European Journal of Operational Research 218 (2012) 124–131

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European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Stochastics and Statistics

On valuing and hedging European options when volatility is estimated directly

Ray Popovic, David Goldsman ⇑School of ISyE, Georgia Tech, Atlanta, GA 30332, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 5 October 2010Accepted 6 September 2011Available online 16 September 2011

Keywords:FinanceRisk analysisVolatility estimationSimulationValuation sensitivities

0377-2217/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.ejor.2011.09.011

⇑ Corresponding author. Tel.: +1 404 894 2365; faxE-mail addresses: r.popovic@gmail.com (R.

(D. Goldsman).

We quantify the effects on contingent claim valuation of using an estimator for the unknown volatility rof a geometric Brownian motion (GBM) process. The theme of the paper is to show what difficulties canarise when failing to account for estimation risk. Our narrative uses a direct estimator of volatility basedon the sample standard deviation of increments of the underlying Brownian motion. After replacing thedirect estimator into the GBM, we derive the resulting distribution function of the approximated GBM forany time point. This allows us to present post-estimation distributions and valuation formulae for anassortment of European contingent claims that are in accord with many of the basic properties of theunderlying risk-neutral process, and yet better reflect the additional uncertainties and risks that existin the Black–Scholes–Merton paradigm.

� 2011 Elsevier B.V. All rights reserved.

1. Introduction

The estimation of volatility is a crucial component in under-standing the time-series properties of financial markets and theclaims they trade, e.g., options markets written on some specifiedequity, foreign exchange rate, or swap/LIBOR rate. This paper usesthe canonical constant-coefficients geometric Brownian motion(GBM) equity model to study the effects of volatility estimation— a source of randomness that permeates all valuation models,but has been given little attention in quantitative finance. Althoughthe volatility of an equity is not constant over the long run, we fi-nesse this problem by making the reasonable assumption that it islocally constant over time periods of interest. Moreover, one canguard against other violations of the GBM’s underlying assump-tions by hedging via the so-called Greeks [12] — for example, thequantity known as vega serves as a stand-in for the sensitivity ofthe equity price to a perturbation in volatility.

In terms of method, the precursors to our paper are Boyle andAnanthanarayanan [3], Butler and Schachter [4], and Ncube andSatchell [10]. These papers place a monetary value on a vanillaEuropean call by using what is informally known as the ‘‘law ofthe unconscious statistician’’ (LUS) [1] to average the classicBlack–Scholes–Merton (BSM) call formula [2,9] with respect toan estimator for volatility. As we show, an implication of strictlyrelying on their use of the LUS methodology — as opposed to ourviewpoint — is that the stage at which the LUS is invoked has

ll rights reserved.

: +1 404 894 2301.Popovic), sman@gatech.edu

consequences for the subsequent calculations of the optionsensitivities.

In applications, the BSM option formula is typically asserted asessentially correct, and then for calibration purposes, a fudge factoris appended to the volatility specification so as to improve the pre-vailing fit-to-market. However, the problem is that the world inwhich economic agents reside is more uncertain than the BSMassumptions (e.g., known volatility and underlying geometricBrownian motion) allow for or that a fudge factor can typicallycompensate for. Our paper moves a little closer to addressing thisproblem by directly estimating the volatility and studying thesmall-sample consequences of such estimation on valuation. Thepaper’s central tenet offers an appropriate valuation strategy thatdeals with a particular type of existing parameter risk. Such a sys-tematic inclusion and resolution of risk is then applied to an assort-ment of European vanilla and exotic option types — those having aclosed-form representation and those lacking an explicit formula,e.g., an Asian call on the arithmetic average of the underlying.

In our set-up, there is — in the eyes of the decision maker — thebasic primary randomness associated with the GBM as well as theadditional evolving perceived risk arising from the attached volatil-ity estimate. The approach we pursue represents one way by whichan individual agent, who attempts to value and hedge a contingentclaim, comes to grips with the uncertainty-risk dichotomy, intro-duced by Knight [7]. According to Knight, agents are placed in acontext of uncertainty when they cannot assign probabilities topotential events, e.g., when an entirely unanticipated change in roccurs to which one cannot react in a preemptory fashion.However, in a situation where risk prevails — such as an antici-pated change in r — the same agents are able to attach probabili-ties to the occurrence of such event types, and so ex ante, mitigate

R. Popovic, D. Goldsman / European Journal of Operational Research 218 (2012) 124–131 125

potential unfavorable outcomes. The case of interest that concernsus most here — where r is unknown but estimable — falls inKnight’s second taxonomic category.

The paper is organized as follows. In Section 2, we review‘‘indirect’’ and ‘‘direct’’ methods for estimating the volatility asso-ciated with the underlying GBM process. Section 3 deals with theconsequences of the direct estimation method. Here we provethat our formulation leads to an unbiased expected value forthe underlying equity price — one that matches the expectationobtained under the risk-neutral measure [12]. Next, we highlightthe effects of our procedure on the valuation and hedging func-tions for a variety of European options. We find that there arecases where our option sensitivities differ from those that wouldbe obtained via application of a strict version of the LUS. Section 4gives conclusions. The proofs of the paper’s main results are inthe Appendix. A separate On-Line Appendix contains supplemen-tary examples and derivations of certain technical or knownresults.

