The Black-Scholes-Merton (BSM) model is a widely used quantitative tool for pricing European-style options on underlying assets, such as stocks, indices, currencies, and commodities. The BSM model is based on the concept of a risk-neutral probability, which allows for the derivation of a mathematical formula for the fair value of an option. The BSM model assumes that the underlying asset follows a lognormal distribution and that the option can be perfectly replicated by a combination of the underlying asset and a riskless bond. The key assumptions of the BSM model are:
1. The underlying asset price follows a lognormal distribution: The BSM model assumes that the underlying asset price follows a random walk process and that the log returns of the asset are normally distributed. This assumption has been criticized for being too simplistic, as it does not account for the fat tails and skewness observed in many asset price distributions.
2. The option is European-style: The BSM model is designed to price European-style options, which can only be exercised at expiration. This assumption allows for the derivation of a closed-form solution for the option price, which is not possible for American-style options that can be exercised at any time prior to expiration.
3. No arbitrage opportunities exist: The BSM model assumes that there are no arbitrage opportunities in the market, meaning that it is not possible to make riskless profits by taking advantage of price discrepancies. This assumption is used to derive the risk-neutral probability measure.
4. The risk-free rate and volatility are constant: The BSM model assumes that the risk-free interest rate and the volatility of the underlying asset are constant over the life of the option. This assumption may not hold in practice, as interest rates and volatility can change over time.
5. No dividends are paid on the underlying asset: The BSM model assumes that the underlying asset does not pay any dividends during the life of the option. This assumption can be relaxed by adjusting the model to account for dividends.
Overall, the BSM model is a powerful tool for pricing options and is widely used by practitioners in the financial industry. However, it is important to understand the key assumptions of the model and their implications for option pricing in practice.