Canonical valuation of options in the presence of stochastic volatility

CANONICAL VALUATION OF

OPTIONS IN THE PRESENCE

OF STOCHASTIC VOLATILITY

PHILIP GRAY*SCOTT NEWMAN

Proposed by M. Stutzer (1996), canonical valuation is a new method forvaluing derivative securities under the risk-neutral framework. It is non-parametric, simple to apply, and, unlike many alternative approaches, doesnot require any option data. Although canonical valuation has great poten-tial, its applicability in realistic scenarios has not yet been widely tested.This article documents the ability of canonical valuation to price derivativesin a number of settings. In a constant-volatility world, canonical estimatesof option prices struggle to match a Black-Scholes estimate based on his-torical volatility. However, in a more realistic stochastic-volatility setting,canonical valuation outperforms the Black-Scholes model. As the volatilitygenerating process becomes further removed from the constant-volatilityworld, the relative performance edge of canonical valuation is moreevident. In general, the results are encouraging that canonical valuation isa useful technique for valuing derivatives. © 2005 Wiley Periodicals, Inc.Jrl Fut Mark 25:1–19, 2005

The authors are grateful for the comments and suggestions of Jamie Alcock, Stephen Gray, PhilipHoang, Egon Kalotay, an anonymous referee, participants at the 16th Australian Finance andBanking conference, and funding from a UQ Business School summer research grant.*Correspondence author, UQ Business School, The University of Queensland, St. Lucia 4072,Australia; e-mail: [email protected]

Received November 2003; Accepted March 2004

� Philip Gray is an Associate Professor at UQ Business School at the University ofQueensland in Brisbane, Australia.

� Scott Newman is with the UQ Business School at the University of Queensland inBrisbane, Australia.

The Journal of Futures Markets, Vol. 25, No. 1, 1–19 (2005) © 2005 Wiley Periodicals, Inc.Published online in Wiley InterScience (www.interscience.wiley.com). DOI:10.1002/fut.20140

2 Gray and Newman

1Although no option prices are strictly required, such data are easily incorporated into canonicalvaluation if desired.

INTRODUCTION

Thirty years after the seminal work of Black and Scholes (1973) andMerton (1973), the benchmark in option pricing continues to be thewidely applied Black-Scholes formula. However, while the strong para-metric assumptions underlying the Black-Scholes model allow a simple,closed-form solution to the price of a European call option, empirical testssuggest that the assumptions are violated in practice. For example, ratherthan being constant, implied volatilities from observed option prices aresystematically related to moneyness and maturity (see Derman & Kani,1994; MacBeth & Merville, 1979; Rubinstein, 1985). There is alsoconsiderable evidence that stock returns are not normally distributed (seeJackwerth & Rubinstein, 1996; Kon, 1994).

In light of these problems, alternative methods have been developedto price options. One such example is canonical valuation. Developed byStutzer (1996), canonical valuation is a nonparametric technique forvaluing derivatives. Unlike the Black-Scholes model, it makes no restric-tive assumptions about the underlying asset’s return generating process;rather, the historical distribution of returns on the underlying asset isused to predict the distribution of future stock prices. A maximum-entropy principle is employed to transform this real-world distributioninto its risk-neutral counterpart, from which option prices follow easilyusing the standard risk-neutral approach. In addition to being relativelysimple to implement, a major advantage of canonical valuation is thatoption price data are not required as input.1

Despite its potential, canonical valuation has only been examined ina handful of papers to date. Using a simulated Black-Scholes world,Stutzer (1996) reports that the accuracy of canonical estimates of optionprices is comparable to Black-Scholes estimates using historical volatility.Foster and Whiteman (1999) modify canonical valuation to incorporate amore sophisticated Bayesian predictive model. In an application to thesoybean futures options market, the modified model performs well withreference to both the simple canonical valuation model using historicalreturns and Black-Scholes. Finally, Stutzer and Chowdhury (1999) applycanonical valuation to bond futures options, with results also suggestingthat the method performs well.

This article explores the potential usefulness of canonical valuationin two directions. First, the analysis of Stutzer (1996) is extended todocument the accuracy of canonical valuation across various levels of

Canonical Valuation of Options 3

maturity and moneyness. Working in a constant-volatility Black-Scholesworld, stock prices are simulated under a geometric Brownian motion sothat true option prices are known. Prices are then estimated using bothcanonical (CAN) and historical-volatility-based Black-Scholes (HBS)methods, and the properties of pricing errors are documented. We alsoexamine the potential to reduce pricing errors by incorporating a tokenamount of option data. The canonical estimator is modified such thatthe risk-neutral density is estimated subject to the constraint that asingle at-the-money option is correctly priced. The performance of theconstrained canonical estimator (CON) is compared to CAN and HBSestimators. The sensitivity of all findings to the number of returns usedto estimate the risk-neutral density is also examined.

