Asset Pricing with Misspecified Models - APJFS

Asset Pricing with Misspecified Models 

Jialin Yu 

Columbia University 

Graduate School of Business 

Finance and Economics 

3022 Broadway Uris Hall 421 

New York NY 10027 

+1-212-854-9140 

jy2167@columbia.edu 

Dongyoup Lee 


Graduate School of Business 

Finance and Economics 

3022 Broadway Uris Hall 4k 

New York NY 10027 

+1-646-530-1303 

dl2135@columbia.edu

Asset Pricing with Misspeci…ed Models 

Jialin Yu y 


Dongyoup Lee z 


August 25, 2008 

Abstract 

This paper provides an asset pricing method to address potential model misspeci…cation. The 

resulting price estimator is consistent irrespective of misspeci…cation. The pricing precision is 

at least that of nonparametric prices, and automatically converges to parametric precision when 

model quality improves. 

The method is applicable to multi-dimensional asset pricing and to 

sensitivity analysis. 

Application to the pricing of CBOT Treasury options suggests that the 

cheapest to deliver practice is an important source of misspeci…cation. 

Potential equilibrium 

implications on bounded rationality are discussed. 

We thank Yacine Aït-Sahalia, Robert Hodrick, Bo Honoré, Wei Jiang, José Scheinkman, and seminar participants 

at Columbia Business School for helpful discussions. All errors are ours. 

y Corresponding author. Address: 421 Uris Hall, 3022 Broadway, New York, NY 10027. Email: jy2167@columbia.edu 

z Address: 4K Uris Hall, 3022 Broadway, New York, NY 10027. Email: dl2135@columbia.edu.

1. Introduction 

Investors constantly face the challenge of imperfect models. For example, a trader of Treasury options 

listed on the Chicago Board of Trade (CBOT) may have learned the state-of-the-art term structure 

model which prescribes an option pricing formula. Over time, the trader starts to observe option 

prices deviate from those predicted by the pricing formula and suspects the model is misspeci…ed 

– just as the Black-Merton-Scholes option pricing formula (Black and Scholes (1973) and Merton 

(1973)) is found by many to have di¢ culty explaining the Black Monday in October 1987. Misspeci…cation 

can take di¤erent forms: the model may be misspeci…ed only along some dimensions of the 

state variables, or the model may be poor along all dimensions. Even in the latter case, the model 

can still provide useful restrictions that may be utilized by some investors. For example, the model 

may approximate the …rst derivative (e.g., the Greek letter delta) well but not the second derivative 

(e.g., the Greek letter gamma). 1 Therefore, misspeci…cation is not a binary concept. Rather, there is 

a continuous middle ground between correct speci…cation and the case of a useless model, the middle 

ground being the more likely scenario in practice than the two polar cases. 

How should the investors deal with models that are misspeci…ed but not completely useless, such 

as those that provide good approximation in some aspects but misleading information in others? 

Further, cautious investors may …nd it desirable to take measures before the latest model is con…rmed 

misspeci…ed. How should such investors act against the possibility of misspeci…cation, the true nature 

of which is unknown as of yet? 

This paper proposes a pricing method (referred to as “robust parametric method” in this paper) 

based on a possibly misspeci…ed model so that the resulting price estimator has the following 

properties: (i) robustness – the price estimator is consistent and the pricing error is at most that 

of the nonparametric rate irrespective of misspeci…cation; (ii) adaptive e¢ ciency –the pricing error 

decreases when the model quality improves, and the pricing error approaches the parametric rate in 

the limit when the model misspeci…cation disappears. 2 In addition, the estimator does not require an 

investor to know the model quality. The appropriate rate of convergence is achieved automatically. 

To see the potential magnitude of improvement from adaptive e¢ ciency, recall that the pricing error 

1 The Greek letter delta refers to the sensitivity of option value to the change in price of its underlying asset. The 

Greek letter gamma measures the rate of change in delta when the underlying asset value changes. 

2 See (2.8) on measuring model quality. 

1

of parametric method, based on a correct model, is in the order of n 1=2 with n being the sample size. 

The pricing error of nonparametric method is in the order of n 2/(4+d) where d is the dimension of 

the state variables. 3 To improve the pricing precision from $0.1 to $0.01, parametric method requires 

100 times the sample size and nonparametric method requires 10,000 times the sample size if d = 4. 

Multi-dimensional state variable is not uncommon in asset pricing which reduces the value added 

of nonparametric methods. For example, option pricing can involve multiple state variables such as 

the underlying asset value, underlying asset volatility, option maturity, strike price, etc. That the 

robust parametric method can, depending on model quality, improve the pricing error towards that 

of the parametric method is a nontrivial contribution. Its advantage relative to parametric methods 

lies in the possibility of model misspeci…cation, in which case the parametric pricing error is di¢ cult 

to quantify. Therefore, the robust parametric method is especially suited if the available model is 

somewhere in between being correct and disastrous. 

To see the intuition of the robust parametric method, consider a pricing model f (X; ), where X 

is the state variable and is the model parameter. Misspeci…cation is de…ned as the nonexistence of 

a parameter such that f (X = x; ) …ts the true model for all values of x. However, misspeci…cation 

does not rule out the existence of a parameter (x) such that f (X = x; (x)) …ts the true model for 

one value X = x only. Tracing out (x) for various x makes f (X; (X)) match the true model. That 

is, if (X) is appropriately chosen, a misspeci…ed model can be turned into a true model. To say that 

the original model is correctly speci…ed amounts to say that (X) is a constant. For example, the 

existence of a volatility smile implies that no single implied volatility number can …t option prices at 

all strike prices using the Black-Merton-Scholes option pricing formula. However, one can …nd one 

implied volatility for each strike so that the Black-Merton-Scholes formula using di¤erent implied 

volatilities …ts the option prices at di¤erent strikes. These implied volatilities, when plotted against 

strikes, constitute the smile curve which is the equivalent of (X). This is one instance where a 

misspeci…ed model is converted into a correct one. Therefore, this paper captures the intuition used 

informally in the investment community. Next, if the true model is continuous, f (X; (x)) may 

provide a good approximation not only for X = x but also for observations in a neighborhood of x. 

The robust parametric method applies parametric method to estimate (x) in this neighborhood of 

3 See Newey and McFadden (1994) and Fan (1992) on the parametric and nonparametric rates of convergence. 

2

…t. The neighborhood is likely small when the model quality is poor, hence only nearby observations 

are used to achieve robustness. When the model quality improves, the neighborhood can be enlarged 

to include more observations to enhance e¢ ciency. As a by-product, this neighborhood (referred 

to as “region of …t” henceforward in the paper) provides information on the quality of the model 

near x. This pricing method is applicable to multi-dimensional pricing problem. For example, in 

a two dimensional case, the region of …t may take the shape of a rectangle where the side along 

the dimension of good model …t is longer to include more distant observations to improve e¢ ciency, 

while the side along the dimension of poor model …t is shorter to utilize only nearby observations to 

achieve robustness. The intuition will be formalized in section 2. 

The intuition of the proposed pricing method relates to Hansen and Jagannathan (1991) and 

Hansen and Jagannathan (1997). 

These two papers show how security market data restrict the 

admissible region for means and standard deviations of intertemporal marginal rates of substitution 

(IMRS) which can be used to assess model speci…cation. 

Speci…cally, Hansen and Jagannathan 

(1991) calculate the volatility bound which is the greatest lower bound on the standard deviation 

of IMRS to price the assets. This bound on the variability of IMRS has a natural connection to 

the variability of parameter estimates in this paper. Here, the parameter (x) is constant within a 

region of …t surrounding x but may di¤er across di¤erent regions of …t. When the model quality is 

poor, regions of …t tend to be smaller to allow greater parameter variability across di¤erent regions 

of the state variables in order to price the assets adequately. This operationalizes the Hansen and 

Jagannathan (1991) volatility bound for investors who know their model is misspeci…ed but have no 

alternative at the time of decision making. 

The robust parametric pricing method can add value even if the correct model is known. For 

example, when the true model is high dimensional and does not admit closed-form pricing formula, estimations 

employing numerical procedures may introduce additional noise when computing resource 

is …nite. In this case, It may improve pricing precision by using an elegant model and explicitly 

adjusting for possible misspeci…cation using the proposed method, compared to parametric estimation 

using the true but complicated model. This echoes the “maxim of parsimony” in Ploberger 

and Phillips (2003) and is consistent with, for example, the widespread practice of using the Black- 

Merton-Scholes option pricing model even when evidence suggests possible misspeci…cation. Section 

3

3.3 illustrates this point using simulation under a realistic setting of Treasury option pricing. 