2. Basics

This section reviews two opposing methodologies to the prob-lem of estimating volatility, and along the way establishes somenotation. In order to focus attention on valuation risk induced byparameter estimation in a simple yet reasonably sophisticated set-ting, we use the well-accepted workhorse of mathematical finance,the GBM constant-coefficients model of the price of an equity,

Sðt; rÞ � s exp r � r2

2

� �t þ rWðtÞ

� �� s exp Nor r � r2

2

� �t;r2t

� �� �; ð1Þ

where s � S(0;r) is the known initial price; ðWðtÞ; t P 0Þ is a stan-dard Brownian motion (BM) process driving the GBM; r > 0 is thevolatility parameter; and r is the risk-free interest rate characteriz-ing the risk-neutral measure attached to the GBM [12]. As S(t ;r) islognormal, the following lemma is repeatedly used in the sequel.

Lemma 1. Suppose Y � seNorða;b2Þ and let /(�) and U(�) denote theNor(0,1) probability density function (p.d.f.) and cumulative distribu-tion function (c.d.f.), respectively. In addition, definex�ðyÞ � 1

b ‘n sy

� �þ a

h i;xþðyÞ � x�ðyÞ þ b; y > 0, and the notation

x+ �max{x,0} for all x. Then Y is lognormal with c.d.f.FYðyÞ ¼ Uðx�ðyÞÞ; y > 0, where FðxÞ � 1� FðxÞ indicates the comple-ment of any generic c.d.f. F(x), and

E½ðY � kÞþ� ¼ seaþb22 UðxþðkÞÞ � kUðx�ðkÞÞ; k P 0: ð2Þ

In particular, for a given t = T, we see that S(T ;r) is lognormalwith c.d.f. FSðT;rÞðyÞ � Uðz�ðs; y;rÞÞ; y > 0, where z�ðs; y;rÞ �

1rffiffiTp ‘n s

y

� �þ r � r2

2

� �T

h i.

2.1. Indirect estimation of r

The indirect approach uses implied volatility [12] as an estimateof r. As described below, implied volatility is somehow ‘‘dis-cerned’’ by surveying a liquid market in options written on anunderlying asset. Our examples are generally restricted to Euro-pean call options, though analogous results typically apply viaput-call parity to puts. The standard ‘‘vanilla’’ European call is acontract dependent on the current equity value s, that permits itsowner to purchase the underlying asset at a pre-agreed strike pricek, at a pre-determined expiry instant T time units in the future.With t � (s,k,T) denoting this discernible vector of market data,the contract at expiry has the random value C(t ;r) � (S(T ;r) � k)+.

What is the contingent claim C(t ;r) worth now? Using (2) withY = S(T ;r), the present value of E[C(t ;r)] at time 0 is

cðt; rÞ � e�rT E½Cðt;rÞ� ¼ sUðzþðs; k;rÞÞ � ke�rTUðz�ðs; k;rÞÞ; ð3Þ

which is the classic formula of BSM [2] giving the value of a call op-tion. In this formula, r is a mystery. The indirect method of resolv-ing what r is depends on the observed market price of the call, saycm, which is thought to incorporate the beliefs of market partici-pants concerning the inherent variability of the underlying GBMover the future [0,T]. In particular, at time 0, given cm and theknown values t and r, the implied volatility is obtained by numer-ically solving cm = c(t ;r) for r. This method, linked to an equilib-rium view of markets, ostensibly allows us to avoid problemsassociated with utilizing historical data in the estimation of r.

Unfortunately, the indirect strategy of obtaining r from (3) of-ten introduces ambiguity for what volatility is, since expiry datesand strike rates provide different values for what is supposed tobe the same r referenced in Eq. (1), i.e., the so-called ‘‘smile orsmirk.’’ Rationalizations for this artifact are that the model is anincorrect representation of economic behavior or that the marketlacks sufficient liquidity at all strike-expiry combinations; and allthis is exacerbated by the asynchronous collection of the involveddata. As a result, much effort has been expended on tweaking var-ious volatility specifications to better fit the formulae to the marketdata, but at the cost of introducing additional — and in most cases— neglected estimation risk.

We now discuss, as a complementary approach, the simplestexplicit accounting of estimation risk in contingent claim valuationformulae.

2.2. Direct estimation of r

The idea is to estimate r using data available in an ‘‘estimationperiod’’ occurring before the present time, say during [�n,0]; andthen at time 0, use the estimate of r to obtain the present dis-counted value of any European contingent claim of interest.

With estimation of r in mind, suppose we model the equityprice during [�n,0] analogously to (1), i.e.,

eSðt;rÞ� eSð�n;rÞexp l�r2

2

� �ðnþ tÞþrfWðnþ tÞ

� �; �n6 t60;

where eSð�n; rÞ is the equity price at time �n,ðfWðnþ tÞ;�n 6 t 6 0Þ is a standard BM, and l is the ‘‘market mea-sure’’ deterministic drift parameter. With no loss in generality, di-vide [�n, 0] into n equal increments, from which we obtain thelog-returns,

Ri � ‘neSð�nþ i;rÞeSð�nþ i� 1;rÞ

!¼ l� r2

2þ ni; for i ¼ 1;2; . . . ;n;

where ni � r½fWðiÞ �fWði� 1Þ� for i = 1,2, . . . ,n. By independent

increments of BM, R1,R2, . . . ,Rn are i.i.d. Nor l� r2

2 ;r2

� �random

variables; and manifestly, any increments from the estimation seg-ment of the underlying BM are independent of the post-estimationsegment ðWðtÞ; t P 0Þ. The task of estimating r2 is then standardunder the GBM model, for in this case, we use the sample varianceof the Ri’s as the point estimator, i.e.,

br2n �

1n� 1

Xn

i¼1

ðRi � RnÞ2 ¼1

n� 1

Xn

i¼1

ðni � �nnÞ2 �r2v2

n�1

n� 1; ð4Þ

where Rn �Pn

i¼1Ri=n and �nn �Pn

i¼1ni=n. Thus, E br2n

¼ r2, so thatbr2

n is unbiased for r2. In addition, it is easy to obtain the related re-sult E½brn� ¼ r

ffiffiffiffiffiffiffi2

n�1

qC n

2

� �=C n�1

2

� �, where C(�) is the gamma function;

this expression converges to r fairly quickly as n increases.

