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

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

This question evaluates numerical root-finding and option pricing competencies, specifically understanding Newton–Raphson convergence properties, implied volatility inference under the Black–Scholes model, and option sensitivity (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

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

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Context

Tasks

Solution

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