QUADERNI DI DIDATTICA - unive.it · Quaderno di Didattica n. 31/2009 Maggio 2009 I Quaderni di...
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Quaderno di Didattica n. 31/2009 Maggio 2009 Department of Applied Mathematics, University of Venice QUADERNI DI DIDATTICA Aleksandra Anusik, Paolo Pianca The research into the impact of the uncertain factors on the Black-Scholes model
Transcript of QUADERNI DI DIDATTICA - unive.it · Quaderno di Didattica n. 31/2009 Maggio 2009 I Quaderni di...
Quaderno di Didattica n. 31/2009 Maggio 2009
Department of Applied Mathematics, University of Venice
QUADERNI DI DIDATTICA
Aleksandra Anusik, Paolo Pianca
The research into the impact of the uncertain factors on the Black-Scholes model
Quaderno di Didattica n. 31/2009 Maggio 2009
I Quaderni di Didattica sono pubblicati a cura del Dipartimento di Matematica Applicata dellUniversit di Venezia. I lavori riflettono esclusivamente le opinioni degli autori e non impegnano la responsabilit del Dipartimento. I Quaderni di Didattica vogliono promuovere la circolazione di appunti e note a scopo didattico. Si richiede di tener conto della loro natura provvisoria per eventuali citazioni o ogni altro uso.
The research into the impact of the uncertain factors
on the Black-Scholes model
ALEKSANDRA ANUSIK
Department of Operations Research
University of Lodz
Department of Operations Research
University of Lodz
90-214 Lodz, Poland
Rewolucji 1905 r., 39
http://www.kbo.uni.lodz.pl/
This work has been partly written during the studies at the Department of Applied
Mathematics of the University Ca Foscari Venice as a part of the scholarship spent
in Italy.
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ALEKSANDRA ANUSIK
The research into the impact of the uncertain factors on the Black-Scholes model
Abstract. This paper discusses the impact of the uncertain factors on the parameters d1 and d2 and the value of option calculated from the Black-Scholes formula. Especially, we pose the question of how the introduction of the disturbances into the Black-Scholes sigma parameter (the volatility of stock prices - ) affects the parameters d1 and d2 and the value of option. To find it out, the stochastic simulation is used. According to the empirical results, the uncertainty of estimates of the disturbed volatility of stock prices has a strong impact on the distributions of d1 and d2 parameters but it causes rather small changes in the value of option. In addition, the results suggest that the intensity of the disturbances introduced to the model play the key role. The bigger the disturbances, the more asymmetric and leptokurtic the distributions of d1 and d2 parameters, and thus, the bigger a dispersion of the disturbed option prices. Keywords: option pricing, Black-Scholes model, d1 and d2 parameters, option value, stochastic simulation. 1 Introduction
Option pricing is one of the most important issues while dealing with this sort of terminal assets. At present, the probabilistic Black-Scholes model is the most popular and widely used to this end. Different approaches have been proposed in the literature in order to analyze the properties of this model. However, the findings of empirical studies are mixed. Some studies show that when estimates of the variance of the underlying asset are high (low), model prices tend to be higher (lower) than market prices. In particular, Black and Scholes [7] find that their model usually overprices options on stocks with high variances and undervalues options on stocks with low variances. They also find that it undervalues deep out of the money options which are near to extinction and overprices deep in the money options. Thus, one can conclude that in general the Black-Scholes model undervalues out of the money options relative to in the money options and overprices in the money options relative to out of the money options. However, contrary to these findings, studies of Macbeth and Merville [21] show that the probabilistic model undervalues in the money options and overprices out of the money options.
What is more, among others, Blattberg and Gonedes [8] and Bollerslev, Chou and Kroner [9] present evidence that volatility changes through time. According to them observed rates of return on common stock can be characterized as independent drawings from a normal population with presumably constant mean but changing volatility. It seems that for this reason many authors have recently checked the influence of stochastic volatility of underlying instruments on the value of option calculated from different formulas.
A number of researchers provided closed-form solutions for the price of European call option when volatility is stochastic. Hull and White [17], Scott [23] and Wiggins [27] were among the first to develop option pricing models based on stochastic volatility. Hull and White, as well as Scott, made the assumption that the risk
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premium of volatility is zero (the volatility risk is not priced in the options market) and that volatility is uncorrelated with the returns of the underlying asset. Hull and White show that when a stochastic volatility is uncorrelated with the security price, the Black-Scholes price overvalues at the money options and undervalues deep in- and out of the money options. When there is a positive correlation between the stock price and its volatility, out of the money options are undervalued by the Black-Scholes formula while in the money options are overpriced. When the correlation is negative, the effect is reversed. As for Wiggins, who also assumed a zero-volatility risk premium, he finds that the estimated option values under stochastic volatility are not significantly different from Black-Scholes values, except for long maturity options. It is also worth noticing that for equity options Lamoureux and Lastarapes [18] offer evidence against the assumption of a zero-volatility risk premium.
There are also many researchers who assume that volatility evolves according to a mean reverting process and permit the correlation between return and volatility to be different from zero (see Bakshi et al [4], Ball and Roma [5]). But it was Heston [16] who first developed a stochastic-volatility option pricing model for European equity and currency options that can be easily implemented, is computationally inexpensive and allows for any arbitrary correlation between asset returns and volatility. The empirical work done on Heston's model find that it is able to generate prices that are in closer agreement with market option prices than those of the Black-Scholes model. However, it is not the case that this model is able to explain all biases of the Black-Scholes formula.
Also, the latest research concentrate on pricing of options with stochastic volatility. In recent years, most researchers improve already existing stochastic volatility models or construct completely new formulas rather than check the properties of the Black-Scholes model itself. For example, Antonelli and Scarlatti [1] propose a new approach for solving the pricing equations of European call options for stochastic volatility models, including the Hull and White and the Heston models (See also Guo and Hung [14], Thavaneswaran et al [26]). On the other hand, Rogers and Veraart [22] present two new stochastic volatility models in which option prices for European vanilla options have closed-form expressions (See also Barone-Adesi et al [6]).
2 The probabilistic Black-Scholes model
For the time being, the Black-Scholes model is the most popular tool used in a calculation of option prices. Its main assumptions are:
option is the European vanilla call option on stocks not paying dividends; the price follows a geometric Brownian motion with constant drift and volatility; it is possible to borrow and lend cash at a known constant risk-free interest
rate; returns on the underlying stock are lognormally distributed; markets are efficient, no commissions are charged, there are no transaction
costs, there are no arbitrage opportunities; there are no restrictions on short selling; all securities are perfectly divisible, there is a continuous trade of these
securities [12]. Short selling is the practice of selling a financial instrument that the seller does not own at the time of the sale. Short selling is done with the intent of later purchasing the financial instrument at a lower price.