126 R. Popovic, D. Goldsman / European Journal of Operational Research 218 (2012) 124–131

2.3. An Organizing identity

We present a simple identity that is applied, in one way or an-other, throughout the paper. The identity motivates us to analyzeoptions and their sensitivities within a BSM market when volatilityis an unknown quantity that can be estimated. Consider a tradedclaim whose value is represented by the random variable X(t ;r),where t is a known constant vector and r is unknown. Weestimate r2 by br2, which has p.d.f. fbr2 ðw;rÞ;w > 0, and whichwe assume to be independent of X(t ;r). Let B represent a knownfunction — a decision rule defined relative to the BSM economy— dependent on the realization of X(t ;r). Set b(t ;r) �E[B(X(t ;r))], where conditioned on a given r, the expectation iswith respect to the perceived risk-neutral measure. Then

hðt;rÞ �Z

bðt;ffiffiffiffiwpÞfbr2 ðw;rÞdw ð5Þ

depends on a realization of X(t ;r), subject to the p.d.f. of the esti-mator br2. The associated hedging rules (comparative statics) canall be obtained by taking the total derivative of (5); e.g., with re-spect to changes in the s component of t and r,

dhðt;rÞ ¼Z@bðt;

ffiffiffiffiwpÞ

@sfbr2 ðw;rÞdw

� �ds

þZ

bðt;ffiffiffiffiwpÞ@fbr2 ðw;rÞ

@rdw

!dr; ð6Þ

where the interchange of integrals and derivatives typically holds inour applications.

The traditional LUS is set out in Eq. (5), which explicitly con-verts uncertainty about the volatility to risk. Eq. (6) provides anextension of the LUS to the known and unknown hedging parame-ters, and is primarily concerned with underlying uncertainty. Thesecond term in (6) is our broadening of the LUS to include the esti-mator of volatility and its dependence on the parameter r. To-gether, Lemma 1 and the above identity explicitly indicate howmarket agents deal with an uncertain environment versus one ofrisk. The sequel considers a constellation of valuation and hedgingexamples, all of which can be decomposed into the components of(6) — though we will often use a more-direct approach to obtain aparticular solution. In any case, we have verified the equivalence ofboth methods for all of our examples; the choice of method isreally a matter of convenience.

Fig. 1. A cornucopia of f ð�Þ p.d.f’s.

3. Consequences of estimating r

This section addresses the consequences encountered in valua-tion and hedging when we incorporate the estimator brn in theclassic BSM valuation model.

3.1. Results concerning the underlying asset

Our first goal is to derive the distribution of the random variableSðT; brnÞ — the equity price at time T reflecting the estimation riskencompassed in brn. The following lemma provides expressionsfor the post-estimation c.d.f. and p.d.f. of the equity process.

Lemma 2. Suppose that br2 is an estimator of r2 that has p.d.f. fbr2 ð�Þand is independent of the underlying BM process WðtÞ. Then the c.d.f.and p.d.f. of SðT; brÞ are

FSðT;brÞðyÞ ¼ Z 1

0Uðz�ðs; y;

ffiffiffiffiwpÞÞfbr2 ðwÞdw; y > 0; and ð7Þ

fSðT;brÞðyÞ ¼ Z 1

0

1yffiffiffiffiffiffiffiwTp /ðz�ðs; y;

ffiffiffiffiwpÞÞfbr2 ðwÞdw; y > 0: ð8Þ

In particular, the direct estimator br2n � r2v2

n�1=ðn� 1Þ and isindependent of ðWðtÞ; t P 0Þ (since br2

n consists of data from timeinterval [�n,0]). We then obtain F

SðT;brnÞðyÞ and f

SðT;brnÞðyÞ by plug-

ging fbr2nðwÞ ¼ n�1

r2 fv2n�1

ðn�1Þwr2

� �into (7) and (8), where fv2

n�1ð�Þ is the

v2n�1 p.d.f. Computationally efficient versions of F

SðT;brnÞðyÞ and

fSðT;brnÞ

ðyÞ are given in the On-Line Appendix as a special case of

Lemma 2.

Example 1. Fig. 1(a) depicts the post-estimation p.d.f.’s fSðT;brnÞ

ð�Þfor the case T = 1/2, s = 10, r = 0.05, and r = 1 using estimates brn

based on n = 3, 4, 10, 30, and BSM (n =1). For large values of n, thedistinction between the post-estimation densities and the BSMp.d.f. becomes inconsequential. On the other hand, we see that forsmall n, the p.d.f.’s differ substantially from the limiting lognormaldensity. Fig. 1(b) plots the p.d.f.’s for the case n = 4, T = 1, s = 10, andr = 0.05, for true values of r = 1/2, 3/4, and 1. Clearly, the value of rsignificantly impacts the shape of the density.

