A Bias in the Volatility Smile

Don M. Chance

Thomas A. Hanson

Weiping Li

Jayaram Muthuswamy

September 29, 2014

Abstract It is well-known that market prices of options produce implied volatilities that vary in a pattern called the volatility smile. We demonstrate that even if options traded with Black-Scholes-Merton pricing and a single volatility, we would observe smiles, skews, and smirks from the algorithms employed and inputs used. While it is theoretically possible to almost eliminate the problem, it is not feasible in practice. It is even more difficult to control the problem when smiles also reflect market imperfections, as it is commonly believed. We empirically estimate the lower bound of this effect at about 18% of the observed smile.

Key words: Black-Scholes-Merton model, option pricing, implied volatility, volatility,

volatility smile, computational finance, options

JEL Classifications: C18, C63, G13 The authors are, respectively, James C. Flores Endowed Chair of MBA Studies and Professor of Finance, Department of Finance, Louisiana State University, 2909 BEC, Baton Rouge, LA 70803, 225-578-0372, dchance@lsu.edu; Visiting Assistant Professor of Finance, Monte Ahuja College of Business, Cleveland State University, 2121 Euclid Avenue, Cleveland, OH 44115, 216-687-3786, , thanson3@kent.edu; Watson Faculty Fellow of Finance and Professor of Mathematics, Department of Finance, Oklahoma State University, MSCS 526, Stillwater, OK, 74078, 405-744-5852, wli@math.okstate.edu; and Associate Professor of Finance, Department of Finance, College of Business Administration, Kent State University, P.O. Box 5190, Kent, OH, 44242, 330-672-1216, jmuthusw@kent.edu. This paper was presented at the 23rd Annual Cornell-FDIC-Houston Derivative Securities and Risk Management Conference, the R/Finance 2013 Conference, the HEC Montreal 2013 Mathematical Finance Days Conference, the 2013 OptionsMetrics Conference, the Southern Finance Association, and at the International Association for Quantitative Finance, Bloomberg, Vanderbilt, Louisiana State University, National University of Singapore, the University of Adelaide, and NYU-Poly. They authors appreciate helpful comments and suggestions of Robert Brooks, Jinsha Zhao, Yisong Tian, Peter Locke, Hatem Ben-Ameur, Paul Kupiec, Phelim Boyle, Jonathan Trowbridge, Paskalis Glabadanidis, Tatjana Puhan, Joseph Leighly, Mike Stanton, Oliver Haynold, Sivlia Muzzioli, and Robert Jarrow.

1

A Bias in the Volatility Smile

A smile is the chosen vehicle for all ambiguities.

Herman Melville (Pierre, 1852)

1. Introduction

Option pricing models are constructed with the objective of deriving the appropriate

price for a given option. The seminal work in the area is the Black-Scholes-Merton (BSM)

model (Black and Scholes (1973), and Merton (1973)), which derives the value of an option in a

frictionless market through the use of a hedge portfolio and arbitrage-free conditions. For an

underlying asset that makes no cash payments or incurs no carrying costs during the option’s

life, the option value is determined by the price of the underyling, the exercise price, the risk-

free rate, the time to expiration, and the volatility of the underlying. Of these five inputs to the

model, only the volatility of the underlying is unobservable.

Soon after the model was introduced, it was recognized that rather than infer option

values from an estimated volatility, one can invert the process.1 Observed market prices for

traded call options could then be inserted into the model to back out an estimate of the

unobserved volatility of the underlying. In this manner, one could infer the volatility of the

underlying that was being used by market participants as they priced and traded options.

Yet this process revealed an empirical enigma: all options on the same underlying with

the same expiration did not yield the same implied volatility.2 This relationship of implied

volatility across strikes has come to be known as the volatility smile, owing to the U-shape that

is often observed. It is now well-established that there are persistent patterns in implied

volatilities with respect both to strike prices and time to expiration, the combined effects of

which are often known as the volatility surface. Yet this smile remains a puzzling phenomenon.3

1The first inversions of the model were apparently done by Latané and Rendleman (1976) and Schmalensee and Trippi (1978). 2Early options empiricists apparently assumed that this result was a reflection of observational inaccuracies in the data, such as might occur if the option and stock prices were not synchronous or if there were so little trading volume that the price would not be a very representative price. The typical approach taken by empiricists to adjust for unequal volatilities has been to take an average implied volatility, in some cases weighting the implied volatilities by trading volume, moneyness, or the options’ vegas. This consolidated volatility is sometimes called the weighted implied standard deviation. 3Mayhew even goes so far as to state, “To even talk about volatility smiles is schizophrenic,” (1995, p. 14).

2

Because the implied volatility represents the volatility of the underlying, which must by

definition be unique, options that differ only by strike should have the same implied volatility.4

Research and trading experience have also uncovered the fact that some, if not most,

patterns are not literally smiles. Some are skewed and some are said to resemble a smirk.

Whether the relationship between implied volatility and exercise price resembles a smile, skew,

or smirk, any variation from the horizontal line is an indication that the model implies multiple

volatilities for the underlying asset.

The obvious explanation that has been discussed extensively in the literature is that the

model’s perfect market assumptions are too far from reality. Actual conditions in the market

certainly are not the same as those assumed, but this issue is common in financial models.

Another explanation is that variations of the statistical process followed by the underlying could

be the source of the volatility smile.

In this paper, we describe a third explanation, one posited on the basis of a simple

experiment. Suppose options trade in perfect markets. If all of the assumptions of the BSM

model are met, in particular constant and known volatility and continuous, costless trading,

would there be a volatility smile? Intuition says that the implied volatilities of options with

different strikes would equal the known constant volatility used to price the options in the first

place. We show here that this seemingly axiomatic result is not true: even if investors price

options in accordance with the Black-Scholes-Merton model using a single volatility, the implied

volatilities across strikes almost always vary and some quite notably. Remarkably, they do so in

the form of smiles, skews, and smirks that greatly resemble the patterns observed in practice

using prices from markets that are far from perfect. We also show that these patterns are more

pronounced the shorter the maturity of the option, which is consistent with empirical

observations based on actual market prices. Of course, this result is puzzling, if not shocking.

How can it be that one can price an option with a known volatility and then be unable to back

that volatility out of the price?

In an extraordinarily deep and broad body of literature, there has been a surprising

absence of attention to the precise details of how implied volatility is obtained. We will show

that the sheer methodology of solving for the implied volatility can lead to remarkably different

implied volatilities of options that vary by strike but have no other economic differences. We

also show that the problem becomes even more difficult to recognize if there are market

4Variations in implied volatilities across time, sometimes known as the term structure of implied volatility, are not particularly enigmatic and are perhaps perfectly normal. The model requires that volatility be constant over the life of the option. This assumption does not preclude different volatilities across different horizons.

