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However, iliis advantage comes ai some cost: tins great flexibility is the result of strong parametric restrictions that each continuous-time process imposes on all its finite-dimensional distributions (see Section 9.1.1 for the definition of a fiiiiie-dimensional dislrihuiion). In particular, the stochastic differentia! equation (9.3.25) imposes independence and normality on all increments of log/(/) and linearity of the mean and variance in the increment interval, hence the continuously compounded return between a Friday and a Monday must have three limes the mean and three times the variance of the continuously compounded return between a Tuesday and a Wednesday, the Tuesday/Thursday return must have the same distribution as the Saturday/Monday return, and so on. In fact, the specification of a continuous-lime stochastic process such as (9.3.25) includes an infinity of parametric assumptions. Therefore, the convenience that continuous-time stochastic processes affords in dealing with irregular sampling should be weighed carefully against the many implicit assumptions that come with it. Coutinuous-llecord Asymplolics Since we can estimate a and a from arbitrarily sampled data, a natural question lo consider is how to sample returns so as lo obtain the best estimator? Are If) annual observations preferable lo 100 daily observations, or should we match die sampling interval to the horizon we are ultimately interested in, e.g., the limc-to-matiirity of the option we wish to value? To address these issues, cousideragainasampleof n+1 prices Pit, l\ , / equally spaced at intervals of length h over the fixed lime span [0, 7] so that l\ = P{kh), к = 0.....n, and 7 = nh. The asymptotic variance of the maximum likelihood estimator of a a- ] is then given by (9.3.7), which may be evaluated explicitly in this case as: VarIй] Vaijcr-l Observe thai (9.3.31) docs not depend on the sampling interval Л. As n increases without bound while 7 is fixed (hence It decreases to 0), a2 becomes more precise. This suggests that die best estimator for cj2, the one with smallest asymptotic variance, is the one based on as many observations as possible, regardless of what their sampling interval is. Interestingly, this result does not hold for the estimator of of, whose asymptotic variance depends on 7 and not n. More frequent sampling within a fixed time span-often called continuous-record asytnptotics-will not increase the precision of a. and the best estimator for a is one based on as long a time span as possible. T ii (9.3.30) (9.3.31) Table 9.2a. Asymptotic standard errors fora. I 1 I I l l l I S Ti! 35 м Ш 55S KR TMi 2 0.2000 0.2828 0.4000 0.5657 0.8000 I.I314 1.6000 2.2627 S.2000 4.525E 4 0.1414 0.2000 0.2828 0.4000 0.5657 0.8000 1.1314 1.6000 2.2627 3.200(1 К 0.1000 0.1414 0.2000 0.2828 0.4000 0.5657 0.8000 1.1314 1.6000 2.2627 16 0.0707 0.1000 0.1414 0.2000 0.2828 0.4000 0.5657 0.8000 1.1S14 1.6000 32 0.0500 0.0707 0.1000 0.1414 0.2000 0.2828 0.4000 0.5657 0.8000 1.1314 64 0.0354 0.0500 0.0707 0.1000 0.1414 0.2000 0.2828 0.4000 0.5657 0.8000 128 0.0250 0.0354 0.0500 0.0707 0.1000 0.1414 0.2000 0.2828 0.4000 0.5657 256 0.0177 0.0250 0.0354 0.0500 0.0707 0.1000 0.1414 0.2000 0.2828 0.4000 512 0.0125 0.0177 0.0250 0.0354 0.0500 0.0707 0.1000 0.1414 0.2000 0.2828 1,024 0.0088 0.0125 0.0177 0.0250 0.0354 0.0500 0.0707 0.1000 0.1414 0.2000 Asymptotic standard error of a for various values of n and h, assuming a base interval of h - 1 year and a - 0.20. Recall lhat T a nh; hence the values n = 64 and h = 1/16 imply a sample of 64 observations equally spaced over 4 years. Tables 9.2a and 9.2b illustrate the sharp differences between a and a2 by reporting asymptotic standard errors for the two estimators for various values of л and h, assuming a base interval of Л=1 year and ex =0.20. Recall that Tsnh, hence ihe values n=64 and h=- imply a sample of 64 observations equally spaced over 4 years. In Table 9.2a the standard error of a declines