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As for the model assumptions mentioned above and the fact that the value of call option depends on the time to maturity, the price of call option is given by:
( ) ( )21 dNeKdNSC Tr = (1) for:
( )T
TrKS
d
++=
)21(ln 2
1 (2)
Tdd = 12 (3) where: C the price of the European call option; S the current price of the stock; K the strike price of the option; T time to the expiration of the option (in years); the volatility of the stock (the standard deviation of stock returns); r the risk-free interest rate; e - the base of the natural logarithm; ln() the natural logarithm of the argument; N(d1), N(d2) the cumulative standard normal distribution for d1 and d2, respectively.
The cumulative standard normal distributions N(d1) and N(d2) are the probabilities of random variables not exceeding the values assigned by d1 and d2 parameters. They fit in the bracket (0,1). Thus, N(d2) can be interpreted as the probability that the final stock price will be above K and the option will be exercised. What is more, if e-rT is a discount rate of the capitalization of the interest rate (r) in time to maturity (T), the whole expression Ke-rT is a current discounted strike price of the option. Finally, the expression SN(d1) is the expected return from call option when it is in the money [14].
As one can see, the main problem in the Black-Scholes model are the estimates of the volatility of stock prices ( ). The precision with which the standard deviation of stock returns is estimated influences the localization of d1 and d2 parameters and consequently the value of option. In this paper to measure price fluctuations over time a historical volatility is used:
( )N
n
zzn
ii
=
=
1 1
2
(4)
for i = 1, 2, ..., n, where the natural logarithmic price change is given by the formula:
=
1
lni
ii S
Sz (5)
where :
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the estimated standard deviation of stock returns; n number of historical stock prices; z mean for the natural logarithmic price change; N number of sessions during one year; Si, Si-1 stock prices in time i and i 1, respectively.
If the value of European non-dividend paying call option is already calculated, for put option of the same expiry (T) and strike price (K) the put-call parity formula can be used. It is assumed then that the investment is risk-free and consists of buying the stock together with put option on this stock and selling call option on the same stock. Consequently the value of put option can be calculated as:
TreKSCP += (6) or
( ) ( )12 dNSdNeKP Tr = (7)
where: P the price of the European put option; rest of the designations as before [28]. 3 Numerical experiments
As it was mentioned before, the purpose of this paper is to show some properties
of the Black-Scholes formula which prices options on stocks not paying dividends. In the Black-Scholes model the volatility of stock prices ( ) is estimated and its value is uncertain. It is worth checking how the introduction of the disturbances to the volatility affects the Black-Scholes parameters d1 and d2 and the value of option. To do this the disturbances are introduced into the estimated volatility. Then the disturbed volatility is used to calculate disturbed d1 and d2 parameters and disturbed prices of call and put options. In such a way it is checked how strong is the impact of the uncertainty of estimates of the disturbed sigma ( ) on the distributions of d1 and d2 parameters and option value as well. 3.1 Description of the experiment
We consider pricing of options on stocks of four chosen companies listed on Eurex Stock Exchange. We assume that the stocks do not pay any dividends and all options have five months to the expiration date. In the experiment it is also assumed that the risk-free interest rate equals r = 0,0388 while the value of sigma ( ) is estimated according to the formula (4).
To find out what the reaction of d1 and d2 parameters to the uncertainty of estimates of the sigma parameter is, disturbances () are introduced into the volatility of stock prices and its 5000 disturbed values )( j + are calculated. The volatility is disturbed under the assumption that the disturbances introduced into it have a normal distribution. To randomise these disturbances, the analytic method of inverting the distribution function is used. Values of random variables (u1) are given by the random number generator with the continuous uniform distribution which is built in the LOS procedure of Microsoft Excel [29].
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The whole stochastic simulation is conducted in Microsoft Excel. Results of the experiment are given for three cases when disturbances have normal distribution N(0; )var( ) and equals consecutively: 1, 0,3 and 0,03. The determines uncertainty of the researcher about the sensitivity of d1 and d2 parameters to the strength of the disturbances. In this paper the sensitivity of d1 and d2 parameters is analyzed mainly for big disturbances (=1). The impact of medium and small disturbances (=0,3, =0,03) is shown only for comparison. As for the symbol )var( it is the variance of the estimator of the variance of stock returns. To estimate it, the sample with n observations was divided into four samples with n/4 observations each and the variance of sigma was estimated in this four samples.
Then, the disturbed volatility of stock prices is used in formulas for d1 and d2. Thus, 5000 disturbed values of both parameters are obtained. In next step the disturbed d1 and d2 parameters are used to calculate the disturbed values of options. In this paper the diversity of the disturbed option prices is described by the range. As it was mentioned above, the maximum and minimum disturbed option values are shown for three cases: when the strength of the disturbances is big, medium or small (=1, =0,3, =0,03). In such a way it is shown how big the deviation of the disturbed values of options from their undisturbed prices is.
Results of the experiment are analyzed and described. To this end, such statistical measures as mean (M), standard deviation (SD), skewness (S) and kurtosis (K) are used. There are also implemented chosen tests for normality of distributions of d1 and d2 parameters. Then, the mentioned distributions are presented on graphs (histograms and normal probability plots). Data about how many observations fit into the bracket: mean +/- ; mean +/- 2 , mean +/- 3 is given as well. Finally, the sensitivity analysis of option prices is made. 3.2 Results of the experiment
Exemplary values of mean (M), standard deviation (SD), skewness (S) and kurtosis (K) for the distributions of d1 and d2 parameters are shown in Table 1. All values are presented for three cases: =1, =0,3 and =0,03. Table 1: Mean (M), standard deviation (SD), skewness (S) and kurtosis (K) for the distributions of d1 and d2 parameters from the Black-Scholes formula on options on stocks of Allianz Se (A), SAP (SAP), Deutsche Bank (DB) and Novartis (N) (=1, =0,3, =0,03). M SD S K M SD S K M SD S K (=1) (=0,3) (=0,03) d1_A 0,06 0,05 -2,47 15,74 0,06 0,02 -0,99 2,56 0,06 0,00 -0,06 -0,01
d2_A -0,13 0,02 -4,76 56,00 -0,12 0,01 -0,16 -0,38 -0,12 0,00 0,00 -0,02
d1_SAP -0,66 0,35 -7,58 98,50 -0,60 0,05 -0,60 0,54 -0,60 0,01 -0,01 -0,15
d2_SAP -0,81 0,33 -8,60 120,28 -0,75 0,04 -0,69 0,74 -0,74 0,00 -0,02 -0,15
d1_DB -0,03 0,12 -5,65 51,15 -0,02 0,03 -1,41 5,40 -0,02 0,00 -0,03 -0,07
d2_DB -0,20 0,08 -9,87 123,62 -0,18 0,01 -7,04 85,06 -0,18 0,00 -0,37 0,11
d1_N -0,91 0,59 -6,39 68,87 -0,79 0,09 -0,88 1,80 -0,78 0,01 -0,04 -0,11
d2_N -1,07 0,56 -7,04 80,67 -0,95 0,08 -1,01 2,34 -0,94 0,01 -0,05 -0,11
The range, not the standard deviation, is analyzed as to estimate the maximum risk carried by investor who is pricing options.
