One basic assumption of Black-Scholes model is that the stock price is log-normally distributed with constant volatility. However, in option market, does this assumption hold?

Interpretation of the results

Using the Excel VBA programme, we got the Black- Scholes lognormal distribution and mixlognormal distribution for these two particular days. The lognormal distribution of Black-Scholes is shown in blue and the mixlognormal distribution is shown in pink.

Figure 1 Mixlognormal distribution and lognormal distribution for Six Continent options on 18th, Feb 2003.

Figure 2

(Figure 1 and 2 are consistent with each other. Both illustrate the option is mispriced by Black-Scholes.)

Comparing these two distributions (i.e. figure 1) of the date of Feb 18th, we find that:

1, the mixlognormal distribution has a fatter left tail and thinner right tail than the lognormal distribution (This phenomenon is also supported by empirical evidence when some people analysed the S&P 500 index), and

2. Both sides of the tails in mixlognormal distribution are longer as well.

3. The right tail is longer than the left tail.

For the first finding, it means that in the more accurate scenario, the volatility used to price a low strike price option (i.e. a deep-out-of-the-money put or a deep-in-the-money call) is higher than that used to price a high-strike-price-option (i.e. a deep-in –the-money put or a deep-out-of-the-money call). The second finding means that the Black-Scholes lognormal distribution chops some small probability events, but in fact, these small probability events do exist in real world.  The explanation of the third finding is that the bid proposal for Six Continent leaked before the announcement date. Investors have taken the probability of takeover into consideration and predict the price will go up. The probability of the share price going up is bigger than the probability of its going down, the right tail is longer.

Figure 3   Mixlognormal distribution and lognormal distribution for Six Continent options on 18th, Feb 2003.

Figure 4

Comparing the mixlognormal figure with the lognormal one (i.e. figure 3) of the date of Feb 20th, we find that:

1. This relationship of the two distributions is similar to those of the date of Feb 18th.
2. The left tail is still a little bit fatter and the right side a little thinner in the mixlognormal than that of lognormal.
3. The right tail is longer than of the left tail.

The interpretations of this figure are the same to the above ones.

From the above comparison and analysis, we can find the reason that Black-Scholes model is often shown to misprice out-of –the money and in-the-money options relative to their at-the-money counterparts: the assumption, which states that future stock price have a lognormal distribution and equivalently future returns have a normal distribution (which both results from the assumption of stock prices following a geometric Brownian motion.), is invalid. If this assumption is neglected, the risk neutral probability density does not necessarily fall into the family of lognormal distributions but can be shaped differently. This implies, that lognormal parametric estimation of risk-neutral probabilities leads to wrong result.

When we move the two mixlognormal distributions (i.e. figure 6) together and also the two Black-Scholes distributions (i.e. figure 5) together, it is obvious to find that beside the mean of the share price moving to high level, the volatility shape changes a little. The left tail of figure before the big event dates is much fatter than the left tail of the figure after the big event date.

Figure 5  Lognormal distributions of Six Continent options on 18th and 20th of Feb 2003.

Figure 6 Mixlognormal distribution of Six Continent options on 18th and 20th of Feb 2003.

This is understandable if we accept the explanation for the volatility smile when concerns leverage. It says that as a company’s share price declines in the value, the company’s leverage increases. Thus the equity of the company becomes more risky and its volatility increases. When the company’s share price increases, leverage decreases, then the equity become less risky and its volatility decreases.

Since the bid for the Six Continents was accepted as favourable news for the shareholders, the share price had a big jump. According to the above explanation, its shares become less risky and thus lower volatility, so the left tail becomes thinner.

Conclusion

We estimated risk neutral density of equity option prices and compared mixlognormal distribution and lognormal distribution before and after big event. We found that the assumption (stock prices are log-normally distributed) of Black-Scholes does not hold in the real world and this hole can make the options mispriced.