The next corollary gives the moments associated with the den-sity f

SðT;brÞð�Þ.Corollary 1. Under the conditions of Lemma 2, E½SjðT; brÞ� ¼sjejrT Mbr2

Tðj2�jÞ2

� �, where Mbr2 ð�Þ is the moment generating function

(m.g.f.) of br2.

Notice that for any estimator br satisfying the conditions of Lem-ma 2, SðT; brÞ inherits the expected value property of the lognor-mally distributed asset price at time T, i.e., E½SðT; brÞ� ¼ serT .Under the risk-neutral measure, this is the no-arbitrage forward

SðT;brn Þ

R. Popovic, D. Goldsman / European Journal of Operational Research 218 (2012) 124–131 127

price of the underlying and is independent of the volatility estima-tion period. Moreover, since the m.g.f. of the v2

m isMv2

mðyÞ ¼ ð1� 2yÞ�m=2 for y < 1/2, we easily obtain moment results

for the direct variance estimator br2n.

Corollary 2. If j P 1 and n P max{2,1 + r2T(j2 � j)}, then

E½SjðT; brnÞ� ¼ sjejrT 1� r2Tðj2 � jÞn� 1

!�n�12

:

In particular, from Corollary 2, the variance of the estimation-augmented equity price is

Var½SðT; brnÞ� ¼ s2e2rT 1� 2r2Tn� 1

� ��n�12

� 1

" #:

An exact recipe for simulating from the post-estimation GBM pro-cess is needed in order to implement some of the subsequent valu-ation examples. The following pseudo-code provides one simulatedrealization of the underlying ðSðt; brnÞ; t P 0Þ at timest ¼ 0; T

m ;2Tm ; . . . ; T , where m P 1 is a ‘‘mesh’’ factor.

Algorithm 1. Simulating a Sample Path of the Post-EstimationUnderlying

1. Initialize n P 2; r; r; T; s; m; and Wð0Þ ¼ 0.2. Generate br2

n r2

n�1 v2n�1.

3. Generate a standard Brownian motion sample path: Fori = 1,2, . . . ,m, set W iT

m

� � W ði�1ÞT

m

� �þ

ffiffiffiTm

qZi; where Z1,Z2, . . . ,Zm

are i.i.d. Nor(0,1) (and independent of br2n).

4. For i = 1,2, . . . ,m, set S iTm ; brn� �

s exp r � 12br2

n

� �iTmþ brnW

iTm

� �� �.

To generate a sample path of S(t ;r), skip Step 2 and use rinstead of brn throughout.

Example 2. Fig. 2 is a sequence of overlaid quantile–quantile (Q–Q) plots to compare the post-estimation c.d.f.’s from Lemma 2 withGBM’s lognormal c.d.f. Two c.d.f.’s describe the same distribution iftheir Q–Q plot coincides with the superimposed diagonal line. Foreach plot (corresponding to n = 4, 10, 30, 1000), we generated 105

replications of ‘nðSðT; brnÞÞ with s = 1, T = 1, r = 1, and r = 1/2. Thegoal is to see how close these logs are to a Nor(0,1) distribution.For small n, the Q–Q plots show that the differences betweenGBM and the estimator-adjusted c.d.f.’s are consequential. But forn = 30, the Q–Q plot is close to the diagonal; and excellent confor-mity exists for n = 1000.

3.2. Results concerning European claims

This section gives a number of examples (vanilla calls, digitalclaims, barrier options, additional exotic types, claims on averages,

Fig. 2. Q–Q plots of ‘nðSðT; brnÞÞ: s = 1; T = 1; n = 4, 10, 30, 1000; r = 1; r = 1/2.

and Greeks) illustrating the relevance of including estimation riskwhen valuing a contingent claim. Our attention is directed at op-tions that correspond to the c.d.f. F

SðT;brnÞð�Þ. We illustrate the wedge

in valuations induced by known versus estimated r. It turns outthat the difference in pricing is often significant — on the orderof few basis points to several hundred basis points — though notso overwhelmingly large as to cast doubt on the underlying BSMmodel.

3.2.1. Vanilla calls and putsThe c.d.f. of the vanilla European call option, Cðt; brnÞ— inclusive

of the volatility estimator brn — is given by

FCðt;brnÞ

ðyÞ � PrðCðt; brnÞ 6 yÞ ¼ FSðT;brnÞ

ðyþ kÞIfyP0g; ð9Þ

where IE is the indicator function for the generic event E. The callhas a point probability at y = 0 equal to F

SðT;brnÞðkÞ — the probability

of being out-of-the-money (OTM) at the time of expiry. Eq. (9)yields the present value of the call Cðt; brnÞ,

cðt; brnÞ � e�rT E½Cðt; brnÞ� ¼ e�rTZ 1

kF

SðT;brnÞðyÞdy; ð10Þ

which, if no closed-form exists, can be solved numerically. The post-estimation put is Pðt; brnÞ � ðk� SðT; brnÞÞþ and still satisfies the put-call parity relation.

Lemma 3. For n P 2, we have pðt; brnÞ � e�rT E½Pðt; brnÞ� ¼cðt; brnÞ � sþ ke�rT .