Because there is considerable evidence that volatility is noncon-stant, Stutzer (1996) foreshadows that the usefulness of canonical valu-ation will be most apparent when we move beyond the Black-Scholesworld. This issue has not previously been examined. The second contri-bution of this article, therefore, is to evaluate the performance of canon-ical valuation in the presence of stochastic volatility. Heston’s (1993)model provides an ideal environment to test this conjecture as it admitsa closed-form solution to the price of a call option under stochasticvolatility. Stock prices are simulated under Heston’s stochastic volatilitymodel, and the performance of CAN and HBS estimates is assessed rel-ative to the true price. Simulation results are reported for a range ofmoneyness and time to maturity. Finally, the sensitivity of results to keyparameters in the stochastic volatility model is examined.

There are several key findings in this article. Not surprisingly, theBlack-Scholes model outperforms canonical valuation in a constant-volatility world. HBS estimates are less biased than standard canonicalestimates, and this performance edge persists even as sample sizeincreases. The practice of incorporating minimal option data into theestimation of the risk-neutral density under canonical valuationproduces significant improvements in pricing performance. The con-strained canonical estimator arguably outperforms HBS estimates,particularly for deep out-of-the-money options which can be difficultto price.

Moving to simulations assuming stochastic volatility, the magnitudeof pricing errors under both CAN and HBS estimators rises markedlyhighlighting the difficulty in pricing real-world options. The out-of-the-money superiority of the HBS estimator over CAN documented underconstant volatility disappears. Plain-vanilla canonical estimationproduces less biased estimates for out-of-the-money options, and the

4 Gray and Newman

constrained canonical estimator performs admirably regardless ofmoneyness. Sensitivity analysis identifies the volatility of the volatilitydynamics as the key parameter impacting on the success of alternativepricing methods.

The remainder of the article is structured as follows. The secondsection, An Overview of Canonical Valuation, reviews other popular non-parametric approaches to pricing derivatives, outlines the potentialadvantages of canonical valuation, and provides a brief overview of thecanonical valuation approach. The next two sections, CanonicalValuation in a Black-Scholes World and Canonical Valuation in aStochastic Volatility World, conduct simulation experiments to docu-ment the properties of alternative pricing methods in constant-volatilityand stochastic-volatility worlds, respectively. The last section is theConclusion.

AN OVERVIEW OF CANONICAL VALUATION

A risk-neutral approach is often adopted to price derivatives. The pri-mary task is to estimate the risk-neutral probability distribution of theunderlying asset, from which the expected payoff to the derivative can becalculated. There are, however, different ways to estimate the requiredrisk-neutral density. Black and Scholes (1973) typify the parametricapproach by specifying the dynamics of the underlying asset from whichthe risk-neutral density (lognormal in this case) is derived. In contrast,Rubinstein (1994), Hutchinson, Lo, and Poggio (1994), Jackwerth andRubinstein (1996), and Aït-Sahalia and Lo (1998) propose nonparamet-ric methods of estimating the risk-neutral density.

Although they make fewer restrictive assumptions over the data-generating process, these nonparametric methods require as input largequantities of market option prices across a range of strike prices. Anadvantage of canonical valuation is that option prices are unnecessary;the method can be implemented merely using a time-series of data forthe underlying asset. Note also that the ability to price options withoutusing observed market prices classifies canonical valuation as an optionpricing theory. The nonparametric methods just cited are best viewed asinterpolation and extrapolation algorithms that predict some optionprices from the observed prices of other options.2

Consider valuing a European call option on a stock expiring at timeT (i.e., T years forward). Obviously the possible option payoffs depend on

2We are grateful to an anonymous referee for elucidating this point.


3In this article, g* is calculated using a single variable optimization routine in Matlab. Alternatively,the optimization is equally simple using Solver in Microsoft Excel.

the distribution of the underlying stock price at time T. Stutzer (1996)begins with the stock’s historical distribution of T-year returns

where returns are expressed as price relatives. An advan-tage of the historical distribution is that it is more likely to capturestylized features of the data (such as skewness and leptokurtosis) thanparametric models. From the returns, the distribution of possible prices,Pi, for the underlying asset T-years forward is constructed:

(1)

where P0 is the current price of the underlying asset. Each possiblefuture price computed in (1) is assigned an equal prior real-world proba-bility such that . These prior probabilities are transformed intotheir risk-neutral counterparts subject to the constraint that the expectedreturn on the stock is the riskless rate; equivalently, the discountedexpected return is unity:

(2)

where denotes the risk-neutral probability of return Ri, and r is therisk-free rate of interest. Employing the maximum entropy principle ofinformation theory, Stutzer (1996) shows that the risk-neutral probabili-ties, are given by:

(3)

where is the Lagrange multiplier, given by the following minimizationproblem:3

(4)

The final step in canonical valuation is to compute the expected dis-countedpayoff to thederivative securityusing the risk-neutral probabilities

g* � arg min g a

n

i�1exp cga Ri

(1 � r)T � 1 b d .