3

imperfections that cause some options to be priced at a premium to others, which, of course, is

the standard explanation for the existence of the smile.

We make no claim that this phenomenon debunks the smile. It is quite real and

generally believed to be driven by the aforementioned market imperfections that are not

captured in pricing models. Yet, it is clear that the problem identified here is not recognized in

the body of knowledge. An obvious question is, therefore, how much of the smile is accounted

for by this problem? We conduct an empirical analysis and find that the lower bound of this

effect is 18% of the market-observed smile, but that for some options at some times, this effect

can be over 90%. We explore some possible explanations and solutions but ultimately find that

this is a problem with no clear fix.

The remainder of the paper is organized as follows. In Section 2 we review the

explanations for the volatility smile that have appeared in the literature. Section 3 provides a

formal specification of the problem and discussion of the known techniques for deriving implied

volatilities and the volatility smile. Section 4 describes our computational specifications and

investigative plan. Section 5 presents our results, which includes not only numerical findings

but also empirical estimates over the period 2004-2010. Section 6 contains our conclusions.

2. Possible Explanations for the Volatility Smile

Figure 1 illustrates a volatility smile produced by the closing ask prices of eBay call

options traded at the Chicago Board Options Exchange on January 18, 2013. The options

expire on February 16, so they have about one month remaining.5 The stock price is $54.21,

and the risk-free rate is 0.05%, based on approximately one-month U. S. Treasury bills. The

option prices are taken from the CBOE’s web site at the close of business. EBay pays no

dividends so these American call options are effectively European call options. The smile is not

completely symmetric and might be more appropriately referred to as a smirk. In general, we

will use the term smile to represent multiple but similar patterns that have been observed. At

certain points, our results will generate smirks and skews, and these variations will be so noted

where relevant.

Observations of volatility smiles have provoked a number of possible explanations and

considerable empirical work. Much of that work has focused on fitting smooth functions for the

smile or surface in order to permit accurate estimation of the implied volatility for options with

5All of our results will focus on call options. The volatility smile is often constructed with put options or a combination of call and put options. The focus of our paper is on problems arising from use of the Black-Scholes-Merton model for European call options in the ideal condition of perfect markets. Similar problems will arise with puts.

4

strikes and expirations not currently traded.6 Our interest differs. In a BSM world, the

existence of a smile implies that after accounting for the effects of the price of the underlying,

the exercise price, the risk-free rate, the time to expiration, and the volatility, the option price is

not what it should be.7 Thus, there is a factor not accounted for that affects the option price.

That factor is, therefore, absorbed into the implied volatility estimate. For example, a U-

shaped smile suggests that after incorporating the five standard factors that differentiate option

prices, options with the highest and lowest strikes are still more expensive than others. Such an

explanation is inconsistent with arbitrage-free pricing in a BSM world. With continuous and

costless trading, any option can completely replicate any other option on the same underlying.

Thus, in a perfect BSM world, no single option can be relatively more or less expensive than any

other option on the same underlying. Otherwise, there would be an arbitrage opportunity.

2.1 Inappropriate Model Assumptions

Previous research has assumed that the implied volatility smile is due to a violation of

one or more of the model’s assumptions. Jump processes are discussed in Rubinstein (1994),

Merton (1976), Bates (1991), Das and Sundaram (1999), while stochastic volatility is posited by

Hull and White (1987) and studied by Heston (1993), Rubinstein (1994), Dumas, Fleming, and

Whaley (1998), and many others. Interestingly, in an empirical study, Bakshi, Cao, and Chen

(1997) find that incorporating stochastic interest rates, stochastic volatility, and stochastic

volatility with random jumps does not eliminate the smile, meaning to return a single volatility.

The possibility of the effect of non-normality on a smile has been raised by Corrado and

Su (1996). The effects of transaction costs (Pena, Rubio, and Serna (1999)), discrete tick size

(Dennis and Mayhew (2009)), net buying pressure and limits to arbitrage (Bollen and Whaley

(2004), Shiu, Pan, Lin, and Wu (2010)), and trader behavior (Vagnani (2009)) have also been

considered. An apparently widely-accepted belief is that unusually high implied volatilities on

low-strike puts is a form of crash insurance, as noted by Rubinstein (1994) in a pattern became

more apparent following the crash of 1987. Finally, there is the rather unsurprising case that a

smile or skew pattern can result when options are priced by one model, but implied volatility is

obtained from another, as illustrated by Shaw (1998). He illustrates how option prices driven

6Smooth implied volatility functions are important to practitioners, because they provide prices that a dealer can quote for options that might not currently be traded, but which could come in demand. In fact, it is common for traders to quote option prices, not by stating the price directly, but by quoting the implied volatility. An investor may be said to buy an option at 20, meaning that the price he pays is not $20 but is determined by inserting 20% as the volatility in the option pricing model. 7Reference to a “Black-Scholes-Merton” world means a world in which all of the assumptions of the model are correct, or in other words, a perfect market. Hence, all prices would conform to the model. Of course, the key assumption here is that the volatility is known and agreed upon by all investors.

5

by the constant elasticity of variance model produce a skew when the implied volatility is

backed out of the BSM model.

There is no reason to believe that these explanations are incorrect. Yet, there is reason

to believe that computational factors could drive at least a portion of the implied volatility. If

computational factors have a significant influence on the implied volatility, it is difficult to

determine the impact of these economic reasons for the smile. Indeed, as we show later, the

interactions of these effects with computational matters makes the problem even more difficult

to detect.

2.2 Computational Considerations

Apparently the only paper to date to address any aspect of computational effects on

implied volatility estimation is Hentschel (2003), who shows that measurement errors in the

model inputs can have significant impact on the implied volatility. He considers finiteness of

quotes, the existence of bid-ask spreads, and non-synchronous prices of the option and

underlying. These slight distortions of certain inputs can lead to significant variation in implied

volatility estimation. Although Hentschel’s study is probably most closely related to ours, the

differences are significant. Hentschel’s focus lies on the error-in-variables problems that would

create biased inputs. In contrast, we assume that all inputs are correctly measured and focus on

the mechanics of how inputs are converted into outputs, taking into account convergence

tolerance, the starting point, and the root-finding algorithm used. In short, we essentially

assume that markets are perfect in the sense that the BSM assumptions are met and that

options are priced according to this model but that measurement processes must operate within

the limits of mathematical knowledge and computational capabilities.

