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Review of Derivatives Research, 4, 263–284, 2000.c© 2001 Kluwer Academic Publishers, Boston. Manufactured in The Netherlands.
Drift Estimation of Generalized Security PriceProcesses from High Frequency Derivative Prices
GURUPDESH S. PANDHER [email protected] of Finance, DePaul University, 1 East Jackson Blvd., Chicago, Illinois 60604
Abstract. This paper presents a framework for using high frequency derivative prices to estimate the drift ofgeneralized security price processes. This work may be seen more generally as a quasi-likelihood approachto estimating continuous-time parameters of derivative pricing models using discrete option data. We developa generalized derivative-based estimator for the drift where the underlying security price process follows anyarbitrary state-time separable diffusion process (including arithmetic and geometric Brownian motion as specialcases). The framework provides a method to measure premia in derivative prices, test for risk-neutral pricing andleads to a new empirical approach to pricing derivative contingent claims.
A sufficient condition for the asymptotic consistency of the generalized estimator is also obtained. A studybased on generating the S&P500 index and calls shows that the estimator can correctly estimate the drift parameter.
Keywords: excess return, market price of risk, risk-neutral pricing, quasi-likelihood estimation, Feynman-Kac,asymptotic consistency.
JEL classification: G13, G14, C13, C52
1. Introduction
This paper presents a framework for using high frequency derivative prices to estimate thedrift of generalized security price processes. Alternatively, since the excess return parame-ter is the security’s drift normalized by the risk-free return, we may view this as estimationof the underlying security’s excess return from derivative prices. Therefore, we use theseterms interchangeably for the purpose of this paper. This work may be seen more generallyas a quasi-likelihood approach to estimating continuous time parameters of derivative pric-ing models using discrete option data. We develop a generalized derivative-based estimatorfor the drift where the underlying security price process follows any arbitrary state-timeseparable diffusion process. Hence, the methodology of this paper gives generalized esti-mators for any diffusion process where the drift and volatility structures are separable intostate-time dependent processes. This includes arithmetic and geometric Brownian motionas special cases. Pandher (2001a) considers estimation of the excess return parameter in thecontext of the underlying security following geometric Brownian motion. The frameworkprovides a way to measure premiums in derivative prices, test for risk-neutral pricing andleads to a new empirical approach to pricing derivative contingent claims.
The finance related literature on derivatives and asset price stochastics has tended to focuson estimation of diffusion parameters (e.g. drift, volatility) from the asset state process X(e.g. index, stock, bond) and the risk neutral density from derivative prices V (X) using max-imum likelihood, moment conditions and non-parametric methods (Banz and Miller, 1978;Breeden and Litzenberger, 1978; Lo, 1989; Sherrick, Irwin and Forster, 1990; Florens-
264 PANDHER
Zmirou, 1993; Dohal, 1987; Hansen and Scheinkman, 1995; Broze, 1997; Ait-Sahaliaand Lo, 1998; Duffie and Glynn, 1998 and others). The estimation of parameters of theunderlying security process from derivative prices, however, has not received much atten-tion. We use high frequency derivative prices V (X) sampled at discrete intervals (e.g. day,intra-day) to estimate the excess return parameter λ of the underlying asset price process X ;note that λ plus the risk-free rate is then the drift estimate. The estimation has interestingempirical derivative pricing applications as market prices of risk (λ/σ) for any state-timeseparable security process can be readily constructed from derivative excess returns andvolatility (Pandher, 2000b, considers the estimation of the underlying security’s volatilityfrom derivative prices). The market price of risk is a key parameter in Girsanov’s formulafor the risk-neutral density critical to valuing contingent claims and this pricing applicationis being explored. A test for measuring divergence from the risk-neutral pricing model bycomparing the excess return parameter estimated from derivative prices V (X) with the sameestimated directly from the underlying asset price process X is discussed in Pandher (2001a).
The excess return parameter is estimated from the derivative’s discounted value process.Here, we exploit the relationship between an arbitrary derivative claim’s partial differentialequation and probabilistic representations (Feynman-Kac theory) and use continuous risk-neutral pricing and quasi-likelihood theory to identify the optimal orthogonality conditionfor the excess return parameter. We show that the generalized excess return estimatoris asymptotically consistent. This result is established in the context of a non-stationaryunderlying security process under a finite conditional second moment assumptions on theunderlying security price process. This requirement is satisfied by a large class of stochasticprocesses. This means that the generalized estimator is robust to distributional assumptionson the security process (e.g. non-Gaussian, Poisson jumps).
There are further implications of this estimation framework for derivatives. First, themethodology is very general and applicable to any arbitrary traded derivative includingcalls, futures and swaps. Second, the estimators inherit the optimality properties of thequasi-likelihood framework (Godambe, 1960; Godambe and Heyde, 1987; Thavaneswaran& Thompson, 1986). This means that the statistical equations used to estimate the excessreturns are i) unbiased and ii) of minimum variance in the class of all linear estimatingequations. Third, the bond pricing models of Vasicek (1977), Brennan and Schwartz (1979),Artzner and Delbaen (1988) and Cox, Ingersoll and Ross (1985) require an “inversion of theterm structure” to remove the market price of risk at the initial step. There are computationaldifficulties in this inversion since bond pricing formulae are highly non-linear. Our approachoffers linear estimation of parameters (market price of risk, excess returns) appearing inhighly non-linear bond prices. The generalized estimation framework of this paper alsopermits estimation of drift parameters in spot interest rate models such as Cox, Ingersolland Ross (1985) from bond prices.
The empirical properties of the estimator are tested and verified using an extensive Monte-Carlo diagnostic study which also enables resolution of important sample design issues.The “S&P500 index” and Black-Scholes call options defined on it are simulated usinghistorical trend and volatility. The volatility and the excess return parameters are estimatedseparately from both the index and call option prices under various scenarios to study theimpact of sample size, strike-level and length of call maturity.
DRIFT ESTIMATION OF GENERALIZED SECURITY PRICE PROCESSES 265
The results of the diagnostic study verify the ability of the proposed estimator to correctlyestimate security excess returns. Estimates of volatility from both the index and calls arevery close for any given sample size. Estimation is also found to be unaffected by thestrike level of the call. Since the vast majority of traded options are of maturities less thanone year, sample size cannot be feasiblely increased by extending the time to maturity. Asampling design which cycles over calls of smaller (non-overlapping) maturities is requiredas a way to reduce the variance of estimators. It is found that sampling from cycles ofshorter maturities with the same effective sample size (cycles times maturity length) yieldsconsistent and stable estimation. This result gives confidence in the applicability of theestimation methodology to market derivative prices where larger samples derived fromcycling over multiple shorter maturities can be used to reduce variance.
