Despite the 2008-2009 financial crisis and other major failures due to the incorrect use of trading models, trading based on mathematical or quantitative models continues to gain momentum.

Complex trading instruments such as derivatives continue to be popular, as do the basic mathematical models of valuation. Although there is no perfect model, understanding its limitations can help make informed trading decisions, reject abnormal situations, and avoid costly mistakes that can lead to huge losses.

## Limitations of the Black-Scholes model

The Black-Scholes model has limitations. It is one of the most popular option pricing models. Some standard limitations of the Black-Scholes model are:

- Assume that the risk-free rate of return and volatility are constant during the term of the option. These may remain the same in the real world.
- Assume continuous and cost-free transactions-ignoring liquidity risks and brokerage fees.
- Assume that stock prices follow a lognormal pattern, such as a random walk (or geometric Brownian motion pattern), thereby ignoring the large price fluctuations that are more frequently observed in the real world.
- Assume that there is no dividend-ignore its impact on valuation changes.
- Assume that it is not exercised early (for example, only suitable for European-style options). This makes this model unsuitable for American options.
- Other assumptions that are operational issues include the assumption that there are no fines or margin requirements for short selling, no arbitrage opportunities, and no taxes. In fact, all of these are not true. Either additional funds are required, or the actual potential profits are reduced.

## The meaning of the Black-Scholes restriction

This section describes how the aforementioned restrictions affect daily transactions and whether any preventive or remedial measures can be taken.Among other problems, the biggest limitation of the Black-Scholes model is that although it provides a calculated price of an option, it still depends on underlying factors

- Suppose to be
**A known** - Hypothesis
**constant**During the validity period of the option

Unfortunately, none of the above is true in the real world. The underlying stock price, volatility, risk-free interest rate and dividends are unknown, and may change in the short term with large variances. This leads to large fluctuations in option prices. It does provide an important profit opportunity for experienced option traders (or those with good luck).

But this comes at a cost to peers-especially novices, speculators or punters, who are often unaware of these restrictions and on the receiving end.

### Avoid disaster

Not only must it be a high-amplitude change; the frequency of this change can also cause problems. Compared with the expected and implied price changes in the Black-Scholes model, large price changes are observed more frequently in the real world.

This higher volatility in the price of the underlying stock leads to large fluctuations in option valuations. This usually leads to disastrous results, especially for short option sellers, who may eventually be forced to close their positions with huge losses due to lack of margin, or be assigned American options when the buyer exercises them.

To prevent any high losses, option traders should always pay attention to changes in volatility and be prepared for a predetermined level of stop loss. Model-based valuation should be supplemented by realistic and predetermined stop loss levels. Intermittent remediation alternatives also include preparing average technology (cost and value in dollars) based on the situation and strategy.

### Real world view

As Black-Scholes assumed, stock prices will never show log-normal returns. The distribution in the real world is biased. This difference causes the Black-Scholes model to significantly underestimate or overestimate options.

Traders who are unfamiliar with this effect may eventually buy overpriced options or short underpriced options, thus incurring losses if they blindly follow the Black-Scholes model. As a precaution, traders should pay close attention to changes in volatility and market developments—try to buy when volatility is in a lower range (for example, as observed in the past duration of the expected option holding period) and when the volatility is in Sell at the low level to get the high range of the maximum premium.

**Coping with fluctuations**

Another meaning of geometric Brownian motion is that the volatility should remain constant during the duration of the option. This also means that the currency of options should not affect implied volatility. For example, ITM, ATM, and OTM options should show similar volatility behavior. But in fact, what is observed is a volatility skewed curve (rather than a volatility smile curve), where the higher the implied volatility, the lower the execution price.

Black-Scholes overestimated ATM options and underestimated deep ITM and deep OTM options. This is why most trades (and the highest open positions) are observed for ATM options and not for ITM and OTM.

Compared with the ITM and OTM options they are trying to take advantage of, short sellers get the maximum time decay value of ATM options (resulting in the highest option premium).

Traders should be cautious and avoid buying OTM and ITM options with a higher time decay value (partial option premium = intrinsic value + time decay value). Similarly, educated traders sell ATM options to obtain higher premiums when volatility is high, and buyers should seek to buy options when volatility is low, resulting in lower premiums paid.

### Extreme event

In short, it is assumed that price changes have absolute applicability and have no relationship or dependence on other market developments or market segments.

For example, the impact of the 2008-09 market crash that caused the overall market collapse due to the real estate bubble burst cannot be explained in the Black-Scholes model (and may not be explained in any mathematical model).

But it did lead to extreme low-probability events where stock prices fell sharply, causing huge losses to option traders. During the crisis, the foreign exchange and interest rate markets did follow the expected price pattern, but they were not completely immune.

**About dividends**

The Black-Scholes model does not consider changes caused by dividends paid by stocks. Assuming all other factors remain the same, a stock with a price of $100 and a dividend of $5 will drop to $95 on the ex-dividend date. Option sellers take advantage of these opportunities to short call options/call put options on the ex-dividend date and close their positions on the ex-dividend date to obtain profits.

Traders who follow Black-Scholes pricing should be aware of this effect and use alternative models, such as binomial pricing, which can explain changes in returns due to dividend payments. Otherwise, the Black-Scholes model should only be used to trade European non-dividend stocks.

The Black-Scholes model does not consider the early exercise of American options. In fact, based on market conditions, few options (such as long put positions) qualify for early exercise. Traders should avoid using Black-Scholes for American options or looking for alternatives, such as binomial pricing models.

## Why is Black Scholes getting so much attention?

There are several convincing reasons:

- It is very suitable for the popular non-dividend stock European option delta hedging strategy.
- It is simple and provides ready-made value.
- In general, when the entire market or most of the market is following it, the price is often calibrated based on the price calculated by Black-Scholes.

## Bottom line

Blindly following any mathematical or quantitative trading model will lead to uncontrollable risk exposure. The 2008-09 financial failure was attributed to the incorrect use of the transaction model.

Despite the challenges, the use of models still exists due to the continuous development of the market, the entry of various tools and the entry of new players. Models will continue to be the main basis for transactions, especially for complex instruments such as derivatives.

A prudent approach has clear insights into the limitations of the model, its impact, available alternatives and remedies, and can lead to safe and profitable transactions.

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