The literature on implied volatility is remarkably void of any discussions of

computational matters. For example, given that there is no direct, closed-form solution, all

estimates of implied volatility are merely estimates and are subject to an error tolerance

specified by the user. This tolerance is seldom specified explicitly in the literature. For

instance, Canina and Figlewski (1993) describe the process of calculating implied volatility as

“solving the pricing model for the volatility that sets the model and market prices equal” (p.

659). Likewise, Rubinstein (1985) notes in a footnote that the Newton-Raphson method

“converges quickly” (p. 460). Chambers and Nawalkha (2001) state that “Implied volatility can

be easily calculated using an iterative search technique” (p. 98). Only a few studies explicitly

describe their tolerance requirements. For instance, Latané and Rendleman (1976) require their

numerical search procedure to provide a resultant price within ± $0.001 of the option’s market

price. Schmalensee and Trippi (1978) utilize an iterative procedure that they terminate when

updates change the estimate by less than 0.001%. In addition, there is little to no mention of

6

starting point estimates or intervals, which are required by every root finding technique. The

overwhelming majority of papers make no mention at all of how the implied volatility is derived.

3. Formal Specification of the Problem

The Black-Scholes-Merton formula is given as

1 22

1

2 1

( ) ( )

ln( / ) ( ( / 2))with

,

rc SN d Ke N d

S K rd

d d

(1)

in which N(·) represents the cumulative normal density function, c is the call option price, S is

the current market price, K is the option’s strike price, τ is time remaining to expiration, r is the

continuously compounded and constant known rate of interest, and σ is the volatility of the

stock’s return, which is formally the annualized standard deviation of the continuously

compounded return. Now let cM represent the price of the option as observed in an active and

liquid market. Setting the model price to the market price, the implied volatility is the implicit

solution to:

1 22

1

2 1

( ) ( )

ˆln( / ) ( ( / 2))with

ˆˆ ,

rMc SN d Ke N d

S K rd

d d

(2)

where ˆ is the implied volatility, defined as the volatility that forces the market price to equal

the model price.

Of course, it is well-known that it is impossible to directly solve (2) for the implied

volatility.8 Some researchers have relaxed certain assumptions and developed closed-form

approximations (Brenner and Subrahmanyam (1988), Chance (1996), Chambers and Nawalkha

(2001), Bharadia, Christofides, and Salkin (1996), Hallerbach (2004), Kelly (2006), Corrado and

Miller (1996), and Li (2005)).9 Closed-form approximations are useful in spreadsheet and

pedagogical applications but have become less critical in large scale research and trading

because of the increased speed of computers.

3.1 Iterative Techniques

8A recent paper by Gerhold (2012) shows more formally that the Black-Scholes-Merton model does not belong to a class of functions for which solutions can be easily found. 9Some of these techniques can be considered moderately iterative in that they use starting points and require at least one guess. And of course, one could always terminate any iterative technique after the initial iteration and call it a closed-form approximation.

7

Iterative methods provide more accurate results by taking an initial estimate or interval

of estimates, guessing the value of the function at the initial estimate or interval, generating a

new estimate, and starting the process over, ultimately and hopefully converging to the final

solution. Consider the nth estimate, n of the implied volatility. This estimate produces a value

of the option, c(n). With the observed market price of the option as cM, the difference between

the market and model price of the option is

1( ) 2( )2

1( )

2 1

( ) ( ) ( )

ln( / ) ( ( / 2))with

.

rn M n n M

nn

n

n

c c SN d Ke N d c

S K rd

d d

(3)

These iterative techniques generally involve the use of a root-finding procedure for the

pricing error equation, c(n) - cM = 0. The particular algorithm employed to solve this problem

will possess certain qualities of robustness and speed of convergence. Furthermore, the tolerance

level, or acceptable error, for calculating the root will influence the estimate of the implied

volatility. As noted earlier, most papers that involve calculations of implied volatility refer

quite obliquely to their root finding technique. A primary contention of this paper is that the

discipline has overlooked the vital importance of the implications of this choice and such factors

as starting points. We will show that these computational considerations have a significant

influence on the existence and shape of the implied volatility smile.

The body of mathematical knowledge is replete with root-finding methods, and root-

finding is a common problem for students of mathematics and computer science. Outside of

solving for the implied volatility, however, root-finding problems are not very common in

finance in general.10 Thus, and unsurprisingly, there has not been much attention paid to

solving such problems in the finance literature.

Kritzman (1991) mentions two methods for deriving the implied volatility, the Newton-

Raphson method and the bisection method. Brown (1990) and Chriss (1997) mention the

bisection method. Other well-known root finding methods, though rarely mentioned in the

finance literature, include the secant method, the regula falsi or false position method, and the

Dekker-Brent method.11

10Finding the internal rate of return on a capital investment decision is one root-finding problem that arises in corporate finance. Most finance problems related to iteration and convergence, however, involve derivatives. For example, convergence issues arise in solving for the binomial option price, and recent research by Muthuswamy and Hanson (2012) suggests that the binomial model might not converge at all. 11Interestingly, the regula falsi technique has been mentioned in cuneiform tables from ancient Babylonian mathematics. See Chabert (1999). Regula falsi is discussed by Fengler (2012), but to our knowledge has not been cited elsewhere in the finance literature. Dekker-Brent is used by Matlab as the default root finder.

8

Probably the best known root-finding iterative technique is the aforementioned Newton-

Raphson method.12 Given its apparent widespread use, we examine it in some detail. In

Appendix B, we will derive a first linear approximation of the implied volatility. We then use

that approximation to examine the behavior of the Newton-Raphson method. The first linear

approximation can be reduced to the following quadratic equation:

21 0 2 0 3

1

2

3

ˆ ˆ 0

2

22

ln

r

r

M

r

q q q

S Keq

S Keq c

Sq S Ke r

K

Of course, the solution is well-known as

22 2 1 3

0

1

2

2

4ˆ

2

2 ln2 22 21

.

r rr

M M

r rr

q q q q

q

SS Ke S KeS Ke rc c

K

S Ke S KeS Ke

(4)

The larger root is given by

2

0 2

2 ln2 22 21

,

r rrm

M t

r rr

SS Ke S KeS Ke rc c

K

S Ke S KeS Ke

which is the solution proposed by Corrado and Miller (1996). Note that the smaller root can be

positive as well when q2 < 0, which is equivalent to 2

r

M

S Kec , and is given by

2

0 2

2 ln2 22 21

.

r rr

M M

r rr

SS Ke S KeS Ke rc c

K

S Ke S KeS Ke

12Chriss (1997) states that due to problems with the bisection method that “industry shops tend to use a faster method, known as the Newton-Raphson method”(p. 330).