The empirical derivative pricing implications of this estimation methodology are beinginvestigated and the proposed estimation is being currently applied to S&P 500 call optionsdata from the Chicago Board of Options Exchange. The remainder of the paper is organizedas follows. Section 2 sets out the probability model and stochastic processes for the arbitraryderivative process and states the estimation problem. Section 3 introduces the main featuresof the quasi-likelihood estimation framework, describes the construction of the excess returnestimator and discusses the consistency and asymptotic normality of the estimator. AMonte-Carlo study is described in Section 4 which verifies the estimation and evaluates theimpact of sample size, strike level, strike replication and maturity length on the estimation.Conclusions follow in Section 5.
2. Derivative & Security Processes and Statement of Problem
This section defines the stochastics for the arbitrary derivative process and its underlyingsecurity process. We plan to estimate the drift parameter of the security process fromderivative prices.
Let X = {Xt ,Ft ; 0 ≤ t ≤ T } represent the stochastic process for the underlying security(e.g. stock, index) on which the derivative claims are defined. Xt is determined (measurable)by the information set (filtration) Ft . We view X as a diffusion following the generalstochastic differential equation (under the empirical measure P):
d Xt = b(t, Xt ) dt + σ(t, Xt ) dWt (2.1)
where b(t, Xt ) = (b1(t, X1t ), . . . , bd(t, Xd
t ))′ is the drift vector, σ(t, Xt ) is the dispersionmatrix (of rank d) and dWt is a d-dimensional Brownian motion with respect to P . Lastly,define a(t, Xt ) = σ(t, Xt )σ
T (t, Xt ) to be the diffusion matrix. Our estimation frameworkwill require that the volatility σ(t, Xt ) and drift b(t, Xt ) processes be “time-state separable”.
Assumption 1 (Time-State Separability of Drift and Volatility Processes) The drift andvolatility processes of the general diffusion (2.1) have the structure b(t, Xt ) = A(Xt )bt andσ(t, Xt ) = D(Xt )σt where A(Xt ) is a diagonal matrix of rank d, bt , is a vector of drifts,σt and D(Xt ) are matrices of rank d.
Let Q be the risk-neutral measure under which expectations of the X process discountedat the risk-free spot interest rate process r = {rt ; 0 ≤ t ≤ T } are Q-martingales where r is
266 PANDHER
the growth rate process of the money market discount factor B(t, T ) = exp(− ∫ Ts=t rs ds).
Risk neutral valuation theory (Harrison and Kreps, 1979; Harrison and Pliska, 1981) assertsthat an attainable contingent claim can then be valued as a discounted expectation underthe measure Q.
The process of making the discounted asset a martingale requires the transformation
dWt = dW̃t − γt (t, Xt ) dt (2.2)
where W̃t is a Brownian motion with respect to the risk-neutral measure Q and the marketprice of risk γ (t, Xt ) satisfies
σt (t, Xt )γt = bt (t, Xt ) − A(Xt )rt . (2.3)
Under Assumption 1, this reduces to
σt D(Xt )γt = A(Xt )(bt − rt ). (2.4)
The relationship between the equivalent measures P and Q is given by Girsanov’s change ofmeasure formula. In the special case of geometric Brownian motion, D(Xt ) = A(Xt ) = Xt ,and (2.4) reduces to σtγt = (bt −rt ). Our framework accommodates any arbitrary state-timeseparable asset price process.
Let V (X) = {V (t, Xt ),Ft ; 0 ≤ t ≤ T } be the generic value process of the derivativeclaim based on the state variable process X where V (t, Xt ) is in the class C2([0, T ]XR
d).Our purpose is to estimate the excess return parameter λ = b − r from high frequencyderivative prices V (X). Note that when D(Xt ) = A(Xt ), the derivative-based market priceof risk can then be constructed from the normalization γ = λ/σ .
3. Estimation of Excess Returns from Derivative Prices
We begin by discussing the structure of the derivative market data. Let the observed pricesfor the derivative security be sampled at points in the sequence {t0, t1, . . . , tn} ∈ [0, T ]with t0 = 0 indexing the start of the sampling period and tn = T represents the time tomaturity. Then, j = tj − tj−1, j = 1, . . . , n, is the length of the period between points inthe term structure. At each sampling point tj , a cross-section of replicate prices may existindexed by k = 1, . . . , m (e.g. calls of different strikes Kk). Further, prices are availableover non-overlapping cycles of maturity times {T1, . . . , Tg, . . . , Tp}. The price data consistsof a sequence of derivative market prices {Vk(tj , Tg) ≡ V (tj , Xtj ; Kk, Tg), g = 1, . . . , p,j = 1, . . . , n, k = 1, . . . , m}. The exact structure of the price sequence Vk(tj , Tg) willdepend on the sample design (e.g. multiple maturities, strike replicates).
The excess return parameter is estimated from the derivative’s discounted value process.We identify the optimal orthogonality condition for the excess return parameter by applyingquasi-likelihood and Feynman-Kac theory (an arbitrary derivative claim’s partial differentialequation “law of motion” has a probabilistic representation) to the continuous risk-neutralpricing model. Before giving the estimator, we need a brief introduction to quasi-likelihoodestimation theory.
DRIFT ESTIMATION OF GENERALIZED SECURITY PRICE PROCESSES 267
3.1. Quasi-Likelihood Framework
Quasi-likelihood (or Estimating Function) theory provides a general framework for param-eter estimation which includes maximum likelihood estimation (MLE) as a special casewhen an exact distribution is specified for the data generating process and incorporatesleast squares estimation for linear models with no distributional assumptions. It borrowsthe strengths of both approaches while eliminating their weaknesses. For example, LSestimation becomes biased when the variance of the dependent process depends on param-eters appearing in the mean. For a further overview of EF theory, see Godambe and Heyde(1987), Godambe & Kale (1991), and Heyde (1989).
Given the derivative sampling scheme described above, the quasi-likelihood estimator ofthe excess return parameter λ is determined from the class of linear estimating functionsgiven by
H ={
H : H =p∑
g=1
n∑j=1
m∑k=1
αjkg(λ)hjkg(λ)
). (3.1)
where αjkg(λ) is determined from information available at time tj−1 (Ftj−1 -measurable) andE(hjkg(λ) | Ftj−1) = 0, j = 1, . . . , n. The optimality criterion of Godambe (1960) canthen be applied to determine the optimal weighting factors αjkg(λ). In relation to GMMestimation (Hansen, 1982), H may be viewed as a particular orthogonality system. Thequasi-likelihood framework, therefore, gives a systematic method for identifying the optimalestimating function starting with a primitive “error” restriction; i.e. E(hjkg | Ftj−1) = 0,j = 1, . . . , n. Meanwhile, GMM starts with any orthogonality restriction and provides atool for imposing further restrictions on the parameter space. If GMM happens to start withthe optimal orthogonality conditions and the number of parameters equal the restrictions,GMM would equal EF estimation.