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As it was expected, the bigger the disturbances introduced into the sigma, the less normal are the distributions of d1 and d2 parameters. Best results are obtained when the disturbances are small (=0,03). In this case all distributions have standard deviation that is near zero. Skewness and kurtosis are rather small too.
Asymmetry and kurtosis are the biggest when the disturbances are big (=1). The measure of skewness range from -9,87 (Deutsche Bank) to -2,47 (Allianz Se) and the measure of kurtosis range from 15,74 (Allianz Se) to even 123,62 (Deutsche Bank). Judging by these results, we may conclude that the disturbances introduced into the Black-Scholes formula have a strong impact on the distributions of d1 and d2 parameters.
In order to check the influence of the disturbed sigma on the distributions of d1
and d2 parameters, chosen tests for normality are conducted. From Table 2 to Table 5 there are statistics of these tests calculated by Statistica for big, medium and small disturbances (=1, =0,3 and =0,03) presented.
Table 2: The staistics of Kolmogorov-Smirnov test with Lilliefors modification (max D) and Shapiro-Wilk test (W) with p-values (p) for d1 and d2 parameters from the Black-Scholes formula on options on stocks of Allianz Se (A), SAP (SAP), Deutsche Bank (DB) and Novartis (N) (=1, =0,3, =0,03).
p-value (KS) p-value (L) max D p-value W =1
d1_A < ,01 < ,01 0,1433 0,00 0,8492 d2_A < ,01 < ,01 0,1611 0,00 0,7230 d1_SAP < ,01 < ,01 0,1913 0,00 0,5269 d2_SAP < ,01 < ,01 0,2251 0,00 0,4647 d1_DB < ,01 < ,01 0,1992 0,00 0,5861 d2_DB < ,01 < ,01 0,3983 0,00 0,2340 d1_N < ,01 < ,01 0,2097 0,00 0,5374 d2_N < ,01 < ,01 0,2353 0,00 0,4864
=0,3 d1_A < ,01 < ,01 0,0626 0,00 0,9571 d2_A < ,01 < ,01 0,0242 0,00 0,9918 d1_SAP < ,01 < ,01 0,0471 0,00 0,9806 d2_SAP < ,01 < ,01 0,0531 0,00 0,9745 d1_DB < ,01 < ,01 0,0661 0,00 0,9281 d2_DB < ,01 < ,01 0,3092 0,00 0,4698 d1_N < ,01 < ,01 0,0502 0,00 0,9639 d2_N < ,01 < ,01 0,0573 0,00 0,9537
=0,03 d1_A > .20 > .20 0,0094 0,34 0,9996 d2_A > .20 > .20 0,0062 0,95 0,9998 d1_SAP > .20 > .20 0,0103 0,28 0,9995 d2_SAP > .20 > .20 0,0102 0,25 0,9995 d1_DB > .20 > .20 0,0097 0,76 0,9997 d2_DB < ,01 < ,01 0,0267 0,00 0,9920 d1_N > .20 > .20 0,0083 0,22 0,9995 d2_N > .20 > .20 0,0089 0,14 0,9994
From the statistics given in Table 2 we can see that for big and medium
disturbances (=1, =0,3) and the significance level =0,05, Kolmogorov-Smirnov test with Lilliefors modification (max D) and Shapiro-Wilk test (W) reject the null hypothesis which says that the distributions of d1 and d2 parameters are normal.
When disturbances are small (=0,03) and the significance level is =0,05, the null hypothesis should be accepted for the Black-Scholes model pricing options on stocks of Allianz Se (A), SAP (SAP) and Novartis (N). However, for the Black-Scholes model pricing options on stocks of Deutsche Bank (DB), Kolmogorov-Smirnov test with Lilliefors modification (max D) and Shapiro-Wilk test (W) reject the
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null hypothesis and accept the alternative hypothesis (for parameter d2). In this case the distribution of d2 parameter is not normal.
Table 3: The statistics of tests for skewness (B1), kurtosis (B2) and Jarque-Bera test (JB) with simple division on the impact of skewness (S) and kurtosis (K) on the normality of the distributions of d1 and d2 parameters from the Black-Scholes formula on options on stocks of Allianz Se (A), SAP (SAP), Deutsche Bank (DB) and Novartis (N) (=1).
B1 B2 JB S K
d1_A -2,47 18,72 56 538,92 5 066,39 51 472,53 d2_A -4,76 58,92 670 420,04 18 874,36 651 545,68 d1_SAP -7,58 101,36 2 063 250,96 47 823,01 2 015 427,95 d2_SAP -8,60 123,11 3 066 927,75 61 626,64 3 005 301,10 d1_DB -5,64 54,08 570 028,50 26 550,87 543 477,63 d2_DB -9,86 126,44 3 255 552,09 81 071,42 3 174 480,67 d1_N -6,39 71,77 1 019 251,95 33 992,69 985 259,25 d2_N -7,04 83,55 1 393 043,23 41 247,69 1 351 795,55
Judging by the statistics presented in Table 3, we may assume that disturbed
values of d1 and d2 parameters do not have normal distributions. In all cases the null hypothesis of Jarque-Bera test should be rejected as the statistics of this test are bigger than the critical value for the significance level =0,05 ( 99,5JB ).
What is more, the statistics of tests for skewness (B1, S) and kurtosis (B2, K) suggest that in all cases the normality of the distributions of d1 and d2 parameters is disturbed mainly by high kurtosis. Table 4: The statistics of tests for skewness (B1), kurtosis (B2) and Jarque-Bera test (JB) with simple division on the impact of skewness (S) and kurtosis (K) on the normality of the distributions of d1 and d2 parameters from the Black-Scholes formula on options on stocks of Allianz Se (A), SAP (SAP), Deutsche Bank (DB) and Novartis (N) (=0,3).