as we move down the table (corresponding to increasing T), increases as we move left (corresponding to decreasing T), and remains the same along the diagonals (corresponding to a fixed 7 ). For purposes of estimatinga, having 2 observations measured at 6-monlh intervals yields as accurate an estimator as having 1,024 observations measured four times per day. In Table 9.2b the pattern is quite different. The entries are identical across the columns-only the number of observations n matters for determining the standard error of a2. In this case, a sample of 1,024 observations measured four times a day is an order of magnitude more accurate than a sample of 2 observations measured at six-month intervals. The consistency of a2 and the inconsistency of a under continuous-record asymptotics, first observed by Merton (1980) for the case of geometric Brownian motion, is true for general diffusion processes and is an artifact of the notiHlifferentiability of diffusion sample paths (see Bertsimas, Kogan,1 and Lo [ 1990] for further discussion). In fact, if we observe a continuous j record of P(t) over any finite interval, we can recover without error the diffusion coefficient er(-), even in cases where it is time-varying. Of course, in Table 9.2b. Asymptotic staitflartl errors finer1.
Asymptotic standard error ol о - lor various values ol* and /i, assuming a lyase inlci vat ol h = 1 year and a = 0.20. Recall dial / = nlr, lience die values н = 04 and h - 1/lli imply a sample ol 1S4 obseivalions equally spaced over 4 years. practice we never observe continuous sample paths-the notion ol continuous time is an approximation, and die magnitude of the approximation error varies from one application lo the next. As the sampling interval becomes liner, other problems may arise such as the effects of the bid-ask spread, nonsynchronous trading, and related market microstructure issues (sec Chapter 3). For example, suppose we -decide to use daily data to estimate a-how should weekends and holidays he treated? Some choose lo ignore diem altogether, which is tantamount to assuming that prices exhibit no volatility when markets are closed, with j the counlcrfactual implication thai Fridays closing price is always equal to iMondays opening price. Alternatively, we may use (9.3.29) in accounting j for weekends, but such a procedure implicitly assumes thai the price process exhibits the same volatility when markets are closed, implying lhat the Fiiday-to-Monday return is three limes more volatile than die Monday-to-Tuesday return. This is also counterfactual, as French and Roll (1980) have shown. The benefits of more frequent sampling must be weighed against the costs, as measured by the types ol biases associated with more finely sampled data. Unfortunately, there are no general guidelines as to how lo make such a tradeoff-il must be made on an individual basis with the particular application and data at hand. .See Hensiiuas, Коцап. and t.o (Hl.lli). l.o and MacKinlay (I.IKVI). Perron (lO.ll). and Shiller and Ierron (I0K5) lor a more deiailed analysis ol die interactions between sampling interval and sample size. VarlG] = -V/(0), - (9.3,34) and V[ may be estimated in the natural way: ot) ttU In much the same way, the precision of the estimators of an options sensitivity to its arguments-the options delta, gamma, dicta, rho, and vega (see Section 9.2.1)-may also be readily quantified. The Hlach-Si holes Case As an illustration of these results, consider the case of Шаек and Scholes (1973) in which /(/) follows a geometric Brownian motion, ilP(l) = / /(/)( + crP(t) ,111(1) (9.3.30) and in which the only parameter of interest is a. Since the maximum likelihood estimator <tl of a- has an asymptotic distribution given by 1 This billows from the principle ol iiivariauec: The maximum likelihood estimator оГ a nonlinear liinciion ol a parameter vector is the nonlinear him lion ol the parameter vei tors iiiasiiiiiiii! likelihood estimator. See, lor example, /.ehn.i (lOtili). / 1. ~) (htanli/ying