9

Interestingly, this other solution has not been considered in previous research.

The numerical approximation ˆ is the solution to

1 2

ˆ ˆ ˆ( ) ( , , , , ) ( , , , , ) 0.rM

F SN d S K r Ke N d S K r c (5)

The Newton-Raphson method is an algorithm that attempts to solve this equation using the

following iteration process,

1

( ),

( )n

n n

n

F xx x

F x (6)

with a choice of initial point x0 to start. Even if F(x) has derivatives everywhere, however, the

Newton-Raphson method may not lead to a root. For example, Newton-Raphson does not

converge to a root for the simple function, F(x) = x3 – 2x + 2 with x0 = 0.

Choosing a good starting point is an important issue for Newton-Raphson. Even with a

seemingly good choice, the local information F(xn) and F´(xn) can lead to slow convergence or

convergence to the wrong root. For example, for the simple expression, F(x) = sin x – x/2, a

starting point of x0 = 1 leads to the largest positive root, a starting point of x0 = 1.01 leads to

the largest negative root, a starting point of x0 = 1.02 leads to negative infinity, and a starting

point of x0 = 1.05 leads to positive infinity. Thus, only slight changes to the starting point can

lead to dramatically different solutions and in some cases, no solutions at all.13

The reflection points rf

of the function F are given by

2ln ,

rf

Sr

K (7)

which occur where either 1 20,

rfd d or

1 2, 0

rfd d . This formula was

discovered by Manaster-Kohler (1982). We explore its properties a little further.

If ˆ0rf , then there are two cases to consider: either (i) d2 < d1 < 0 when

2ln

rf

Sr

K, which occurs when ln 0

Sr

Kor (ii) 0 < d2 < d1 when

2ln

rf

Sr

K, which occurs when ln 0

Sr

K. Without loss of generality, we

take the second case for discussion. Therefore, 2

20

ˆ

Fprovided ˆ0

rf .

13The reader might recognize this sensitivity to initial conditions in a nonlinear process as related to chaos theory and nonlinear dynamics. Recent surveys of economic and financial applications are provided by Hseih (1991) and Barnett, Serletis, and Serletis (2006).

10

If ˆrf

, then there are also two cases: either (i) 22 10

rfd d when

2ln

rf

Sr

K, or (ii) 2

2 1rfd d when

2ln

rf

Sr

K. In either

case, we have 2

20

ˆ

Fprovided ˆ

rf.

From Equation (5), we build the Newton-Raphson method with initial point 0 rfx

from the explicit formula given by the roots of the quadratic equation. We have the vega and

vomma,

1

21 2

2

( ) 0,ˆ

,ˆ ˆˆ

FSn d

d dF F

where d1 and d2 are functions of the implied volatility with

1 2 2 1ˆ ˆ ˆ ˆ( ) ( ) ( ) ( )

, .ˆ ˆ ˆ ˆ

d d d d

If an approximated solution exists, then the solution is unique by the strictly positive vega.

This result reduces the convergence issue of the Newton-Raphson method to a simpler one: if

the algorithm converges, it must converge to the correct solution. These results are formally

stated in the following theorem.

Theorem 1. Let rf be the reflection point of the implied volatility from the Black-Scholes-

Merton formula. Then we choose the starting point 0 rfx for the Newton-Raphson method.

We have 01 0

0

( )

( )

F xx x

F x.

(i) If ˆ0rf , then we have 1

ˆn n rfx x for all n. Thus, the Newton-Raphson

method overestimates the implied volatility ˆ . The algorithm is convergent to the implied

volatility, ˆlimnnx , and the rate of convergence is quadratic provided

2 21 1 1 1 2

ˆˆ ˆ ˆ( ) ( ) ( ) ( )exp 1.

ˆ2 2

rfd x d d d

(ii) If ˆ0rf , then we have 1

ˆn nx x for all n. Thus, the Newton-Raphson

method underestimates the implied volatility ˆ . The algorithm is convergent to the implied

volatility, ˆlimnnx , and the rate of convergence is quadratic provided

11

2 21 21 1 1

ˆˆ ˆ( ) ( )ˆ( ) ( )exp 1.

2 2

rf

rf

d dd x d

(iii) If 0rf

, then the Newton-Raphson method underestimates the implied volatility ˆ .

The algorithm is convergent to the implied volatility ˆlimnnx . Moreover, the rate of

convergence is quadratic provided

2 21 21 1 1

1

ˆ ˆ( ) ( )ˆ( ) ( )ˆexp 1.

2 2

d dd x d

x

Proof of this theorem is provided in Appendix A.

The algorithm itself cannot guarantee convergence so we choose a special starting

point.14 The overestimates and underestimates are due to the convexity of the BSM function in

the implied volatility. Note that it is possible that the iteration of the Newton-Raphson method

produces a slow convergence sequence for the implied volatility. The convergence conditions in

the theorem guarantee only that the rate of convergence is quadratic. While we have presented

a general way of viewing the problem, the convergence conditions can be simplified for special

cases. For example, if2( ) 0rf

d in case (i), we can replace the x1 by rf

in the convergence

inequality. The condition is purely theoretic and involved with the implied volatility. In

practice, convergence is rapid in both cases due to the strictly increasing and decreasing

properties. If t̂ rf , then we have x0 = xn for all n, and the starting point is the implied

volatility.

If we select different starting points, then we would have either 0ˆ

rfx or

0ˆ

rfx , which are not covered in the theorem. For the first case, we have x1 > x0 and

1ˆ x . Therefore, there are two possibilities for the location of x1: (a) when the first iteration

leads to 1 rfx , we obtain the same result as in the theorem; and (b) when the first iteration

leads to 1 rfx , we have x2 < x1 and can be located at either side of ˆ . Thus, the Newton-

Raphson method is inconclusive unless ˆ n rfx for some n. The same phenomena occurs for

the second case, 0ˆ

rfx unless 1

ˆrfx for some n, which leads to the same result as

covered in our theorem.

The Newton-Raphson method can also fail for other reasons. The objective function

may not be differentiable at all points. While this problem does not exist for the continuous

BSM model, price rounding, whereby the value of an option perceived by a trader is obtained to

14Hallerbach (2004) notes that the special case in which the strike is set at the forward rate will not guarantee convergence. Thus, the implied volatility of an at-the-money forward option would not necessarily converge using the Manaster-Kohler starting point. This special case, however, is covered by the Brenner-Subrahmanyam formula.

12

many digits but then rounded to the penny in order to make a trade, injects a degree of

discreteness that can lead to non-differentiability.

