About Black-Sholes formula, volatility, implied volatility and math. statistics. Mark Ioffe Abstract We analyze application Black-Sholes formula for calculation of implied volatility from point of view of mathematical statistics. We can consider implied volatility as estimation of unknown distribution option prices parameter based on 1 observation. For calculation one uses math. expectation of observed values as function of unknown parameter of distribution (Black-Sholes formula). Such method of assessment is not conventional for math. statistics. We show that volatility calculated by using classical maximum-likelihood method better represents market prices than volatility calculated by using Black-Sholes formula. The simple question is what does one calculates by formulas Black-Scholes. The seemingly simple, comprehensive and full answer is: the Black-Scholes formulas for European (call and put) vanilla options for given values of today s price, for example, stock (spot) S0, strike k, domestic rate rd, dividend rate rf, volatility σ0, and time to expiration t one calculates non- arbitrage option s premium. Unfortunately, from a mathematical point of view this definition has some vagaries. It is not full clear what is a non-arbitrage option s premium. Obviously, for example, call option s premium (Pr) may be determined uniquely at the time of expiration on given value of stock price S (intrinsic value) by the formula: (1) Unfortunately, value S is unknown at the moment of calculation option s premium, therefore it is necessary to make some assumptions about it s future s value. The natural and obvious assumption is S(t) is a stochastic process, that is, at time of expiration S(t) is stochastic variable, for example, it has continuous density distribution function, and the non- arbitrage option s premium is the mathematical expectation of this function, determined by formula (1). Because value S is positive, next natural and obvious assumption is its logarithm has normal (Gaussian) distribution, i.e. value S has the lognormal (Gaussian) distribution and density distribution function as determined by (2) (2) Where parameters of distribution σ и μ are calculated from initial parameters by (3)
(3) This assumption is direct consequence of hypothesis that stochastic process S(t) is geometric Gaussian random walk. A Call option s premium Pr is determined by (1) as function S, and has following density distribution function Pr(y) and cumulative distribution function FPr(y), (4) L order moments of stochastic variable y ml are determined by (5). (5) Our assumption is the option s premium is a mathematical expectation, i.e. first moment of distribution function m1, and after substitution in (5) l=1 and other known values (S0,k,rd,rf,σ,t) we will find out the Black-Scholes formula. In general case, if we have the sample of market s option s premium, then by using the foregoing formulas, we can perform a statistical tests for any hypothesis used for the Black-Scholes formula derivation. Because we observed only positive market price we must use conditional probability density Pr(y price>0) and cumulative distribution function FPr(y price>0), i.e. (6) It follows from (4) that for any option price u and strike k
(7) The big question is: if it is possible testing of statistical hypothesis based the above mentioned Black-Schools formula derivation on and if what set of the data we can consider as a sample for testing. For example, let us analyze OEX(S&P 100 Index) option by using Delayed Quotes Classic from CBOE site. We get 2 information types about option s price: Last Sale and Bid and Ask. Data for Sep 13 2007 @ 15:11 ET Call option with expiration times September 2007, November 2007 and December 2008 are shown in Table1, Table2 and Table3. Spot price is 696.4 Table 1 7-Sep Strike Last Price (Bid+Ask)/2 IVol/Last IVol/Av IVolM/Last IVolM/Av 675 24.5 24.2 22.20663 21.1036663 2.04340766 1.7702267 680 20.5 20.7 22.0371 22.6364717 2.9531646 3.1349654 685 16.6 16.7 21.30594 21.5723038 3.99444392 4.0942138 690 13 13.7 20.53052 22.2241049 6.16506331 6.9713266 695 9.75 10 19.68781 20.2648649 7.50784668 7.739208 700 7.2 7.1 19.44354 19.2127042 7.85705566 7.7646096 705 4.8 5 18.42953 18.917923 7.25074628 7.4587342 710 3.1 3.1 17.9047 17.9046969 6.47731668 6.4773167 715 1.85 1.85 17.34263 17.3426304 5.48461263 5.4846126 720 0.925 0.95 16.37978 16.5019817 4.30216708 4.302561 725 0.4 0.5 15.509 16.2723522 5.21904965 5.2206471 730 0.2 0.2 15.51156 15.5115623 6.13500933 6.1350093 735 0.05 0.05 14.26054 14.2605362 7.04578733 7.0457873 740 0.025 0.05 14.63681 15.8486176 7.95265432 7.9530512 745 0.025 0.05 16.10098 17.4082794 8.85407904 8.8544734 750 0.025 0.05 17.54015 18.9520264 9.74973873 9.7501307 760 0.025 0.05 20.36163 21.9383469 11.5240619 11.524449 765 0.15 0.05 26.83332 23.4013824 12.4047943 12.403256 770 0.025 0.05 23.08081 24.8473663 13.2762073 13.27659 780 0.15 0.2 31.59074 32.8571167 15.0086247 15.00938 Vindex 13.56523 13.5821892 19.53469 19.9489467 Table 2 7-Nov Strike Last Price (Bid+Ask)/2 IVol/last IVol/Av IVolM/last IVolM/Av 650 60.1 60.1 23.455948 23.45594788 2.55870992 2.5587099 660 52.1 43.7 22.801092 11.60125303 3.23041523 0 670 44.4 44.6 22.047726 22.2486167 4.04706161 4.120939 680 37 36.4 21.159049 20.59659147 5.56250341 5.2814946 690 30.1 24.3 20.271508 15.08292532 6.99375801 4.7416052 700 23.8 22.8 19.400379 18.53763461 7.49084827 7.1553495 710 18.1 15.6 18.473407 16.31160665 7.375101 6.5215881 720 13.3 13.4 17.683558 17.77325678 6.94251633 6.9795294 730 9.05 8.5 16.6563 16.11398458 6.16411303 5.9265494 740 5.85 5.7 15.82201 15.65046263 5.30556398 5.2267193 760 1.9 1.2 14.261389 12.73682499 3.95372914 3.9497557 780 0.6 0.25 13.84277 11.99464989 5.25180997 5.2498352 Vindex 5.266502 5.24323294 18.822928 16.84197954