Suppressing the indices k and g without loss of generality, the optimal choice of αj (λ) isgiven by Godambe (1960)
α∗j =
(E
∂hj
∂λ| Ftj−1
)′ (Ehj h
′j | Ftj−1
)−1, j = 1, . . . , n. (3.2)
This choice of weights minimizes the variance of the estimating equation and maximizesits sensitivity function to departures from the true parameter value.
The quasi-likelihood estimator then follows by setting H(λ) = 0 as the sample analogueof E(H(λ)) = 0 and solving for λ. Therefore, the estimator depends on the sample designused: i) single strike and maturity, ii) multiple strike replicates on a single maturity andiii) replicates on multiple non-overlapping maturity cycles.
3.2. The Generalized Excess Return Estimator
To construct the estimator, we first identify a function of the data and parameter whoseexpectation given the information up to the time is zero (conditional martingale difference
268 PANDHER
equation or cmde). This is done by constructing an Ito-expansion of the discounted deriva-tive process between two given sampling intervals under the risk-neutral measure. Wenext apply the Feynman-Kac result (Karatzas & Shreve, 1991, p. 366) to reduce terms inthis expansion and introduces the parameter of interest (excess return) by switching to theempirical measure. This theorem shows that any contingent claim has a representation asa partial differential equation and as a corresponding probabilistic representation under therisk-neutral measure. Once the cmde is constructed, the optimal orthogonality restrictionon the cmde is obtained from quasi-likelihood theory. A discrete “feasible” estimator isnext developed from this procedure in which all quantities are known (measurable) withrespect to information available at each sampling point. The asymptotic consistency of thisfeasible estimator is established in Section 3.3.
We now derive the estimating functions (cmde) in the context of our generalized dif-fusion process of Assumption 1. We will consider sampling options with a single strikeand multiple non-overlapping maturity cycles. Pandher (2000a) finds that inclusion ofstrike replicates introduces strong dependence between calls within the same maturity cy-cles due to the common Brownian motion. Therefore, multiple strike replicates do notincrease efficiency and we will ignore them to reduce notational burden. This estimator,in the special case of geometric Brownian motion, is given in Proposition 4 of Pandher(2000a).
Proposition 1 (The Estimating Equation for the Excess Return λ) Let g(t, Xt ) andr(t, Xt ) be continuous bounded functions defining the dividend and risk-free returns, re-spectively. Then, the estimating functions hjg(λ), j = 1, . . . , n, are given by (d = 1)
hjg(λ) = Yjg − λZ̃ jg = V (tj , Tg)B(tj−1, tj ) − V (tj−1, Tg) +∫ s
u=tg(u, Xu)
×B(tj−1, u) du −[∫ tj
tj−1
V X (u, Tg)′ A(Xu)B(tj−1, u) du
]λ (3.3)
where λ = b − r is the excess return, V X (u, Tg) = ∂V (u,Tg)
∂ X is the option “delta” andE(hgj | Ftj−1) = 0. Further, the conditional martingale difference equation hjg(λ) has the“error-side”
Mjg ≡∫ tj
u=tj−1
B(tj−1, tj )V X (u, Tg)′ D(Xu)σudWu . (3.4)
The financial interpretation of the estimating equation (3.3) is as follows. The first threeterms of hjg(λ) represent the dependent “Y” observation in the regression sense while thelast term represents the corresponding independent “X” variable. The first two terms givethe change in the discounted value of the contingent claim observed over the samplinginterval. The third term adds back the discounted dividends paid out over this period. Thefourth integral term involving the “delta” of the derivative claim represents the amount ofthe underlying asset held to replicate the change in the claim’s value over the interval (plusdividends). Therefore, the net change in the value of the claim minus its hedge replication
DRIFT ESTIMATION OF GENERALIZED SECURITY PRICE PROCESSES 269
should be approximately zero. Also, note that higher derivatives-order terms, includinggamma and vega, do not enter the estimating equation (3.3) whose conditional expectationis zero. Application of Feynman-Kac theory allows us to eliminate these terms in the Ito-expansion of the discounted derivative process over the two sampling points (see proof fordetails).
Proof of Proposition 1: Without any loss of generality, we reduce the notational burdenand obtain the conditional Martingale difference equation hj (λ) for the case of a singlematurity cycle (this allows us to drop the index g). We also pursue the development in thegeneral setting d > 1. First, define the expected instantaneous change in V (t, Xt ) underthe risk-neutral measure Q by the second order differential operator
(At V )(x) ≡ lims→0
E Q(V (t + s, Xt+s) − V (t, Xt ) | Ft )
s
= 1
2
d∑j=1
d∑k=1
aj,k(t, x)∂V (x)
∂xj∂xk+
d∑j=1
rt A(xj )∂V (x)
∂xj.
Applying the “chain-rule” version of Ito’s formula to the functional f (Vs, Bs) = V (s, Xs)
B(t, s) with t, s ∈ [0, T ], and s > t yields
V (s, Xs)B(t, s) − V (t, Xt ) =∫ s
u=t
∂ f
∂ Bd Bu +
∫ s
u=t
∂ f
∂VdVu (P.1)
In the above expansion all second derivatives of the functional f (Vs, Bs) are zero and hencethe corresponding quadratic variation terms drop out. Next substitute the following expres-
sions into the right hand side of (P.1): dVs = ∂V∂s ds + As V ds +
(∂V (s,T )
∂ X
)′σ(s, Xs) dW̃s
and d Bs = −Bsrs ds. Applying the Feynman-Kac result (Karatzas & Shreve, 1991, p. 366)to the expansion yields
V (s, Xs)B(t, s) − V (t, Xt ) =∫ s
u=tVu Buru du +
∫ s
u=t
[∂V
∂u+ Au V
]Bu du
+∫ s
u=tBu
(∂V (u, T )
∂ X
)′σ(u, Xu) dW̃u
=∫ s
u=t
[∂V
∂u+ Au V − ru V
]Bu du + M̃(t, s)
= −∫ s
u=tgu Bu du + M̃(t, s) (P.2)
where M̃(t, s) ≡ ∫ su=t Bu
(∂V (u,T )
∂ X
)′σ(u, Xu) dW̃u is a Q stochastic integral.
