[Verse 1] Picture a stock that's priced today Tomorrow it moves just two ways Up with a multiplier we call "u" Down with a "d" that's less than you'd choose The future's uncertain but we'll make it clear With math that turns chaos into something we can hear [Chorus] Up times down equals one, that's the binomial way Risk-neutral probability lights up the gray Replicate the payoff, match it dollar for dollar One-period model makes the complex holler S times u, S times d Price the option, set it free [Verse 2] Start with one hundred, up factor's one-point-two Down factor's point-eight, splits the future in two One-twenty or eighty, those are your only bets Now we price derivatives without breaking a sweat The stock tree branches like a fork in the road Each path has a payoff that we need to decode [Chorus] Up times down equals one, that's the binomial way Risk-neutral probability lights up the gray Replicate the payoff, match it dollar for dollar One-period model makes the complex holler S times u, S times d Price the option, set it free [Bridge] Build a portfolio, delta shares of stock Borrow at risk-free rate, make the prices lock In the up state and down state, payoffs must align That's replication magic, crossing every line Hedge ratio calculated, risk completely gone Arbitrage-free pricing from dusk until dawn [Verse 3] Risk-neutral measure, not the real-world odds Q equals risk-free minus d, divided by the difference that nods Multiply by up payoff, one minus q times down Discount at risk-free rate, the option price is found No matter your risk preference, the answer stays the same One-period binomial, master of the game [Outro] From stock to bond to option fair The tree reveals what's really there Up and down, the only way To price derivatives today
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