# Black-Scholes Model Assumptions

This page discusses the assumptions behind the Black-Scholes option pricing model and confronts them with the real world. When using the model to make actual trading decisions, it is important to be aware of the differences between the model and reality, and their implications.

## The Black-Scholes World

The model works with three different assets:

• The risky asset (such as a stock)
• The riskless asset (such as cash)
• The option on the risky asset, whose fair value we want to find

Each of the above trades on a market that has certain properties.

## Four Groups of Assumptions

In light of the above, we can classify the Black-Scholes model assumptions into four groups:

• Assumptions about the risky asset
• Assumptions about the riskless asset

## Assumptions about the Risky Asset

Most papers, books, and websites discussing the model, including the original 1973 paper by Fisher Black and Myron Scholes, refer to the risky asset as a “stock” and I will stay consistent with that, but keep in mind that the model applies to any kind of risky asset – stocks, bonds, currencies, commodities… – generally anything whose future price is uncertain and anything that can have options traded on it.

### 1. Random Walk

The first assumption, which may be familiar to you from various other financial models, is that the direction of the stock’s price can’t be consistently predicted and is completely random.

Price makes the so called random walk, a movement pattern also known as Wiener process (from mathematics) or Brownian motion (from physics). At any moment, the next move can be up or down, but we don’t know which way.

### 2. Constant Volatility

One thing we do know about the stock’s future moves though, is volatility – the general amplitude of the moves, or in other words how big or small moves we can generally expect. Under the Black-Scholes model, volatility is constant (doesn’t change in time) and known in advance.

This assumption is of course very problematic in the real world (volatility is neither constant nor known in advance). At the same time, volatility is one of the inputs for the model which have the greatest effect on the resulting option price.

### 3. Normal Distribution of Returns

As a result of the random walk price path (assumption 1 above), returns on the risky asset are normally distributed.

Probabilities of future percentage changes in the stock’s price over a given time period follow the so called bell curve, with small price changes close to the mean having relatively high probabilities and more extreme positive or negative price changes much less likely, with probabilities gradually approaching zero as you get further towards the tails.

Due to the relationship between returns and prices, when returns are normally distributed it means that the future stock price at a given point of time must be lognormally distributed.

Knowing the shape of price distribution and the standard deviation – the volatility (assumption 2), you can compute the exact probability of the stock ending up above a certain price at the option’s expiration.

The normal distribution assumption is also quite problematic in reality, as most risky assets’ return distributions show some degree of skewness and kurtosis. For instance, equity returns across different time horizons are typically negatively skewed (big downside moves are more likely than big upside moves) and have fat tails (extreme moves to either direction are more likely than normal distribution would suggest). This can be a serious problem for the practical use of the Black-Scholes model, as it would lead to underpricing of some options (such as more distant downside strikes) and overpricing of others. There are ways to address this issue – among the simplest and most popular fixes is the use of different volatility (standard deviation) inputs for different strikes (the so called volatility skew or smile).

### 4. No Dividends

Last assumption concerning the risky asset is that is pays no dividends or other distributions during the option’s life.

Dividends are paid to shareholders, but not to option holders, and share price typically decreases as a stock goes ex dividend (which decreases the attractiveness of call options and increases the attractiveness of puts). Therefore the value of an option on a dividend paying stock is different from the value of the same option if dividends are not paid.

The no dividend assumption would of course severely limit the usability of the original Black-Scholes model in the real world, especially for equity options, as many stocks do pay dividends (quarterly in the US). This shortcoming (among others) was addressed by Robert C. Merton in his 1973 paper, where he expanded the Black-Scholes model to also work with dividends. For his contribution Merton received the Nobel Prize in 1997 alongside Scholes (Black died in 1995).

## Assumptions about the Riskless Asset

Under the Black-Scholes model (and many other financial models), the riskless asset has two roles:

Firstly, it is an investment alternative to the risky asset or the option. If you have cash, you can use it to buy the stock, buy the option, or deposit it in a bank and earn interest. If you short the stock or the option, you receive cash from the buyer, deposit it and earn interest until you use it to buy back the stock or option.