g*

p̂*i �

exp ag* Ri

(1 � r)Tb

an

i�1exp ag* Ri

(1 � r)Tb

p̂i*,

p̂i*

1 � an

i�1p̂*i a Ri

(1 � r)Tb

p̂i � 1np̂i,

Pi � P0 Ri, i � 1, p , n

Ri, i � 1, p , n,

6 Gray and Newman

calculated in (3). The price of a European call option expiring at T with anexercisepriceofX is simply:

(5)

To summarize, canonical valuation uses the historical time-series ofprices on the underlying asset to estimate the future distribution of theasset price. A maximum-entropic technique transforms the distributioninto the required risk-neutral density. Most importantly, no option pricedata is required. If desired, the risk-neutral probabilities can be esti-mated subject to an additional constraint that they correctly price one ormore traded options (see Stutzer, 1996, Equations 11 and 12). Thisarticle also explores the usefulness of this marginally more complexprocedure.

CANONICAL VALUATION IN ABLACK-SCHOLES WORLD

To investigate its applicability in the most basic setting, canonical valua-tion is applied to price European call options across a range of money-ness and maturities in a simulated Black-Scholes world. Under theseconditions, stock price follows a geometric Brownian motion:

(6)

where St is the stock price at time t, m is the average stock return, s isthe constant instantaneous volatility of the process and dzt is the incre-ment in a standard Wiener process. Under this model, the distribution ofcontinuously compounded T-year returns is normal:

(7)

For each time to maturity T, 200 returns are drawn from the normaldistribution (7), and the distribution of possible future stock prices isconstructed as per Equation (1). Risk-neutral probabilities and areestimated from Equations (3) and (4), respectively. Canonical optionprices follow from Equation (5). The simulation employs a drift m of 10%and annual volatility s of 20%. The riskless rate of interest is assumed tobe a constant 5% continuously compounded.4 These values are consistent

g*

ln(Ri) � N ((m � 12 s2)T, s2T).

dSt � mSt dt � sSt dzt

p̂i*

C � an

i�1amax(P0 Ri � X, 0)

(1 � r)T bp̂i*.

4To ensure comparability, the discrete compounding equivalent of this continuously compoundedrate is employed in Equations (2)–(5) for canonical estimates.


5Results are not reported for deep out-of-the-money options with just one week to expiration. Thetrue price for this option is $0.0001, therefore even the slightest pricing error results in anenormous percentage error.

with those used in simulations performed by Hutchinson et al. (1994)and Stutzer (1996).

Canonical estimates of option price (CAN) are compared to Black-Scholes model prices (HBS), using a historical estimate of volatility fromthe simulated data, thus ensuring that estimates under each method relyon the same data. The accuracy of CAN and HBS estimates is then eval-uated with reference to the true Black-Scholes price calculated using theknown volatility. The simulation procedure is repeated 10,000 times, andthe properties of estimates are tabulated.

Table I reports results for various combinations of moneyness andmaturity.5 The top and middle numbers in each cell represent themean percentage error (MPE) of HBS and CAN call option estimatesrelative to the true Black-Scholes price. Without exception, the HBS

TABLE I

MPE of Canonical and Black-Scholes Estimates in a Black-Scholes World

MoneynessTime to expiration (years)

(spot/strike) 1�52 1�13 1�4 1�2 3�4 1

Deep out-of-the-money n/a 0.0155 �0.0020 �0.0028 �0.0013 �0.0022(0.90) n/a �0.0272 �0.0245 �0.0292 �0.0321 �0.0372

n/a �0.0212 �0.0057 �0.0037 �0.0030 �0.0024

Out-of-the-money 0.0003 �0.0021 �0.0022 �0.0020 �0.0009 �0.0015(0.97) �0.0098 �0.0077 �0.0111 �0.0158 �0.0191 �0.0234

�0.0052 �0.0008 �0.0001 �0.0001 �0.0002 �0.0001

At-the-money �0.0012 �0.0014 �0.0015 �0.0007 �0.0009 �0.0011(1.00) �0.0032 �0.0042 �0.0078 �0.0120 �0.0150 �0.0191

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

In-the-money 0.0000 �0.0005 �0.0009 �0.0010 �0.0005 �0.0009(1.03) �0.0005 �0.0020 �0.0052 �0.0091 �0.0118 �0.0155

�0.0003 �0.0004 �0.0004 �0.0004 �0.0003 �0.0003

Deep in-the-money 0.0000 0.0001 0.0000 �0.0001 0.0000 �0.0002(1.125) 0.0000 �0.0001 �0.0012 �0.0033 �0.0052 �0.0077

�0.0001 �0.0001 �0.0007 �0.0009 �0.0010 �0.0011

Note. Canonical and HBS estimates are compared to the true Black-Scholes call price, where stock prices are simulatedby a geometric Brownian motion with m� 0.1 and s� 0.2. Each cell represents a particular combination of moneyness andmaturity. The top and middle numbers reported for each combination are the mean percentage error (MPE) of the HBS andCAN estimates respectively over 10,000 simulations. The bottom number is a modified canonical estimate (CON)constrained to price an at-the-money option correctly.

