Suppose that the volatility of an asset will be 20 % from...

Suppose that the volatility of an asset will be $20 \%$ from month 0 to month $6,22 \%$ from month 6 to month 12 , and $24 \%$ from month 12 to month 24 . What volatility should be used in Black-Scholes-Merton to value a 2-year option?

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Key Concepts

Effective Volatility

Effective volatility is the constant volatility input that, when used in the Black-Scholes-Merton model, produces the same option price as that obtained using a variable volatility structure. It consolidates the impact of different volatility levels experienced over various time periods into a single representative value, ensuring that the aggregated risk over the option's life is accurately captured.

Weighted Variance Aggregation

Weighted variance aggregation is the technique used to compute the overall variance when an asset exhibits different volatilities over distinct time segments. This approach involves calculating the variance for each separate period, weighting each by the time duration it applies, then summing them up, and finally taking the square root of the time-averaged total variance to obtain the effective volatility utilized in models like Black-Scholes-Merton.

Black-Scholes-Merton Model

The Black-Scholes-Merton model is a foundational framework in financial mathematics for pricing European-style options. It assumes constant volatility, a constant risk-free interest rate, and a lognormal distribution for asset prices, allowing the derivation of a closed-form solution for option pricing under these assumptions.

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