To introduce the excess return parameter, we now reverse the transformation in Brownianmotion defined by (2.3)–(2.4) in going from the risk-neutral measure to the empirical
270 PANDHER
measure and obtain
M̃(t, s) =∫ s
u=tBu
(∂V (u, T )
∂ X
)′[A(Xu)(bu − ru)] du
+∫ s
u=tBu
(∂V (u, T )
∂ X
)′D(Xu)σu dWu . (P.3)
where Wt is a d-dimensional Brownian motion with respect to the empirical probability
measure P . Defining M(t, s) ≡ ∫ su=t Bu
(∂V (u,T )
∂ X
)′D(Xu)σu dWu this is the “error-side”
of the cmde) and combining (P.2) and (P.3) yields
M(t, s) = V (s, Xs)B(t, s) − V (t, Xt ) +∫ s
u=tgu Bu du
−∫ s
u=tBu
(∂V (u, T )
∂ X
)′[A(Xu)(bu − ru)] du. (P.4)
Note that E(M(t, s) | Ft ) = 0, hence it is a conditional martingale difference function andthe right hand side of (P.4) is zero in conditional expectation. Therefore, the right hand sidedefines the “data-side” of the cmde (replacing s and t by the sampling times tj−1 and tj ,respectively):
hj (λ) ≡ M(tj−1, tj ) = V (tj , T )B(tj−1, tj ) − V (tj−1, T ) +∫ s
u=tgu Bu du
−[∫ s
u=tBu
(∂V (u, T )
∂ X
)′A(Xu) du
]λ, j = 1, . . . , n, (P.5)
where λ = b − r .
Proposition 1 identifies the estimating function hj (λ). To construct the quasi-likelihoodestimator for λ, we also need the weighting functions
α∗jg =
(E
∂hjg
∂λ| Ftj−1
)′ (Ehjgh′
jg | Ftj−1
)−1, j = 1, . . . , n. (3.5)
This choice of weights minimizes the variance of the estimating equation and maximizesits sensitivity to departures from the true value. We then obtain the feasible EF estimatorby solving H∗(λ) = 0 and replacing unknown intra-sampling quantities in the interval[tj−1, tj ] by their known (Ftj−1 -measurable) surrogates as defined in Proposition 2.
Proposition 2 (The Generalized Feasible Estimator for Excess Returns) The general-ized feasible EF estimator for λ and its variance are
λ̂ =[
p∑g=1
n∑j=1
Z ′jgW −1
jg Z jg
]−1 p∑g=1
n∑j=1
Z ′jgW −1
jg Yjg (3.6)
DRIFT ESTIMATION OF GENERALIZED SECURITY PRICE PROCESSES 271
and
V̂ar(λ̂) =[
p∑g=1
n∑j=1
Z ′jgW −1
jg Z jg
]−1
σσ ′. (3.7)
where Yjg = V (tj , Tg)B(tj−1, tj ) − V (tj−1, Tg) + ∫ tj
u=tj−1g(tj−1, Xtj−1)B(tj−1, u) du,
Z jg =∫ tj
u=tj−1
V X (tj−1, Tg)′ A(Xtj−1)B(tj−1, u) du
Wjg =∫ tj
u=tj−1
V X (tj−1, Tg)′ D(Xtj−1)D(Xtj−1)
′V X (tj−1, Tg)B2(tj−1, u) du,
and V X (u, Tg) ≡ ∂V (u,Xu ;K ,Tg)
∂ X is the option’s delta.1
Note that the volatility parameter σ is not required in the estimation of λ̂. To compute thevariance estimator, however, we need an estimate of the volatility parameter. The variancesand covariances in the matrix σσ ′ can be estimated from the state price process X usingstandard methods (Campbell et al., 1997, p. 36):
σ̂ 2d = 1
N
n∑j=1
(ln(Xdk) − ln(Xdk−1) − αd̂j )2
σ̂dl = 1
N
n∑j=1
(ln(Xdk) − ln(Xdk−1) − αd̂j )(ln(Xlk − ln(Xlk−1) − αl̂j )
where αd̂ = 1N
∑nj=1(ln(Xdk) − ln(Xdk−1)) and N =∑n
j=1 j .The estimator (3.5) is then a weighted least squares regression of discounted deriva-
tive and underlying asset prices defined by the quantities “Y”, “X” and “W”. The op-timal weighting is identified by applying the quasi-likelihood theory to the risk-neutralderivative pricing framework. The weighting scheme (3.2) is very crucial: the underlyingstate process in non-stationary! The generalized quasi-likelihood estimator adapts itselfto the structure of the underlying diffusion process. To see this, note that the drift A(Xtj )
and volatility D(Xtj ) functions appearing in the estimator (3.6) depend on the underly-ing price process. For example, under geometric Brownian motion, these functions areA(Xtj ) = D(Xtj ) = Xtj while under arithmetic Brownian motion they become A(Xtj ) =D(Xtj ) = 1.
3.3. Properties: Consistency and Asymptotic Normality
The randomness of the V (X) process is driven by stochastic integrals with respect toBrownian motion. The feasible EF estimator of Section 3.2 is a discretized approximationto the exact estimator implied by the continuous risk neutral pricing framework: unknownquantities between sampling points were replaced by known surrogates. Therefore, it iscritical that the feasible EF estimator be consistent. We establish this property under a
272 PANDHER
conditional second moment assumption on the security process X . Therefore, the proposedestimator is robust to distributional assumptions on the security price process. Moreover,due to the adaptive nature of the generalized estimator, the consistency result of Proposition 4below holds under all state-time separable diffusions. The main results are stated below.
Proposition 3 obtains a bound in L2 norm (E(|λ̂n|2)1/2) for the sequence of estimators{λ̂n} which is then used in Proposition 4 to establish consistency. To reduce notationalburden, we derive the results under the case of a univariate X process (d = 1) without lossof generality.
Proposition 3 (A bound for the sequence {λ̂n} in L2 norm) The generalized EF estima-tor for λ defined in Proposition 2,
λ̂n =[
n∑j=1
Zj W−1j Z j
]−1 n∑j=1
Zj W−1j Yj , is bounded in L2 norm by
E(λ2n) ≤ 1
n2
{λ2 E
(k1
n∑j=1
E(A(Xtj )2 | Ftj−1)
D(Xtj−1)2
)+E
(k2
n∑j=1
E(D(Xtj )2 | Ftj−1)
D(Xtj−1)2
)}(3.8)
for some positive constants k1 and k2.
Proof: See Appendix B.