8 Gray and Newman

6Results for the CON estimator differ from Stutzer (1996) who constrained an out-of-the-moneyoption to be correctly priced (S�X � 0.95). This article constrains the risk-neutral probabilities toprice an at-the-money option correctly.

estimate outperforms the CAN estimate for each combination of mon-eyness and time to expiration, and this performance edge is mostnoticeable for deep out-of-the-money options. The accuracy of bothHBS and CAN estimates increases monotonically with moneyness.Although HBS estimates show no discernible pattern with maturity,CAN exhibits persistent negative pricing error (i.e., underpricesoptions) for the vast majority of combinations, and this negative biasincreases with maturity.

Stutzer (1996) reports that the performance of canonical valuationimproves when a small amount of option data is used. Specifically,additional constraints can be incorporated into the estimation of risk-neutral probabilities such that one or more options are correctly priced.In Table I, the bottom number in each cell represents this constrainedcanonical estimate (CON), calculated to correctly price a single at-the money option. This practice of augmenting historical return datawith a minimal amount of option data appears highly advantageous. TheMPE for CON estimates is a significant improvement over CANestimates and, in some cases, even outperforms HBS estimates.Curiously, like the CAN estimator, the CON estimator also consistentlyunderprices options, although the magnitude of negative bias is notablysmaller.

The MPE is a useful statistic in that it shows the average pricingeffectiveness of alternative methods (in this case, averaged over 10,000simulations) and documents the direction of errors. However, themagnitude of pricing errors can be masked to the extent that positive andnegative errors cancel each other out. Table II reports an alternative per-formance measure, mean absolute percentage error (MAPE), adopted byStutzer (1996). The tenor of the results is similar to MPE in Table I. HBSconsistently outperforms CAN, although the performance edge is lessdramatic under this alternative metric; in many cases, the MAPE of CANis only marginally higher than that for HBS. Again, CON produces thelowest MAPE in most cases. MAPE decreases monotonically inmoneyness, and it decreases (increases) in maturity for out-of-the-money(in-the-money) options. These findings are largely consistent with asimilar analysis performed by Stutzer (1996) over a narrower band ofmoneyness and maturities.6


TABLE II

MAPE of Canonical and Black-Scholes Estimates in a Black-Scholes World


(spot/strike) 1�52 1�13 1�4 1�2 3�4 1

Deep out-of-the-money n/a 0.2272 0.1088 0.0763 0.0635 0.0559(0.90) n/a 0.3603 0.1252 0.0847 0.0724 0.0660

n/a 0.3355 0.0878 0.0464 0.0326 0.0254

Out-of-the-money 0.1214 0.0701 0.0498 0.0419 0.0381 0.0354(0.97) 0.1491 0.0765 0.0543 0.0462 0.0439 0.0427

0.1095 0.0339 0.0142 0.0090 0.0068 0.0055

At-the-money 0.0379 0.0369 0.0341 0.0318 0.0302 0.0289(1.00) 0.0410 0.0396 0.0378 0.0356 0.0354 0.0353

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

In-the-money 0.0074 0.0174 0.0227 0.0238 0.0238 0.0235(1.03) 0.0095 0.0199 0.0260 0.0274 0.0284 0.0292

0.0069 0.0086 0.0067 0.0052 0.0044 0.0038

Deep in-the-money 0.0000 0.0007 0.0051 0.0087 0.0107 0.0118(1.125) 0.0000 0.0015 0.0072 0.0116 0.0141 0.0158

0.0000 0.0014 0.0055 0.0072 0.0074 0.0073

Note. Canonical and HBS estimates are compared to the true Black-Scholes call price, where stock prices are simulatedby a geometric Brownian motion with m� 0.1 and s� 0.2. Each cell represents a particular combination of moneyness andmaturity. The top and middle numbers reported for each combination are the mean absolute percentage error (MAPE) ofthe HBS and CAN estimates respectively over 10,000 simulations. The bottom number is a modified canonical estimate(CON) constrained to price an at-the-money option correctly.

Overall, the simulation results reported in this section support theuse of canonical valuation for options. In an environment simulatedprecisely under the Black-Scholes assumptions, HBS estimates mightbe expected to outperform nonparametric techniques such as canonicalvaluation. However, the performance of the CAN estimate is onlyslightly inferior to HBS, whereas incorporation of a token amount ofoption data results in CON estimates that clearly outperform the HBSestimate. In practice, of course, it is impossible to verify any specificfunctional form for the stock price process, and it is this impossibilitythat motivates the use of nonparametric techniques such as canonicalvaluation.

One final issue explored in this section relates to the sample sizerequired to successfully implement canonical valuation. The simula-tions conducted to date utilize 200 returns—the HBS option estimateuses these returns to estimate the volatility of the stock, whereas theCAN estimate uses the returns to construct the predictive distributionof stock price. That HBS outperforms CAN suggests that 200 returns

10 Gray and Newman

do a better job at estimating s (and consequently the correspondinglognormal density assumed by the HBS estimate) than they do atapproximating the full return density required by the nonparametriccanonical valuation method. One might question whether the perform-ance edge of HBS over CAN diminishes as the number of returnsincreases.