We then apply the robust parametric pricing method to the pricing of Treasury options traded on 

the CBOT. In both in-sample analysis and out-of-sample performance, the robust parametric method 

consistently performs better than the nonparametric method and the parametric method based on 

models in which the short rate follows an a¢ ne term structure model. This suggests that option 

pricing formulas based on such models are misspeci…ed, though they still contain useful information 

so that the robust parametric prices perform better than nonparametric prices. Analysis of the region 

of …t indicates that these option pricing formulas have poor …t along the dimensions of short rate 

and bond maturity but provide useful economic restriction along the dimension of option maturity. 

Such information provided by the region of …t facilitates future development of asset pricing models. 

Speci…cally, it suggests that the cheapest to deliver (CTD) practice in the CBOT Treasury options 

market is an important source of model misspeci…cation which is typically ignored in bond option 

pricing formulas based on term structure models. Jordan and Kuipers (1997) document an interesting 

event where CTD a¤ected the pricing of those Treasuries used in futures delivery. Results in this 

paper suggest that CTD is also an important feature in day-to-day Treasury options pricing. 

In addition to being an important issue in asset pricing, model misspeci…cation is an important 

topic in the econometrics literature and has motivated study of speci…cation tests (e.g., Hausman 

(1978) and Gibbons, Ross, and Shanken (1989)), nonparametric estimation (e.g., Fan and Gijbels 

(1996) and Bandi and Phillips (2003)). Nonparametric estimation alleviates the concern of robustness. 

However, a misspeci…ed model may nonetheless contain useful information. Given the increasing 

popularity of multi-dimensional models, the e¢ ciency loss of nonparametrics from omitting valid 

model restrictions can be nontrivial (the “curse of dimensionality” problem illustrated previously). 

To improve e¢ ciency, nonparametric pricing can be conducted under shape restrictions implied by 

economic theory (Matzkin (1994), Aït-Sahalia and Duarte (2003)). 

There is also a literature on 

semiparametric estimation reviewed by Powell (1994). 

However, shape restriction and semiparametric 

estimation apply only to selected classes of models so far. Further, they do not achieve the 

parametric rate of convergence and may in some cases lose their robustness when, for example, the 

shape restriction is misspeci…ed. Gozalo and Linton (2000) propose to replace the local polynomial 

in nonparametric estimation with an economic model and show that the resulting estimator is con- 

4

sistent and retains the nonparametric rate of convergence. This paper builds on their insight and 

shows that incorporating the economic restrictions of a model also improves estimation e¢ ciency 

towards that of the parametric rate when the model quality improves, hence constituting a continuous 

middle ground between parametric and nonparametric estimations. This paper focuses on the 

estimation of asset price, which is the conditional expectation function of discounted future payo¤ 

given current state. In the context of likelihood estimation, quasi-maximum likelihood estimator 

(White (1982)) and local likelihood estimator (Tibshirani and Hastie (1987)) have been proposed to 

address misspeci…cation. When the model is correctly speci…ed, the maximum likelihood estimator 

is optimal under fairly general conditions (e.g., Newey and McFadden (1994)). When the model is 

misspeci…ed, the quasi-maximum likelihood estimator minimizes the Kullback-Leibler Information 

Criterion (KLIC) which is the distance between the misspeci…ed model and the true data-generating 

process measured by likelihood ratio. However, minimal distance measured by likelihood ratio does 

not translate into minimal distance in price (i.e., conditional expectation function) if the model is 

misspeci…ed. This also applies to the local likelihood estimator. 

This paper is organized as follows. 

Section 2 details the proposed robust parametric pricing 

method and its properties. Section 3 uses simulation to demonstrate its performance. Section 

4 studies the pricing of Treasury options traded on CBOT using the robust parametric method. 

Section 5 concludes. The appendix contains the proofs and collects the various Treasury options and 

Treasury futures pricing formulas used in the simulation and empirical analysis. 

2. Asset pricing with misspeci…ed models 

Consider an asset whose price is P (X) where X is a d-dimensional state variable. We assume an 

investor has an economic model which implies a possibly misspeci…ed pricing formula f (X; ). is a 

p-dimensional parameter. The data consist of observations fx i ; y i g n i=1 where y i = P (x i ) + " i . " has 

zero mean and can capture the market microstructure e¤ects (see Amihud, Mendelson, and Pedersen 

(2005) for a recent review) or sampling errors. 

As motivated in the introduction, a misspeci…ed model f (X; ) can be turned into a true model 

5

if there exists a function (X) such that 

P (X) = f (X; (X)) (2.1) 

for all X. For example, the potentially misspeci…ed Black-Merton-Scholes option pricing formula can 

be used to …t option prices over di¤erent strikes by using the volatility smile curve which corresponds 

to (X). Correct speci…cation is equivalent to (X) being constant. Assuming continuity, a Taylor 

expansion implies that for X near x, 

 

P (X) = f (X; (x)) + b 1 (x) (X x) + (X x) T b 2 (x) (X x) + o kX xk 2 (2.2) 

i.e., f (X; (x)) approximates P (X) at X near x. Therefore, we propose to estimate (x) using 

observations in a neighborhood of x, 

b (x) = argmin 

 

kx i 

X 

xkh 

[y i f (x i ; )] 2 (2.3) 

The reason we include observations at X 6= x in the presence of misspeci…cation is that the 

additional observations likely reduce estimation noise as long as the misspeci…cation is not severe. 

This creates a trade-o¤ between estimation e¢ ciency and robustness which is represented in the choice 

of h in (2.3). We will refer to h as “region of …t” in this paper. When the model misspeci…cation 

is minor, one can a¤ord to use a larger region of …t to improve e¢ ciency. On the contrary, if model 

misspeci…cation is severe, one might want to use a smaller region of …t to ensure robustness. We will 

discuss the optimal choice of region of …t shortly. For now, assuming an estimate b (x) is obtained 

using the optimal region of …t, we estimate P (X = x) by 

bP (X = x) = f 

 

x; b 

(x) : 

The (infeasible) optimal choice of region of …t, denoted h , can be determined by minimizing the 

integrated mean squared pricing error 

h 

h = argmin E P (X) 

h 

f 

 

X; b (X)i 2 

: (2.4) 

6

Equation (2.4) cannot be directly applied because the true expectation is unknown. 

In this 

paper, we follow a method similar to the crossvalidation in nonparametric bandwidth choice. The 

crossvalidation procedure is asymptotically optimal with respect to the criterion function in (2.4), see 

Härdle and Marron (1985) and Härdle, Hall, and Marron (1988). 4 

Speci…cally, the crossvalidation 

method involves two steps. First, for a given candidate h, we obtain a …rst-step estimate b i;h (x i ) 

of (x i ) using all observations less than h away from x i except x i itself, 5 

b i;h (x i ) = argmin 

 

0 0. The lower bound in the order of n 1=(4+d) is the nonparametric rate of bandwidth 

choice. The upper bound, when ! is close to zero, is allowed to decrease at a very slow rate in the 

case of a good model. The propositions in this paper will be proved for the feasible choice of region 

of …t b h instead of for the infeasible h . In general, b h depends on the sample size n. However, the 

dependence is not made explicit to simplify notations. 

Proposition 1 (Consistency) Under Assumptions 1-5, whether or not the model f is correctly spec- 

4 There is a large statistics literature on choosing the optimal smoothing parameter h. See Härdle and Linton (1994) 

for a review. 

5 If x i itself is included in the crossvalidation, it will result in a mechanical downward bias in the h estimator because 

a perfect …t is possible by choosing a very small region of …t so that only x i is included to …t itself. 

7

i…ed, when n ! 1, we have 

b 

p 

(x) ! (x) 

 

f x; b p! 

(x) 

P (x) 

if b h n!1 ! 0 and n b h d n!1 ! 1. 

Given the consistency, we next study the asymptotic distribution of the price estimate. 

The 

asymptotic distribution of b (x) varies with the quality of the model. (2.2) implies that any model 

can always locally match the level of the true pricing formula. Therefore, in this paper, model quality 

is measured by the mismatch between the true model and f (X; (x)) when the state variable X 

moves away from x, which relates to how well the derivatives of f (X; (x)) match those of the 

true model. f (X; (x)) is capable of matching the true model at X far from x if its derivatives 

approximate those of the true model well. We say a model matches the true model up to its 2k-th 

derivative if, for any x, (using univariate notation for simplicity) 

 

P (X) = f (X; (x)) + b 2k+1 (x) (X x) 2k+1 + b 2k+2 (x) (X x) 2k+2 + o kX xk 2k+2 (2.8) 

for X near x. Let n x;h denote the number of observations less than h away from x. When X is 

d-dimensional, the number of observation is in the order of 

n x;h = O p 

nh d (2.9) 

when n ! 1 and h ! 0. When the model …ts reality well at state variables away from x, we can 

a¤ord to use a larger region of …t h. This uses more observations and lowers estimation noise. This 

intuition is formalized in proposition 2 below. 

