the asset price takes before expiration.

Binomial Option Pricing Model

The binomial option pricing model is a discrete-time framework used to evaluate options by modeling the underlying asset's price movements as a series of up or down steps over successive time intervals. At each node in the tree, the asset price moves either up or down by a specific percentage, and the option price is determined by working backwards from the final payoffs using risk-neutral probabilities.

Risk-Neutral Valuation

Risk-neutral valuation is a fundamental concept in financial mathematics that assumes all investors are indifferent to risk. Under this assumption, the expected return of the underlying asset is equal to the risk-free rate. This allows the use of risk-neutral probabilities to calculate the expected payoff of an option, which can then be discounted back to present value to obtain its price.

Continuous Compounding

Continuous compounding refers to the mathematical limit of the compounding process, where the interest is compounded an infinite number of times per period. In finance, it is often used to model the accumulation of interest in a smooth and mathematically tractable way. The risk-free rate with continuous compounding is applied to discount future cash flows or option payoffs back to their present value.

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