Example 3. Fig. 3 plots, as a function of the current equity price s,call values cðt; brnÞ;n ¼ 3;4;10;30, and BSM (n =1), using k = 10,T = 1/2, r = 0.05, and r = 1. For this example, the inclusion ofestimation risk underprices the call relative to BSM, with theunderpricing progressively decreasing as we move further in-the-money (ITM) or OTM. For instance, the at-the-money (ATM)valuations are cðt; br3Þ ¼ 2:523; cðt; br4Þ ¼ 2:624; cðt; br10Þ ¼ 2:774;cðt; br30Þ ¼ 2:829, and the BSM value c(t ;r) = 2.854. Note that ifprices from the post-estimation valuation schedule are input intoclassic BSM for the purpose of obtaining implied volatility, thenone will conclude that a non-constant volatility is indicated — afake smile effect — even though r is in fact constant.

In the case of a vanilla call valuation, the post-estimation Eq.(10) gives the same value as when the LUS is applied directly tothe BSM formula (3) (as in [3,4,10]). The results are summarizedby the next proposition, proven in the On-Line Appendix.

Fig. 3. BSM vs. post-estimation cðt; brnÞ : t ¼ ðs; k; TÞ ¼ ðs;10;1=2Þ; r ¼ 0:05;r ¼ 1;

n ¼ 3;4;10;30;BSM; and max{s � 10,0} is the expiry valuation profile.

Table 2Barrier probabilities for s = 10, T = 1/4, r = 1.5, and r = 0.05.

Barrier k 11 12 13 14 15

FMðT;br4Þ

ðkÞ 0.879 0.780 0.697 0.627 0.567

FMðT;rÞðkÞ 0.896 0.809 0.736 0.673 0.618

Table 1OTM probabilities for s = 10, T = 1/4, r = 1.5, and r = 0.05.

k 5 10 15

FSðT;br4Þ

ðkÞ 0.240 0.672 0.845

FS(T;r)(k) 0.442 0.694 0.813

128 R. Popovic, D. Goldsman / European Journal of Operational Research 218 (2012) 124–131

Proposition 1. cðt; brnÞ ¼ E sUðzþðs; k; brnÞÞ � ke�rTUðz�ðs; k; brnÞÞ

;

where the right-hand side is the calculation via the LUS.Though they yield the same vanilla call value, we stress that the

general methods are not equivalent, as will be demonstrated whenwe consider this call’s sensitivities in Section 3.2.6.

3.2.2. Digital claimsThe simplest of European options, and one that directly makes

use of the post-estimation c.d.f., is the digital claim. The digital ispredicated on the occurrence of an event E, e.g.,E ¼ fSðT; brnÞ > kg (a digital call) or the complementary event E

(a digital put), and pays a ‘‘coupon’’ of $1 if the event occurs. Sym-bolically, a digital has value

dðt; brnÞ � e�rT E½IE � ¼ e�rT PrðEÞ:

Example 4. Using Eq. (9), we calculate the OTM probabilityF

SðT;brnÞðkÞ of a European digital call when using brn in place of r.

Table 1 illustrates an example for s = 10, r = 1.5, r = 0.05, and T = 1/4; and we display the resulting probabilities for strike values k = 5,10, 15, and n = 4 and 1 (BSM). Note that for a well-ITM option(k = 5), the probability of being OTM at expiry is much smallerunder c.d.f. F

SðT;br4Þð�Þ than for the standard BSM c.d.f. FS(T;r)(�). To

obtain digital option values dðt; brnÞ, simply multiply the proba-bilities in Table 1 by e�r T = 0.9876.

3.2.3. Barrier optionsHere we calculate the value of a digital barrier option. Let

M(T ;r) �max06t6TS(t ;r) record the maximum value of the GBMprice path observed up to time T. Our choice of claim is the digital‘‘knock-in,’’ having payoff D(t ;r) � I{M(T;r)Pk}. If S(t ;r) hits thebarrier k by time T, the payoff is $1; otherwise, the claim paysnothing. To determine the fair value of D(t ;r), we calculatePr(M(T ;r) P k) and then discount by the risk-free rate. The c.d.f. ofM(T ;r) is [12] (cf. the On-Line Appendix)

FMðT;rÞðkÞ ¼ PrðMðT; rÞ 6 kÞ

¼ Uðz�ðs; k;rÞÞ � ks

� � 2rr2�1

Uðz�ðk; s;rÞÞ: ð11Þ

Thus, when the volatility is known, the fair value of D(t ;r) isdðt;rÞ � e�rT FMðT;rÞðkÞ. For unknown r, we employ the LUS directlyvia Eq. (5) to obtain

dðt; brnÞ � e�rT PrðMðT; brnÞP kÞ

¼ n� 1r2

Z 1

0dðt;

ffiffiffiffiwpÞfv2

n�1

ðn� 1Þwr2

� �dw:

Example 5. Table 2 gives representative barrier probabilities fromthe two complementary c.d.f.’s F

MðT;br4ÞðkÞðn ¼ 4Þ and

FMðT;rÞðkÞðn ¼ 1Þ for the case T = 1, s = 10, r = 0.05, with the truevalue of r = 1.5. We see that as the barrier k is raised, the differencein values is monotonically increasing.

