Implied Volatility and Volatility Smile › Key Parameters

Key parameters for implied volatility computation and convergence analysis include:

• $S$: Spot price of the underlying asset, used in all Black-Scholes calculations. Observable from market quotes.
• $K$: Strike price of the option. Determines moneyness and convergence properties. Deep options present numerical challenges because vega is small.
• $T$: Time to expiration in years. Short-dated options are numerically challenging because vega becomes very small near expiration.
• $r$: Risk-free interest rate affecting present value calculations. Use rates matching the option's maturity, typically from Treasury yields or SOFR.
• market_price: Observed option price, the target value to match. Must lie within no-arbitrage bounds.
• initial_guess: Starting volatility estimate for Newton-Raphson, typically 0.2. Good starting points accelerate convergence.
• tolerance: Convergence criterion, typically 1e-8. Smaller values require more iterations but yield more accurate results.
• max_iterations: Maximum iterations before failing, typically 100. Prevents infinite loops.
• sigma_low and sigma_high: Bracketing bounds for bisection method. Must contain the true solution. The lower bound should be small and positive, and the upper bound should be large enough for any realistic volatility.