B1 B2 JB S K
d1_A -0,99 5,55 2 168,58 813,40 1 355,18 d2_A -0,16 2,61 52,53 21,63 30,90 d1_SAP -0,60 3,54 358,12 297,16 60,96 d2_SAP -0,69 3,73 508,81 396,79 112,02 d1_DB -1,41 8,39 7 715,11 1 658,93 6 056,18 d2_DB -7,03 87,94 1 544 405,01 41 239,39 1 503 165,63 d1_N -0,88 4,80 1 312,83 639,57 673,26 d2_N -1,01 5,34 1 988,14 850,93 1 137,20
When the disturbances introduced into the volatility of stock prices are medium
(=0,3), for the significance level =0,05 Jarque-Bera test rejects the null hypothesis for all distributions examined in the experiment. In this case the distributions of d1 and d2 parameters are not normal party because of high kurtosis (Allianz Se, Deutsche Bank, Novartis) and partly because of big asymmetry (SAP). As we can see in Table 4, the distributions of d1 and d2 parameters are disturbed the most in the Black-Scholes model pricing options on stocks of Deutsche Bank.
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Table 5: The statistics of tests for skewness (B1), kurtosis (B2) and Jarque-Bera test (JB) with simple division on the impact of skewness (S) and kurtosis (K) on the normality of the distributions of d1 and d2 parameters from the Black-Scholes formula on options on stocks of Allianz Se (A), SAP (SAP), Deutsche Bank (DB) and Novartis (N) (=0,03).
B1 B2 JB S K
d1_A -0,04 2,98 1,18 1,06 0,11 d2_A -0,46 3,25 189,29 175,79 13,50 d1_SAP -0,01 2,85 5,07 0,11 4,95 d2_SAP -0,02 2,85 5,27 0,33 4,94 d1_DB -0,03 2,93 1,74 0,60 1,14 d2_DB -0,37 3,11 114,44 112,02 2,42 d1_N -0,04 2,88 4,47 1,67 2,79 d2_N -0,05 2,89 5,22 2,49 2,73
Generally, when the disturbances are small (=0,03), the null hypothesis of
Jarque-Bera test may be accepted. Only in two cases (Allianz Se and Deutsche Bank) the distributions of d2 parameter are not normal. They are disturbed mainly by big asymmetry (See statistics S and K in Table 5).
In figures from 1 to 48 there are histograms and probability plots for d1 and d2 parameters shown for all examples discussed in this paper and disturbances that are big, medium or small (=1, =0,3 or =0,03). In histograms empirical distribution (E) and normal distribution (N) are compared. Thanks to the normal probability plots, one can assess whether or not a data set is approximately normally distributed. In such a graph the standardized values of the normal distribution are plotted on the Y-axis while the observed residuals are plotted on the X-axis. Figures 1-2: Histograms for d1 and d2 parameters (Allianz Se, =1)
Histogram
0
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-0,518 -0,318 -0,118 0,082
(D1_A)
Fre
qu
ency
E
N
Histogram
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90010001100
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1300
-0,529 -0,429 -0,329 -0,229 -0,129
(D2_A)
Fre
qu
ency
E
N
Figures 3-4: Normal probability plots for d1 and d2 parameters (Allianz Se, =1)
Normal P-Plot: d1_A
-0,6 -0,5 -0,4 -0,3 -0,2 -0,1 0,0 0,1 0,2 0,3
Value
-4
-3
-2
-1
0
1
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3
4
Exp
ecte
d N
orm
al V
alue
Normal P-Plot: d2_A
-0,55 -0,50 -0,45 -0,40 -0,35 -0,30 -0,25 -0,20 -0,15 -0,10 -0,05
Value
-4
-3
-2
-1
0
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Exp
ecte
d N
orm
al V
alue
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Figures 5-6: Histograms for d1 and d2 parameters (Allianz Se, =0,3) Histogram
0
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-0,090 -0,040 0,010 0,060 0,110
(D1_A)
Fre
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ency
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Histogram
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-0,155 -0,145 -0,135 -0,125 -0,115 -0,105 -0,095 -0,085
(D2_A)
Fre
qu
ency
E
N
Figures 7-8: Normal probability plots for d1 and d2 parameters (Allianz Se, =0,3)
Figures 9-10: Histograms for d1 and d2 parameters (Allianz Se, =0,03)
Histogram
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0,0550 0,0570 0,0590 0,0610 0,0630 0,0650 0,0670
(D1_A)
Fre
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ency
E
N
Histogram
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-0,1250 -0,1230 -0,1210 -0,1190
(D2_A)
Fre
qu
ency
E
N
Figures 11-12: Normal probability plots for d1 and d2 parameters (Allianz Se, =0,03)
Normal P-Plot: d1_A
-0,12 -0,10 -0,08 -0,06 -0,04 -0,02 0,00 0,02 0,04 0,06 0,08 0,10 0,12
Value
-4
-3
-2
-1
0
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Exp
ecte
d N
orm
al V
alue
Normal P-Plot: d2_A
-0,16 -0,15 -0,14 -0,13 -0,12 -0,11 -0,10
Value
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Exp
ecte
d N
orm
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alue
Normal P-Plot: d1_A
0,054 0,056 0,058 0,060 0,062 0,064 0,066 0,068
Value
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ecte
d N
orm
al V
alue
Normal P-Plot: d2_A
-0,126 -0,125 -0,124 -0,123 -0,122 -0,121 -0,120 -0,119 -0,118
Value
-4
-3
-2
-1
0
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Exp
ecte
d N
orm
al V
alue
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For big disturbances (=1) the distributions of d1 and d2 parameters are not normal. They are disturbed both by high kurtosis and big asymmetry. In histograms (Figures 1-2) we can observe that the distributions mentioned above are more leptokurtic than the standard normal distribution. Normal probability plots (Figures 3-4) show a strongly nonlinear pattern which suggests that the normal distribution is not a good model for these data. What is more, the patterns in the normal probability plots are the signature of a significantly left-skewed data set and a presence of many observations that are far from the mean. As it was expected, the normal distribution appears to be a good model for the distributions of d1 and d2 parameters only when the disturbances are small (Figures 9-12). Figures 13-14: Histograms for d1 and d2 parameters (SAP, =1)
Histogram
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-7,627 -5,627 -3,627 -1,627 0,373
(D1_SAP)
Fre
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ency
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Histogram
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-7,640 -5,640 -3,640 -1,640 0,360
(D2_SAP)
Fre
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ency
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N
Figures 15-16: Normal probability plots for d1 and d2 parameters (SAP, =1)
Figures 17-18: Histograms for d1 and d2 parameters (SAP, =0,3)
Histogram
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-0,850 -0,750 -0,650 -0,550 -0,450
(D1_SAP)
Fre
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Histogram
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250
300
350
-0,960 -0,860 -0,760 -0,660
(D2_SAP)
Fre
qu
ency
E
N
While judging the goodness of fit of empirical distributions to the normal one we should concentrate mainly on the analysis of normal probability plots. They seem to be more precise than the histograms made in Microsoft Excel.