Proposition 2 (Bias-variance trade-o¤ ) Under Assumptions 1-5, if the model f matches the true 

model up to its 2k-th derivative as in (2.8) for some k 0, when n ! 1, b h ! 0 and n b h d ! 1, 

 

Bias b (x) 

 

Var b (x) 

 

= O bh 2k+2 + n 1 b h 

d 

(2.10) 

 

= O n 1 b h 

d 

: 

8

This proposition shows the trade-o¤ between estimation e¢ ciency and robustness. When the 

region of …t b h is larger, more observations are used which results in lower variance in the estimate. 

However, if the model is misspeci…ed (i.e., k is …nite), increasing the region of …t can lead to a larger 

bias. When the model quality improves (k increases), the bias becomes smaller. In the limit when 

the model is correctly speci…ed, k ! 1, both bias and variance inversely relate to the region of …t 

hence the optimal choice is to use all observations as in parametric estimation. The next proposition 

shows the estimator will, depending on model quality, automatically select an appropriate region of 

…t b h to balance e¢ ciency and robustness. 

Proposition 3 (Model quality) Under Assumptions 1-5, when the model f matches the true model 

up to its 2k-th derivative as in (2.8) for some k 0, 

b h 

1 

= O p 

n 1=(4+4k+d) (2.11) 

 

P (x) = f x; b 

(x) + O p 

n (2+2k)=(4+4k+d) 

Note that n (2+2k)=(4+4k+d) ! n 1=2 when k ! 1. 

When k = 0 (i.e., if the model can only locally match the level but none of the variations of the 

true model with respect to the state), the estimator automatically achieves the nonparametric rate 

of convergence n 2=(4+d) . 6 

When the model gives a better …t in the sense that k increases, the rate of convergence of the 

proposed method automatically improves towards that of the parametric rate n 1=2 . Therefore, a 

continuous middle ground between nonparametric and parametric estimation is achieved depending 

on the quality of the model. The e¢ ciency gain comes from a better economic model. When k 

increases, (2.11) implies that the region of …t b h decreases at a slower rate. Recall that (2.7) implies 

an upper bound n ! for the region of …t. When the model is so good that the region of …t implied 

by (2.11) exceeds the upper bound, further e¢ ciency gains hence full parametric rate of convergence 

cannot be achieved. This e¢ ciency loss relative to the full parametric rate of convergence is necessary 

because we need h ! 0 to ensure robustness in case the model is misspeci…ed. However, ! can be 

6 See Fan (1992) on the nonparametric rate of convergence. This is intuitive because standard nonparametric 

estimators (e.g., Nadaraya-Watson kernel estimator) do not place any restrictions on how the true model varies with 

the state variable to achieve robustness. 

9

made arbitrarily small to make the rate of convergence arbitrarily close to the parametric rate in 

the case of a good model. Further, if one views most models as reasonable approximations (i.e., 

misspeci…ed) rather than literal descriptions of the reality, this e¢ ciency loss associated with ! > 0 

is likely a small price to pay in practice to ensure robustness. 

This improved e¢ ciency is achieved without introducing additional parameters. This contrasts 

with the Taylor expansion used in local polynomial estimators (see Fan and Gijbels (1996)) in which 

smaller bias can be achieved by using a higher-order polynomial to approximate the true model. 

However, this leads to increased variance due to increased number of parameters. 

For example, 

going from a local linear model to a local quadratic model can double the asymptotic variance 

for typical kernels (Table 3.3 in Fan and Gijbels (1996)). On the contrary, the improved rate of 

convergence in this paper comes from a better economic model. 

(2.3) weighs observations equally for ease of illustration and does not explicitly discuss the possibility 

of weighting the observations as in, for example, GMM estimation (Hansen (1982)) or LOWESS 

nonparametric estimation (Fan and Gijbels (1996)). This is similar to using a uniform kernel in nonparametric 

estimation where it is known that the choice of kernel is not crucial (Härdle and Linton 

(1994)). Equal weighting is also technically convenient. When the model is correct and the sampling 

errors are homoskedastic, we would like the estimator to use all observations with equal weight just 

like the parametric nonlinear least-squares estimation. To achieve this using a kernel with unbounded 

support (such as normal), h ! 1 is required which is inconvenient in numerical implementation. 

However, weighting implicitly occurs in this paper through the region of …t –observations outside of 

the region of …t receive zero weight. 

2.1. Sensitivity analysis 

Sometimes one may be interested in estimating derivatives of the pricing formula. Examples include 

the various Greek letters of the option pricing formula or other sensitivity analyses. Recall that (X) 

satis…es 

P (X) = f (X; (X)) 

10

for any X. Taking derivative with respect to the state variable implies 

P 0 (X) = f X (X; (X)) + f (X; (X)) 0 (X) : 

To simplify notation, f X is used to denote 

@ 

@X f, similarly for f . 

In order to estimate P 0 (x), 0 (x) needs to be estimated. Otherwise there is a bias if f X (X; (X)) 

alone is used to estimate sensitivity when the model is misspeci…ed (when the model is correctly 

speci…ed, there is no bias because (X) is a constant). To estimate the …rst derivative of (X), we 

can use the following augmented model 

f (X; 0 (x) + 1 (x) (X 

x)) 

to approximate the true model at X near x and the estimation can then proceed in the same way 

 

as previously discussed using this augmented model. When the parameter estimates b0 (x) ; b 1 (x) 

are obtained, the derivative of the true model P 0 (X) at X = x is estimated as 

f X 

x; b 

0 (x) + f x; b 

0 (x) b 1 (x) : 

The estimation of higher-order derivative is similar. 

Counterparts to Proposition 1 – 3 exist 

for derivative estimation. These propositions and their proofs are very similar to Proposition 1 –3, 

except that the nonparametric rate of convergence is slightly modi…ed to re‡ect derivative estimation 

which is standard in the nonparametric literature. These propositions and proofs are omitted for 

brevity and are available from the authors upon request. 

2.2. Multivariate pricing models 

The robust parametric pricing method is well suited for multivariate models. In fact, proposition 1 - 3 

are derived for the general case of d-dimensional state variables. Contrary to nonparametric methods, 

there need not be a “curse of dimensionality”problem as long as an investor has a good model. As 

shown in Proposition 3, when the model quality improves (i.e., k ! 1 in the proposition), the 

estimation e¢ ciency approaches that of the parametric rate which is not a¤ected by dimensionality. 

11

In this section, we show that the region of …t can be re…ned in a multivariate model to re‡ect 

the possibility that a model may …t well along certain dimensions of the state variables but …t 

poorly along other dimensions. In (2.3), the estimator is obtained using observations x i satisfying 

kx i 

xk h. I.e., we consider the pricing formula to have a good …t for observations less than h 

away from x. When the state variable is multi-dimensional, we can apply a separate region of …t for 

each dimension. 

We illustrate this using a two dimensional example where the state variable can be written as 

x = x (1) ; x (2) . (2.3) can be modi…ed so that the parameters are estimated from 

b (x) = argmin 

 

X 

 

 

x (1) 

i x (1) h (1) 

 

 

x (2) 

i x (2) h (2) 

[y i f (x i ; )] 2 : (2.12) 

The region of …t now takes the shape of a rectangle. The interpretation is that the pricing formula 

is considered to …t well for those observations that are less than h (1) away from x in the …rst 

dimension and are less than h (2) away from x in the second dimension. 

This re…nement can be 

used to re‡ect di¤erent scales of the state variables – e.g., measured in di¤erent currencies, or to 

re‡ect the di¤erent degrees of misspeci…cation along various model dimensions – those with less 

misspeci…cation are associated with larger regions of …t. The estimation then proceeds in the same 

way and the conclusions in Proposition 1 - 3 remain the same. 

2.3. Numerical implementation 

The estimation in (2.3) and (2.5) involves nonlinear least squares which is programmed in many 

statistical software packages. Nonlinear least squares estimation is fairly quick because it is typically 

implemented as iterated linear least squares, see Greene (1997). Nonetheless, when the dataset has 

a large number of observations and when the state variable has many dimensions, there is room for 

faster implementation of the proposed pricing method. 

A potential bottleneck of the robust parametric pricing method lies in the crossvalidation step 

(2.5) for …nding the optimal region of …t which, in principal, is estimated for all possible candidates 

of region of …t h at all observations x i to evaluate the model quality at various regions. However, this 

12

is not necessary –evaluations can be done at fewer h and x i to trade e¢ ciency gain for computational 

cost savings. 

First, one can restrict the choice of h by searching over the following grid 

h 1 = n 1=(4+d) ; h 2 = h 1 + ; h 3 = h 1 + 2; ; h m = n ! (2.13) 

where ! is a small positive number discussed in (2.7). The grid size is = (h m h 1 ) = (m 1). m 

can be increased when additional computational resource is available. The downside from searching 

over fewer candidates is that the chosen region of …t may deviate from the optimal choice b h in 

proposition 3, which can reduce (though not eliminate) the e¢ ciency gain associated with a good 

model. However, the price estimator remains consistent and achieves at least the nonparametric rate 

of convergence. For a multivariate model discussed in 2.2, the grid for region of …t can be applied 

separately to each dimension. 