The next Proposition establishes a sufficient condition for the strong consistency of thegeneralized EF estimator λ̂n .
Proposition 4 (Strong Consistency of Generalized Excess Return Estimator) The gen-eralized excess return estimator of Proposition 2 is strongly convergent
λ̂n =[
n∑j=1
Zj W−1j Z j
]−1 n∑j=1
Zj W−1j Yj → λ, a.s., on the sets
{k1
n2
n∑j=1
E(A(Xtj )2 | Ftj−1)
D(Xtj−1)2
→ 0
}and
{k2
n2
n∑j=1
E(D(Xtj )2 | Ftj−1)
D(Xtj−1)2
→ 0
}(3.9)
Proof: See Appendix C.
Proposition 4 gives a sufficient condition for the consistency of the generalized excessreturn estimator. Any price process with finite conditional second moments leads to strongconsistency. The proof of the above proposition is based on showing that terms in theexpansion of the sequence {λ̂n} are conditionally bounded in L2 norm by E(A(Xtj )
2 | Ftj−1)
and E(D(Xtj )2 | Ftj−1). If these conditional second moments are finite, then the generalized
estimator converges to its true value in large samples. It is easy to check that the sufficientconditions of Proposition 4 are met in the special case where the price process followsarithmetic Brownian motion (A(Xtj )
2 = D(Xtj )2 = 1). Pandher (2000a) shows that
in the case of geometric Brownian motion (A(Xtj )2 = D(Xtj )
2 = X2tj), when the drift
DRIFT ESTIMATION OF GENERALIZED SECURITY PRICE PROCESSES 273
and volatility functions of the stochastic differential equation (2.1) of X satisfy the globalLipschitz and linear growth conditions (see Karatzas and Shreve, 1991, p. 289), the sufficientconditions in Proposition 4 are satisfied. These conditions ensure that the price process doesnot explode.
3.4. Estimation of λ from the Security Process X
In the Monte-Carlo study of Section 4, we will compare the derivative-based estimator λV
with its counterpart λX based on the underlying price process. This, along with comparisonto the true parametric values used to generate the price process and Black-Scholes calls,gives verification of the accuracy of the derivative estimator λV . The generalized estimatorλX for an arbitrary state-time separable diffusion (Assumption 1) is given in Proposition 4below. It is obtained by first obtaining the cmde for the discounted price process anddetermining the optimal weights (3.2) as in Proposition 1–2. These two estimators can alsobe used to measure additional premia in derivative securities and test for the hypothesis ofrisk-neutral pricing. If no-arbitrage pricing holds, then the estimate λV − λX should bestatistically near zero (for details on the testing procedure, see Pandher, 2001a).
Proposition 5 (X -based Feasible Estimator of Excess Return) Let {X (tj ), j = 1, . . . ,
n} define the sequence of market prices on the underlying security. Then, the generalizedX-based feasible EF estimator for λ and its variance are
λ̂X =[
n∑j=1
Zj W−1j Z j
]−1 n∑j=1
Zj W−1j Yj and
V̂ar(λ̂X ) =[
n∑j=1
Zj W−1j Z j
]−1
σσ ′ (3.10)
where
Yj = A(Xtj )B(tj−1, tj ) − A(Xtj−1), Zj =∫ tj
u=tj−1
D(Xtj−1)B(tj−1, u) du, and
Wj =∫ tj
u=tj−1
D(Xtj−1)D(Xtj−1)′ B2(tj−1, u)du.
4. Monte-Carlo Study
The empirical properties of the estimator are investigated using an extensive Monte-Carlostudy. We also resolve certain sample design issues such as the impact of differing strike-levels (moneyness) and shorter maturity cycles. In the market setting, most traded calls areof maturities less than one year. Therefore, the sample size cannot be feasiblely increasedby extending the time to maturity. We evaluate the performance of an alternative sam-pling design which cycles over calls of shorter (non-overlapping) maturities for variancereduction.
274 PANDHER
4.1. Generation of “S&P500” Index and Call Options
In the diagnostic study, a synthetic S&P500 index is generated using historical trend andvolatility parameters. Estimates of the annualized historical volatility over a 260 day (onetrading year) trading period over 1999–2000 fluctuated in the range 16–25%. For theempirical study, the annualized volatility was set at σ = 20%. From a similar examinationof the S&P500 index, an annualized price appreciation of b = 25% was chosen, the risk-freereturn was set at r = 6% and the starting value of the index was set at X0 = 1000 (estimationholds uniformly for other parameter values for drift and volatility). The S&P index processsimulated as a discrete geometric Brownian motion using the recursive formula
X j ≡ Xtj = X j−1 exp
{(b − σ 2
2
)j + σ
√j Z
}, j = 1, . . . , n (4.1)
where j = 1/260 and Z ∼ N (0, 1) is a standard Normal random variate (generated by arandom number generator). The initial 1000 recursions of (4.1) were discarded to remove“start-up” problems in the series and the simulation size was set at 1000.
Call options {Vk(tj , Tg) ≡ V (tj , Xtj ; Kk, Tg), g = 1, . . . , p, j = 1, . . . , ng , k =1, . . . , mg} on the S&P500 index were generated by using (4.1) in the Black Scholesformula:
Vk(tj , Tg) ≡ Vk(tj , Xtj ; Kk, Tg) = Xtj N (d1) − K exp(−rτj )N (d2)
d1 =ln(
Xtj
K
)+ (r + 1
2σ 2)τj
σ√
τj, d2 = d1 − σ
√τj (4.2)
where τj = ∑ng
i= j−1 j , j = 1, . . . , ng , is the time to maturity for each non-overlappingmaturity cycle with lengths {T1, . . . , Tg, . . . , Tp}. A random draw of the index and at-themoney-calls is plotted in Figure 4.1 (the calls are scaled by adding the starting index value).
4.2. Performance and Sample Size
The performance of the V and X -based excess return estimators in correctly estimating thetrue excess return λ = b − r = 19% (or equivalently the market price of risk γ = λ/σ =0.45) over differing sample sizes is first examined. Due to the discretization involved (with260 trading days per year) in generating the index recursively with formula (6.2), the actualλ is slightly different from the continuous theoretical λ (19%). This should be kept in mindwhen comparing the performance of the estimator with the “true value”.
Sample sizes varying from 100 to 10,000 trading days with single maturity and strike(at the money: K = X0) were used in the first experiment. Results averaged over 1000simulations are reported in Table 6.1. It is clear that estimates produced by the EF estimatorsfrom both the call and index prices are very close to the true value of λ (= 19%) at eachsample size.