Normal P-Plot: d1_SAP
-8 -7 -6 -5 -4 -3 -2 -1 0 1
Value
-4
-3
-2
-1
0
1
2
3
4
Exp
ecte
d N
orm
al V
alue
Normal P-Plot: d2_SAP
-8 -7 -6 -5 -4 -3 -2 -1 0
Value
-4
-3
-2
-1
0
1
2
3
4
Exp
ecte
d N
orm
al V
alue
11
Figures 19-20: Normal probability plots for d1 and d2 parameters (SAP, =0,3)
Figures 21-22: Histograms for d1 and d2 parameters (SAP, =0,03)
Histogram
0
50
100
150
200
250
300
350
-0,6140 -0,6040 -0,5940 -0,5840
(D1_SAP)
Fre
qu
ency
E
N
Histogram
0
50
100
150
200
250
300
350
-0,7580 -0,7530 -0,7480 -0,7430 -0,7380 -0,7330 -0,7280
(D2_SAP)
Fre
qu
ency
E
N
Figures 23-24: Normal probability plots for d1 and d2 parameters (SAP, =0,03)
For the Black-Scholes model pricing options on stocks of SAP the biggest impact on the distributions of d1 and d2 parameters have disturbances which are big (=1). As we can see in figures 13-16, for big disturbances empirical distributions are leptokurtic and left-skewed while the normal probability plots show a strongly nonlinear pattern. There are many deviations from the line fit to the points on the probability plots. We should underline, however, that kurtosis and asymmetry of the distributions are far smaller for medium and small disturbances (=0,3, =0,03). This can be easily observed in Figures 17-24. Such observations confirm earlier conclusions. Firstly, disturbances introduced into the volatility of stock prices have a strong impact on the normality of the distributions of d1 and d2 parameters. Secondly, the bigger the disturbances and the uncertainty of the estimates of the sigma, the less normal the distributions of Black-Scholes d1 and d2 parameters.
Normal P-Plot: d1_SAP
-0,90 -0,85 -0,80 -0,75 -0,70 -0,65 -0,60 -0,55 -0,50 -0,45 -0,40
Value
-4
-3
-2
-1
0
1
2
3
4
Exp
ecte
d N
orm
al V
alue
Normal P-Plot: d2_SAP
-1,00 -0,95 -0,90 -0,85 -0,80 -0,75 -0,70 -0,65 -0,60
Value
-4
-3
-2
-1
0
1
2
3
4
Exp
ecte
d N
orm
al V
alue
Normal P-Plot: d1_SAP
-0,615 -0,610 -0,605 -0,600 -0,595 -0,590 -0,585 -0,580 -0,575
Value
-4
-3
-2
-1
0
1
2
3
4
Exp
ecte
d N
orm
al V
alue
Normal P-Plot: d2_SAP
-0,760 -0,755 -0,750 -0,745 -0,740 -0,735 -0,730 -0,725
Value
-4
-3
-2
-1
0
1
2
3
4
Exp
ecte
d N
orm
al V
alue
12
Figures 25-26: Histograms for d1 and d2 parameters (Deutsche Bank, =1) Histogram
0
200
400
600
800
1000
1200
1400
-1,558 -1,058 -0,558 -0,058
(D1_DB)
Fre
qu
ency
E
N
Histogram
0
200
400
600
800
1000
1200
1400
1600
-1,568 -1,068 -0,568 -0,068
(D2_DB)
Fre
qu
ency
E
N
Figures 27-28: Normal probability plots for d1 and d2 parameters (Deutsche Bank, =1)
Figures 29-30: Histograms for d1 and d2 parameters (Deutsche Bank, =0,3)
Histogram
0
50
100
150
200
250
300
350
400
450
-0,300 -0,200 -0,100 0,000 0,100
(D1_DB)
Fre
qu
ency
E
N
Histogram
0
100
200
300
400
500
600
700
800
900
1000
-0,320 -0,270 -0,220 -0,170 -0,120
(D2_DB)
Fre
qu
ency
E
N
Figures 31-32: Normal probability plots for d1 and d2 parameters (Deutsche Bank, =0,3)
Normal P-Plot: d1_DB
-1,8 -1,6 -1,4 -1,2 -1,0 -0,8 -0,6 -0,4 -0,2 0,0 0,2
Value
-4
-3
-2
-1
0
1
2
3
4
Exp
ecte
d N
orm
al V
alue
Normal P-Plot: d2_DB
-1,8 -1,6 -1,4 -1,2 -1,0 -0,8 -0,6 -0,4 -0,2 0,0
Value
-4
-3
-2
-1
0
1
2
3
4
Exp
ecte
d N
orm
al V
alue
Normal P-Plot: d1_DB
-0,30 -0,25 -0,20 -0,15 -0,10 -0,05 0,00 0,05 0,10
Value
-4
-3
-2
-1
0
1
2
3
4
Exp
ecte
d N
orm
al V
alue
Normal P-Plot: d2_DB
-0,34 -0,32 -0,30 -0,28 -0,26 -0,24 -0,22 -0,20 -0,18 -0,16
Value
-4
-3
-2
-1
0
1
2
3
4
Exp
ecte
d N
orm
al V
alue
13
Figures 33-34: Histograms for d1 and d2 parameters (Deutsche Bank, =0,03) Histogram
0
50
100
150
200
250
300
350
-0,0300 -0,0250 -0,0200 -0,0150
(D1_DB)
Fre
qu
ency
E
N
Histogram
0
50
100
150
200
250
300
350
-0,1800 -0,1795 -0,1790 -0,1785 -0,1780 -0,1775 -0,1770
(D2_DB)
Fre
qu
ency
E
N
Figures 35-36: Normal probability plots for d1 and d2 parameters (Deutsche Bank, =0,03)
For the Black-Scholes model pricing options on stocks of Deutsche Bank we can
observe the same tendency as in both examples described earlier. It is easily observed when we analyze Figures from 25 to 36. For big and medium disturbances (=1, =0,3) we can reasonably conclude that the normal distribution does not provide an adequate fit for these data sets. In these cases, as we can see in histograms (Figures 25-26; 29-30), the distributions of d1 and d2 parameters are non-normal, non-symmetric and rather long-tailed. The points on normal probability plots (Figures 27-28; 31-32) form a nonlinear pattern which also indicates that the normal distribution is not a good model for these data sets. However, we should underline that the smaller the disturbances, the less disturbed the distributions of d1 and d2 parameters.
