Answer-first summary for fast verification
Answer: If the assumptions of the BSM model hold, the implied volatility of a longer-term option and the implied volatility of a shorter-term option on the same underlying asset will be the same.
## Explanation Let's analyze each option: **Option A**: Incorrect. Both the Black-Scholes-Merton (BSM) model and binomial tree approach typically use implied volatility as an input. Historical volatility is not commonly used in either model for option pricing. **Option B**: Incorrect. Both models assume that the expected return from the underlying asset is the risk-free rate. This is a fundamental assumption in risk-neutral valuation used by both approaches. **Option C**: Incorrect. In the binomial tree approach, delta (the hedge ratio) changes at each node as the option price and underlying asset price change. Delta is not constant throughout the tree. **Option D**: Correct. If all the assumptions of the BSM model hold (including constant volatility), then the implied volatility should be the same for options of different maturities on the same underlying asset. This is because the BSM model assumes volatility is constant over time. Therefore, statement D is the correct one as it accurately reflects the constant volatility assumption in the BSM model.
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In comparing the two models, which of the following statements is correct?
A
The BSM model uses an underlying asset's implied volatility as an input but the binomial tree approach uses its historical volatility.
B
The binomial tree approach, but not the BSM model, assumes that the expected return from the underlying asset is the risk-free rate of interest.
C
In the binomial tree approach, delta is equal at each node since the probabilities of the price moving up or down during a period are constant and equal for both the underlying asset and the option.
D
If the assumptions of the BSM model hold, the implied volatility of a longer-term option and the implied volatility of a shorter-term option on the same underlying asset will be the same.