Next, one can restrict the number of observations x i at which (2.5) is evaluated whose output 

is then used in (2.6) as a sample analog of (2.4). For the purpose of consistently estimating the 

expectation in (2.4) using its sample analog, the number of evaluations should increase asymptotically 

towards in…nity though the rate of increase can be lower than that of the sample size. This can 

be implemented, for example, by estimating (2.5) at randomly selected n v observations for some 

0 < v 1. When v is bigger, the expectation in (2.4) is estimated more precisely which results in 

a more precise choice of region of …t at the cost of computational resource. When (2.5) is evaluated 

at selected observations (i.e., v < 1), (2.6) needs to be adjusted to include only these observations. 

As long as v > 0, the price estimator remains consistent irrespective of misspeci…cation. 

3. Simulation –Treasury options pricing 

This section uses simulation to illustrate the proposed robust parametric pricing method in realistic 

samples, comparing its performance to parametric and nonparametric methods. When a true model 

is complicated, we illustrate that it may improve matters to use a simpler model and explicitly adjust 

for misspeci…cation using the robust parametric method. 

We illustrate in the context of pricing Treasury options. 

Speci…cally, we price call options 

13

C (; T; X) on Treasury zero-coupon bonds, where is time to option expiration, T is bond maturity 

at option expiration, and X includes other state variables such as the prevailing interest rate, 

the strike price, etc. This is a multivariate simulation example in that the option pricing formula 

needs to be estimated along the dimensions of option maturity, underlying bond maturity, and other 

state variables. 

We assume that the true data generating process follows the Cox, Ingersoll, and Ross (1985) 

model (CIR model) under the risk-neutral probability 

dr t = k ( r t ) dt + p r t dW t (3.1) 

where r t is the instantaneous short rate at time t. 

The short rate mean reverts to its long-run 

mean at a speed governed by k. The standard Brownian motion W drives the random evolution 

of the short rate. The instantaneous volatility of the short rate is determined by the parameter 

and the square root of the short rate (hence the process is also known as the square root process). 

Under the CIR model, the Treasury zero-coupon bond option has a closed-form expression (detailed 

in the appendix). Alternatively, the Treasury option price under CIR model can be computed via 

numerical integration using the method in Du¢ e, Pan, and Singleton (2000), which we later use in the 

simulation to quantify the performance of parametric estimator when the true model is complicated 

so that closed-form pricing formula is unavailable and numerical method is used instead to obtain 

prices. 

To implement the robust parametric method, we assume that an investor is aware that the Vasicek 

(1977) model delivers a closed-form Treasury option pricing formula (detailed in the appendix), but 

this same investor has yet to adopt the CIR model. In the Vasicek model, the evolution of the short 

rate under the risk-neutral probability is assumed to follow 

dr t = k ( r t ) dt + dW t : (3.2) 

That this investor uses the Vasicek model but not the CIR model may happen if this investor 

has studied Vasicek (1977) but the Cox, Ingersoll, and Ross (1985) paper is either not published 

yet or has not caught this investor’s attention when he faces the pricing problem. 

However, the 

14

investor is aware that the Vasicek option pricing formula may be misspeci…ed and addresses such 

misspeci…cation using the robust parametric method. 

We compare the robust parametric method to four other estimation methods: (i) parametric estimation 

using the CIR model (parametric estimation using correct model); (ii) parametric estimation 

using the Vasicek model (parametric estimation using misspeci…ed model); (iii) nonparametric estimation; 

(iv) parametric estimation using the correct CIR model but applying numerical integration 

method to obtain option prices instead of relying on the closed-form CIR option pricing formula, 

which allows us to compare estimation using a correct but complicated model which relies on numerical 

methods to estimation using a simpler model and explicitly adjusting for misspeci…cation using 

the robust parametric method. The estimation performance is measured by the sample analog of the 

root integrated mean squared error 

v 

u 

RIMSE t 1 n 

nX 2 

bCi C i (3.3) 

i=1 

where C b and C are, respectively, the estimated and the true Treasury option prices in each simulation. 

RIMSE captures the average goodness of …t with smaller RIMSE indicating better …t. 

The simulation is designed to re‡ect the characteristics of the Treasury option contracts traded 

on the Chicago Board of Trade (CBOT). The simulation draws 100 sample paths of short rate, 

each sample path being equivalent to 5 years of weekly observations. Such samples are commonly 

encountered in practice, see for example Du¢ e and Singleton (1997). For each sample path, Treasury 

call option prices are generated according to the CIR model for the option maturity =1, 2, 3, 6, 

9, 12, 15 months, underlying bond maturity T =2, 5, 10, 30 years. The …rst short rate is drawn 

randomly from the stationary distribution of CIR process. To simplify the summary of the simulation 

result, we consider only at-the-money options which tend to be the most liquid contracts in practice. 

As a result, we study the simulation performance of various estimators for three-dimensional state 

variables –option maturity, underlying bond maturity, and short rate. In the simulation, the true 

CIR parameters are set to the estimates in Aït-Sahalia (1999) 

k = 0:145; = 0:0732; = 0:06521 (3.4) 

15

and we add to the true option price a zero-mean normally distributed random variable whose standard 

deviation equals to one percent of the true option price to generate the observed option price in the 

simulation. This extra random variable is intended to capture market microstructure e¤ects such as 

the bid-ask bounce. At the true parameter, the bond option prices average to around $1 hence the 

pricing errors can be interpreted either as dollar pricing errors or as proportional pricing errors. 

3.1. Simulation result: parametric and nonparametric prices 

Table 1 panel A shows the performance of the various option price estimators. When an investor 

knows correctly the underlying data generating process, parametric option price estimator performs 

the best, generating an average pricing error of only 0.022 cents with …ve years of weekly observations. 

7 

However, the accuracy of the option prices depends crucially on the validity of the model. 

When the model is misspeci…ed, the parametric prices result in an error of 4.1 cents which is about 

200 times that when the model is correct. This calls for caution in practice with parametric prices 

when the validity of the pricing model is in question. 

Nonparametric prices, on the other hand, 

do not depend on any model and avoid misspeci…cation. In the simulation, nonparametric prices 

register an average pricing error of 1.3 cents, about 70% less than the parametric prices when the 

model is wrong. 8 

However, the nonparametric prices ignore all model information (whether correct 

or not) and perform much worse than parametric prices from a correctly speci…ed model. 

3.2. Simulation result: robust parametric prices 

The pricing method proposed in this paper aims to achieve a continuous middle ground between 

parametric and nonparametric prices. Table 1 panel A shows that option prices from the robust 

parametric method have an average pricing error of 0.15 cents, about 7 times larger than that of the 

parametric prices using the correct model yet 27 times smaller than the parametric estimates from a 

wrong model. The error of 0.15 cents is also an order of magnitude smaller than the nonparametric 

errors. 

To see the intuition of improvement, let us turn to panel B in Table 1 and Figures 1 and 2. In 

7 We use nonlinear least squares to …nd the parameter estimates. 

8 We use the Nadaraya-Watson nonparametric estimator with uniform kernel and cross-validation bandwidth selection 

in the simulation, see Härdle and Linton (1994) for more details. 

16

panel B of Table 1, the regions of …t along the option maturity and bond maturity dimensions are 

both zero, indicating that the wrong model provides a poor …t of the true bond option prices along 

these two dimensions. 9 Figures 1 and 2 further illustrate this. Figure 1 plots the true and estimated 

option prices at various option maturities. 

The robust parametric price estimate at one month 

maturity, which is estimated using observations whose option maturity is around one month, …ts 

the true one-month option price almost exactly. This con…rms that even a wrong model can provide 

good local …t (see (2.1)). However, the local …t does not extrapolate well to option maturities slightly 

di¤erent from one month. 10 

Similarly around the other option maturities shown in Figure 1. This 

indicates severe model misspeci…cation along the dimension of option maturity. Figure 2 illustrates 

that the Vasicek option price …ts the CIR option price poorly along the dimension of bond maturity, 

too. 

That the Vasicek option price …ts the CIR option price poorly along option maturity and bond 

maturity is the reason why the robust parametric pricing method improves over parametric estimation 

using a misspeci…ed model. 

When the model …ts poorly along these two dimensions, the robust 

parametric method sets the regions of …t to zero along both dimensions and conducts estimation using 

only observations with the same bond maturity and option maturity to address misspeci…cation. 

The result is di¤erent along the dimension of short rate. Panel B of Table 1 shows that the region 

of …t is 0.026 along this dimension. I.e., if one is estimating the option price at short rate 7%, the 

robust parametric estimator will use all observations whose short rates are between 4.4% and 9.6%. 