Figures 37-38: Histograms for d1 and d2 parameters (Novartis, =1)
Histogram
0
200
400
600
800
1000
1200
1400
1600
-9,185 -7,185 -5,185 -3,185 -1,185 0,815
(D1_N)
Fre
qu
ency
E
N
Histogram
0
500
1000
1500
2000
2500
-9,267 -7,267 -5,267 -3,267 -1,267 0,733
(D2_N)
Fre
qu
ency
E
N
Normal P-Plot: d1_DB
-0,032 -0,030 -0,028 -0,026 -0,024 -0,022 -0,020 -0,018 -0,016 -0,014 -0,012 -0,010
Value
-4
-3
-2
-1
0
1
2
3
4
Exp
ecte
d N
orm
al V
alue
Normal P-Plot: d2_DB
-0,1800-0,1798
-0,1796-0,1794
-0,1792-0,1790
-0,1788-0,1786
-0,1784-0,1782
-0,1780-0,1778
-0,1776-0,1774
Value
-4
-3
-2
-1
0
1
2
3
4
Exp
ecte
d N
orm
al V
alue
14
Figures 39-40: Normal probability plots for d1 and d2 parameters (Novartis, =1)
Figures 41-42: Histograms for d1 and d2 parameters (Novartis, =0,3)
Histogram
0
50
100
150
200
250
300
350
-1,458 -1,258 -1,058 -0,858 -0,658
(D1_N)
Fre
qu
ency
E
N
Histogram
0
50
100
150
200
250
300
350
400
-1,552 -1,352 -1,152 -0,952 -0,752
(D2_N)
Fre
qu
ency
E
N
Figures 43-44: Normal probability plots for d1 and d2 parameters (Novartis, =0,3)
Figures 45-46: Histograms for d1 and d2 parameters (Novartis, =0,03)
Histogram
0
50
100
150
200
250
300
350
-0,8090 -0,7990 -0,7890 -0,7790 -0,7690 -0,7590 -0,7490
(D1_N)
Fre
qu
ency
E
N
Histogram
0
50
100
150
200
250
300
350
-0,9680 -0,9580 -0,9480 -0,9380 -0,9280 -0,9180
(D2_N)
Fre
qu
ency
E
N
Normal P-Plot: d1_N
-14 -12 -10 -8 -6 -4 -2 0 2
Value
-4
-3
-2
-1
0
1
2
3
4
Exp
ecte
d N
orm
al V
alue
Normal P-Plot: d2_N
-14 -12 -10 -8 -6 -4 -2 0
Value
-4
-3
-2
-1
0
1
2
3
4
Exp
ecte
d N
orm
al V
alue
Normal P-Plot: d1_N
-1,5 -1,4 -1,3 -1,2 -1,1 -1,0 -0,9 -0,8 -0,7 -0,6 -0,5
Value
-4
-3
-2
-1
0
1
2
3
4
Exp
ecte
d N
orm
al V
alue
Normal P-Plot: d2_N
-1,6 -1,5 -1,4 -1,3 -1,2 -1,1 -1,0 -0,9 -0,8 -0,7
Value
-4
-3
-2
-1
0
1
2
3
4
Exp
ecte
d N
orm
al V
alue
15
Figures 47-48: Normal probability plots for d1 and d2 parameters (Novartis, =0,03)
From the analysis of Figures 3744 we can conclude that disturbances introduced
into the sigma also change the distributions of Black-Scholes d1 and d2 parameters calculated for the model pricing options on stocks of Novartis. The distributions of both parameters are disturbed the most when the disturbances are big or medium (=1, =0,3). Then kurtosis is high and data sets are left-skewed. As in the rest cases described earlier in this paper, the mentioned asymmetry and leptokurticity disappear almost completely when disturbances are small (=0,03) (See Figures 45-48).
In the next step, we check how many observations fit into the bracket: mean +/- ; mean +/- 2 and mean +/- 3 . Then, the results for empirical distributions are compared with the results that are obtained for a standard normal distribution. Table 6: Number of observations that fits into the bracket: mean +/- , mean +/- 2 and mean +/- 3 for d1 and d2 parameters from the Black-Scholes formula on options on stocks of Allianz Se (A), SAP (SAP), Deutsche Bank (DB) and Novartis (N) (=1, =0,3, =0,03).
For a standard normal distribution:
about 68,0% of all observations fit into the bracket: (x - ;
x + );
about 95,5% of all observations fit into the bracket: (x - 2 ;
x + 2 );
about 99,7% of all observations fit into the bracket: (x - 3 ;
x + 3 ).
Judging by the data presented in Table 6, we can conclude that especially for big
disturbances (=1) the distributions of d1 and d2 parameters are more leptokurtic than for a standard normal distribution. It is well seen for the Black-Scholes model pricing options on stocks of SAP, Deutsche Bank and Novartis.
1 2 3 1 2 3 1 2 3
(=1) (=0,3) (=0,03)
d1_A 75,14 97,02 98,76 73,76 95,92 99,14 68,00 96,08 99,84
d2_A 73,84 97,02 99,38 66,42 96,98 99,90 70,14 95,82 99,78
d1_SAP 89,84 97,80 98,58 67,40 96,08 99,52 83,76 96,18 99,82
d2_SAP 84,00 97,90 98,70 70,82 95,98 99,36 66,80 95,76 99,74
d1_DB 91,26 97,60 98,46 71,44 95,86 76,38 67,46 95,90 99,76
d2_DB 72,96 97,92 98,74 54,56 82,22 94,58 76,24 98,30 99,74
d1_N 93,00 97,38 98,46 70,06 95,96 99,06 67,62 95,30 99,84
d2_N 93,10 97,38 98,48 67,98 96,08 99,02 68,00 95,40 99,86
Normal P-Plot: d1_N
-0,81 -0,80 -0,79 -0,78 -0,77 -0,76 -0,75 -0,74
Value
-4
-3
-2
-1
0
1
2
3
4
Exp
ecte
d N
orm
al V
alue
Normal P-Plot: d2_N
-0,97 -0,96 -0,95 -0,94 -0,93 -0,92 -0,91
Value
-4
-3
-2
-1
0
1
2
3
4
Exp
ecte
d N
orm
al V
alue
16
What is more, in most cases the smaller the disturbances, the less disturbed are the empirical distributions. However, exceptions to this are the distribution of d1 parameter for options on stocks of SAP and the distribution of d2 parameter for options on stocks of Deutsche Bank. For these two cases it seems that the distributions are disturbed the least for =0,3 and not =0,03.