DRIFT ESTIMATION OF GENERALIZED SECURITY PRICE PROCESSES 275
Figure 4.1. Random draw of generated S&P500 Index and calls. The graph depicts a random Monte-Carlo drawof the generated S&P500 index and corresponding Black-Scholes call option prices. The S&P index processsimulated as a discrete geometric Brownian motion using the recursive formula
X j ≡ Xtj = X j−1 exp
{(b − σ 2
2
)j + σ
√j Z
}, j = 1, . . . , n
where the annualized volatility and drift is set at σ = 20% and b = 25%, respectively. The sampling intervalis fixed at j = 1/260 and Z ∼ N (0, 1) is a standard normal random variate. The starting value of the indexis set at X0 = 1000. The simulation length is set at 1000 trading periods. At-the-money calls option prices aregenerated using the Black-Scholes formula and are re-scaled by adding the initial index value.
Table 4.1. Sample size and performance of λ̂V and λ̂X . The performance of theEF estimator of excess returns is reported with respect to sample size (subscriptsV and X refer to the data used—call and index, respectively). The true parametervalue is λ = 19%. Sample sizes vary from 100 to 10,000 trading days. Black-Scholes calls are generated with a single maturity with strikes at the money:K = X0. Results are averaged over 1000 simulations.
Sample Size (Trading Days)
Data Estimator 100 500 1000 2000 5000 10,000
Call-V λ̂V .1900 .1937 .1907 .1925 .1904 .1908
SE(γ̂V ) .3224 .1442 .1020 .07212 .04560 .03224
Index-X λ̂X .1909 .1936 .1906 .1925 .1904 .1908
SE(γ̂X ) .3224 .1442 .1020 .07212 .04560 .03224
276 PANDHER
Table 4.2. Impace of “moneyness” on λ̂V . The performance of the excessreturn estimator λ̂V is reported with respect to the strike-level (“moneyness”)of the calls. The sample size is fixed at 2000. Black-Scholes calls aregenerated at the strikes K = Xom, m = .8, 1, 1.2. The true parameter valueis λ = 19%. Results are averaged over 1000 simulations.
“Moneyness”: Strike K = Xomhline Estimator m = .8 m = 1 m = 1.2
λ̂V .1925 .1925 .1925
SE(γ̂V ) .07211 .07211 .07211
4.3. Effect of Call “Moneyness”
The results so far were obtained using strikes “at the money”. The impact of changing thestrike-level (“moneyness”) of calls is now investigated. Black-Scholes calls are generatedat the strikes K = Xom, m = .8, 1, 1.2. The sample size is set at 2000 and the results areconsistent at other sample sizes. Table 6.2 demonstrates that the strike-level of the call hasabsolutely no impact on the estimation the EF excess return estimator. The reason for thisis that movements in call value and “delta” V X
k (tj−1, Tg) move in a parallel fashion acrossdifferent strikes—see Figure 4.2. Although the underlying price process is random, callsof different strikes defined on it are highly dependent.
One implication of the high dependence among calls of different strikes over the samematurity is that this replication of data does not lead to variance reduction as a result of thedependence structure. Hence, strike replicates do not serve as additional observations, atleast in theory. In application, however, it may be of interest to estimate the excess returnparameter of calls by moneyness or perhaps even pool across moneyness. The estimator andits variance under a sample design using strike replicates is stated in Pandher (2000a) undergeometric Brownian motion. The estimator can be easily extended to handle generalizeddiffusions with the benefit of Proposition 2.
4.4. Effect of Shorter Maturity Cycles
The vast majority of traded calls are of maturities less than one year. Therefore, the samplesize cannot be feasibly increased by extending the time to maturity. An alternative samplingdesign is considered which cycles over calls of shorter non-overlapping maturities forvariance reduction. Performance of the volatility and excess return estimators is evaluatedat an effective sample size of 500 and 2000 trading days by varying the size of the maturitycycles. At an effective sample size (pn) of 500, maturity cycles (p) of sizes 1, 10 and 31are paired with maturity lengths (n) 500, 50 and 15, respectively. At the effective samplesize of 2000, maturity cycles 1, 40 and 125 are paired with maturity lengths 2000, 50 and16, respectively. The last 20% of the observations are ignored for numerical stability inthe estimation: as time to maturity approaches, the value and delta of the call changes verysharply, causing weighting factors in the EF estimator to fluctuate rapidly. The drop rate
DRIFT ESTIMATION OF GENERALIZED SECURITY PRICE PROCESSES 277
Figure 4.2. Calls of different moneyness. The graph shows a random Monte-Carlo draw of three Black-Scholescall option calls of differing moneyness m = K/X0: m = 0.8, 1, 1, 2. The underlying index is simulated as adiscrete geometric Brownian motion using the recursive formula
X j ≡ Xtj = X j−1 exp
{(b − σ 2
2
)j + σ
√j Z
}, j = 1, . . . , n
where the annualized volatility and drift is set at σ = 20% and b = 25%, respectively. The sampling interval isfixed at j = 1/260 where Z ∼ N (0, 1) is a standard Normal random variate. The starting value of the index isset at X0 = 1000. Each draw runs over 1000 trading intervals.
was set at 20% because we wish to consider estimation also at the one month maturity. Thisgives around 20 trading days with the last 4 dropped (20%) for the reasons sited above.Dropping only the last 2 or 3 observations still caused numerical problems for this shortermaturity. Therefore, to be consistent, the drop rate was fixed at a constant 20% for allmaturities. The results are reported in Table 4.3.
It is found that sampling from cycles of shorter maturities with the same effective samplesize (cycles times maturity) yields similar estimates of volatility as single maturities oflarger duration. The estimation is stable in moving to the shorter maturity cycles. Thisresult gives confidence in the applicability of the proposed estimation to market derivativeprices where larger samples derived from cycling over multiple shorter maturities can beused to reduce variance.
278 PANDHER
Table 4.3. Effect of smaller maturity cycles on excess return estimation. The effect of shorter maturitycycles on the performance of λ̂V and λ̂X is reported at effective sample sizes pn of 500 and 2000trading days. Maturity lengths (n) are 500, 50 and 16 trading days. The true parameter value isλ = 19%. Results are averaged over 1000 simulations.