Figure 3 con…rms that the Vasicek price estimates can be extrapolated to nearby short rates and 

still …t the true option prices reasonably well (the two option price curves almost overlap). This 

allows the robust parametric method to enlarge the region of …t to include more observations which 

improves the estimation e¢ ciency. This is the intuition why the proposed robust parametric method 

performs better than nonparametric method –it retains valid model restrictions. 

9 To be exact, the region of …t for option maturity averages to 0.01. However, because the observations come in 

weekly and the interval between successive observations of option maturity is at least 1=52 0:02, the region of …t for 

option maturity is essentially zero. 

10 Using the notation in section 2, extrapolation here refers to using f (X; (x)) to estimate option prices at state 

variable X di¤erent from x. 

17

3.3. Simulation result: comparison with numerical methods 

The proposed robust parametric method can add value even when the correct model is known. A 

true model is likely complicated. 

A common situation is that the closed-form pricing formula is 

unavailable as a result of the complexity. For example, many continuous-time term structure models 

do not render closed-form bond option pricing formula. The Vasicek and CIR models used in the 

simulation, along with a handful of other models, constitute the exception. Sometimes, numerical 

methods can be used to obtain the option prices. For example, Du¢ e, Pan, and Singleton (2000) 

show that option prices can be obtained via numerical integration of characteristic functions. 

In this section, we compare the performance of the proposed pricing method to the performance of 

parametric estimation which uses true model and numerical method to get option prices. Speci…cally, 

the robust parametric estimator still uses the closed-form Vasicek option pricing formula which is 

misspeci…ed. On the contrary, the parametric estimator uses the true CIR model but pretends that 

this is a model so complicated that closed-form option pricing formula is unavailable. 

To control the magnitude of the numerical error in this study, we …rst model numerical errors by 

C NUM = C (1 + ") 

where C is the true option price from the closed-form CIR pricing formula. " is set to be a uniformly 

distributed random variable over [ 

!; !]. C NUM is assumed to be the option prices obtained from 

numerical methods. In this simulation, we do not actually use a numerical method but rather we 

start from the closed-form option price and let ! proxy the degree of numerical error. When ! = 0, 

numerical error disappears and we return to the case of parametric estimation using the closed-form 

formula. A larger ! indicates larger numerical error. We use ! = 0:01%, 0:1%, 0:2%, 0:3%, 0:5%, and 

1% in the simulation and the results are shown in panel C of Table 1. The proposed robust parametric 

method using the misspeci…ed Vasicek model is comparable in performance to parametric estimation 

using the true model when the numerical error is between 0.2% and 0.3%. 

This is remarkable 

because Vasicek option prices are grossly misspeci…ed relative to CIR option prices. 

Parametric 

estimation using the Vasicek model results in an error that is 200 times that when the model is 

correctly speci…ed (see panel A). Panel B further shows that the Vasicek model does not …t CIR 

18

model along the dimensions of bond maturity and option maturity at all. Nonetheless, adjusting for 

misspeci…cation using the proposed robust parametric method improves the estimation performance 

to the equivalent of parametric estimation using true model with a numerical error of around 0.25%. 

Next, to see what the magnitude of numerical error can be in practice, we follow Du¢ e, Pan, 

and Singleton (2000) and generate option prices using the correct CIR model by actual numerical 

integration. The estimation RIMSE is shown in panel D of Table 1. The result is comparable to 

the 1% error in panel C. In practice, numerical integration precision can be improved at the cost 

of longer computing time. Therefore, the result in panel D should be taken with caution. However, 

even with a relatively tractable model like CIR, there are already non-trivial issues with numerical 

integration. For example, Carr and Madan (1999) point out that poor numerical precision can result 

from the highly oscillatory nature of the characteristic function in the integrand. When the true 

model becomes more complicated, the numerical errors are likely more di¢ cult to understand and 

control, especially when the option prices need to be evaluated at all parameter values searched 

in the estimation procedure. This implies that sometimes it may be desirable to use a simple but 

misspeci…ed model and explicitly adjust for misspeci…cation using the proposed robust parametric 

method. 

4. Empirical application –Treasury options pricing 

We next apply the robust parametric method to the pricing of Treasury options traded on CBOT, 

examine its pricing performance both in terms of in-sample …t and out-of-sample performance, and 

see what if any model misspeci…cation it can help investors infer. 

We collect weekly call option closing price data from CBOT. The sample period is from May 

1990 to December 2006. CBOT lists options on 2-, 5-, 10-, and 30-year Treasuries. These options are 

more precisely options on Treasury futures. However, those option maturities with the most trading 

volume (March, June, September, and December) coincide with futures expiration. Therefore, upon 

option exercise, the delivery is essentially made in the underlying Treasuries. We focus on the option 

maturities of March, June, September, and December and will refer to the options as Treasury 

options for simplicity. The 2-year option does not have much trading volume and is excluded from 

the analysis. To reduce data error, we eliminate those observations where the recorded option price is 

19

less than the intrinsic value, i.e., if C < max(F 

K; 0) where C, F , and K are the observed Treasury 

call option price, observed Treasury futures price, and option strike, respectively. Further, for each 

option contract, we use only data for the at-the-money contract (contract whose F is closest to K) 

which tends to have the most trading volume. There are a few instances where CBOT supplies a 

closing option price but indicates a trading volume of zero. Such observations are eliminated. 

As in section 3, we apply the robust parametric pricing method using the possibly misspeci…ed 

Vasicek (1977) model which allows for closed-form bond option pricing formula (see appendix F.2). 11 

The Vasicek (1977) option pricing formula assumes that a zero-coupon bond underlies the option. 

This di¤ers from the cheapest to deliver practice of CBOT listed options where the delivery may not 

be in the form of a zero coupon bond. Because we do not have information on the cheapest Treasury 

for delivery, we use the following procedure to adjust for unknown coupon of the bond issue used 

for delivery. Speci…cally, we convert the delivery bond into a zero coupon bond by assuming that 

all the coupons are paid at bond maturity. This assumption ignores the time value between coupon 

payment and bond maturity. It is an imperfect way to model the cheapest to deliver practice and 

we will discuss more on this issue later. Now the problem of unknown coupon size is translated to 

the new problem of unknown par value which we can back out using the observed Treasury futures 

price from CBOT. Speci…cally, let M denote the unknown par value, then M can be computed from 

M = 

F 

F (; T; r) 

where F is the observed CBOT Treasury futures price, F (; T; r) is the Vasicek (1977) implied futures 

price on a zero coupon bond with par=$1 (see appendix F.2). 12 This allows us to construct a pricing 

formula for the option 

C adj (; T; r; K) = M C(; T; r; K M ) (4.1) 

where C adj is the call option price adjusted for the cheapest to delivery practice, C is the Vasicek 

(1977) pricing formula for call option on a Treasury zero coupon bond with $1 par, is the option 

11 We have alternatively estimated a model in which the short rate follows the Cox, Ingersoll, and Ross (1985) process. 

The result is similar. It is suppressed for brevity and available from the authors upon request. 

12 The CBOT Treasury futures price data are from Datastream. 

20

maturity, T is the bond maturity, r is the short rate which is measured by one month Treasury bill 

rate, and K is the option strike price. 

We compare both the in-sample …t and out-of-sample performance of three pricing methods: the 

robust parametric method proposed in this paper, the parametric method, and the nonparametric 

method. 13 

4.1. Misspeci…cation of Treasury option pricing models 

We use the root integrated mean squared error (RIMSE) de…ned in (3.3) to measure the in-sample 

performance of various estimators. The result is shown in panel A of Table 2. In the sample, the 

model is so misspeci…ed that the nonparametric prices which completely ignore the model provide 

better …t than parametric prices. 

Nonetheless, the model remains useful because the proposed 

robust parametric method which uses the same model but adjusts for possible misspeci…cation does 

better than either parametric or nonparametric methods. The robust parametric pricing method 

also produces the highest R-square in the regression of observed option prices on …tted option prices 

–90.2% versus 49.8% and 74.4% from parametric and nonparametric estimators, respectively. The 

improvement in R-square is consistent with the scatter plots shown in Figure 4. 

The intuition of the improved …t is the same as that observed in section 3: the robust parametric 

method imposes the model only in the region where the model provides a good approximation of 

reality and discards the model elsewhere. Figure 5 shows the RIMSE for various regions of …t along 

the dimensions of option maturity, bond maturity, and short rate. The robust parametric method 

selects a region of …t separately for each dimension to minimize the RIMSE (see section 2.2). In the 

sample period from May 1990 to December 2006, the Vasicek (1977) model does not …t bond option 

prices well along the dimensions of bond maturity and short rate. Therefore, regions of …t are set to 

minimum along these two dimensions –0 for bond maturity and 0.5% for short rate (we set minimum 

possible region of …t along the short rate to 0.5% to make sure there are su¢ cient observations in the 

estimation). This implies that, to estimate option prices at bond maturity T and short rate r, the 

robust parametric method uses only those observations at which the options are written on a T -year 

13 We use nonlinear least squares in the parametric and robust parametric estimations. We use the Nadaraya-Watson 

nonparametric estimator with uniform kernel and cross-validation bandwidth selection, see Härdle and Linton (1994) 

for more details. 

