Key assumptions of the Black-Scholes model

By roma, 17 August, 2020

The model has some important simplifying assumptions about the world around us. However, in periods of imbalance and panic in the financial market, these assumptions are completely meaningless, which distorts the fair value of options. Read the “Intrinsic and time value of an option”. Below we will discuss in more detail the main assumptions of the Black Scholes model.

 

Assumption #1: Normal return distribution for the underlying asset

The return (hourly, daily, weekly, etc.) of the underlying asset has a normal distribution, shaped like a bell. However, many option traders and analysts note the presence of some asymmetry (right tail is longer than left), especially in the US stock market. This means that in practice there is a higher probability of negative returns than is assumed by the normal distribution function. Therefore, the Black-Scholes model may in some cases overestimate the value of call options and underestimate put options.

Bell Bell

 

Assumption #2: Continuous random price volatility of the asset

The profitability of the underlying asset behaves according to the theory of continuous random walk theory, where the last movement (rise or fall) of the asset price has no relation to the future price movement of the asset. In this case, the model does not take into account possible price surges on the underlying asset, which often occur during important news and between the closing and opening of trading on the exchange. Especially such leapfrogging behavior falls on crisis periods (2008 - 2009).

 

Assumption #3: Volatility is constant and known in advance

The model assumes that the future price volatility of the underlying asset is known in advance and remains unchanged until the option is exercised. However, history shows that during market collapses, panic sets the tone and implied option volatility can increase significantly.

In practice, option traders can compensate for the model's limitations by changing the volatility at which they are willing to buy or sell options. For example, if the underlying asset is illiquid and has a wide bid-ask spread, hedging the delta becomes unprofitable. To compensate for potential losses from delta hedging (i.e. trading the underlying asset to reduce the delta), a trader may increase the selling price of an option by increasing the implied volatility, or reduce the option's purchase price by reducing the implied volatility. Also, a trader can develop a more complex pricing model that takes into account fluctuations, changes in volatility, and other limitations of the Black Scholes model.

 

Assumption #4: No transaction fees and perfect liquidity

Delta hedging, which is done by selling and buying the underlying asset, is possible without transaction costs and liquidity problems. However, in the real world, traders are required to pay transaction costs, and liquidity difficulties often arise.  Therefore, permanent continuous hedging of the delta is impossible. Typically, professional participants in the options market hedge the delta only when it reaches a certain level relative to the size of gamma.

You can read about practical aspects of delta hedging strategy in Delta Hedging and gamma of options portfolio.

On the night of December 15-16, 2014, the Central Bank of Russia raised the key interest rate by 6.5% to 17%, which disconcerted the market. As a result, liquidity disappeared from almost all markets, and hedging became much more expensive. It should be noted that over the last ten years, global investors have increasingly come across non-standard situations

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