How does the Black-Scholes model function?
The Black-Scholes options pricing model proposes that financial instruments like stock or futures contracts will possess a lognormal price distribution following a random walk alongside volatility and constant drift. By employing this conjecture and taking into account other useful variables, the equation deduces the price of a European-based call option.
This model needs 6 variables: an option (call or put), volatility, the underlying asset’s price, the option’s strike price, the expiration of the option, and a risk-free rate of interest. Using these variables, it is theoretically viable for options sellers to establish reasonable prices for the options they are trying to sell.
Additionally, the model forecasts that the price of frequently traded assets adheres to a geometric Brownian motion with consistent volatility and drift. When it is applied to a stock option, the BSM model uses the constant price movement of the stock, the money’s time value, strike price of the option, and option’s expiration date.
Black-Scholes model assumptions
The following are the assumptions made by the Black-Scholes model:
- Zero dividends are paid during the lifespan of the given option.
- No transaction expenses during the purchase of the option.
- Markets are random, i.e., movements in the market cannot be forecasted.
- The volatility and risk-free rate of the underlying asset are known and unchanging.
- The underlying asset’s returns are distributed normally.
- The given option is a European-style option and can only be exercised on the date of expiry.
Initially, the Black-Scholes options pricing model did not factor in the impact of dividends paid during the option ‘s lifetime. However, the model is often customised to consider dividends by computing the underlying asset’s ex-dividend date value. It is also altered by several option sellers to include the outcomes of options that can be exercised before expiry date.
Additional read: Options and derivatives
The Black-Scholes model formula
The Black-Scholes model formula is given by:
C(S, t) = N(d1)S – N(d2)Ker t
Where:
d1 = (ln (S/K) + (r +( σ_v^2)/2)t)/(σ_s √t)
And d2 = d1 - σ_s √t
Also, C = Call option price
S = current stock price (or underlying price)
K = Strike price
r = Risk-free interest rate
t = Time to maturity
N = Normal distribution
What are the advantages of the Black-Scholes model?
The Black-Scholes model (BSM) has been instrumental in transforming options pricing, offering a range of benefits for traders and investors. Its advantages include:
1. Structured model:
The Black-Scholes model offers a theoretical framework for options pricing. This enables individuals to compute fair prices of options based on a framework that uses a defined and sound methodology.
2. Risk management:
Investors can mitigate their portfolio risk by determining the theoretical value derived from the Black-Scholes model. Therefore, it is productive in not only assessing possible returns but also in gauging any weaknesses in the portfolio and investment gaps.
3. Portfolio optimisation:
The BSM model provides a measure for predicated returns and risks with varying options which gives investors a chance to optimise portfolios maximising their opportunities for profits and minimise their risk.
4. Market efficiency:
The Black-Scholes model has contributed to an enhanced efficiency and transparency in capital markets as participants are able to price and trade options in a more reliable manner.
5. Streamlined pricing:
Since the BSM model is widely acknowledged and applied by market participants, it ensures a high level of comparability and consistency across diverse markets.
Additional read: Shares
What are the limitations of the Black-Scholes model?
Despite its widespread application, the Black-Scholes model has several limitations that may restrict its accuracy in real-world scenarios:
1. Restricted application:
The Black-Sholes model can only be used for pricing European-oriented options and does not take into account varying option instruments of countries like the United States that can be exercised before their expiry.
2. Limited cash flow flexibility:
While the model assumes risk-free interest rates and dividends as constants, this might not be necessarily accurate in real-world applications. Ergo, the BSM model falls short when it comes to precise reflection of cash flow of an investment in the future owing to its rigid framework.
3. Constant volatility:
Likewise, the model assumes that the volatility remains unchanged throughout the option’s lifespan. However, in reality, this assumption does not hold a practical approach, as the volatility keeps oscillating with the demand and supply forces.
4. Additional misleading assumptions:
Besides the aforementioned assumptions, the model also includes assumptions such as risk-free interest rate, no transaction costs, or no riskless arbitrage probabilities. Derived prices may hence deviate from the actual results due to each of these assumptions.
Additional read: OTC derivatives
Closing thoughts
The Black-Scholes model remains a pivotal concept in contemporary financial theories. A mathematical calculation, it approximates the potential value of derivatives by factoring in additional investment instruments and considering the effects of time and various risks. While this model has multiple benefits, including portfolio optimisation and high levels of efficiency, it comes with some critical setbacks due to its rigid framework and limited application. Therefore, investors and traders should carefully study this model to ensure its correct application to amplify gains and cut down on losses.
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