21

ond and the short rates are within [r 

0:005; r + 0:005] to minimize the impact of misspeci…cation, 

which improves its performance relative to the parametric method. 

The improvement of the robust parametric method over nonparametric method is due to the 

model providing some useful restrictions along the dimension of option maturity. In Figure 5, the 

RIMSE bottoms out when the region of …t is set to 3 weeks for option maturity. This implies that the 

Vasicek option pricing formula provides a good approximation for observations with adjacent option 

maturity –when estimating option prices at option maturity , the robust parametric method will 

use observations whose option maturities are within 3 weeks of . 14 

The information provided by the regions of …t along various dimensions of the state variable 

can be used to triangulate model misspeci…cation which is useful for future development of asset 

pricing models. In this case, the model …ts well along the dimension of option maturity but not 

along bond maturity and short rate. Pinpointing the exact cause of bond options misspeci…cation 

requires a separate study, though the evidence is suggestive that the cheapest-to-deliver (CTD) 

practice associated with the CBOT Treasury futures/options plays a role. CTD refers to the right of 

the short party to deliver any Treasuries designated eligible by CBOT. For example, for the 10 year 

contracts, deliverable grades include US Treasury notes maturing at least 6 1/2 years, but no more 

than 10 years, from the …rst day of the delivery month. To address the fact that Treasuries vary 

in their coupon, maturity, and other features, CBOT uses a system known as the conversion factor 

to equalize various bonds. According to CBOT, the conversion factor is the price of the delivered 

note ($1 par value) to yield 6 percent and the invoice price equals the futures settlement price times 

the conversion factor plus accrued interest. The conversion system usually makes some bonds less 

costly to deliver than others, which is not captured by the typical bond option pricing formula based 

on term structure models. 

The actual cheapest-to-deliver bond varies across contracts involving 

di¤erent bond maturities and across di¤erent interest rate environments (see, for example, Kane 

and Marcus (1984) and Livingston (1987)) which is consistent with the misspeci…cation along the 

dimensions of bond maturity and short rate indicated by the regions of …t. The region of …t for option 

maturity, on the contrary, shows good …t up to 3 weeks. Observations less than 3 weeks apart are 

14 The optimal region of …t along the dimension of option maturity is 2 weeks if the Cox, Ingersoll, and Ross (1985) 

process is used instead of the Vasicek (1977) process to model the short rate. The optimal regions of …t along bond 

maturity and short rate remain the same. This suggests better …t of Vasicek (1977) process for the purpose of modeling 

CBOT Treasury option prices. 

22

likely consecutive weekly observations of the same contract for which the cheapest-to-deliver bonds 

are likely similar or even identical. Therefore, the evidence suggests that the cheapest to deliver 

practice is an important source of misspeci…cation for Treasury option pricing which is ignored by 

typical bond option pricing formulas based on term structure models. 

4.2. Out-of-sample performance 

To con…rm that the improved …t is not due to over…tting and can be extrapolated out of the sample, 

panel B of Table 2 shows the out-of-sample comparison of the proposed robust parametric method 

to parametric and nonparametric methods. Speci…cally, model parameters are estimated using …ve 

years of weekly observations which are then used to compute RIMSE and regression R-square in 

the subsequent year. Because the sample period starts in May 1990, the …rst year of out-of-sample 

comparison is 1996. Panel B shows the RIMSE and the R-square in the regression of observed option 

prices on predicted option prices for each year separately. The robust parametric pricing method has 

the lowest out-of-sample error in all years. Overall, the robust parametric method has a reduction 

of 46.6 and 33.9 percent in RIMSE, and an increase of 39.6 and 16.5 percentage points in R-square 

relative to parametric and nonparametric methods, respectively. 

5. Conclusion 

Misspeci…ed pricing formula confronts most investors. 

This paper proposes a robust parametric 

pricing method which utilizes information in a model yet explicitly controls for possible misspeci…- 

cation. The resulting price estimator is consistent irrespective of misspeci…cation and it provides a 

continuous middle ground between parametric and nonparametric estimators in terms of e¢ ciency. 

It can improve pricing precision over parametric methods when the model is misspeci…ed. Because 

it retains valid information supplied by a model, the robust parametric pricing method does not 

su¤er as much from the “curse of dimensionality” problem faced by nonparametric methods and is 

applicable to high dimensional pricing problems or sensitivity analyses. 

Model restrictions also help to alleviate the concern of over…tting. As pointed out by Campbell, 

Lo, and MacKinlay (1997) (page 524), “... perhaps the most e¤ective means of reducing the impact 

of over…tting and data-snooping is to impose some discipline on the speci…cation search by a priori 

23

theoretical considerations.” The estimator proposed in this paper does exactly that – it confronts 

the data with an a priori model. This is con…rmed by the out-of-sample performance in section 4.2. 

Using an approximate (i.e., misspeci…ed) model may also provide other advantages. For example, 

the true model can be complicated and it may sometimes be preferable to use a simple yet misspeci…ed 

model, as pointed out by Fiske and Taylor (1991) (page 13), “... 

People adopt strategies that 

simplify complex problems; the strategies may not be normatively correct or produce normatively 

correct answers, but they emphasize e¢ ciency.” Interestingly, one of the simulations shows that 

applying the proposed estimator on a good parsimonious model and explicitly adjusting for possible 

misspeci…cation can sometimes do better than fully parametric estimation using a complicated model 

even if the complicated model is correctly speci…ed. 

This echoes the “maxim of parsimony” in 

Ploberger and Phillips (2003) and opens up additional application areas for the estimator when the 

true model is complicated and a good parsimonious (though misspeci…ed) model is available. 

The robust parametric pricing method can have interesting implications on asset pricing equilibrium 

based on bounded rationality. According to Simon (2008), “The term ‘bounded rationality’ 

is used to designate rational choice that takes into account the cognitive limitations of the decision 

maker –limitations of both knowledge and computational capacity. Bounded rationality is a central 

theme in the behavioral approach to economics, which is deeply concerned with the ways in which 

the actual decision-making process in‡uences the decisions that are reached.”If using a simple model 

and explicitly adjusting for possible misspeci…cation can achieve good results and if the economy is 

populated with investors with su¢ cient sophistication to act accordingly, it can mitigate the e¤ect 

of cognitive costs associated with complex reality on the equilibrium outcome. On the other hand, 

the robust parametric method itself introduces an additional layer of complexity into the economy 

(e.g., the uncertainty on if and when other investors adopt the method) and can amplify the e¤ect 

of cognitive limitation. Such implications have interesting potentials and await future studies. 

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27

Appendix: Assumptions, Proofs, Option Pricing Formulas 

A. Assumptions 

First, we collect the regularity conditions assumed in this paper. Recall that we want to estimate the pricing 

formula P (X) where X 2 R d is the state variable. We assume an investor has an economic model which 

implies a possibly misspeci…ed pricing formula f (X; ) for P (X). 2 R p . 

Assumption 1 There exists a unique function (X) such that f (X; (X)) = P (X). The range of (X) is 

in a compact set . 

When the model is correctly speci…ed, (X) is a constant. For all practical purposes, knowing f (X; (X)) 

amounts to knowing the true pricing formula P (X). 

Assumption 2 P (X) and f (X; ) are bounded and thrice-continuously di¤erentiable with respect to X and 

with bounded derivatives. 

Assumption 3 (Sample) The sample consists of independent and identically distributed observations fx i ; y i g n i=1 

where 

y i = P (x i ) + " i : 

" i is an independent random error satisfying E [" i j X = x i ] = 0, Var[" i j X = x i ] = v (x i ) > 0. v () is continuous 

and bounded. 

Assumption 4 inf X; kf (X; ) f T (X; )k > 0. For any x 2 R d and any 2 , there exists an H > 0, a 

neighborhood N of , a non-random function G (; h) continuously di¤erentiable at 2 N and 0 h H, 

and random variables Z h N (0; (h)) indexed by h where (h) is continuous at h = 0 such that 

sup 

2N;0hH 

n 1 

sup 

0hH 

n X 

i=1 

n 1=2 

f x h i ; f T x h i ; G (; h) 

= O p 

n X 

i=1 

f x h i ; " i x h 

i 

Z h 

= o p (1) 

n 1=2 

where the sequence of independent random variables x h i 

n 

i=1 satis…es x 

h 

i x h for all i. 

Other than requiring uniformity over h H, Assumption 4 is standard in large sample asymptotics (see 

Newey and McFadden (1994)). Restricting to x 

h 

i 

x h is because the model will be estimated locally 

using observations less than h away from x when the model is misspeci…ed. 

28

Let p (X) denote the probability density function of X. 

Assumption 5 p (x) > 0 for all x 2 R d , p () is twice-continuously di¤erentiable. 