To summarize, in the Black-Scholes model the uncertainty of estimates of the volatility of stock prices has a very strong impact on the distributions of d1 and d2 parameters. It is worth checking, how it affects the value of option calculated from the Black-Scholes formula. 3.3 Sensitivity analysis
In tables presented below we show cases when the strike price of the option (K) is constant and the current price of the stock (S) changes (Table 7) and when the current price of the stock (S) is constant and the strike price of the option (K) is changeable (Table 8). In two top rows of Table 7 there are results for call and put options when the stock price equals the strike price of the option (S=K) presented. In the next two rows there are results obtained in a situation when stock prices are higher than strike prices of options (S>K). Analogically, in the last two rows results for stock prices smaller than strike prices of options are shown (SS). Results for strike prices of options smaller than stock prices are presented in the last two rows of the table (K
17
Table 8: Values of call and put options on stocks of Allianz Se for different strike prices of the option (K) and the stock price S=141,88 (=1, =0,3, =0,03).
K Call/ Put Undisturbed
values Disturbed values
(=1) Disturbed values
(=0,3) Disturbed values
(=0,03) min max min max min max
C 7,94 0,00 18,94 0,76 12,50 7,43 8,48 150,00 P 13,66 5,71 24,66 6,48 18,22 13,14 14,20 C 6,24 0,00 17,13 0,19 10,71 5,75 6,76 155,00 P 16,88 10,63 27,77 10,82 21,34 16,38 17,40 C 15,18 9,04 24,59 9,28 19,38 14,71 15,67 135,00 P 6,14 0,00 15,55 0,24 10,34 5,67 6,63 C 18,35 13,96 27,94 14,00 22,21 17,93 18,79 130,00 P 4,39 0,00 13,97 0,03 8,25 3,97 4,83
As we can see in Table 7, when the stock price of Allianz Se (S) equals the strike
price of the option (K), the undisturbed value of call option is 11,44 . Disturbed prices of such an option vary the most for big disturbances (=1) and change within the bracket . In this case, the probabilistic model of Black-Scholes tends to underestimate the value of option. Maximum disturbed value of option is 11,25 and it is lower than the undisturbed value of this option which is, as we mentioned before, C=11,44 . It is also worth noticing, that for call option which is at the money (S=K) the smallest deviations of disturbed prices from the undisturbed price of this option are for small disturbances (=0,03). Then, the disturbed value of call option on stocks of Allianz Se fits into the bracket . Table 9: Values of call and put options on stocks of SAP for different stock prices (S) and the strike price of the option K=35,67 (=1, =0,3, =0,03).
S Call/ Put Undisturbed
values Disturbed values
(=1) Disturbed values
(=0,3) Disturbed values
(=0,03) min max min max min max
C 2,37 0,59 2,83 1,84 2,82 2,32 2,42 35,67 P 1,80 0,02 2,26 1,27 2,25 1,75 1,85 C 3,90 2,90 5,14 3,44 4,32 3,86 3,94 38,00 P 1,00 0,00 2,23 0,54 1,42 0,95 1,04 C 5,47 4,90 6,56 5,13 5,82 5,43 5,50 40,00 P 0,56 0,00 1,66 0,23 0,92 0,53 0,60 C 0,35 0,00 1,20 0,12 0,62 0,32 0,37 30,00 P 5,45 5,10 6,30 5,22 5,71 5,42 5,47 C 0,12 0,00 0,70 0,02 0,18 0,11 0,14 28,00 P 7,22 7,10 7,80 7,12 7,28 7,21 7,24
Table 10: Values of call and put options on stocks of SAP for different strike prices of the option (K) and the stock price S=35,67 (=1, =0,3, =0,03).
K Call/ Put Undisturbed
values Disturbed values
(=1) Disturbed values
(=0,3) Disturbed values
(=0,03) min max min max min max
C 1,39 0,00 2,70 0,87 1,85 1,34 1,44 38,00 P 3,11 1,72 4,42 2,59 3,57 3,06 3,16 C 0,83 0,00 2,04 0,41 1,24 0,79 0,87 40,00 P 4,52 3,69 5,73 4,10 4,93 4,48 4,56 C 3,32 2,22 4,53 2,86 3,74 3,28 3,37 34,00 P 1,11 0,00 2,32 0,65 1,53 1,06 1,15 C 4,72 4,18 5,72 4,41 5,05 4,69 4,76 32,00 P 0,54 0,00 1,54 0,23 0,87 0,51 0,58
18
For the probabilistic model pricing options on stocks of SAP, the introduction of the disturbances into the volatility of stock prices changes the value of call and put options. The biggest changes in the disturbed value of option are for big disturbances (=1) and options that are out of the money. In this case, for the strike price of the option K=35,67 and the stock price S=30,00 , the disturbed value of call option fits into the bracket while the disturbed value of put option varies from 5,10 to 6,30 (See Table 9). In these ranges investor could not be sure whether such options are overestimated or rather underestimated. However, for the undisturbed value of call option which equals 0,35 and the undisturbed value of put option that is 5,45 , it seems that the uncertainty of estimates of the sigma do not have a very strong impact on the value of the option (it is true especially for put options). We should also notice that for the Black-Scholes model pricing options on stocks of SAP the value of options is the least sensitive to the disturbances when put options are in the money (S
19
Table 13: Values of call and put options on stocks of Novartis for different stock prices (S) and the strike price of the option K=34,18 (=1, =0,3, =0,03).