3.2.4. Other exotics with closed formsThere are many non-standard options to which our methodol-

ogy can be applied. One that readily fits into our paradigm is theforward start call [12]. With 0 < T0 6 T, the forward start is(S(T ;r) � xS(T0 ;r))+, x > 0, and can be interpreted as having a strikevalue k = xS(T0 ;r) — now a random variable dependent on a futureoutcome of the underlying. From the On-Line Appendix, we obtainthe option value

e�rT E½ðSðT; rÞ � xSðT 0;rÞÞþ� ¼ e�rðT�T 0ÞE½Cðs; xs; T � T 0;rÞ�; ð12Þ

i.e., use replacements k ? xs and T ? T � T0 in Eq. (3). Aside fromthe indicated adjustment of the parameters, the BSM formula isthe same as for a vanilla option.

With a little ingenuity, other claims can be valued (see On-LineAppendix B). The general idea is straightforward: Obtain the jointlaw governing the relevant process, and then use the pre- orpost-estimation c.d.f. to determine the fair price.

3.2.5. Asian optionsThis section outlines relevant results for a variety of Asian op-

tions, i.e., options based on certain averages of the equity priceas it evolves over time. An interesting property of some of theseclaim types is that no closed-form formulae exist for pricing orhedging. For these we use simulation to provide valuations. Thereare many types of Asian options, but in the current paper, we dealwith contingent claims on ‘‘continuously’’ monitored averages. Amore-extensive discussion dealing with the finer points of bothcontinuous and discrete options on averages can be found in [11].

3.2.5.1. Geometric average with known r. For discrete monitoringover [0,T] at m equally spaced times, T

m ;2Tm ; . . . ; T , the geometric

average based on the underlying is

Ymi¼1

SiTm

;r� � !1=m

¼ s exp1m

Xm

i¼1

r � r2

2

� �iTmþ rW iT

m

� � �( )

!D s exp1T

Z T

0r � r2

2

� �t þ rWðtÞ

�dt

� �ð13Þ

¼ exp1T

Z T

0‘nðSðt;rÞÞdt

� �� GðT; rÞ;

where !D denotes convergence in distribution as m ?1, andG(T ;r) is the continuously monitored version of the geometricaverage of the equity price. Since

VarZ T

0WðtÞdt

� �¼Z T

0

Z T

0CovðWðtÞ;WðuÞÞdt du

¼Z T

0

Z T

0minðt;uÞdt du ¼ T3

3;

Eq. (13) implies that

GðT;rÞ � s exp Nor r � r2

2

� �T2;r2T

3

� �� �: ð14Þ

Thus, G(T ;r) is lognormal, and it follows that we can directly applythe BSM formula to price a call on the geometric average. By (2)with Y = G(T ;r), the BSM valuation of the continuously monitoredgeometric average option CG(t ;r) � (G(T ;r) � k)+ is

cGðt;rÞ � e�rT E½CGðt; rÞ�

¼ s e� rþr26

� �T2U zG

þðs; k;rÞ� �

� k e�rTU zG�ðs; k;rÞ

� �;

:

Fig. 4. BSM #ðt;rÞ; #ðt; brnÞ, and LUS vega: s = 10; k = 10; T = 1/2; r = 0.05; r = 1;n = 4.

Table 3Option values with t = (s,k,T) = (10,k,1/6), r = 0.05, r = 1, n = 4, and m = 176.

k 8 9 10 11 12

c(t ;r) 2.706 2.126 1.653 1.274 0.977cG(t ;r) 2.093 1.398 0.879 0.523 0.297�cAðt; rÞ 2.201 1.490 0.956 0.586 0.346

(0.007) (0.006) (0.005) (0.004) (0.003)cðt; br4Þ 2.651 2.023 1.523 1.151 0.882

cGðt; br4Þ 2.098 1.357 0.804 0.465 0.275�cAðt; br4Þ 2.206 1.451 0.885 0.533 0.329

(0.007) (0.006) (0.006) (0.005) (0.004)

R. Popovic, D. Goldsman / European Journal of Operational Research 218 (2012) 124–131 129

where

zGþðs; k; rÞ �

‘n sk

� �þ r þ r2

6

� �T2

rffiffiT3

q and zG�ðs; k;rÞ �

‘n sk

� �þ r � r2

2

� �T2

rffiffiT3

q

3.2.5.2. Geometric average with unknown r. By comparing the dis-tributions of S(T ;r) and G(T ;r) from (1) and (14), and then carry-ing out the same manipulations as those leading to the c.d.f. ofSðT; brnÞ given in (7) of Lemma 2, we obtain for the continuouslymonitored case the c.d.f. of GðT; brnÞ,

FGðT;brnÞ

ðyÞ � n� 1r2

Z 1

0U zG

�ðs; y;ffiffiffiffiwpÞ

� �fv2

n�1

ðn� 1Þwr2

� �dw; y > 0:

With substitution analogous to (10), we choose to compute the callnumerically via

cGðt; brnÞ � e�rT E½CGðt; brnÞ� ¼ e�rTZ 1

kF

GðT;brnÞðyÞdy: ð15Þ

3.2.5.3. Arithmetic average with known r. We next turn to optionsbased on the arithmetic average of GBM with known r. For discretemonitoring at times T

m ;2Tm ; . . . ; T , the arithmetic average is

1m

Xm

i¼1

SiTm

;r� �

!D AðT;rÞ � 1T

Z T

0Sðt; rÞdt; as m!1;

where A(T ;r) denotes the continuously monitored version [12]. Asthis functional of GBM lacks a closed-form representation, a BSM-type formula cannot be obtained; so in what follows, we use simu-lation to price the call CA(t ;r) � (A(T ;r) � k)+.