The majority of implied volatility studies in the literature appear to employ the Newton-

Raphson method. The likely widespread use of Newton-Raphson is summarized by Jackel

(2006) who states that “Conventional wisdom has it that the best all-around method of choice

for the root-finding of smooth functions is Newton’s algorithm. Alas, this is not always so.” (p.

2). While our focus is on Newton-Raphson, we do consider other algorithms.

3.2 Numerical Precision

Estimating an option pricing model with a computer model necessarily involves the use

of discrete, floating point arithmetic to approximate continuous processes. Rarely is this issue

made explicit in the literature, but computer software generally operates in at most double

precision, with one rigorous example provided by Li and Lee (2011).

A second instance for which numerical precision is vital is the required approximation of

the Gaussian cumulative density function. Given that the Gaussian density cannot be

integrated, a computer algorithm is necessary to provide this critical input to the option pricing

model, and the result will be correct only to a desired degree of decimal accuracy. There are

several well-known polynomial approximations, many of them described in the classic

Abramowitz and Stegun (1972, pp. 932-933), and some are discussed in books on options (e. g.,

Chriss (1997, pp. 88-89), Wilmott (1998, p. 110), Haug (1998, pp. 187-188), and Whaley (2006,

p. 252)). In Appendix B we show that failure to accurately estimate the normal probability can

also lead to variations in the implied volatility across strikes. We suspect that most computer

programs estimate the normal probability to a sufficient degree to render this problem relatively

minor. We examine our own algorithm in light of this concern and find no reason to suspect

that inaccuracy in the normal probability is a source of computational error in our results. This

may not be the case in all other papers and applications in which users are unaware of these

computational issues.

4. Computational Methods

All calculations and simulations for this study are conducted using the statistical

computing program R and its contributed package, Rmpfr (Maechler (2011)). The calculations

use at least quadruple precision arithmetic (128-bit storage). This package allows full control

over the precision of the decimal representation of all numbers.15

In general, the methodology is a four-step process. First, we make assumptions about

the numerical values of the inputs and enter these inputs into the BSM formula to obtain the

15The R code is available from the authors upon request.

13

price of an option to an arbitrary level of precision. The software accepts the standard inputs

to the model, as well as the number of bits requested for numerical storage, and produces the

option’s value and vega. These outputs are calibrated with estimates from the function

GBSOption in the contributed R package fOptions (Wuertz et al. (2013)). Second, we vary the

exercise prices across a wide range of moneyness and repeat the calculations of BSM option

prices. These prices are then taken as the assumed market prices of options on the same

underlying with known volatility, equivalent expirations, and various strikes. Third, we reverse

the process by assuming that the BSM prices are the market prices and use those prices to

estimate the implied volatilities using one of the various methods mentioned above in

conjunction with a specified quotation unit and error tolerance. Because the market prices are

perfect in the BSM sense, the resulting implied volatilities should be the same across strike

prices. Fourth, we generate a graph of the implied volatilities across the range of strikes

covering four expirations.

We consider several factors and computational issues that influence the shape of the

implied volatility smile. First, we vary the numerical precision of the algorithm. As discussed

previously, the numerical procedure to calculate implied volatility terminates when the

iterations yield an estimated price that is within a preset tolerance of the market price. Because

this investigation can use perfect BSM prices to a high degree of precision, the tolerance can be

set far below a single penny. We examine tolerances of 0.01, 0.00001, and machine epsilon of 2-

52.16

We also examine a second factor under consideration, the initial input for the Newton-

Raphson method and the initial interval for the other techniques. While previous research has

made little mention of this parameter, it can result in significant changes in the algorithm’s

output. The baseline cases use the initial estimate of Manaster and Koehler (1982) given in

equation (7). Other techniques require an interval. We start with -1 as the lower point of the

interval and the Manaster-Kohler starting point as the upper point of the interval.17

We also examine the quotation unit. Most studies of the implied volatility smile have

been on exchange-listed equity options. Prior to the year 2000, exchange-listed equity options

traded in units of sixteenths for options less than $3, and in units of eighths for options at $3 or

16Tolerance does not guarantee convergence, and in fact, and as we previously discussed, the solution can diverge significantly from the solution. We consider the algorithm to have failed if successive iterations clearly move toward positive or negative infinity. We also impose a maximum of 1,000 iterations, at which point the algorithm stops and the implied volatility is considered to be the final estimate, even though that estimate might not be within the specified tolerance. 17If a particular iteration results in volatility of zero, the values of d1 and d2 are set to positive or negative infinity, resulting in normal probabilities of one or zero, depending on moneyness. Note that starting at an interval with an endpoint below zero is not a problem. If that interval is too wide, it simply slows down the process. We obtain fewer failures to converge with one endpoint negative.

14

more. Around the year 2000, equity options began trading in units of $0.05 for options less than

$3 and $0.10 for options at $3 or more. Starting in 2007, the CBOE went to its penny pilot

program in which certain options traded in pennies. Over time, more and more options have

been added to the penny program. Recall that the procedure is that we calculate the BSM

model price and then feed that price back into the model to back out the implied volatility. We

calculate the option price and then round to sixteenths and pennies.18 We also feed non-

rounded prices back into the model. These prices are effectively in quadruple precision, so they

are carried to 32 decimal places. 19 We will refer to these non-rounded prices as continuous,

though strictly speaking they are not continuous in a pure mathematical sense.20 We recognize

that price rounding of any degree is a relaxation of the perfect market assumptions. We view

the notion of a perfect market as a characterization of the conditions of continuous, costless

trading with no taxes, and constant volatility and risk-free rate.

Finally, we also use the five aforementioned approaches to solving non-linear equations

and examine how they perform in the case of implied volatility. These methods are the Newton-

Raphson, bisection, Dekker-Brent, secant, and Regula Falsi. We reverse the pricing process and

generate the implied volatility with the computed option prices entered into the model. The

resulting implied volatility should be the one used to price the options in the first place.

The most critical part of this experiment to understand is that we price our options

under the perfect market conditions of the BSM world with a known volatility and then reverse-

engineer the process in an attempt to arrive back at the volatility that was used to price the

options. Intuitively, it would seem that surely we would arrive back at the original volatility.

But in fact, often we do not. Appendix C provides a step-by-step example showing how an

original inputted volatility of 20% comes out as an implied volatility of 28%, an overestimate of

slightly more than 40%, using the Newton-Raphson method and is even further off with some

other methods.

5. Results

18Interestingly, while penny pricing is generally regarded as an improvement in reducing bid-ask spreads, we will show that penny pricing, with other factors, can still lead to volatility smiles. 19Our emphasis is on implied volatilities from prices on exchange-listed markets, which specify the standard trading unit. Customized over-the-counter (OTC) contracts can use any trading unit the parties agree on. Due to the large notional sizes, OTC transactions are sometimes carried out to more decimal places, such as four or six but never thirty-two 20In a few cases, price rounding can push the price below its lower bound. For example, a 30-day option struck at $82.50 has a BSM price of $17.7712 with a lower bound of $17.7708. The BSM price would be rounded to $17.77, which is below the lower bound. In an actual market, its price would likely be the lower bound rounded up, in this case, to $17.78. The implied volatilities are 25.22% (BSM non-rounded price), 24.94% (price rounded below lower bound), and 27.03% (price set lower bound rounded up to closest penny). These differences are not sufficiently large to be of concern. Of the 84 combinations of expirations and strikes examined, this lower bound violation occurs in only five other cases. In no cases are the implied volatilities materially affected.

15

The spot price is set at $100, while the strike price is varied from $75 to $125 in

increments of $5. The interest rate is 4%, the options expire in 7 days, 30 days, 90 days, and

182 days, corresponding to times to expiration of 0.0192, 0.0822, 0.2466, and 0.4986,

respectively.21 The volatility is set at 20%. Recall that we generate the option prices based on

these parameters.

5.1 Volatility Smiles Under Various Conditions

Our initial focus is on the Newton-Raphson method, as it is likely to be the most widely

used. The starting value for the Newton-Raphson method is the formula of Manaster and

Koehler (1982), equation (7). We also conducted tests using 100% as the starting point, given

that implied volatility would rarely be above 100% for equities. These results led to widespread

failures to converge.

Figure 2 shows the volatilities by strikes using the Newton-Raphson method under the

various assumptions and considerations above. We see that with tolerance of 0.01, all three

quotation methods show a similar picture, the familiar volatility smile. The smile is most

pronounced for the shortest maturities and is virtually non-existent for the longest maturity,

which is consistent with observations from studies using market data. Moreover, even though

the exercise prices are symmetric around the at-the-money case, the implied volatilities are

much higher for the deep out-of-the-money options than the correspondingly deep in-the-money

options, which is consistent with empirical observations based on market prices.

With the coarsest prices of pennies and 16ths, decreasing the tolerance to 0.00001 and to

machine epsilon produces seemingly erratic graphs. Yet in fact these graphs display the

characteristics of smirks and skews that are often observed with market data. With continuous

prices, lower tolerance shows no real improvement, except for machine epsilon tolerance in

which case the smile nearly disappears. But of course, as a practical matter, transactions

cannot occur at continuous prices, and machine epsilon tolerance is very unlikely to be used in

most algorithms. Thus, we see that the Newton-Raphson method does not generally return the

single volatility that was used to price the options across strikes. It not only fails to return the

volatility that was used in the first place, the volatilities it returns show familiar patterns.

In addition, for the Newton-Raphson method we have conducted tests based on the

starting points of Bharadia et al, Corrado and Miller, and Li. We continue to find essentially

the same results.

21Using multiple expirations provides a rough approximation of the effect of the term structure of volatility, and hence, combined with the smile effect, we could approximate the volatility surface. Our primary focus, however, is on the impact of different exercise prices, so we will not formally construct volatility surfaces, which require some degree of smoothing or a large number of expirations.

16

As noted, we examined four other techniques for solving for the implied volatility. These

remaining techniques all require a starting interval. For this initial set of tests, we used -1 and

the Manaster-Kohler starting point as the interval. The results are shown in Figures 3, 4, 5,

and 6. The results for the secant method in Figure 3 are virtually indistinguishable from those

of Newton-Raphson. The benchmark result of an implied volatility of 0.20 across strikes occurs

only when option prices are quoted continuously and with machine epsilon tolerance and even

then a discrepancy occurs for the lowest strike.

The regula falsi method, Figure 4, produces smiles and mostly skews and smirks for

prices quoted in sixteenths for all degrees of tolerance, and lower tolerance provides virtually no

improvement. Sixteenths prices produce skews and smirks for all tolerances. Penny prices

produce a smile for tolerance of 0.01, but the smile turns to skews and smirks for lower

tolerance. Continuous option prices produce a fairly consistent smile regardless of tolerance.

The bisection method, illustrated in Figure 5, displays many skews and smirks, and

lower tolerance seems to exacerbate the problem. Continuous prices all provide no notable

improvement and produce another usual result of some implied volatilities being well below the

true volatility.

Figure 6 shows that the Dekker-Brent method produces skews and smirks for prices in

sixteenths with no real improvement with lower tolerance. For prices in pennies, smiles occur

for tolerance of a penny and 0.00001, while skews and smirks occur for machine epsilon

tolerance. Continuous option prices generate smiles for tolerance of 0.01 and 0.00001, but

return nearly a constant implied volatility for machine epsilon tolerance. These results are also

similar to Newton-Raphson and the secant method.

One might reasonably ask if the volatility smiles reported here start far too late in

relation to those smiles observed in empirical data, which seem to start very near at-the-money.

After attempting a variety of parameters, we find that it is relatively easy to obtain smiles that

start around 5% beyond the money using Newton-Raphson with very short-term options.

Estimates are about 22% for a 20% known volatility, or an error of 10%.

5.2 Basic Discussion and Analysis of Numerical Results

To recap, we find volatility smiles from merely attempting to derive the implied

volatility of options priced according to the BSM model. We relax the model’s assumptions

only modestly and then by merely rounding to pennies and sixteenths. We find that the

Newton-Raphson, the bisection method, and Dekker-Brent produce very similar visual results,

but that finding is hardly comforting. For prices quoted in 16ths and pennies, there are distinct

volatility smiles, with the highest volatilities occurring with the lower strikes.

17

One source of computational error could be the starting point. It is well known that

poor choices can slow down the estimation process and alter the outcome. With so many

possible choices, however, it is difficult to assess the impact. Moreover, in the controlled

experiment we conduct, we are required to know the solution. Hence, our choice of the solution

could bias our starting point estimate. We are consequently subject to a variant of the

Heisenberg Uncertainty Principle: one cannot measure something without altering it. Thus, the

starting point could well be an even greater problem than we can assess at this time.

Tolerance clearly has an important effect as well. Holding everything else constant, a

smaller tolerance can eliminate some of the smile, but it cannot eliminate all of it. Penny

tolerance, which we suspect is commonly used in research and in practice, is probably not small

enough. We seriously doubt that many users require tolerance of machine epsilon, but they

probably should.