Table III reports the sensitivity of MAPE under alternative methodsto the number of returns used in estimation. All numbers are for a long-dated call option with one year to expiration. The accuracy of all estima-tion methods improves with sample size. Yet, perhaps surprisingly, HBSmaintains its performance edge over CAN even as the number of returnsemployed approaches 500. In a simulated Black-Scholes world, there-fore, it appears that the canonical approach requires the assistance of atleast some option data to compete with the standard Black-Scholesmodel.

TABLE III

Sensitivity of Long-Dated Option Estimates to Number of Returns

MoneynessNumber of returns used in estimation

(spot/strike) 60 100 200 300 400 500

Deep out-of-the-money 0.1025 0.0777 0.0559 0.0449 0.0392 0.0351(0.90) 0.1145 0.0882 0.0660 0.0557 0.0503 0.0475

0.0482 0.0363 0.0254 0.0208 0.0179 0.0162

Out-of-the-money 0.0650 0.0492 0.0354 0.0284 0.0248 0.0223(0.97) 0.0748 0.0573 0.0427 0.0359 0.0324 0.0304

0.0105 0.0080 0.0055 0.0046 0.0039 0.0035

At-the-money 0.0530 0.0402 0.0289 0.0232 0.0202 0.0182(1.00) 0.0622 0.0476 0.0353 0.0298 0.0267 0.0250

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

In-the-money 0.0430 0.0326 0.0235 0.0188 0.0165 0.0148(1.03) 0.0517 0.0395 0.0292 0.0246 0.0221 0.0206

0.0073 0.0055 0.0038 0.0031 0.0026 0.0024

Deep in-the-money 0.0215 0.0163 0.0118 0.0095 0.0083 0.0074(1.125) 0.0284 0.0217 0.0158 0.0132 0.0118 0.0110

0.0144 0.0107 0.0073 0.0059 0.0051 0.0046

Note. This table reports the sensitivity of the MAPE of HBS, CAN, and CON estimates to the number of returns used inestimation. Canonical and HBS estimates are compared to the true Black-Scholes call price, where stock prices aresimulated by a geometric Brownian motion with m� 0.1 and s� 0.2. The call option has a maturity of one year. The top andmiddle numbers reported for each cell are the mean absolute percentage error (MAPE) of the HBS and CAN estimatesrespectively over 10,000 simulations. The bottom number is a modified canonical estimate (CON) constrained to price anat-the-money option correctly.


7See, for example, Heston (1993), Hull and White (1987), Melino and Turnbull (1990), Scott(1987), Stein and Stein (1991), and Wiggins (1987).8One-day time steps are used, and 253 trading days per year are assumed.

CANONICAL VALUATION IN A STOCHASTICVOLATILITY WORLD

There is a weight of empirical evidence to suggest that the varianceof stock returns is stochastic, in direct contrast to the assumptions ofBlack-Scholes used in the previous analysis. The practical usefulnessof a nonparametric technique like canonical valuation, therefore, is bestassessed in an environment where the constant-volatility assumption isrelaxed.

Numerous stochastic volatility models have been suggested in theliterature.7 However, with the exception of Heston (1993) and Stein andStein (1991), these models do not admit closed-form solutions to optionprice. To document the properties of alternative pricing methodologies,knowledge of the true option price under stochastic volatility is required.For this purpose, Heston’s model is employed in the following simula-tions because, in addition to giving a closed-form solution, it also accom-modates correlation between the random shocks in the volatility andstock processes.

Heston (1993) derives the solution to the price of a European calloption assuming that the stock price, St, follows the process:

(8)

where the stochastic variance of the stock return vt, is generated by thefollowing Ornstein-Uhlenbeck process:

(9)

where k is the speed of mean-reversion, u is the long-run mean variance,j is the volatility of the volatility generating process, and dz1,t and dz2,t areincrements in Wiener processes, which have a correlation coefficient of r.

Using Euler discretizations of (8) and (9), data are simulated underHeston’s stochastic-volatility world out to the option’s expiration time T.8

For consistency with the Black-Scholes world simulations in the previoussection, the drift of the stock return, m, and the long-run mean, u, areassumed to be 10% and 4%, respectively. The remaining parameters k, j,

dvt � k(u � vt) dt � j1vt dz2,t

dSt � mSt dt � 1vtSt dz1,t

12 Gray and Newman

9We are grateful to an anonymous referee for suggesting the IBS estimate.

and r are assumed to be 3, 0.40, and �0.50, respectively. These valuesare typical of estimates from actual market data (see Lin, Strong, & Xu,2001; Zhang & Shu, 2003). This simulation is repeated 200 times togenerate the distribution of T-year returns.

