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Real-time theoretical pricing model and risk sensitivities (Delta, Gamma, Vega, Theta, Rho) for quantitative finance research and options trading interviews.
Rate of change of option price with respect to underlying spot price (∂V/∂S). Market makers use delta to construct delta-neutral hedges by buying/selling underlying shares.
Sensitivity of delta to moves in spot price (∂²V/∂S²). Highest for at-the-money (ATM) options approaching expiration, causing convexity risks for short option desks.
Sensitivity of option price to changes in implied volatility (∂V/∂σ). Long options profit from volatility expansion; options with longer maturities hold higher Vega.
Decay of option value over time as expiration approaches (∂V/∂t). Known as time decay. Options lose extrinsic value each passing day, accelerating near expiry.
Sensitivity of option value to shifts in the risk-free interest rate (∂V/∂r). Critical for interest-rate sensitive derivatives desks and long-dated LEAPS.
The fundamental no-arbitrage relationship connecting European call and put prices with identical strike and expiration: