Binomial Pricing of a European Call Option

This question introduces a one-period binomial model for pricing a European call option on a single stock. The setup is deliberately simple: the stock can move to one of two possible future prices over a fixed horizon, and the risk-free rate is given. The candidate is asked to determine the fair value of a call option with a specified strike, using this discrete-time framework. Variants of this problem often appear in entry-level quant, trading, and risk interviews, as well as in junior derivatives roles, because it connects a very basic market model to a concrete pricing task without requiring heavy mathematics.

The solution leans on core ideas from arbitrage-free pricing: constructing a replicating portfolio, or equivalently, switching to a risk-neutral measure. It requires comfort with linear payoffs, solving simple two-equation systems, and discounting expected payoffs at the risk-free rate. Interviewers look for an understanding that real-world probabilities are irrelevant for pricing, the ability to derive risk-neutral probabilities or hedge ratios correctly, and a clear explanation of why the resulting price eliminates arbitrage opportunities.

What it tests

In pricing derivative securities like options, the key is to value them under the risk-neutral measure, not the real-world probability distribution. The risk-neutral measure is constructed so that the expected return of all traded assets, when discounted at the risk-free rate, matches their current market price. This ensures no arbitrage opportunities exist in the market. The risk-neutral probability is not a prediction of actual outcomes but a mathematical tool that adjusts for risk preferences and market completeness. This approach works because, under no-arbitrage, any replicable payoff must have a unique price, regardless of investors' subjective beliefs.