[Verse 1] Picture a coin flip, heads or tails tonight Stock moves up or down, just two paths of flight Binomial tree branches, splitting the price One period forward, we calculate twice Risk-neutral world where investors don't care Expected returns match the risk-free share [Chorus] Call plus present value of strike equals put plus stock today Put-call parity shows us the arbitrage way Delta measures speed, gamma curves the bend Vega loves volatility, theta time's cruel friend Rho responds to rates while implied vol reveals What the market's pricing in these option deals [Verse 2] Black-Scholes-Merton needs assumptions clean Constant volatility and rates serene No dividends paid, continuous trade European style, no early exercise played Stock price follows geometric Brownian motion Log-normal distribution, that's the mathematical potion [Chorus] Call plus present value of strike equals put plus stock today Put-call parity shows us the arbitrage way Delta measures speed, gamma curves the bend Vega loves volatility, theta time's cruel friend Rho responds to rates while implied vol reveals What the market's pricing in these option deals [Bridge] Replicating portfolios hedge the risk away Buy delta shares and borrow cash to stay Volatility smile shows implied vol's not flat Out-of-money options trade fatter than that Greeks are sensitivities, partial derivatives true Measuring how option prices shift on cue [Verse 3] Two-period binomial builds complexity higher Backward induction, working through the entire Tree of possibilities, each node has its worth Risk-neutral probabilities give each branch its girth Implied volatility tells a different story Than historical measures of market glory [Outro] From binomial basics to Black-Scholes fame Options pricing models play the valuation game Greeks guide the traders, parity keeps things fair In the world of derivatives dancing in the air
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