Table 3 8-Dec Strike Last Price (Bid+Ask)/2 IVol/last IVol/av IVolM/last IVolM/Av 640 106.2 111.6 14.2308023 16.88390422 0 1.1413863 660 92.8 86 14.7011272 11.54262006 1.28302223 0 680 80.3 76.6 14.9647535 13.53417832 2.16579709 1.7301054 700 68.3 66.8 14.9405969 14.41591978 3.50780059 3.2605221 720 57.2 39.2 14.815338 8.826249242 4.87124139 2.4049621 740 47.1 39 14.6272365 12.04079604 5.39820099 4.4584171 780 30 28.8 14.0961738 13.70426899 5.3242992 5.1730009 VIndex 3.86705 3.27625484 14.6251469 12.99256238 Columns 2 (Last Price) and 3(average value of Bid and Ask) in Tables are samples taken at random option s price. According to statistical procedure this samples we must use for estimation of parameters of distribution and testing of statistical hypothesis. In our case statistical distribution of observed (market) option s price accordingly (7) has 2 parameters µ and σ. We suppose that riskless rate r= rd- rf is given and we don t know only parameter σ. For estimation we use classical maximum-likelihood method [see, for example, Mathematical Methods of Statistics by H. Cramer]. According to (7) likelihood function L(s ) equals (8) Where k(i)=i strike price y(i)=i option price. According with maximum-likelihood method likelihood function get maximum when its argument σ equals sought quantitys. Number s is the root of an equation
(9) We solved equation (9) by bisection method. The results of calculation are shown in last row (volatility index (VIndex)) in Table1, Table2 and Table3 in 2 and 3 columns for appropriate samples. We used value rd=0.05 and rf=0. In financial mathematics the problem of volatility estimation is solved with help of implied volatility. This is quote from Wikipedia: Implied volatility of an option contract is volatility implied by the market price of the option based on an option pricing model. In other words, it is volatility that given a particular pricing model yields a theoretical value for the option equal to current market price. A particular pricing model for European Call option is Black-Sholes formula (first moment of distribution function): (10) Implied volatility σ0 for given values S0,K,rd, rf,t and Pr is root of equation (11) (11) Because price Pr is a monotonically increasing function for σ0 root of equation (11) always exists and single. We solved equation (11) also by bisection method. The results of calculation, implied volatility for 2 sample, last and average prices, are shown in columns 4 and 5 Tables 1,2,3. Because option price is a stochastic variable, implied volatility also is stochastic variable. This implied volatility is best regarded as rescaling of options prices which makes comparisons between different strikes, expirations, and underlying easier and more intuitive. (see. Wikipedia). We can consider implied volatility as estimation of unknown distribution parameter σ0 based on 1 observation. For calculation we use math. expectation of observed values as function of unknown parameter of distribution (Black-Sholes formula). Such method of assessment is not conventional for math. statistics. Implied volatility is widely used for building all kinds of volatility indexes. For example, The Chicago Board Options Exchange Volatility Index (VIX) is calculated using a weighted average of implied volatility of At- The-Money and Near-The-Money options on the S&P500 Index or IV Index Call -a specially designed vega weighted average of implied volatility using only call options (see www.ivolatility.com). In any case volatility index must be estimation of unknown distribution parameter σ0. Simplest method of volatility index building is calculation of arithmetic average of implied volatilities for all strikes. If each implied volatility is unbiased estimation of unknown parameter σ0 then arithmetic average is also unbiased estimation with much lesser deviation The results of arithmetic averages calculation are shown in last row (volatility index (VIndex)) in Table1, Table2 and Table3 in 4 and 5 columns for appropriate samples. Above we used classical maximum-likelihood method for estimation unknown distribution parameter σ0 by using total, for all strikes, samples. But we also can use classical maximum-likelihood method for each strike, i.e. can calculate implied volatility by using not math. expectation but likelihood function for one observation. Results of calculation of implied volatility based on likelihood function are shown in columns 6 and 7 in Table1, Table2 and Table3. Comparing volatility indexes in Tables 1, 2 and3 calculated by 2 methods we see essential differences in numbers. What numbers are more useful? The criterion of usefulness is how volatility index represents market option prices. We will use volatility index as parameter σ0. For given σ0 cumulative distribution function FPr(y price>0) can be calculated by formula (6). For given cumulative distribution function we can calculate stochastic variable having this distribution (method Monte Carlo). Stochastic variable is observed market option price. So defined observed market option price is calculated by formula:
(11) Where Rnd()- function returns a random value less than 1 but greater than or equal to zero. Lp-1()-inverse Laplace function. Fig. 1 Fig. 2 Fig. 3 On Fig.1,2 and 3 Series 1-market last option prices Series 2-MC prices, calculated by using classical maximum-likelihood method Series 3-MC prices, calculated by using Black-Sholes formula. As it followed from Fig. 1,2,3 volatility calculated by using classical maximum-likelihood method better represent market prices than volatility calculated by using Black-Sholes formula.