S Call/ Put Undisturbed
values Disturbed values
(=1) Disturbed values
(=0,3) Disturbed values
(=0,03)
min max min max min max C 2,50 0,56 2,71 1,56 2,71 2,44 2,57 34,18 P 1,95 0,01 2,16 1,01 2,16 1,89 2,02 C 2,99 1,37 4,55 2,08 3,62 2,93 3,06 35,00 P 1,63 0,00 3,18 0,71 2,25 1,56 1,70 C 4,37 3,37 5,93 3,63 4,95 4,32 4,43 37,00 P 1,00 0,00 2,56 0,27 1,58 0,95 1,07 C 1,88 0,01 3,49 0,96 2,50 1,82 1,95 33,00 P 2,51 0,64 4,12 1,59 3,13 2,45 2,58 C 0,75 0,00 2,09 0,16 1,24 0,70 0,80 30,00 P 4,38 3,63 5,73 3,79 4,87 4,33 4,43
Table 14: Values of call and put options on stocks of Novartis for different strike prices of the option (K) and the stock price S=34,18 (=1, =0,3, =0,03).
K Call/ Put Undisturbed
values Disturbed values
(=1) Disturbed values
(=0,3) Disturbed values
(=0,03) min max min max min max
C 2,12 0,06 3,78 1,16 2,76 2,05 2,19 35,00 P 2,38 0,32 4,04 1,42 3,02 2,31 2,45 C 1,36 0,00 3,01 0,49 1,98 1,30 1,43 37,00 P 3,59 2,23 5,23 2,72 4,21 3,53 3,66 C 3,14 1,71 4,71 2,28 3,74 3,08 3,21 33,00 P 1,43 0,00 3,01 0,57 2,03 1,37 1,50 C 4,44 3,68 5,81 3,84 4,94 4,39 4,49 31,00 P 0,76 0,00 2,13 0,16 1,26 0,72 0,82
The sensitivity analysis of prices of options on stocks of Novartis confirms
conclusions that have been drawn for all three examples described above. In this case too, the value of option is disturbed the least when the Black-Scholes model prices in the money call options (S>K). The strongest reaction to the disturbances introduced into the volatility of stock prices is noticed when the probabilistic model prices call and put options that are out of the money (See Table 13 and Table 14).
To sum up, after the sensitivity analysis of option prices calculated from the Black-Scholes formula we should conclude that the value of option reacts to the uncertainty of estimates of the disturbed sigma. However, although the disturbed volatility of stock prices has a strong impact on the distributions of d1 and d2 parameters, it causes rather small changes in the value of options. 4 Concluding remarks
The experiment described in this paper enables the answer to the question whether and to what extend disturbances introduced into one of the Black-Scholes variables ( ) influence d1 and d2 parameters calculated in this model as well as the value of option itself. The results obtained suggest that:
the uncertainty of estimates of disturbed volatility of stock prices has a strong impact on the distributions of d1 and d2 parameters;
the strength of the disturbances (=1, =0,3, =0,03) introduced into the sigma affects the results of the experiment in a significant way. The bigger the disturbances, the more asymmetric and leptokurtic the distributions of d1 and d2 parameters are. Big asymmetry and high kurtosis may cause bigger
20
diversity of option prices. For these reasons, model prices could be more pessimistic than the real ones (market prices);
although the uncertainty of estimates of disturbed volatility of stock prices has a strong impact on the distributions of d1 and d2 parameters, it causes rather small changes in the value of options calculated from the Black-Scholes formula;
there are situations when the probabilistic model tends to underestimate the value of stock options that are at the money and out of the money.
What is more, from the sensitivity analysis of option prices calculated from the
Black-Scholes formula it results that when the disturbances are: big (=1), the Black-Scholes model is the most sensitive to the disturbances
when out of the money call and put options are priced. The least sensitive to the disturbances is the probabilistic model pricing put options which are in the money. It suggests that this model is good in pricing options of in the money type. In this case, the precision with which the volatility of stock prices is estimated has rather a small impact on the precision of option pricing;
medium (=0,3), the least sensitive to the disturbances is the probabilistic model pricing in the money call options. The Black-Scholes model pricing these types of options is more resistant to the uncertainty of estimates of the disturbed sigma than model pricing options that are out of the money;
small (=0,03), the value of option changes slightly regardless of the option type (call or put, at-, in- or out of the money).
5 Summary and possible further research
In this paper we considered pricing of options on stocks of four chosen companies listed on Eurex Stock Exchange. For the calculation of option prices the Black-Scholes formula was used. We assumed that the stocks do not pay any dividends and all options have five months to the expiration date. It was also assumed that the risk-free interest rate equals r = 0,0388.
Firstly, it appears that there are situations when the probabilistic model underestimates the disturbed value of at the money and out of the money options relative to their undisturbed values. What is more, although the uncertainty of estimates of disturbed volatility of stock prices has a strong impact on the distributions of d1 and d2 parameters, it causes rather small changes in the value of options calculated from the Black-Scholes formula. Put options that are out of the money are the most sensitive to the disturbances introduced into the sigma. The most resistant to these disturbances are call and put options that are in the money.
It is also worth noticing that during the experiment we checked whether the chosen research procedure was right. Thus, it was examined whether measures of skewness and kurtosis of the distributions of d1 and d2 parameters react to the number of replications. It appears, however, that relative to the results of the experiment for 5000 replications results do not change in a significant way both for 1000 and 10000 replications.
In this situation it was checked whether the random number generator drew values with a normal distribution and whether the standard deviation of stock returns was well estimated. Tests of normality confirmed that the distribution of the drawn disturbances is normal. Thus, it seems that the random number generator was properly chosen. As to check the quality of estimates of the sigma, the Parkinson
21
formula was used to estimate the volatility of stock prices. Then, the estimates obtained in this way were compared to the ones obtained for the historical volatility estimated in a classical way (see formula (4)). Because results of both estimates were very similar, we assume that the way in which the sigma is estimated do not affect further results of the experiment.
Certainly, the research described above does not fulfil all possible further studies. They can be continued in many different directions. For example, the sensitivity of the Black-Scholes model pricing options on stocks that pay dividends could be checked [15]. Apart from calculating minimum and maximum disturbed values of options (range), one can also check the frequency of the disturbed option values or calculate how wide is a bracket: mean +/- standard deviation of the disturbed values. However, these problems are not discussed in this paper. As it was mentioned before, we analyzed here the range of the disturbed values of options on stocks not paying dividends. In this way, we underlined the maximum risk carried by investor who is pricing options and showed how strong may be reactions of option values calculated from the Black-Scholes formula to the disturbances introduced into the volatility of stock prices ( ). References [1] Antonelli F., Scarlatti S. (2009) Pricing options under stochastic volatility: a
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