Secondly, it is a source of financing. If you don’t have cash and want to buy a stock or option, you borrow cash and pay interest for that. This is technically the same thing as “going short” the riskless asset – you borrow it (cash) and exchange it for a different asset (stock or option).

The return earned (when long) or paid (when short) on the riskless asset is of course the risk-free interest rate.

### 5. Constant Risk-Free Interest Rates

The fifth assumption of the Black-Scholes model is that the risk-free interest rate is constant and known in advance.

In the real world this assumption appears to be much more realistic than constant and known volatility (assumption 2), but it is not that simple.

By “constant” the model means that the interest rate is exactly the same for borrowing and lending. In reality, the interest rate you pay when taking a loan is of course considerably higher than the interest rate you receive on a bank deposit or T-bill with the same maturity.

Furthermore, the model assumes the same interest rate regardless of amount. In reality, a large bank or fund borrowing or lending \$10,000,000 will most likely have access to more favorable interest rates than a retail investor with \$1,000.

That said, the effect of the interest rate input on most option prices is much smaller than the effect of volatility and other factors (unless interest rates are in the double digits like they were in the 1980’s). Therefore, the problems with the constant risk-free rate assumption are usually not as serious as the problems with the constant volatility or normal distribution assumptions.

### 6. European Options

The original Black-Scholes model was designed for options of European style, i.e. those that can be exercised only at expiration.

There are two main types of options: American and European. Their only difference is that the former can be exercised at any time until and at expiration, while the latter can only be exercised at expiration. This means an American options gives its holder the same rights as an otherwise identical European option, plus some more (the right to exercise early). Therefore, the price of an American option can never be lower than the price of an identical European option. It can be greater (if the right to exercise early has some value) or equal (if the right to exercise early is worthless).

In reality, the value of the early exercise right, and thereby the difference in American and European option prices, is often zero or very small. This allows the Black-Scholes model to also be used for pricing most American options with good accuracy.

There are exceptions: particularly deep in the money put options, or call options on high dividend stocks. These options can, under some circumstances, have negative time value and it can make sense to exercise them early. Therefore Black-Scholes model may be inaccurate for some American put options and for some American calls on dividend paying stocks – it may undervalue these options by some amount, depending on the combination of factors such as moneyness, time to expiration, interest rate, dividend yield and volatility.

The last group of assumptions concern the operation of the markets where all the three assets (the risky asset, the riskless asset, and the option) are traded.

### 7. No Transaction Costs

The Black-Scholes model assumes zero transaction costs.

Trades in the stock, the option, as well as all cash operations are subject to no commissions, no transfer fees, no option exercise or assignment fees.

This is again not the case in the real world, although the effect of transaction costs varies across markets and across market participants.

### 8. Perfect Liquidity

Another factor that varies across the real world markets and goes hand in hand with transaction costs is liquidity. The model assumes all markets are perfectly liquid.

It is possible to buy or sell any quantity at any time. It is even possible to trade fractions of shares – like 0.4317 shares of a stock. This is of course not the case in the real world.

### 9. No Restrictions to Short Selling

The above also concerns short selling, which the model assumes to be allowed and available on any asset at any time. To create a negative exposure to an asset’s price (profit from decline), you can borrow that asset, sell it in the market, and at a later time buy it back and return it to the lender.

In the real world short selling is subject to various restrictions. For instance, some markets have the so called uptick rule in place, which only allows initiating a short sale if the last trade in the stock has been on an uptick (last price move must have been upwards). In some countries and on some assets short selling may not be available at all. Even in cases when short selling is legal and theoretically possibly, it may be difficult to find a willing counterparty.

### 10. No Arbitrage Possible

I have left this assumption to the very end (because it fits under the market assumptions category), but it is really the main, underlying idea from which the entire model is derived.

It is assumed that the markets are efficient and therefore it is not possible to create a position that would have zero risk and sure gain greater than the return on the riskless asset. In other words, any riskless position must earn the risk-free interest rate. This assumption is very common in various derivative pricing models, which are sometimes referred to as no-arbitrage models.

In reasonably liquid markets, this assumption is actually quite realistic. Finding true arbitrage opportunities in the modern, electronically traded markets, is extremely hard.