Estimator Asymptotic Variance

These asymptotic variances arc based on ihe assumption that tile variance estimator a* is the maximum likelihood estimator which has asymptotic distribution ,/ii(rf- - п-( ~ tf (II. in). hence Jn{t!{o*) - /(a )) ~ Nu.).la4{ilF\,a)/ila*)2) where /(a2) is the option sensitivity. Following standard conventions, the expressions reported in die table are the asymptotic variances of Уй(/ (<}1) - / (г/2)) and must be divided by die sample si/.e н to obtain the asymptotic variances of the (unnormalizcd) sensitivities Fici ).

the drift does not enter into the PDE (0.2.15), for purposes of pricing options it may be set to any arbitrary function or constant without loss of generality (subject to some regularity conditions). In particular, under the equivalent martingale measure in which all asset prices follow martingales, the options price is simply the present discounted value of its expected payoff al maturity, where the expectation is computed with respect to the risk-neutralized process / (():

dl-tl) = rl V)dt + ol (t)dB (0.3.44)

(/log/J*(/) s ( /(()= [r - dl + a dB. (0.3.45)

; Although the risk-neutralized process is not empirically observable,11 it is nevertheless an extremely convenient tool for evaluating the price of an option on the slock with a data-generating process given by /()

Moreover, the risk-neutral pricing approach yields the following implication: as long as the diffusion coefficient for the log-price process is a fixed

I lowever, under certain conditions it can lie estimated; see, tor example, Ait-Sahalia and l.o>( 1УШ)).Jackwerth and Rubinstein (Hl.ir.), Rubinstein (НИМ), Shiuiko (l!Ki:i), and Section IZ.M ol diaptet VI.

WW 4

l-X4<hd, -1)-

.Y(7-/)r- - (/,0((/,)

(V.f, ((,)-

constant ci, then the Black-Scholes formula yields the correct option price regardless of the specification and arguments of the drill. This holds more generally for any derivative asset which can be priced purely by arbitrage, and where the underlying assets log-price dynamics is described by an I to diffusion with constant diffusion coefficient: the derivative-pricing formula is functionally independent of the drift and is determined purely by the diffusion coefficient and the contract specifications of the derivative asset.

1 his may seem paradoxical since two stocks with the same a but dilfereiil drifts will yield the same Black-Scholes price, yet the slock with the larger drill has a larger expected return, implying that a call option on that stock is more likely to be in-the-money al maturity than a call option on the stock with the smaller drift. The resolution of this paradox lies in the observation lhat although the expected payoff of the call option on the larger-drift slock is indeed larger, it must be discounted al a higher tale of return-one that is commen.su rate with the risk and expected relurn of its underlying slock- and this higher discount rate exactly offsets the larger expected payoff so dial the present value is the same as the price of the call on the stock with lb- smaller drift. Therefore, the fact that the drift plays no direct role in the Black-Scholes formula belies its importance. The same economic forces thai determine the expected returns of slocks, bonds, and oilier financial assets are also at work in pricing options.

These considerations are less important when the drift is assumed lo be constant through lime, but if expected returns are lime-varying so (hat stock returns are predictable: to some degree, this predictability must be taken into account in estimating a.

The Treading Oruslcin-Uhleubeck Process

To see how a lime-varying drift can influence option prices, we follow l.o and Wangs (1095) analysis by replacing the geometric Brownian motion assumption (A3) of the Black-Scholes model with the following stochastic differential equation for the log-price piocess p(t):

dp(l) = {-y(p(D - it I) + n)dt + cr dl}. (9.3.40)

where

у > О, /ДО) = p,. I e [O.oo).

Unlike the geometric Brownian motion dynamics of the original Black-Scholes model, which implies thai log-prices follow an arithmetic random walk with 111) normal increments, this log-price process is the sum of a zero-mean stationary autoregressive Gaussian process-an Oi nslein-Uhlenbcck process-and a deterministic linear trend, so we call this the trending Oil process. Rewriting (9.3.40) as

d(p(t)-nl) = -y(lU) --nt)dl Ia dll (9.3.47)

Table 9.4. Asymptotic variances of Hlack-Scholes call price sensitivity estimators.

shows lhat when /,(t) cli-viaii-s from its trend щ, it is pulled hack at a rale proportional to its deviation, where у is the speed of adjustment. This reversion lo the trend induces predictability in the returns of this asset.

