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In actuarial economics, binary models occupy a central role in the valuation of financial derivatives, particularly options. They provide a discrete-time framework where the underlying asset price can evolve either upward or downward at each step, which gives them the name “binary.” Within this framework, two broad classes are recognized: the single period binary model and the multi-period binary model. While both are grounded in the same theoretical principles, their scope, application, and computational complexity differ significantly. Understanding these distinctions and the formulation of binary models is essential for determining the fair or “right” price of options and other contingent claims. The single period binary model is the most basic representation. It assumes that within one discrete time interval, typically denoted by Δt, the underlying asset price can move in only two possible directions: either up by a factor $u$ or down by a factor $d$. Let the current price of the asset be $S_0$. At the end of the single period, the price will either be $S_0u$ or $S_0d$. This simplicity makes the single period model extremely useful for illustrating the principles of arbitrage pricing, risk-neutral valuation, and hedging. For example, if one wants to price a European call option expiring at the end of this single period, the option payoff is contingent on whether the price moves up or down. The option payoff would be $\max(S_0u - K, 0)$ in the up-state and $\max(S_0d - K, 0)$ in the down-state, where $K$ is the strike price. The core idea is to determine the fair present value __________ _____ _____ __________ _______ __________ __________ ____ _______ _____.
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