Sample Size (Trading Days)
500 2000
Data Estimator n p n p n p n p n p n p
500 1 50 10 16 31 2000 1 50 40 16 125
Call-V λ̂V .1937 .1937 .1925 .1925 .1923 .1923
SE(γ̂V ) .1442 .1442 .1448 .07212 .07211 .07211
Index-X λ̂X .1936 .1932 .1914 .1925 .1919 .1911
SE(γ̂X ) .1442 .1442 .1448 .07212 .07211 .07211
5. Conclusions and Further Work
This paper presents an econometric framework for estimating a security’s drift parameterfrom derivative prices where the asset price process follows a generalized state-time sepa-rable diffusion process. Since the excess return parameter is the security’s drift normalizedby the risk-free return, we may also view this as a method for estimating the underlyingsecurity’s excess return from high frequency derivative prices. The proposed frameworkprovides a way to measure premiums in derivative markets, test for risk-neutral pricing anda new empirical approach to pricing derivative contingent claims.
The drift parameter is estimated from the discounted derivative process. We exploitthe relationship between an arbitrary derivative claim’s partial differential equation andprobabilistic representations (Feynman-Kac theory) and use continuous risk-neutral pricingand quasi-likelihood theory to identify the optimal orthogonality condition for the excessreturn parameter. The estimation framework leads to an interesting empirical derivativepricing applications as market prices of risk (λ/σ) can be readily constructed from derivativeexcess returns and volatility. Estimation of volatility from high frequency derivative pricesis considered in Pandher (2000b). The market price of risk is a key parameter in Girsanov’sformula for the risk-neutral density critical to valuing contingent claims. The estimation canalso be used to test for divergence from the risk-neutral pricing model by comparing excessreturns estimated from derivative prices V (X) with the underlying security process X .
The proposed approach is applicable to any arbitrary derivative security, does not requireestimation of the risk-neutral probability measure, inherits the optimal efficiency propertiesof quasi-likelihood theory and has application to spot-rate bond pricing models where itoffers linear estimation of parameters (excess returns, market price of risk) in highly non-linear bond formulae.
The diagnostic study based on generating the S&P500 index and calls verifies the abilityof the proposed method to correctly estimate security excess returns from derivative prices.Sample design issues are also resolved. Estimators are found to be invariant to the strike-
DRIFT ESTIMATION OF GENERALIZED SECURITY PRICE PROCESSES 279
level and shorter maturity calls with the same effective sample size yield consistent andstable estimation. The empirical study gives confidence in the applicability of the estimationmethodology to market derivative prices where larger samples derived from cycling overshorter maturities can be used to reduce variance. Collectively, this framework enables directestimation of the market price of risk from market derivative and bond prices. Current workis directed at investigating the empirical pricing implications of this work and applying theestimation to S&P 500 options data from the Chicago Board of Options Exchange.
Appendix A: Proof of Proposition 2 Feasible EF Estimator for λ
For the linear class of orthogonal estimating functions defined by in (3.1), the optimalestimating equation follows from choosing αj (λ) according to (3.2):
α∗j =
(E
∂hj
∂λ| Ftj−1
)′ (Ehj h
′j | Ftj−1
)−1
=(∫ tj
u=tj−1
E(V X (tj−1, Tg)
′ A(Xu) | Ftj−1
)B(tj−1, u)du
)′
×[∫ tj
u=tj−1
E(V X (tj−1, Tg)
′ D(Xu)σuσ′u D(Xu)
′V X (tj−1, Tg) | Ftj−1
)B2(tj−1, u) du
]−1
= E Z̃ ′j [EW̃j ]
−1
EW̃j above is obtained from the “error-side” representation of hj given by the stochasticintegral Mj = ∫ tj
u=tj−1V X (u, T )′ D(Xu)σ B(tj−1, u)dWu (see Proposition 1), where E(h2
j |Ftj−1) = ∫ tj
u=tj−1E(V X (u, Tg)
′ D(Xu)σuσ′u D(Xu)
′V X (u, Tg) | Ftj−1)B2(tj−1, u)du ≡ EW̃j
follows from the isometry property of squared stochastic integrals.The direct EF estimator λ̂∗
a follows from solving H∗(λ) = 0:
λ̂∗a =
[n∑
j=1
E Z̃ j (EW̃ )−1j Z̃ j
]−1 n∑j=1
E Z̃ j (EW̃ )−1j Yj .
Its variance V (λ̂∗a) follows from applying the variance operator in two steps, first condi-
tioning on the “stochastic regressors” Z̃ = (Z̃1, . . . , Z̃n) (g = 0 w.l.g):
V (λ̂∗a) = EZ
[V (λ̂∗
a | Z̃)]
+ VZ
[E(λ̂∗
a | Z̃)]
= EZ
[n∑
j=1
E Z̃ j (EW̃ )−1j E Z̃ j
]−1
+ VZ [λ].
The feasible EF estimator λ̂ and its variance estimator V̂ (λ̂) are finally obtained byreplacing the integrals and expectations in λ̂∗
a and V (λ̂∗a) with their best known (Ftj−1 -
measurable) surrogates, as defined in Proposition 2, yielding (3.5) and (3.6).
280 PANDHER
Appendix B: Proof of Proposition 3 A Bound for the Sequence {λ̂n} in L2 Norm
To keep the exposition simple, we will pursue the proof in the univariate case of X (d = 1)
without loss of generality. Note that Yj = hj +λZ̃ j = hj +λZj +λ(Z̃ j − Zj ). This allowsλ̂n to be written as
λ̂n − λ =[
n∑j=1
Z ′j W
−1j Z j
]−1
λ
n∑j=1
Z ′j W
−1j (Z̃ j − Zj ) +
[n∑
j=1
Z ′j W
−1j Z j
]−1
×n∑
j=1
Z ′j W
−1j h j = λan + bn. (B.1)
Z̃ j − Zj = ∫ tj
u=tj−1
(V X (u, T )A(Xu) − V X (tj−1, T )A(Xtj−1)
)B(tj−1, u)du where for u >
tj−1 the following bound holds
V X (u, T )A(Xu) − V X (tj−1, T )A(Xtj−1) ≤ A(Xu) (B.2)
because 0 < V X (u, T ) ≤ 1 for Xu > 0. This follows directly from the fact that derivativeoption prices are bounded above by the value of the underlying asset: V (u, Xu; K , T ) ≤ Xu .Therefore, using the fact that A(Xu)
2(tj−1 < u ≤ tj ) is an increasing process, we can write
E((Z̃ j − Zj )
2 | Ftj−1
) ≤ E
(∫ tj
u=tj−1
A(Xu)B(tj−1, u)du
)2
| Ftj−1
≤ E
A(Xtj )2
(∫ tj
u=tj−1
B(tj−1, u)du
)2
| Ftj−1
= Wj
E(
A(X2tj
| Ftj−1
)V X (tj−1, T )2 D(Xtj−1)
2
[∫ tj
u=tj−1Budu
]2
[∫ tj
u=tj−1B2
u du] (B.3)
= WjE(
A(Xtj )2 | Ftj−1
)V X (tj−1, T )2 D(Xtj−1)
2k1
where k1 =[∫ tj
u=tj−1Bu du
]2[∫ tj
u=tj−1B2
u du
] . Also, we have
Z ′j W
−1j Z j = k1
A(Xtj−1)2
D(Xtj−1)2
(B.4)
for a positive constant k1. Substituting the conditional second moment bound (B.3) and(B.4) in an yields
E(a2n) = E
[ n∑j=1
Z ′j W
−1j Z j
]−2 n∑j=1
Zj W−1j E
((Z̃ j − Zj
)2 | Ftj−1
)W −1
j Z j
DRIFT ESTIMATION OF GENERALIZED SECURITY PRICE PROCESSES 281
≤ E
[ n∑j=1
Z ′j W
−1j Z j
]−2 n∑j=1
Zj W−1j Z j
E(
A(Xtj )2 | Ftj−1
)V X (tj−1, T )2 D(Xtj−1)
2k1
= E
[ n∑j=1
k1 A(Xtj−1)2
D(Xtj−1)2
]−2 n∑j=1
k1 A(Xtj−1)2
D(Xtj−1)2
E(
A(Xtj )2 | Ftj−1
)V X (tj−1, T )2 D(Xtj−1)
2k1
≤ E
(k2
n2
n∑j=1
E(
A(Xtj )2 | Ftj−1
)D(Xtj−1)
2
)(B.5)
for some positive constant k2 where the last inequality follows from the fact that condition-ally A(Xtj−1)
2, D(Xtj−1)2 and V X (u, T ) are Ftj−1 -measurable and bounded functions.