Certainly another factor is price rounding. Consider a price carried out to any degree of

precision higher than the quotation unit. By forcing a trader to round a price, clearly a given

rounded price can derive from many non-rounded prices. For example, suppose trade prices are

quoted in pennies, and to simplify, the theoretical model produces prices rounded to three

decimal places. A trade price of $10.25 can arrive from non-rounded (theoretical) prices from

$10.245 to $10.254. A price rounded up will lead to a higher implied volatility, while a price

rounded down will lead to a lower implied volatility. This explanation can explain only a

portion of the smile, however, since it would imply that the implied volatilities would not be

uniformly high or low. The evidence shown here suggests that biased high implied volatilities

are much more common than biased low implied volatilities. Moreover, the difference in results

from continuous prices and rounded prices is not particularly large. So price rounding cannot

explain all of the effect.

5.3 Further Analysis: The Vega Effect

We extend our analysis a step further by examining a possible source of the problem. If

the source can be identified, it could lead to a recommendation on how to address this issue

when one is working with actual option prices and the true volatility is not known. We focus on

the results from the Newton-Raphson method, inasmuch as it is widely used and gives very

similar results to several other methods.

The vega of an option is the partial derivative of the price with respect to the volatility.

In solving for the implied volatility using Newton-Raphson, the movement required to converge

is determined by the vega in relation to the distance of the function from zero. Thus, the vega

18

may provide a clue as to where the problem lies.22 In Figure 7 we show a slightly more granular

reproduction of the first graph in Figure 2C, the case of no price rounding and tolerance of

$0.01. The strikes in Figure 7 are spaced closer ($2.50 apart) but cover the same range, $75 to

$125. Panel A shows the smile itself, while Panel B shows the vegas across the corresponding

degrees of moneyness. Keep in mind that as a result of this being an experiment, we know the

true vegas because we know the true volatility. In practice, not knowing the vegas would be a

real problem, as we address momentarily. Perhaps not surprisingly, the vegas are extremely low

for the same degrees of moneyness in which the implied volatilities are quite high. Low vegas

suggest that when iterating, large volatility changes are required to move a price by a small

amount.

A casual examination of the vegas shows that when they are below 0.5, the implied

volatilities deviate substantially from the true volatilities. Thus, we may have stumbled on a

useful rule of thumb. One might reasonably argue that when vega < 0.5, the option does not

provide sufficient confidence to trust the implied volatility. We might say that the option price

and implied volatility are not reliably informative (NRI). An option that is NRI should

probably be discarded. The options retained are called reliably informative (RI). Nonetheless,

there are two non-trivial problems with any such cutoff.

In this experiment the true volatility is known, so we can easily determine the true

vegas. In practice, the true volatility would not be known, so we cannot determine the true

vegas. Proxy vegas could, however, be obtained from the implied volatilities after they have

been derived. Figure 8 shows the difference between the true vegas and the proxy vegas for all

of the expirations. With the exception of the at-the-money case being zero, all differences are

negative, implying that the proxy vegas are larger. In some cases, such as the highest and

lowest strikes, they are considerably larger. Thus, in general proxy vegas are more likely to pass

the 0.5 cutoff. Therefore, we increase the risk that options that are NRI based on the unknown

true volatility are deemed RI based on the proxy vegas. But there is no choice. The true vegas

would not be known.

Regardless of which volatility is used to calculate the vegas, the evaluation of a

particular option to determine if it is NRI based on its vega requires circular reasoning as

follows:

Step 1: Calculate the implied volatility.

Step 2: Calculate the vega using the implied volatility.

22The vega is used directly in the Newton-Raphson method, but not in the other four methods. Those methods, however, do incorporate discrete estimates of the relationship between the price and the volatility. Thus, they are influenced by the concept captured by the vega.

19

Step 3: If the vega is below the cutoff, conclude that the implied volatility is not reliably

informative (NRI).

But if the implied volatility is NRI, how can one use it compute the vega in Step 2? The vega

would, therefore, have to be NRI. And if the vega is NRI, then how can one conclude that the

implied volatility is NRI? And so on.

Thus, it is tempting to conclude that because this problem occurs only with very low

vegas, one could use a minimum vega as a signal of when this problem exists. This method is

even used by OptionMetrics, which does not report an implied volatility if the vega is below

0.5.23 We now know that this approach is circular. For the time being, however, let us ignore

the circular reasoning and expose another, perhaps even more critical, problem.

To date, the literature has treated the volatility smile as though it implies pricing

discrepancies that are thought to arise from market imperfections not incorporated into the

BSM model. Suppose we impose an arbitrary price premium on our option prices that can

represent this type of price discrepancy. For example, let us conservatively inject a very small

premium of 1% into the option price, an amount less than the bid-ask spread. Figure 9 parallels

Figure 7, but in the former the option prices have been amplified by a factor of 1.01. Figure 9

clearly looks extremely similar in shape to Figure 7, but the magnitudes of the implied

volatilities and vegas are quite different. For example, note the 30-day (expiration = 0.0822)

75-strike option. Without the premium, its implied volatility is 36.87% and its vega is 0.2236.

It, thus, appears NRI. With a 1% pricing premium, however, its implied volatility shoots up to

58.98% and its vega is 2.2417, which would make it RI.

Now, consider the implication of this result. We have a deep out-of-the-money option

that we know from this controlled experiment is priced at a volatility of 20%, with the price not

rounded off as it would be in practice. Its implied volatility at 37% is almost twice the true

volatility, and, disregarding the circular logic, its vega is so low that one should reject it as

being NRI. But if the option were priced at a small 1% premium over BSM, as it would be if

other factors help create the smile, its implied volatility is now almost three times the true

known volatility and its vega is sufficiently high that it is deemed RI. It is quite taxing to

believe that imperfections could make an option price and volatility reliable. And while one

might reasonably assume that there is a price premium to account for market imperfections, a

price premium could also occur due simply to anomalous pricing. Thus, we are left to conclude

that an option price and its implied volatility can be NRI when it trades in a perfect market

23It is not clear how OptionMetrics justifies not reporting an implied volatility when vega is less than 0.5, when it has to have a volatility to estimate the vega.

20

and is valued precisely according to a rational model, but it becomes NRI in an imperfect

market or one with an arbitrary pricing inefficiency.

