Assumptions of the Black-Scholes-Merton Option Valuation Model

The Black-Scholes-Merton (BSM) model is an optional pricing model. Under this model, the underlying share prices evolve in continuous time and are characterized at any point in time by a continuous distribution rather than a discrete distribution.

The following key assumptions underpin the BSM model:

1. The price of the underlying share follows a geometric Brownian motion. This implies that there are no jumps in share prices.
2. There are no risk-free arbitrage opportunities.
3. The risk-free rate of interest is constant, equal for all maturities, and identical for borrowing or lending.
4. The volatility of the return of the underlying is known and constant.
5. Unlimited short selling of the underlying is permitted.
6. No taxes or transaction costs are payable.
7. The underlying share can be traded continuously and in very small numbers of units.
8. Early exercise of the options is not allowed (BSM, therefore, can only be used to value European options).

These assumptions result in a complete market.

Question

Which of the assumptions of the Black-Scholes-Merton Model is least accurate:

1. There are no taxes or transaction costs.
2. The risk-free rate of interest is known and constant. It is the same for all maturities, borrowing, and lending.
3. Unlimited short selling is not allowed.

Solution

The correct answer is C.

Unlimited short selling is permitted. This means that we can sell securities that we do not own. This is a necessary assumption because to hedge a derivative whose price is positively correlated with that of the underlying asset – e.g., a call option, which will have a positive delta – we need to hold a negative quantity of the underlying asset.

A and B are assumptions of the BSM model. 

Reading 34: Valuation of Contingent Claims

LOS 34 (f) Identify assumptions of the Black–Scholes–Merton option valuation model.