When the Black-Scholes-Merton and binomial tree models are used to value an option on a non-dividend-paying stock, which of the following is true?

Question

BlackTom AI · Accepted Answer

The question asks about the relationship between the Black-Scholes-Merton (BSM) model and the binomial tree model when pricing an option on a non-dividend-paying stock.
Option 1: 'The binomial tree price converges to a price slightly above the Black-Scholes-Merton price as the number of time steps is increased.' This is not correct in the standard convergence behavior: as the number of steps grows, the binomial price approaches the BSM price, not necessarily staying above it. The direction of convergence is not guaranteed to be a persistent slight premium.
Option 2: 'The binomial tree price converges to a price slightly below the Black-Scholes-Merton price as the number of time steps is increased.' Similar to Option 1, this asserts a persistent bias below the BSM price. In general, the binomial model converges to the BSM value in the limit, and any bias (above or below) is not a theoretical guarantee; the convergence can approach from either side depending on discretization, but the standard result is convergence to the BSM price.
Option 3: 'Either A or B can be true.' This option claims that the direction of convergence is uncertain and could be above or below as steps increase. While finite-step binomial approximations can be biased depending on implementation details, the conventional and widely cited result is convergence to the BSM price in the limit of infinite steps, and there isn’t a universal rule that it must be above or below for finite steps.
Option 4: 'The binomial tree price converges to the Black-Scholes-Merton price as the number of time steps is increased.' This aligns with the fundamental theorem that the binomial model approximates the BSM model when the number of time steps goes to infinity (and thus, the binomial price converges to the BSM price for a non-dividend-paying stock under standard assumptions).
In summary, the correct understanding is that the binomial tree price converges to the BSM price as the number of time steps increases. The other options describe specific biased directions of convergence or claim ambiguity that is not generally supported by the standard convergence result.

2025FallDYN-T-FIN530-86763-86762 Quiz 6 - "Black-Scholes-Merton Model and Dynamic Hedging"

When the Black-Scholes-Merton and binomial tree models are used to value an option on a non-dividend-paying stock, which of the following is true?

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