Four estimators are examined under the stochastic volatility world.The HBS estimate is again the Black-Scholes model price using historicalvolatility estimated from the simulated data. Canonical estimates ofoption prices (both with and without a constraint) are computed using themethodology outlined in the second section. In addition to the HBS esti-mate, a modified estimate is obtained by backing out the Black-Scholesimplied volatility from the at-the-money option, assumed to be correctlypriced. The implied volatility is then used with the Black-Scholes formulafor other levels of moneyness to obtain an implied Black-Scholes estimate(IBS). The IBS estimate allows a fairer comparison with the CON esti-mate because both use the same information (i.e., that the at-the-moneyoption is correctly priced). Thus, the relative comparison between CONand IBS is their respective ability to correct in-the-money and out-of-the-money bias.9 All estimates are compared to the true Heston price for eachcombination of moneyness and expiration. The entire process is repeated10,000 times.

Tables IV and V report the simulation results. In practice, theBlack-Scholes model is known to overprice (underprice) out-of-the-money (in-the-money) options. One explanation is that, in the presenceof stochastic volatility, the distribution of stock returns has a fatter lefttail than the lognormal assumed by Black-Scholes. Table IV reports thatMPE for HBS and IBS is indeed positive (negative) for out-of-the-money(in-the-money) options. It is important to note that MPE for CANestimates is significantly lower than HBS estimates for deep out-of-the-money options. Thus, the canonical technique presents itself as a usefulalternative for pricing such options. It appears to better capture the fatleft tails of stock return distributions under stochastic volatility. Notealso that the constrained canonical estimator (CON) clearly outperformsHBS and CAN for all combinations of moneyness and maturity.

Although the IBS estimator performs notably better than HBS forout-of-the-money options, the latter hold their own in-the-money. Infact, surprisingly, as the option moves into the money, HBS pricing errorsare little different and arguably lower than for CAN estimates. Theresults for MAPE in Table V are qualitatively similar to those for MPE.


TABLE IV

MPE of Canonical and Black-Scholes Estimates in a Stochastic-Volatility World


(spot/strike) 1�52 1�13 1�4 1�2 3�4 1

Deep out-of-the-money n/a 0.5990 0.3804 0.2538 0.1823 0.1429(0.90) n/a �0.3138 �0.0970 �0.0768 �0.0738 �0.0670

n/a �0.1736 �0.0059 0.0033 0.0021 0.0017n/a 0.8420 0.3170 0.1761 0.1185 0.0868

Out-of-the-money �0.1678 0.0115 0.0607 0.0629 0.0542 0.0484(0.97) �0.3298 �0.0979 �0.0523 �0.0488 �0.0491 �0.0464

�0.1267 �0.0139 �0.0010 0.0006 0.0006 0.00060.1218 0.0652 0.0403 0.0270 0.0200 0.0156

At-the-money �0.0863 �0.0268 0.0134 0.0264 0.0266 0.0264(1.00) �0.0945 �0.0490 �0.0363 �0.0382 �0.0401 �0.0389

0.0000 0.0000 0.0000 0.0000 0.0000 0.00000.0000 0.0000 0.0000 0.0000 0.0000 0.0000

In-the-money �0.0224 �0.0275 �0.0087 0.0048 0.0088 0.0113(1.03) �0.0165 �0.0225 �0.0244 �0.0294 �0.0324 �0.0324

�0.0011 �0.0003 �0.0004 �0.0008 �0.0007 �0.0007�0.0071 �0.0154 �0.0175 �0.0148 �0.0120 �0.0100

Deep in-the-money 0.0000 �0.0035 �0.0146 �0.0150 �0.0126 �0.0096(1.125) �0.0001 �0.0013 �0.0064 �0.0121 �0.0161 �0.0177

�0.0001 �0.0003 �0.0008 �0.0015 �0.0018 �0.0017�0.0001 �0.0032 �0.0169 �0.0225 �0.0222 �0.0205

Note. Canonical, HBS and IBS estimates are compared to the true Heston call price, where stock prices are simulated inHeston’s (1993) stochastic volatility world (Equations (8) and (9)). Each cell represents a particular combination of money-ness and maturity. The first and second numbers reported for each combination are the mean percentage error (MPE) of theHBS and CAN estimates respectively over 10,000 simulations. The third number is a modified canonical estimate (CON)constrained to price an at-the-money option correctly. The fourth number is an estimate (IBS) from the Black-Scholes formulausing the implied volatility of the at-the-money option which is assumed to be correctly priced.

Overall, there is evidence that canonical valuation is increasinglyuseful when the dynamics of the underlying asset exhibit stochasticvolatility. Modifying canonical valuation to incorporate a single marketoption price is clearly the preferred valuation technique. In both Black-Scholes and Heston worlds, the constrained canonical estimatorconsistently outperforms CAN, HBS, and IBS estimates.