Suppose that we have at our disposal ‘ independent simu-lated replications of the sample path, denoted by(Sj(t ;r),0 6 t 6 T), for replications j = 1,2, . . . ,‘. Further, letAjðT;rÞ � 1

T

R T0 Sjðt;rÞdt and CA

j ðt;rÞ � ðAjðT;rÞ � kÞþ for replica-tion j. In order to estimate the price of the call, we must deter-mine cA(t ;r) � e�rTE[CA(t ;r)]. To do so, we use the crude MonteCarlo (MC) estimator �cAðt;rÞ � e�rT

‘

P‘j¼1CA

j ðt;rÞ, which averagesthe CA

j ðt;rÞ’s over the ‘ replications and accounts for the timevalue of money.

3.2.5.4. Arithmetic average with unknown r. Finally, we consider anAsian call CAðt; brnÞ � ðAðT; brnÞ � kÞþ, where AðT; brnÞ is the arith-metic average of GBM incorporating brn over the time interval[0,T]. In the absence of a closed-form expression for

cAðt; brnÞ � e�rT E½CAðt; brnÞ�, we appeal to the crude MC estimator�cAðt; brnÞ � e�rT

‘

P‘j¼1CA

j ðt; brn;jÞ, where br2n;j is sampled from an appro-

priately scaled chi-squared c.d.f. on the jth path.

Example 6. We value a variety of vanilla and Asian callsusing Algorithm 1, where necessary, to simulate ‘ independentreplications of the sample path of the equity price, (Sj(t ;r),0 6 t 6 T), for j = 1,2, . . . ,‘ = 105. The other input parameters areT = 1/6, s = 10, r = 0.05, and r = 1, with the estimator brn based onn = 4. We discretize the two-month (T = 1/6) time period bytaking m = 176 (essentially continuous averaging) equallyspaced equity price observations — 4 daily observa-tions � 22 days � 2 months. Table 3 gives results for strike pricesk = 8, . . . ,12.

The c(t ;r) and cðt; br4Þ rows of Table 3 respectively providethe exact BSM pre- and post-estimation vanilla call values.Similarly, the cG(t ;r) and cGðt; br4Þ rows give the exact pre-and post-estimation geometric average call values. The �cAðt;rÞand �cAðt; br4Þ rows give analogous arithmetic average call valuesobtained by crude MC, with standard errors in parentheses.Observe that:

Due to the ‘‘averaging’’ of the underlying, vanilla calls are moreexpensive than ‘‘average’’ calls. The results on Asian call values conform with the geometric-

arithmetic average inequality, i.e., the geometric average is alower bound for the arithmetic average.

Our paper [11] details assorted performance improvements con-cerning the above simulations on the Asian claims.

3.2.6. Vanilla GreeksThe ‘‘Greeks’’ are price sensitivities accounting for an unantic-

ipated change in some structural parameter, and are typicallyused in devising and monitoring hedging strategies. For example,in the case of a vanilla BSM call, when the stock is non-dividend paying and r is known, the most-frequently usedGreeks are delta, gamma, theta, rho, and vega [12]; in the currentpaper, we will deal with the BSM delta and vega:dðt;rÞ � @cðt;rÞ

@s ¼Uðzþðs;k;rÞÞ and #ðt;rÞ � @cðt;rÞ@r ¼ s

ffiffiffiTp

/ðzþðs;k;rÞÞ.In the strict BSM paradigm, d(t ;r) indicates how many addi-tional units of the underlying one needs to go short or long soas to balance out in value a portfolio consisting of the call, thestock, and a money market account. For the unknown r case,the corresponding LUS versions of delta and vega are, via Eq.(6), E½Uðzþðs;k; brnÞÞ� and s

ffiffiffiTp

E½/ðzþðs;k; brnÞÞ�, respectively.Our post-estimation (unknown r) Greeks are given in the fol-

lowing proposition. We see that for the vanilla call, the post-esti-mation delta is the same as the corresponding LUS version; theinteresting news is that the versions of vega differ since now thechange in r is unanticipated and therefore categorized asuncertain.

Table 4Delta and vega convergence in n: k = 10; T = 1/2; r = 0.05; r = 1.

s dðt; brnÞ #ðt; brnÞ

5 10 15 5 10 15

n = 4 0.232 0.645 0.862 1.010 2.363 2.287n = 10 0.259 0.649 0.842 1.112 2.527 2.524n = 30 0.271 0.650 0.835 1.160 2.588 2.619n = 1000 0.277 0.651 0.832 1.183 2.615 2.662BSM 0.277 0.651 0.832 1.184 2.615 2.663

130 R. Popovic, D. Goldsman / European Journal of Operational Research 218 (2012) 124–131

Proposition 2. For n P 2,

dðt; brnÞ �@cðt; brnÞ

@s¼ E½Uðzþðs; k; brnÞÞ�; ð16Þ

#ðt; brnÞ �@cðt; brnÞ

@r¼ s

ffiffiffiTp

rE brn /ðzþðs; k; brnÞÞ

: ð17ÞDue to its lengthy technical nature, the proof is relegated to the

On-Line Appendix.

Example 7. Fig. 4 compares the BSM #(t ;r), our post-estimation#ðt; br4Þ, and the LUS version of vega. The operating parametersare set at s = 10, k = 10, T = 1/2, r = 0.05, and r = 1, with n = 4.(See On-Line Appendix B for the analogous delta sensitivities.)Table 4 shows numerically that the post-estimation delta andvega converge to the classical BSM Greeks as n becomes large.There is a substantial difference in the sensitivities for lowvalues of n; but by the time n = 30, these differences havedissipated.

