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: **A. Incorrect** - Both BSM and binomial tree models typically use implied volatility as an input, not historical volatility. Historical volatility is backward-looking while option pricing models use forward-looking volatility expectations. **B. Incorrect** - Both BSM and binomial tree models assume that the expected return from the underlying asset is the risk-free rate. This is a fundamental assumption in risk-neutral pricing used in both models. **C. Incorrect** - In the binomial tree approach, delta changes at each node as the option's sensitivity to the underlying price changes. Delta is not constant throughout the tree. **D. Correct** - If the assumptions of the BSM model hold perfectly (including constant volatility), then the implied volatility should be the same for options with different maturities on the same underlying asset. This is because BSM assumes volatility is constant over time. The correct answer is D because under the BSM assumptions, volatility is assumed to be constant, so implied volatility should not vary with time to expiration.
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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.
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