the Precision of Option Price Estimators Once the maximum likelihood estimator <7 ol ihe imdcrlving assets parameters is obtained, the maximum likelihood estimator ol the option price 0 may be cousiruelcd by inserting 0 into the option-pricing formula (or into the numerical algorithm that generates the [nice).1 Since 0 contains estimation error, C, = ( (/(/), 0) will also contain estimation error, and for trading and hedging applications it is imperative that we have some measure of its precision. This can be constructed by applying a first-order Taylor expansion lo (,(0) lo calculate its asymptotic distribution (see Section A.I of the Appendix) s/Ti(C. - С.) ~ N(().\,(0)) (9.3.32) л ;)(;(/(/), ) . <)( ;(/>(,).0) V/{0) = - -I-(fl)---J-L, ( ,.3.33) at) at) where C, == G(l(t), 0) and 1(0) is die information matrix defined in (9.3.7). Th-.тсfore, for large n the variance of С may be approximated by: j. tJCIIUUtlll . , ii Dig Aloitels Jii(a- - о-) ~ /V(0. 2(7) (sec Section 9.3.2), the asymptotic distribution of lite Black-Scholes call-option price estimator G is s/wlG-G) ~ J\f(i), V/), V, = /*(/)а-у(,/,), (9.3.37) where </>( ) is the standard normal probability density function and d{ is given in (9.2.17). From the asymptotic variance V/ given in (9.3.37), some simple comparative static results may be derived: -Pay/T-l</>-(<V) il,. i) V/ P1 i).V ау/Т-1ф-<<1\)<1\ (9.3.38) i i l + KИ-О У /1ой(/7Х)\- lhe following iiie<pialities may then be established: where Щ i)P i)X i) V/ il(V-n - () iff 0 iff - i o, X * - > 0 if л, < I, , (- -0 ( К > + (9.3.39) (9.3.40) (9.3.41) (9.3.42) (9.3.43) Inequality (9.3.10) shows thai the accuracy of G decreases with the level of the stock price as long as die ratio ol the slock price to the strike price is less than C. However, as the stock price increases beyond AYj, the accuracy of 9.3. Implementing Parametric Option lacing Models 369 -i, < Table 9.3. Cutoff values for comparative statics ofVj.
<5 begins to increase. Inequality (9.3.41) shows a similar pattern for Vj with! respect to the strike price. I Interestingly, inequality (9.3.43) does not depend on either the stock o* strike prices, and hence for shorter maturity options the accuracy of G will increase with the time-to-maturity T-l. But even if (9.3.43) is not satisfied, the accuracy of G may still decline with T-t if (9.3.42) holds. Table 9.3 reports values of c\ through q for various values of T-t assuming an annual interest rate of 5% and an annual standard deviation of 50%, corresponding to weekly values of r = log(1.05)/52 and a = 0.50/\/52. Given the numerical values of ci and 1/ег, (9.3.42) will be satisfied by options that are far enough in- or out-of-the-money. For example, if the stock price is $40, then options maturing in 24 weeks with strike prices greater than $42.09 or less than $38.02 will be more precisely estimated as the time-to-maturity declines. This is consistent with the finding of MacBeth and Merville (1979, 1980) that biases of in- and out-of-the-money options decrease as the time-to-maturity declines, and also supports the observation by Gultekin, Rogalski, and Tinic (1982) that the Black-Scholes formula is more accurate for short-lived options.18 Th rough first-order Taylor expansions (see Section A.4 of the Appendix), the accuracy of the options sensitivities (9.2.24)-(9.2.28) can also be readily derived, and thus the accuracy of dynamic hedging strategies can be measured. For convenience, we report the asymptotic variances of these quantities in Table 9.4. 9.3.4 The Effects of Asset Return Predictability The martingale pricing method described in Section 9.2.2 exploits the fact that the pricing equation (9.2.15) is independent of the drift of P(t ). Since IHThere air, ol course, oilier possible explanations for such empirical regularities, such as the presence of stochastic volatility or a misspecilicatum of the stock price dynamics. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 [ 62 ] 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 |