We next prove the propositions. Recall that n x; b h 

denotes the number of observations less than b h away 

 

from x. When X is d-dimensional, n x; b h 

= O p n b 

h d when n ! 1, b h ! 0, and n b h d ! 1. 

B. Proof of Proposition 1 

See Theorem 1 in Gozalo and Linton (2000). 

C. Proof of Proposition 2 

Using the standard large sample asymptotics argument (see for example Newey and McFadden (1994)), 

= 

p 

nx; b h 

 

b (x) 

0 

@ 1 

n x; b h 

kx i 

X 

xk b h 

(x) 

 

F i F T i 

1 

A 

1 

n 1=2 

x; b h 

kx i 

X 

xk b h 

F i (y i 

 

f (x i ; (x))) + O p n 1=2 

x; b + b h 2k+2 

h 

(C.1) 

To simplify notation, F i f (x i ; (x)). Without model misspeci…cation (i.e., if k ! 1), this is the standard 

asymptotics result. With misspeci…cation, (C.1) is modi…ed slightly because y i 

f (x i ; (x)) does not have 

mean zero for x i 6= x. The magnitude of the bias 

E 0

D. Proof of Proposition 3 

The crossvalidation criterion function 

By (2.10) and (2.9), 

CV (h) = 1 n 

= 1 n 

+ 2 n 

nX 

i=1 

h 

y i 

f 

nX 

" 2 i + 1 n 

i=1 

 

x i ; b i 2 

i;h (x i ) 

nX 

i=1 

h 

P (x i ) 

nX 

" i 

hP (x i ) f 

i=1 

f 

 

x i ; b i 2 

i;h (x i ) 

 

x i ; b i 

i;h (x i ) : 

1 

n 

nX 

i=1 

h 

P (x i ) 

f 

 

x i ; b i 2 

i;h (x i ) = Op h 4k+4 + nh d 1 

: 

(D.1) 

We will later prove the following lemma. 

Lemma 1 Under the conditions of Proposition 3, 

1 

n 

nX 

" i 

hP (x i ) f 

i=1 

= o p 

1 

n 

nX 

i=1 

h 

P (x i ) 

 

x i ; b i 

i;h (x i ) 

f 

! 

 

x i ; b i 2 

i;h (x i ) : 

(D.2) 

Lemma 1 and (D.1) imply 

CV (h) = 1 nX 

" 2 i + O p 

h 4k+4 + nh d 1 

: (D.3) 

n 

i=1 

b h, which minimizes CV (h), satis…es 

b h = n 1=(4+4k+d) : 

It can then be calculated using (2.10) that the resulting estimation error is 

P (x) = f 

 

x; b 

(x) + O n (2+2k)=(4+4k+d) : 

30

E. Proof of Lemma 1 

Recalling P (x i ) = f (x i ; (x i )), Taylor expansion around gives 

 

P (x i ) f x i ; b 

i;h (x i ) 

 

= f T (x i ; (x i )) (x i ) b i;h (x i ) + O p (xi ) b i;h (x i ) 2 

= F T i 

= 

2 

4 

0

nX 

The …rst term 1 n 

" i T h (x i ) has a zero mean and its variance is 

i=1 

V ar 

" 

1 

n 

# 

nX 

" i T h (x i ) 

i=1 

= 1 X 

n 

n 2 V ar [" i T h (x i )] + 1 n 2 

i=1 

n X 

X 

Cov [" i T h (x i ) ; " u T h (x u )] 

i=1 u6=i 

(E.6) 

First, we bound the variance terms in (E.6). 

V ar [" i T h (x i )] = E " 2 h 

i E 

= E " 2 

i 

= O p 

 

0

Next, we bound the second term in (E.5) involving U h . Expand w (x i ; x j ) around x j = x i , 

w (x i ; x j ) = w (x i ; x i ) + w 2 (x i ; ex j ) (x j x i ) : 

where w 2 (; ) denotes derivative of w (; ) with respect to the second argument. ex j is in between x i and x j . 

From (E.2), 

w (x i ; x i ) = O p 

 

w 2 (x i ; ex j ) = O p 

 

nh d 1 

nh d 1 : 

(E.12) 

Recall the model matches P (X) up to its 2k-th derivative (see (2.8)), 

U h (x) = 

= 

0

variance bounded by (recall that each remainder term depends only on observations less than h away from x i 

so the remainder terms are independent of each other if x i and x u are more than 2h apart) 

O p 

1 

n 2 n 

h 4k+4 + nh d 1 2 

+ 

1 

n 2 n nhd h 4k+4 + nh d 1 2 

= O p 

 

h 4k+4 + nh d 1 2 

n 1 + h d 

which implies that the sum of the remainder terms in (E.5) is of order 

O p 

h 4k+4 + nh d 1 h d 1=2 : (E.16) 

That n 1 = o p h d is because the crossvalidation criterion is minimized over h that satis…es nh d ! 1. 

Combining (E.11), (E.15) and (E.16), (E.5) becomes (notice the fact that 2 jabj a 2 + b 2 ) 

1 

n 

 

= O p 

nX 

" i 

hP (x i ) f 

i=1 

 

x i ; b i 

i;h (x i ) 

nh d 

1=2 

n 1=2 + h 2k+2 + nh d 1=2 

n 1=2 + 

h 4k+4 + nh d 1 

h d 1=2 

h 4k+4 + nh d 1 h d 1=2 

= O p 

h 2k+2 n 1=2 + 

 

= O p h 2k+2 h d 1=4 

n h d 1=2 1=2 

+ h 4k+4 + nh d 1 

h d 

1=2 

= O p 

 

h 4k+4 

h d 

1=2 

+ n h d 1=2 1 

+ h 4k+4 + nh d 1 

h d 

1=2 

= O p 

 

h 4k+4 + nh d 1 h d 1=2 

which, together with (D.1), proves Lemma 1. 

F. Option pricing formula 

F.1. 

CIR model 

When the short rate follows the CIR model in (3.1), the price of a call option with maturity and strike price 

K on a T -year Treasury zero-coupon bond with par $1 is given by Cox, Ingersoll, and Ross (1985), 

C (; T; r 0 ; K) = B (r 0 ; T ) 2 2r [ () + B(T )] ; 4 

2 ; 

KB (r 0 ; ) 2 

2r [ () + ] ; 4 

2 ; 2 ()2 r 0 e 

() + 

! 

2 () 2 r 0 e 

() + B(T ) 

! 

34

where r 0 is the short rate at the time of option pricing and 2 (; n; c) denotes the cumulative probability 

distribution function of a non-central Chi-square distribution with degree of freedom n and non-centrality 

parameter c. The other terms used in the option pricing formula are 

B (r 0 ; T ) = A (T ) exp (B (T ) r 0 ) 

A (T ) = 

 

2 exp 1 2 

(k + ) T 

(k + ) (exp (T ) 1) + 2 

B (T ) = 

2 (exp (T ) 1) 

(k + ) (exp (T ) 1) + 2 

! 2k 

2 

p k 2 + 2 2 

 

r 1 A (T ) 

= 

B (T ) log K 

2 

() = 

2 (e 1) 

= + 

2 : 

F.2. 

Vasicek model 

When the short rate follows the Vasicek model in (3.2), the price of a call option with maturity and strike 

price K on a T -year Treasury zero-coupon bond with par $1 is given by Jamshidian (1989), 

C (; T; r 0 ; K) = B (r 0 ; T ) (z 1 ) KB (r 0 ; ) (z 2 ) 

where r 0 is the short rate at the time of option pricing and () denotes the cumulative probability distribution 

function of a standard normal random variable. The other terms used in the option pricing formula are 

B (r 0 ; T ) = exp [A (T ) + B (T ) r 0 ] 

A (T ) = 

B (T ) = 

2 

4k B (T )2 (T + B (T )) 

1 

k 1 e kT 

 

 

2 

2k 2 

35

z 1 = 1 B (r0 ; T ) 

log 

p B (r 0 ; ) K 

z 2 = 1 p 

log 

B (r0 ; T ) 

B (r 0 ; ) K 

 

+ p 

2 

 

p 

2 

s 

(1 e 2 ) 1 e (T ) 2 

p = 

2 3 : 

Observing r 0 , the price of a treasury future that delivers a T -year zero coupon bond in years can be 

calculated according to Chen (1992), 

F (; T; r 0 ) = exp [C (; T ) + D (; T ) r 0 ] 

where 

C (; T ) = A (T ) + 1 

4k B (T ) e 2k e k 1 B (T ) 2 + e k B (T ) 2 + 4k 

D (; T ) = e k B (T ) : 

36

Table 1. Simulation 

This table reports the Treasury option pricing simulation result comparing four estimation methods: the 

parametric estimator using the correct model (Cox, Ingersoll, and Ross (1985) process), the parametric estimator 

using a misspeci…ed model (Vasicek (1977) model), the robust parametric estimator proposed in this 

paper which uses the misspeci…ed Vasicek (1977) model but explicitly adjusts for misspeci…cation, and the 

nonparametric estimator. The simulation is iterated 100 times and each simulation sample path corresponds 

to …ve years of weekly observations. 

r 

Panel A shows the average root integrated mean squared error (RIMSE) 

P 2 

1 n 

de…ned as RIMSE 

bCi 

n i=1 

C i where C b and C are, respectively, the estimated and the true 

Treasury option prices in the simulation. Panel B shows the average regions of …t (h in (2.12)) in the proposed 

method over which parametric …t is used. Panel C shows the estimation RIMSE for parametric estimation 

using the correct CIR model where the closed-form option price C is perturbed to C (1 + ") where " is 

uniformly distributed over [ !; !]. This models potential numerical error if a numerical method instead of 

the closed-form formula is used to compute the option prices in the estimation. Panel D shows the RIMSE 

for a simulation using parametric estimation based on the correct CIR model and numerical integration to 

obtain option prices. 