Sensitivity to Heston Parameters

The two key parameters in Heston’s (1993) model that drive the volatilitygenerating process are the speed of mean reversion k and the volatility ofthe volatility process j. Different values of these parameters can signifi-cantly influence the dynamics of volatility, and are therefore likely toaffect the resulting canonical and HBS estimates. For example, as j

14 Gray and Newman

TABLE V

MAPE of Canonical and Black-Scholes Estimates in a Stochastic-Volatility World


(spot/strike) 1�52 1�13 1�4 1�2 3�4 1

Deep out-of-the-money n/a 0.6317 0.3855 0.2580 0.1872 0.1489(0.90) n/a 0.5649 0.1859 0.1248 0.1053 0.0922

n/a 0.5807 0.1358 0.0664 0.0444 0.0333n/a 0.8420 0.3170 0.1761 0.1185 0.0868

Out-of-the-money 0.1893 0.0789 0.0805 0.0768 0.0674 0.0616(0.97) 0.3344 0.1140 0.0708 0.0628 0.0603 0.0564

0.1671 0.0396 0.0161 0.0098 0.0072 0.00580.1218 0.0652 0.0403 0.0270 0.0200 0.0156

At-the-money 0.0874 0.0453 0.0429 0.0465 0.0447 0.0434(1.00) 0.0954 0.0564 0.0476 0.0473 0.0478 0.0461

0.0000 0.0000 0.0000 0.0000 0.0000 0.00000.0000 0.0000 0.0000 0.0000 0.0000 0.0000

In-the-money 0.0225 0.0302 0.0283 0.0309 0.0316 0.0319(1.03) 0.0172 0.0268 0.0319 0.0357 0.0379 0.0376

0.0075 0.0083 0.0065 0.0050 0.0041 0.00350.0071 0.0154 0.0175 0.0148 0.0120 0.0100

Deep in-the-money 0.0000 0.0035 0.0149 0.0176 0.0179 0.0178(1.125) 0.0000 0.0023 0.0092 0.0149 0.0185 0.0201

0.0000 0.0022 0.0058 0.0066 0.0066 0.00640.0000 0.0032 0.0169 0.0225 0.0222 0.0205

Note. Canonical, HBS and IBS estimates are compared to the true Heston call price, where stock prices are simulated inHeston’s (1993) stochastic volatility world (Equations (8) and (9)). Each cell represents a particular combination of money-ness and maturity. The first and second numbers reported for each combination are the mean absolute percentage error(MAPE) of the HBS and CAN estimates respectively over 10,000 simulations. The third number is a modified canonicalestimate (CON) constrained to price an at-the-money option correctly. The fourth number is an estimate (IBS) from theBlack-Scholes formula using the implied volatility of the at-the-money option which is assumed to be correctly priced.

increases, the random component of the volatility generating process hasmore weight, causing greater movement in volatility. Similarly, as kdecreases, volatility mean reverts at a slower rate, also causing volatilityto be driven more by the random component. Thus, when k is low and jis high, volatility is more random and the stock generation process isfurther removed from the Black-Scholes world. In such a case, it is likelythat HBS/IBS estimates are less accurate than when k is high, and/or j islow. A nonparametric technique like canonical valuation may be betterequipped to accommodate greater instability in volatility.

To examine the sensitivity of pricing errors under alternativemethods, the simulation procedure described in the previous section isrepeated for assorted combinations of k and j. Values of k range from 2to 10, and j ranges from 0.10 to 0.50. For each combination, 200returns are generated under Heston’s model to form the density of future

FIGURE 1Sensitivity of pricing errors to parameter values. This figure graphs the mean pricing

errors (MPE) for HBS, CAN, CON, and IBS estimates over 5,000 simulations.Values of k range from 2 to 10, and values of j range from 0.10 to 0.50. Thefour rows represent MPE for HBS, CAN, CON, and IBS, respectively. The

three columns represent out-of-the-money options (0.97), at-the-moneyoptions (1.00), and in-the-money options (1.03), respectively.


stock prices, from which the price of a call option with three months toexpiration is estimated using HBS, IBS, and both canonical approaches.Estimates are compared to the true Heston price to calculate the meanpercentage errors across 5,000 simulations.10

Figure 1 shows the sensitivity to values of k and j of the mean per-centage error (MPE) under HBS (first row), CAN (second row), CON(third row), and IBS (fourth row) estimates. The three columns of

10Fewer simulations are used in this subsection because of the computational demands in runningsimulations over a wide grid of parameter values. However, even with 5,000 simulations, the graph-ical results display a sufficient degree of smoothness.

16 Gray and Newman

11Note that, by construction, the constrained canonical estimate perfectly prices the at-the-moneyoption. The same comment applies to at-the-money IBS estimates.

Figure 1 represent out-of-the money options (0.97), at-the-moneyoptions (1.00), and in-the-money options (1.03).

