Black-Scholes Option Pricing Model Case Study

Introduction
In 1973, Fischer Black and Myron Scholes first published the Black-Scholes Model in the paper, “The Pricing of Options and Corporate Liabilities”, published in the Journal of Political Economy. From this model, the Black-Scholes option pricing Model (BSM) was deduced as a means to price European options.
The simplicity of the use of the BSM allowed traders to effectively price and trade options and derivatives in markets all over the world. It is still widely used today, although with some modifications, by market participants.
This paper seeks to evaluate the strengths and weaknesses of the Black-Scholes option pricing model.

Options
To better understand the strength and weaknesses of the BSM, an understanding of how options
The VIX and S&P 500 Index (Bloomberg 2015)

Fig 1 shows the VIX index in White and the S&P 500 Index in Yellow over a 2 day period (23-24 Oct 2008). It can be clearly seen that the VIX is hardly constant, even in the short term, swinging from 67.80 to 89.53 (32.05% increase) from close of 23 Oct 2008 to the open of 24 Oct 2008. While the reaction in the VIX could be attributed to Alan Greenspan commenting that the financial crisis was a “once-in-a-century credit tsunami” in 2008, the movement in the VIX can make trading across borders and time-zones challenging.

• European Exercise terms
As mentioned earlier, a European option can only be exercised on the exercise date. This limitation is not a major concern, as few American calls are exercised before the exercise date, thereby forfeiting the time value. However towards the end of the life of an American call, the remaining time value is small, while intrinsic value is the same as a European option (Kwan, 2014)

• No commission is charged
The BSM assumes that there are no fees in trading, but in reality traders and brokers charge spreads on such trades. Occasionally, these fees can be large enough to skew the result of the BSM.

• No dividends
Limitations of the BSM in pricing Convertible Bonds
The BSM offers a good framework to begin in valuing such an option. However, it must be noted that CBs often make use of options with American terms, which the original BSM is unable to price. Han and Wu (2003) suggest a modification of the BSM to a standard forward diffusion equation over an infinite domain.

Another assumption used in the BSM is that volatility is known and constant. However, if the company that issued the CB is young, such volatility data might not be readily available and reliance on historical volatility may not be appropriate in real world applications. Also, as shown in Fig 1 above, implied volatility is not constant and can vary greatly even in one day.

Conclusion
When the BSM was developed in 1973, there were only 16 stocks on which calls were traded on. The market was also “young” and many of today’s structured products that made use of derivatives did not exist. The volume of derivative trades was also relatively limited and as such, the assumptions taken by the BSM may have been relevant