Next consider the second term of (B.1). From the proof of Proposition 1, the “error-side”of hj is given by the stochastic integral hj = ∫ tj
u=tj−1V X (u, T )D(Xu)σ B(tj−1, u)dWu .
Using Proposition 2, it can be bounded in L2 norm as follows
E(h2
j | Ftj−1
) =∫ tj
u=tj−1
E(V 2
X (u, Xu)D(Xu)2σ 2 | Ftj−1
)B2
u du
≤∫ tj
u=tj−1
E(D(Xtj )
2 | Ftj−1
)σ 2 B2(tj−1, u)du (B.6)
= WjE(D(Xtj−1)
2 | Ftj−1
)(D(Xtj−1)V X (tj−1, T )
)2 .
Using (B.6) and repeating the steps of (B.5) yields
E(b2
n
) ≤ E
∑n
j=1 Zj W−1j E
(h2
j | Ftj−1
)W −1
j Z j(∑nj=1 Zj W
−1j Z j
)2
≤ E
(k3
n2
n∑j=1
E(D(Xtj )
2 | Ftj−1
)D(Xtj−1)
2
)(B.7)
for some positive constant k3.Note that E
(hj (Z̃ j − Zj ) | Ftj−1
) = 0 due to the Brownian motion in hj . Therefore, the
cross-moment of (B.1) is zero. Finally, we have the desired bound for λ̂n in L2 norm:
E(λ2
n
) ≤ [λ2 E(a2
n
)+E(b2
n
)]≤ 1
n2
{λ2 E
(k1
n∑j=1
E(
A(Xtj )2 | Ftj−1
)X2
tj−1
)+E
(k2
n∑j=1
E(D(Xtj )
2 | Ftj−1
)X2
tj−1
)}. (B.8)
Appendix C: Proof of Proposition 4 Strong Consistency of Generalized Excess ReturnEstimator λ̂n
Let Sn = λ̂n − λ and En = {ω: |Sn| > ε} for some ε > 0. The event of departing the truevalue λ by ε in large samples satisfies the following set relationships: limsupn→∞ En =↓
282 PANDHER
limm{supn>m [|Sn| > ε]
} ⊆ {supn>1 [|Sn| > ε]}
because{supn>m [|Sn| > ε]
}is a decreas-
ing sequence of sets in m. Therefore, we have
P(limsupn→∞ |Sn| > ε
) ≤ P
(supn>1
|Sn| > ε
)= lim
n→∞ P
(max
1< j<n
∣∣Sj
∣∣ > ε
)≤ lim
n→∞1
ε2E(|Sn|2
)= lim
n→∞1
ε2
(λ2 E
(a2
n
)+ E(b2
n
))≤ 1
n2
{λ2 E
(k1
n∑j=1
E(
A(Xtj )2 | Ftj−1
)D(Xtj−1)
2
)
+E
(k2
n∑j=1
E(D(Xtj )
2 | Ftj−1
)D(Xtj−1)
2
)}→ 0 (C.1)
as n → ∞ by hypothesis of Proposition 4. The third inequality of (4.3) follows fromKolgomorov’s inequality (Heyde and Hall, 1987, p. 14), the fourth equality is obtainedin the proof of Proposition 4 (see Appendix B, equation B.8). This establishes the resultλ̂n → λ, a.s., on the sets{
1
n2
n∑j=1
k1E(
A(Xtj )2 | Ftj−1
)D(Xtj−1)
2→ 0
}and
{1
n2
n∑j=1
k1E(D(Xtj )
2 | Ftj−1
)D(Xtj−1)
2→ 0
}.
Acknowledgments
The author would like to thank Robert Jarrow, Fred Arditti, Avi Bick, Ren-Raw Chen, RickDurret, Thomas Epps, Ali Fatemi, Yongmiao Hong, Dongcheol Kim, Peter Klein, HitaoLi, Carl Luft and Lorne Switzer for useful comments and discussions. Special thanks alsoto seminar participants at Cornell University, Concordia University (Montreal), DePaulUniversity, Simon Fraser University, University of Virginia and Rutgers University.
Notes
1. An anonymous referee pointed out the implications of stochastic volatility on the estimator. The estimatoris still consistent but loses efficiency. With time-varying volatility, the weighting functions become time-
dependent taking the form wjg =∫ tj
u=tj−1V X (tj−1, Tg)′ D(Xtj−1 )σtσ
′t D(Xtj−1 )
′V X (tj−1, Tg)B2(tj−1, u) du.
An exogenous estimate of this process (e.g. GARCH, implied volatility) is then required as input to the estimatorin Proposition 2. This modification, however, does not effect the strong consistency result of Proposition 4because only the weighting functions are effected. An analogy to regression is helpful here: OLS estimationis still consistent in the presence of heteroskedasticity but is less efficient than WLS estimation.
DRIFT ESTIMATION OF GENERALIZED SECURITY PRICE PROCESSES 283
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