A. Performance of the option price estimators 

RIMSE 

Parametric $0:00022 

Parametric (using misspeci…ed) $0:041 

Nonparametric $0:013 

Proposed (using misspeci…ed) $0:0015 

B. Robust parametric estimator: region of …t (h) along various dimensions 

C. Simulate numerical error 

h 

Interest rate 0:026 

Option maturity 0:01 

Bond maturity 0 

! RIMSE 

0:01% $0:00023 

0:1% $0:00061 

0:2% $0:0012 

0:3% $0:0017 

0:5% $0:0028 

1% $0:0056 

D. Performance of parametric estimation using correct model and numerical integration 

RIMSE 

Parametric (Numerical) $0:0063 

37

Table 2. CBOT Treasury option pricing 

This table reports the Treasury option pricing result using data of Treasury options traded on CBOT in 

the sample period from May 1990 to December 2006. Three pricing methods are compared: the parametric 

estimator, the robust parametric estimator, and the nonparametric estimator. Both the parametric and the 

robust parametric estimators use the possibly misspeci…ed option pricing formula (4.1) which assumes that the 

short rate follows the Vasicek (1977) 

r 

process. Panel A shows the average root integrated mean squared error 

P 2 

1 n 

(RIMSE) de…ned as RIMSE 

bCi 

n i=1 

C i where C b and C are, respectively, the estimated and 

the observed Treasury option prices. Also shown in panel A is the R-square in the regression of observed call 

option price on predicted option price. The estimation in panel A uses observations in the entire sample period. 

Panel B shows the out-of-sample RIMSE and R-square comparisons of the three estimation methods. The 

out-of-sample estimation uses …ve years’observations to obtain parameter estimates and then measures the 

RIMSE and R-square in the subsequent year using the estimated parameters. The …rst year of out-of-sample 

comparison is 1996. 

A. In-sample pricing performance 

B. Out-of-sample pricing performance 

Parametric Nonparametric Proposed 

RIMSE 0.476 0.383 0.212 

R 2 0.498 0.744 0.902 

RIMSE R 2 

Parametric Nonparametric Proposed Parametric Nonparametric Proposed 

1996 0.468 0.379 0.164 0.523 0.788 0.941 

1997 0.485 0.375 0.221 0.476 0.727 0.947 

1998 0.543 0.495 0.324 0.487 0.623 0.834 

1999 0.430 0.347 0.149 0.578 0.794 0.955 

2000 0.395 0.292 0.148 0.482 0.798 0.930 

2001 0.418 0.325 0.199 0.559 0.804 0.933 

2002 0.540 0.444 0.247 0.582 0.793 0.912 

2003 0.632 0.481 0.296 0.485 0.761 0.922 

2004 0.549 0.385 0.322 0.596 0.817 0.932 

2005 0.538 0.432 0.414 0.532 0.804 0.971 

2006 0.401 0.405 0.399 0.527 0.654 0.907 

Average 0.491 0.396 0.262 0.530 0.760 0.926 

38

ond option price 

Figure 1. Compare option prices along the dimension of option maturity. This …gure 

compares the option prices of CIR model (true model in simulation) and Vasicek model along the option 

maturity dimension. Prices from Vasicek model are shown in neighborhoods around option maturity of 1 

month, 3 months, 6 months, and 1 year. The parameter for CIR process is set to that in (3.4). The parameters 

for Vasicek process are set to those estimated in section 3, which di¤er across the four Vasicek price curves 

shown. The underlying bond maturity is set to 10 years and the short rate is set to 7% (approximately the 

mean interest rate) in the simulation. 

1.8 

Vasicek vs CIR bond option prices 

1.6 

1.4 

1.2 

τ = 0.5 

↓ 

↑ 

τ = 1 

1 

0.8 

τ = 0.25 

↓ 

0.6 

τ = 1 month 

0.4 

↑ 

τ = 0.083 

τ = 3 months 

τ = 6 months 

0.2 

τ = 1 year 

CIR (true) 

0 

0 0.2 0.4 0.6 0.8 1 1.2 

τ: option maturity 

39


Figure 2. Compare option prices along the dimension of bond maturity. This …gure 

compares the option prices of CIR model (true model in simulation) and Vasicek model along the bond 

maturity dimension. Prices from Vasicek model are shown in neighborhoods around bond maturity of 2, 5, 

10, and 30 years. The parameter for CIR process is set to that in (3.4). The parameters for Vasicek process 

are set to those estimated in section 3, which di¤er across the four Vasicek price curves shown. The option 

maturity is set to 3 months and the short rate is set to 7% (approximately the mean interest rate) in the 

simulation. 

1 

0.9 

0.8 


← T = 5 

← T = 10 

T = 2 

T = 5 

T = 10 

T = 30 

CIR (true) 

0.7 

0.6 

0.5 

← T = 2 

0.4 

0.3 

T = 30 → 

0.2 

0 5 10 15 20 25 30 

T: bond maturity 

40


Figure 3. Compare option prices along the dimension of short rate. This …gure compares 

the option prices of CIR model (true model in simulation) and Vasicek model along the short rate dimension. 

Prices from Vasicek model are shown in neighborhoods around short rate of 0.04, 0.07, and 0.1, which are 

approximately the mean and mean plus/minus one standard deviation of the short rate. The parameter for 

CIR process is set to that in (3.4). The parameters for Vasicek process are set to those estimated in section 

3, which di¤er across the three Vasicek price curves shown. The option maturity is set to 3 months and the 

bond maturity is set to 10 years. 

1.1 

1 

0.9 

0.8 


← r = .07 

← r = .10 

0.7 

0.6 

← r = .04 

0.5 

0.4 

r = .04 

r = .07 

r = .10 

CIR (true) 

0.02 0.04 0.06 0.08 0.1 0.12 

r: short rate 

41

obust prices 

nonparametric prices 

parametric prices 

Figure 4. Scatter plots of observed and estimated Treasury option prices. This …gure 

shows the scatter plots of observed Treasury option prices against option prices estimated, respectively, using 

parametric methods, nonparametrics, and the robust parametric pricing method which addresses possible 

misspeci…cation (labeled “robust prices”in the plot). The estimation covers the sample period from May 1990 

to December 2006. 

5 

observed vs parametric bond option prices 

4 

3 

2 

1 

0 

0 1 2 3 4 5 

observed prices 

observed vs nonparametric bond option prices 

4.5 

4 

3.5 

3 

2.5 

2 

1.5 

1 

0.5 

0 

0 1 2 3 4 5 


4 

observed vs robust bond option prices 

3.5 

3 

2.5 

2 

1.5 

1 

0.5 

0 

0 1 2 3 4 5 


42

RIMSE 

RIMSE 

RIMSE 

Figure 5. RIMSE for various regions of …t. This …gure shows root integrated mean squared error 

(RIMSE) in CBOT Treasury option pricing for various regions of …t along the dimensions of option maturity, 

bond maturity, and short rate. The robust parametric pricing method selects a region of …t separately for each 

dimension to minimize the RIMSE. In the plot for bond maturity, the horizontal axis refers to the number of 

nearest bond maturities. For example, 1 means using the nearest 1 bond maturity –bond maturities of 5, 10, 

and 30 years are included in 10-year Treasury option pricing. The sample period is May 1990 to Dec 2006. 

0.26 

RIMSE and region of fit for option maturity 

0.25 

0.24 

0.23 

0.22 

0.21 

0.2 

0.19 

0 5 10 15 20 

region of fit for option maturity (weeks) 

0.4 

RIMSE and region of fit for bond maturity 

0.35 

0.3 

0.25 

0.2 

0 1 2 

region of fit for bond maturity 

0.23 

RIMSE and region of fit for short rate 

0.225 

0.22 

0.215 

0.21 

0.205 

0.2 

0.195 

1 2 3 4 5 6 7 8 

region of fit for short rate (%) 

43

Asset Pricing with Misspecified Models - APJFS

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