Examining row 1, HBS estimates are generally positively biased forout-of-the-money and negatively biased for in-the-money options. Thisconclusion holds across the range of parameter values. A spread of posi-tive and negative pricing errors arises for at-the-money options.Irrespective of moneyness, mispricing is clearly a positive function of j,particularly when k is small. For low k, the volatility of volatility j is thedominant parameter. The higher j, the higher volatility and consequently,the higher the Black-Scholes estimate of option price. Thus, row 1 ofFigure 1 shows that, as j increases, HBS overpricing of out-of-the-moneyoptions worsens, while HBS underpricing of in-the-money optionsimproves.

In row 2 of Figure 1, the (unconstrained) CAN estimates are alsohighly sensitive to the value of j. As j increases, the magnitude of nega-tive bias increases sharply. The MPEs of CAN estimates are less sensitiveto k. Although difficult to read from the graphs, CAN pricing errorsimprove as k increases.

Comparing HBS and CAN estimates, CAN outperforms HBS acrossthe grid of parameter values for out-of-the-money options. This is againencouraging as out-of-the-money options are notoriously difficult toprice using the Black-Scholes parametric approach. However, movinginto the money, the magnitude of CAN pricing errors is larger than forHBS. This finding is preempted by Table IV, which reports that HBSarguably outperforms CAN as moneyness increase.

Figure 1 row 3 clearly illustrates the superior performance of theCON estimator. For out-of-the-money and in-the-money options, themagnitude of CON pricing errors is significantly smaller than HBS andCAN estimates.11 Most important, the CON estimator is also highlyrobust to varying dynamics of the underlying asset. With the possibleexception of the low k high j combination, CON pricing errors areextremely small across the grid of parameter values.

Turning to the IBS estimator, Figure 1 row 4 is interesting. Table IVshows that MPE for IBS is very respectable, particularly as moneynessincreases. However, Figure 1 reveals that IBS performance is not robustacross parameter values. MPE is highly sensitive to j, with positive (neg-ative) bias increasing with j for out-of-the-money (in-the-money) options.


Several final general comments can be made regarding Figure 1.First, the magnitude of pricing errors across all estimation methodsdecreases with moneyness. This is a common finding in option-pricingliterature. Second, of the two parameters, all methods are more sensitiveto the volatility of the volatility generating process j, rather than thespeed of mean reversion k. Pricing errors are almost invariant to k values,especially when j is low.

CONCLUSION

This study investigates the merit of a nonparametric risk-neutral methodfor valuing derivative securities known as canonical valuation. Althoughthe Black-Scholes model remains a popular benchmark for the pricing ofoptions because of its ease of implementation, its restrictive parametricassumptions over stock price dynamics give rise to empirical inconsisten-cies with observed option prices. There are many methods thataccommodate these inconsistencies; however, widespread use of suchmodels remains stifled because of their complexity and the requirementof a large quantities of option data as input. Canonical valuation requiresno input of option data and is not computationally demanding.

This article conducts simulation experiments in Black-Scholes andHeston worlds to explore the potential usefulness of canonical valuation.In a constant-volatility Black-Scholes world, canonical valuation pricescall options with less accuracy than an analogous implementation of theBlack-Scholes model. The performance edge of HBS estimates is attrib-utable to the fact that the data are simulated precisely in accordancewith the Black-Scholes assumptions. However, when estimation of therisk-neutral density under canonical valuation is constrained to correctlyprice a single at-the-money option, CON clearly outperforms the HBSestimator for most combinations of moneyness and maturity. With theassistance of some token option data, therefore, the canonical approachis able to compete with the Black-Scholes model, even under conditionsideal for the latter.

A more useful test of canonical valuation is undertaken in a settingwhere volatility is stochastic over time. Stock returns are simulatedaccording to Heston’s (1993) stochastic volatility model and canonicaland Black-Scholes estimates of option prices are benchmarked againstthe true Heston price. In this case, the CAN estimator outperforms HBSfor deep out-of-the-money options. Perhaps surprisingly, HBS estimatesperform well as moneyness increases. Again, the constrained canonicalestimator dominates other approaches.

18 Gray and Newman

In summary, the simulation results suggest that, relative to Black-Scholes estimation using historical volatility, the canonical approach hasmerit. This is particularly the case when a simple constraint is imposedwhen estimating the risk-neutral density and when the dynamics of theunderlying asset depart from Black-Scholes’ constant-volatility assump-tion. Sensitivity analysis confirms that the accuracy of HBS estimatesdeteriorates as the asset dynamics become further removed from theBlack-Scholes world. Pricing errors are most sensitive to increasing val-ues for the volatility of the volatility process j, yet relatively insensitive tothe strength of mean reversion in volatility k.

Although the departure of reality from the Black-Scholes worldremains an empirical question, this article documents the viability of theconstrained canonical estimator even in a constant-volatility setting. Anobvious avenue for future research is to investigate the performance ofcanonical valuation in pricing traded derivatives. In addition, the hedg-ing performance under canonical valuation is another area warrantinginvestigation. Finally, further work is required to explain the consistentnegative bias in canonical estimates of option price.

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Canonical valuation of options in the presence of stochastic volatility