4. Conclusions

Our purpose in this paper was to highlight the existence andconsequences of estimation risk in financial modeling. To do sowe focused on the well-known and accepted BSM view of optionmarkets. Our results typically hold at any given time point, and de-pend on both the market structure (BSM technology) and howindividuals view the risks associated with their limited knowledgeof the market parameters (estimator choice) they face. The conclu-sions are in line with a general proposition from Lucas [8] —namely, within our purview, the BSM formula ‘‘is derived fromdecision rules (demand and supply functions) of agents in theeconomy and these decisions are, theoretically, optimal given thesituation in which each agent is placed.’’ In other words, peopleuse information optimally — in their view, at least — when consid-ering the decisions they make.

The perturbation of the BSM model that we study herein shouldbe viewed as a calibration more in line with reality — one that willbe of concern to institutions dealing with the valuation and hedg-ing of a portfolio marked-to-market at many billions of dollars.Surprisingly, when it comes to gauging a ‘‘model’s fit,’’ great atten-tion is paid by practitioners to a few basis points, yet little concernis placed on formally incorporating the risk attached to the funda-mental parameters of a model and to what the consequences ofthat risk are. Model fit may be improved by adding parameters,but at the cost of increased out-of-sample variability. For purposesof prediction, neglecting the variability of the available data used inthe calibration of a model is analogous to failing to incorporate forfriction or wind effects when calculating the trajectory of a missile— it can be consequential.

Finally, in addition to providing new results on estimation-dependent BSM contingent claim values, our working model issuggestive of approaches that can be pursued to extend the studyof estimation risk to other more-complex set-ups. For example, in

models utilizing stochastic programming [13] or VaR analysis [5],it would be interesting to know the distributional effects of learn-ing and updating of the associated VaR covariance matrix. A furtherapplication of our methodology dealing with firm financing poli-cies [6] would also be insightful. Finally, we believe that it wouldbe interesting to study regimes that incorporate economic behav-ior subjected to a set of intermittent volatility shocks drawn fromsome probability law that is more-or-less well-known by marketagents. Market participants will be confronted by a vector of un-known, but estimable parameters. In turn, they proceed to makeand update their estimates of the unknowns, thereby convertingsituations of uncertainty to those that are characterized by degreesof risk.

Acknowledgments

We thank Paul Griffin, Steve Hackman, Bob Kertz, Alex Shapiro,and the anonymous referees for their comments and suggestions.

Appendix A

This appendix proves the various new results we introduce inthe body of the paper.

Proof of Lemma 1. The p.d.f. of Y follows by the definition of thelognormal. Then

E½ðY � kÞþ� ¼Z 1

0ðy� kÞþfY ðyÞdy ¼

Z 1

kðy� kÞ /

‘n sy

� �þ a

b

0@ 1A 1yb

dy

¼Z x�ðkÞ

�1ðs ea�xb � kÞ/ðxÞdx where x ¼ 1

b‘n

sy

� �þ a

�� �¼ s eaþb2

2

Z x�ðkÞ

�1

1ffiffiffiffiffiffiffi2pp exp

�ðxþ bÞ2

2

( )dx� kUðx�ðkÞÞ: �

Proof of Lemma 2. Since br2 is independent of WðTÞ, the law oftotal probability implies F

SðT;brÞðyÞ ¼ R10 FSðT;ffiffiffiwpÞðyÞfbr2 ðwÞdw. h

Proof of Corollary 1. Let fðwÞ � ‘nðsÞ þ ðr � w2ÞT. Starting at (8), we

find that

E½SjðT; brÞ� ¼ Z 1

0yjf

SðT;brÞðyÞdy ¼Z 1

0yjZ 1

0

1yffiffiffiffiffiffiffiwTp /ðz�ðs; y;

ffiffiffiffiwpÞÞfbr2 ðwÞdwdy

¼Z 1

0fbr2 ðwÞ

Z 1

0yj�1 1ffiffiffiffiffiffiffi

wTp /

‘nðyÞ � fðwÞffiffiffiffiffiffiffiwTp

� �dydw

¼Z 1

0fbr2 ðwÞ

Z 1

0yj�1 1ffiffiffiffiffiffiffiffiffiffiffiffiffi

2pwTp exp �ð‘nðyÞ � fðwÞÞ2

2wT

( )dy dw

¼Z 1

0fbr2 ðwÞ efðwÞjþwTj2

2

Z 1

�1

1ffiffiffiffiffiffiffiffiffiffiffiffiffi2pwTp exp

�ðz� ðfðwÞ þwTjÞÞ2

2wT

( )dzdw

¼ sjerTjZ 1

0fbr2 ðwÞ e

T2ðj

2�jÞw dw;

where the penultimate step follows upon setting y = ez and com-pleting the square; the final step follows after noting that the inte-rior integrand is a normal p.d.f. h

Proof of Lemma 3. If we denote S � SðT; brnÞ, then by Corollary 2,

serT � k ¼ E½S� k� ¼ E½ðS� kÞþ � ðk� SÞþ�¼ erT ½cðt; brnÞ � pðt; brnÞ�: �

Appendix B. Supplementary material

Supplementary data associated with this article can be found inthe online version at doi:10.1016/j.ejor.2011.09.011.

R. Popovic, D. Goldsman / European Journal of Operational Research 218 (2012) 124–131 131

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