###### Calculating Expected Returns from the ...

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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:

- The price of the underlying share follows a geometric Brownian motion. This implies that there are no jumps in share prices.
- There are no risk-free arbitrage opportunities.
- The risk-free rate of interest is constant, equal for all maturities, and identical for borrowing or lending.
- The volatility of the return of the underlying is known and constant.
- Unlimited short selling of the underlying is permitted.
- No taxes or transaction costs are payable.
- The underlying share can be traded continuously and in very small numbers of units.
- 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:

- There are no taxes or transaction costs.
- The risk-free rate of interest is known and constant. It is the same for all maturities, borrowing, and lending.
- 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.*