To develop further intuition for the properties of (9.3.46), consider its explicit solution:

pit) = lu + e-rlb + o / е-* -* dli(s), (9.3.48)

Irom which wc can obtain the unconditional moments and co-moments of continuously compounded т-period returns r,(r) = pit)-pit-r):2

l-li(r)] = Itr (9.3.49)

Yarr,(r)j = Z [l -> ], r > () (9.3.50)

Cov, (r). r,.(r) = -1 ,->- --.->[! -,-УГ2,

i 4- r < I., (9.3.51)

Con ,(r), iM,(r) e= /),(r) = -1 [l - c 1 ]. (9.3.52)

Since (9.3.4f>) is a Gaussian process, the moments (9.3.49)-(9.3.51) completely characteri/.e the finite-dimensional distributions of r, (t) (see Section 9.1.1 for the definition of a finite-dimensional distribution). Unli.e the arithmetic Brownian motion or random walk which is nonstationary and olten said to be difprrnce-stationnry or a stochastic trend, the trending O-U process is said to be trend-stationary since its deviations from trend follow a stationary process.21

- Since we liave <nnditlonctl on /,((1) = /, j defining ihe detrended log-price process, il is л slight abuse ol terminology ю rail these moinenis inicoiirtiiional . However, in ibis rase ihe disiinc lion is primal ily senianiic since the conditioning variable is more of an initial rondiiion than an information variable-if we define ihe beginning of time as / = 0 and the fully observable slatting value of/,(()) as/ then (!I.M!l)-(il..1.ri2) are iiiiroiKliiion.il inomenLs relative lo these initial conditions. We shall adopt this definition of an nniondiiional moment thiouglioiii the remainder ol ibis chapter.

-An implication of tienil-si.uionariiy is that the variance of r-period returns has a finite limn as x increases without hound, in this case а-/у, whereas ibis variance increases linearly with r under a random walk. While ire nil-stationary processes are often simpler to estimate,

they have been ciitici/cil as tuiicalistic t.....lels ol financial asset prices since they do not accord

well with Ihe common iuiuiiion lhat longer-hori/oii asset relurnsexhibil more risk or that price loiriasts exhibit more unceitainiy as the forecast hoi./on grows. However, ilthe source of such intuition is empirical observation, it may well be consistent with licntl-slationaiily since il is now well-known that for any finite set ol data, treiul-sialionarity and dillerence-slaiionarilv are virtually indistinguishable (see, lor example. Section 2.7 in Chapter 2, Campbell and IVrion I II.H , llainilli.i, i I.W-I. Chapters I7-IK, and the many other unit toot papers tbev cite).

Note that the first-order autocorrelation (9.3.52) of the trending O-U increments is always less than or equal to zero, bounded below by -, and approaches - as r increases without bound. These prove to be serious restrictions for many empirical applications, and they motivate the alternative processes introduced in Lo and Wang (1995) which have considerably more flexible autocorrelation functions. But as an illustration of the impact of serial correlation on option prices, the trending O-U process is ideal: despite the differences between the trending O-U process and an arithmetic Brownian motion, both data-generating processes yield the same risk-neutralized price process (9.3.44), hence the Black-Scholes formula still applies to options-on stocks with log-price dynamics given by (9.3.46).

However, although the Black-Scholes formula is the same for (9.3.46), the a in (9.3.46) is not necessarily the same as the a in the geometric Brownian motion specification (9.2.2). Specifically, the two data-generating processes (9.2.2) and (9.3.46) must fit the same price data-they are, after all, two competing specifications of a single price process, the true DGP. Therefore, in the presence of serial correlation, e.g., specification (9.3.46), the numerical value for the Black-Scholes input a will be different than in the case of geometric Brownian motion (9.2.2).

To be concrete, denote by r,(r), s2[r,(x)}, and Pi(r) the unconditional mean, variance, and first-order autocorrelation of r,(r)}, respectively, which may be defined without reference to any particular data-generating process.22 The numerical values of these quantities may also be fixed without reference to any particular data-generating process. All competing specifications for the true data-generating process must come close to matching these moments to be plausible descriptions of that data (of course, the best specification is one that matches all the moments, in which case the true data-generating process will have been discovered). For the arithmetic Brownian motion, this implies that the parameters (д, о2) must satisfy the following relations:

7лЧ) = pir (9.3.531)

/l>,(r)] = cr2r (9.3.54)

p,(r) = 0. (9.3.55)

From (9.3.54), we obtain the well-known result that the Black-Scholes input a2 may be estimated by the sample variance of continuously compounded returns r,(r)}.

Nevertheless, l.o and Wang (109Г>) provide a generalization of the trending O-U process thai contains stochastic trends, in which case the variance of returns will increase wilh ihe holding period r.

Of course, il must be assumed lhat the moments exist. However, even if iheydo not, a similar but more involved argument may be based on location, scale, and association parameters.

