[Verse 1] Picture a stock price branching like a tree Two paths each period, up or down it goes From today's value, multiply by factors we see U for upward jumps, D when it slows Work backwards from the end, that's the secret key Calculate option values as the timeline flows [Chorus] Two periods, double the branches Up-up, up-down, down-up, down-down Backward induction never abandons The risk-neutral probabilities we've found Delta hedging through each node's chances Present value discounted, safe and sound [Verse 2] Start at expiration, intrinsic value's clear Max of stock minus strike, or zero if it's less Move one step backwards, discount what we hear Weight by probability, that's our success Early exercise? American options we revere Compare immediate payoff to our time-based guess [Chorus] Two periods, double the branches Up-up, up-down, down-up, down-down Backward induction never abandons The risk-neutral probabilities we've found Delta hedging through each node's chances Present value discounted, safe and sound [Bridge] Terminal nodes reveal the final picture Each path's payoff crystal clear to see But stepping backwards, we become the scripter Of fair value through probability European waits, but American's quicker Exercise early if optimal, that's the key [Verse 3] Risk-free rate discounts our future gains Volatility shapes our up and down Through every node, the no-arbitrage reigns Portfolio replication wears the crown Four final outcomes, but one price remains The present value we have tracked down [Chorus] Two periods, double the branches Up-up, up-down, down-up, down-down Backward induction never abandons The risk-neutral probabilities we've found Delta hedging through each node's chances Present value discounted, safe and sound [Outro] From simple single steps to complex trees The binomial model grows with ease
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