One basic assumption of Black-Scholes model Essays

One basic assumption of Black-Scholes model Essays One basic assumption of Black-Scholes model Essay One basic assumption of Black-Scholes model Essay 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? In our paper, we try to show how wrong Black-Scholes is by challenging this assumption and illustrate the difference between Black-Scholes and real world. Method used to exam mispricing problem of Black-Scholes model About Mixlognormal: The probability distribution of the stock price might be made up of a mixture of two lognormal distributions, one for the possibility of an increase in share price and the other one of a decrease. In this way, we can capture the empirical distribution of stock price; its shape must be more accurate and accordingly more likely to be the same as the distribution of the companys share price in the real world. (Precisely, if we can employ more lognormal distributions to obtain the possibility for the movement of share price, we would get better distribution to describe real world.) Therefore, we could simply test the accuracy of Black-Scholes model by comparison. About Data: We chose Six Continents as our target company to do our analysis and made a comparison between its mixlognormal distribution and Black-Scholes lognormal distribution. We chose Feb 19th as the big event date for the company, because there was a takeover bid from the management of Pizza Express on that day. The share price of Six Continents jumped more than 10% on Feb 19th and the trading volume increased more than 200%. The dates before and after the big event date are Feb 18th and Feb 20th respectively. Following this way, the mispricing drawback of Black-Scholes might be detected more easily due to the noise of big event. 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 companys share price declines in the value, the companys leverage increases. Thus the equity of the company becomes more risky and its volatility increases. When the companys 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. Appendix 1: The raw data of lognormal distribution for Six Continent options on 18th, Feb 2003. Trade 18-Feb-03 Maturity 16-Apr-03 r 0 T 0.1561644 F 554.5 sigma 0.4091905 mkt call strikes BS theory Sq Error moneyness weigted sq error 196 360 194.58 2.01 194.5000 0.0103 166.5 390 164.89 2.59 164.5000 0.0157 138 420 135.86 4.57 134.5000 0.034 101.5 460 99.5 3.98 94.5000 0.0421 69 500 67.93 1.14 54.5000 0.0211 38 550 37.88 0.01 4.5000 0.0032 17.5 600 18.77 1.62 45.5000 0.0356 7 650 8.34 1.8 95.5000 0.0188 2.5 700 3.36 0.75 145.5000 0.0051 1 750 1.25 0.06 195.5000 0.0003 0.5 800 0.43 0 245.5000 0 18.53 0.1862 Appendix 2: The raw data of lognormal distribution for Six Continent options on 20th, Feb, 2003. Trade 37672 Maturity 37727 r 0 T 0.15068493 F 615.5 sigma 0.36813434 mkt call strikes BS theory Sq Error moneyness weigted sq error 256.5 360 255.5013903 0.997221354 255.5 0.003903019 227 390 225.5132704 2.210364876 225.5 0.009802062 197 420 195.5835263 2.006397745 195.5 0.010262904 157.5 460 156.0819138 2.010968563 155.5 0.012932274 119.5 500 118.0715084 2.040588233 115.5 0.017667431 76.5 550 75.70383997 0.633870793 65.5 0.009677417 42.5 600 42.91727918 0.174121912 15.5 0.011233672 21 650 21.38061239 0.144865789 34.5 0.004199008 9 700 9.401761762 0.161412513 84.5 0.001910207 3.5 750 3.688784898 0.035639738 134.5 0.000264979 1 800 1.308782808 0.095346823 184.5 0.000516785 0.5 850 0.425705755 0.005519635 234.5 2.35379E-05 10.51631797 0.082393295 Appendix 3: The raw data of mixlognormal distribution for Six Continent options on 18th, Feb, 2003. Trade 18-Feb-03 Maturity 16-Apr-03 r 0 T 0.1561644 F 554.5 F1 613.42684 sigm1 0.0050015 F2 545.13413 sigma2 0.4331614 p 0.1371431 Cmarket X Cimplied (Cmarket-Cimplied)^2 Implied sigma by BS 196 360 194.6650299 1.782145105 0.605651935 166.5 390 165.1475618 1.82908915 0.53766516 138 420 136.4577704 2.378472184 0.502173454 101.5 460 100.7421585 0.574323782 0.454991983 69 500 69.60810609 0.369793012 0.424942163 38 550 38.86792249 0.753289445 0.410555737 17.5 600 17.00376787 0.246246324 0.393289282 7 650 6.928636732 0.005092716 0.38547187 2.5 700 2.914632833 0.171920386 0.382378979 1 750 1.14301875 0.020454363 0.394114927 0.5 800 0.422795028 0.005960608 0.417111769 8.136787075 Appendix 4: The raw data of mixlognormal distribution for Six Continent options on 20th, Feb 2003 Trade 20-Feb-03 Maturity 16-Apr-03 r 0 T 0.1506849 F 615.5 F1 631.64945 sigm1 0.005 F2 613.47651 sigma2 0.3942533 p 0.1113462 Cmarket X Cimplied (Cmarket-Cimplied)^2 Implied sigma by BS 256.5 360 255.5039835 0.992048772 0.684121728 227 390 255.5291768 2.163320768 0.634591317 197 420 195.6487533 1.825867624 0.544725925 157.5 460 156.3364319 1.353890697 0.457835271 119.5 500 118.6654525 0.696469529 0.411053745 76.5 550 76.50974266 9.49E-05 0.380098628 42.5 600 42.7445391 0.059799374 0.363618501 21 650 20.42471433 0.330953606 0.363939585 9 700 9.572591543 0.327861075 0.362150688 3.5 750 4.077689599 0.333725272 0.363395708 1 800 1.597153827 0.356592693 0.351554917 0.5 850 0.581926491 0.00671195 0.376206304 8.447336279