In the case of the trending O-U process, ihe parameters (/(. y, a2) must satisfy

м7) = ЦТ (9.3.50)

s2[,;(t)] = ~[\-e-y]. г > 0 (9.3.57)

Pi(r) = --[1 -<- ]. (9.3.58)

bscrvc that tlicse relations must hold for the population values of the parameters if the trending O-U process is to he a plausible description of the PGP. Moreover, while (9.3.56)-(9.3.58) involve population values of the parameters, they also have implications for estimation. In particular, under the trending O-U specification, the sample variance of continuously Compounded returns is clearly not an appropriate estimator for a2. j Holding the unconditional variance of returns fixed, the particular value of <72 now depends on y. Solving (9.3.57) and (9.3.58) for у and h2 yields:

у = --log(l +2Pi(r)) (9.3.59)

a2 = s2{r.) K(l - -Г = ~-[ YT(l - г )-1 ]. (9.3.00)

vhich shows the dependence of a2 on у explicitly.

In the second equation of (9.3.00), a2 has been re-expressed as the pVoduct of two terms: the first is the standard Black-Scholes input under the assumption that arithmetic Brownian motion is the data-generating process, and the second term is an adjustment factor required by the trending O-U specification. Since this adjustment factor is an increasing function of y, as returns become more highly (negatively) autocorrelated, options on the slock will become more valuable ceteris paribus. More specifically, (9.3,00) may be rewritten as the following explicit function ofpi(r):

.. s2[r,{T)} loiO -f 2/Mr)) ,

o2 = -- h / . P,(r) e (-i.01. (9.3.01)

r 2p,(r)

Holding fixed the unconditional variance ol returns s2 [ ,(r)), as the absolute value of the attlocorrelation increases from 0 to , the value olci2 increases without bound.2 This implies that a specification error in the dynamics of s(t) can have dramatic consequences for pricing options.

2{V\V locus on ihe absolute value ol ilic auioroirclaiion to avoid confusion it) making comparisons between results lor negatively autocorrelaled and positively autocorrelated asset returns. See lxi and Wang (liltiri) lor ftirtlier details.

As (he relurn interval r decreases, il can be shown that the adjustment factor to s21 i)(t ) /r in (9.3.01) approaches unity (use l.llopitals rule). In the continuous-time limit, the standard deviation of continuously compounded returns is a consistent est i mat or for rr and die el I eels of predictability on cj vanish. The intuition comes from ihe lacl thai ci is a measure ol local volatility-the volatility of inlinilesimal price changes-and there is no predictability over any inlinilesimal lime interval by construction (see Sec-lion 9.1.1). Therefore, the influence of predictability on estimators for cr is an empirical issue which depends on the degree of predictability relative to how finely the data is sampled, and must be addressed on a casc-by-case basis. However, we provide a numerical example in the next section in which the magnitude of these effects is quantified for the Black-Scholes case.

Adjusting Ihe Illach-Scholes Formula for Ireditlability

Expression (9.3.01) provides the necessary inpiii lo the Black-Scholes formula for pricing options on an asset with the trending O-U dynamics. 11 (he unconditional variance of daily returns is ,r />( I) , and if the first-order autocorrelation of г-period returns is pt (r), then die price of a call option

is given by:

(;, (()... K. 7, r,<7) = /(/)44<M- 4M</..). (9.3.02) where

., .v-r,(l) log(l +2/),(r>)

(II --2p,(T)l1 - 1)

/Mr) fc (-,.( !. (9.3.03)

and d\ and d2 are defined in (9.2.17) and (9.2.18), respectively.

Expression (9.3.02) is simply the Black-Scholes formula with an adjusted volatility input. The adjustment factor multiplying .v r,( 1) /r in (9.3.03) is easily tabulated (see l.o and Wang [ 19951); hence in practice il is a simple matter to adjust the Black-Scholes formula for negative autocorrelation of the form (9.3.58): Multiply the usual variance estimator s2\i,(\)]/t by (Inappropriate factor from Table 3 ofl.o and Wang (1995), and use this as a- in the Black-Si holes formula.

Note lhat for all values of/>,(r) in (-n-01. die factor multiplying л - [ r, (I )/r in (9.3.03) is greater than or equal to one and increases in the absolute value of the first-order autocorrelation coefficient. This implies that option prices under the trending O-U specification are always greater than or equal lo prices under the standard Black-Scholes .specification, and that option prices are an increasing function of the absolute value of the first-order autocorrelation coefficient. These are purely features of the trending O-U process and do not generalize to oilier specifications of the drift (see l.o and Wang [ 1995] for examples of other patterns).