Analyze Newton-Raphson Convergence, Choose Sigma, and Derive Vega

Q: Analyze Newton-Raphson Convergence, Choose Sigma, and Derive Vega

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Question

Context

You are given the market price of a European option and must infer the Black–Scholes implied volatility. This requires solving for the volatility parameter (\sigma) such that the Black–Scholes price equals the observed market price. Define the root-finding objective as

(f(\sigma) = P_{BS}(\sigma) - P_{mkt}), where (P_{BS}(\sigma)) is the Black–Scholes price (call or put) and (P_{mkt}) is the observed market price.

Tasks

Explain the Newton–Raphson method for solving (f(\sigma)=0) in this setting, and detail when it fails to converge (objective/derivative properties, step-size issues, edge cases).
Describe how to choose an initial volatility seed before running Newton–Raphson, including practical heuristics and justifications.
Write the closed-form Vega for a European option under Black–Scholes, define all variables and units, and explain how Vega’s magnitude affects Newton–Raphson updates.

Analyze Newton-Raphson Convergence, Choose Sigma, and Derive Vega

Context

Tasks

Solution (Locked)

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