Moreover, if an arbitrary vega cutoff were a useful rule of thumb, it should hold up

under more accurate computations. Unfortunately, it does not. Applying the rule of thumb

with tolerance of 0.00001 results in the rejection of numerous cases of options in which the

implied volatility equals the known implied volatility. For example, with 90-day options, we

would reject strikes of 77.50 and 75, even though their implied volatilities are 0.20. For 30-day

options, we would reject strikes of 85 and below even though their volatilities are almost

precisely 0.20. For 7-day options, we would reject options with strikes at 90 or below even

though several of those also have implied volatilities of 0.20. Moreover, imposing an arbitrary

premium continues to increase the implied volatilities and vegas so that they become RI, when

they were NRI in the absence of an arbitrary pricing premium.24

An obvious question is whether there is an alternative model that would work better. It

seems unlikely that models with more parameters, such as jump-diffusion and stochastic

volatility models, could possibly be free of this problem. Perhaps a less-restrictive model would

be better. It is possible that the model-free implied volatility, which solves for the implied

volatility from the implied probabilities without passing through a formal model, could be a

better alternative. Unfortunately, Jiang and Tian (2005) identify some computational concerns

in that model as well. Moreover, model-based and model-free implied volatilities both require

the existence of a liquid market in a broad range of strikes, which can be a problem for

individual equity options.

5.4 Empirical Estimates of the Impact

To this point we have shown how purely hypothetical options assumed to be priced in a

perfect market by the BSM model with a known volatility produce a volatility smile merely

through the inability of computational methods to return the original volatility. It would be a

valuable exercise to see if we can estimate how much of the smiles observed in practice are

accounted for by this computational problem. The problem, however, is that until this point we

have used hypothetical options with a known volatility. When we turn to the real world of

options trading, however, we run into the problem that the true volatility is not known.

Suppose we use the implied volatility of the option that is closest to at-the-money as a proxy for

the known volatility. At-the-money options are typically the most widely traded, so there are

24We also consider one further possibility for neutralizing the problem. Instead of calculating the implied volatility, it might be better to calculate the implied variance. Unfortunately, the variance vega is also lowest for deep in- and out-of-the-money. It equals the ordinary vega divided by two times the volatility. Thus, for high volatility stocks, the variance vega would be even lower.

21

far more eyes looking at these options. Their implied volatilities should most accurately reflect

the market consensus, even though they may not necessarily represent the true volatility.

Recall that earlier we illustrated the empirical smile for a sample of eBay options.

Figure 10 shows the implied volatilities based on the market prices of the eBay call options

illustrated earlier in comparison to the implied volatilities based on model price of the eBay

options. As noted, we use the implied volatility of the approximately at-the-money option as

the known volatility. The market implied volatilities are higher for the lower strikes, but the

shapes are remarkably similar, particularly at the points at which the curve starts turning

upward both above and below the middle strike.

We now have two sets of volatility smiles, so we can estimate the proportion of the

empirical smile accounted for by computational effects. Suppose that we define the magnitude

of the smile as the sum of the squared deviations of the implied volatilities from the known

volatility based on market prices. Then we define the magnitude of the smile that arises from

this computational effect as the sum of the squared deviations of the computed implied

volatilities from the known volatility based on model prices. We then express the ratio of the

latter to the former and obtain: 2

,1

2

,1

ˆ( )

ˆ( )

N

c ii

N

m ii

For these eBay options, turns out to be approximately 40%. Thus, for these quotes, we

estimate that the computational effect accounts for about 40% of the observed smile.

We now undertake a more formal and extensive empirical analysis. Data are collected

from the OptionsMetrics data set on a quarterly basis (January, April, July, and October) from

2004 to 2010 for options on the current thirty stocks in the Dow Industrial Average as well as

the S&P 500 and Nasdaq indexes, yielding a total of 896 underlying-quarter combinations. All

of these options have approximately one month remaining to expiration. To this point, we have

not addressed American-style, i.e., early exercise, issues and how they might affect these

computational concerns. Nonetheless, it is impossible to provide empirical results without

accounting for the early exercise feature in American-style options, which are the norm for

options on individual stocks. Thus, we compute their values using the binomial model, which

incorporates the effect of early exercise that is driven by dividends. Of course, the S&P 500 and

Nasdaq index options are European-style. All prices are based on the midpoint of the bid-ask

spread.

22

To allow for comparability across the underlying stocks as well as to avoid stale prices

due to nontrading of deep in- or out-of-the-money options, the strike prices are truncated at the

prices closest to 20% above and below the contemporaneous price of the underlying stock. In

some cases, options did not trade in a range wide enough to reach the 20% point, and in those

cases the stock was omitted from the analysis for that quarter. This omission occurred in just

20 of the 896 cases.

We have reason to believe that our estimate of is a lower bound. Recall that in our

hypothetical options, the greatest computational effect occurs well away from the at-the-money

exercise price. But when working with empirical data, if these options are not sufficiently

traded to produce reliable prices, we cannot use them. Truncating our strikes at 20% above and

below will result in loss of some observations for which the computational effect is likely to be

the greatest. Thus, our estimates are clearly lower bounds.

Again, we assume that the implied volatility of the option closest to at-the-money

represents the true volatility. This implied volatility is calculated through the NR algorithm

with a tolerance of one penny and the MK initial estimate. All of the options are then priced

using a binomial tree model with 1,000 time steps. American exercise is taken into account, and

LIBOR is used as the interest rate. This procedure provides a model-based price for each of the

options.

The mid-point of the closing bid-ask spread is used as the market price of the option.

The implied volatility is then calculated for both the model and market prices, again using the

NR algorithm with penny tolerance and the MK initial estimate. These results are used to

calculate the -statistic to provide an estimate of how much of the empirical volatility smile is

due to computational issues.

Table 1 presents the mean, standard deviation, and maximum -metrics for each

underlying. The pooled average is 17.96% with maximum values at times running over 80%.

6. Conclusions

In this paper, we examine a seemingly superfluous question. Setting market

imperfections aside, can one use a known volatility to price an option and then reverse-engineer

the original volatility from that price? The answer would appear obvious: one can surely

recover the original volatility. As we show here, however, recovering the original volatility is not

so straightforward. In fact, recovered volatilities display patterns across strikes that are

remarkably similar to smiles, smirks, and skews. Moreover, these patterns are most pronounced

for short-term options, as is the case for the smiles observed in research and in practice. We

show that while the choice of root-finding technique, tolerance, starting point or interval, and

23

the method of price rounding can produce these patterns, no single explanation dominates the

others.

It is tempting to argue that this effect is a chimera, merely a collateral byproduct of

computational decisions but having no meaningful significance. Yet a review of the literature

shows a remarkable absence of attention to computational specifications, and the effect does not

appear to have been noted by researchers. Suggestions that one could simply discard

observations that are untrustworthy due to excessively low vegas ignore the fact that one must

apply circular logic: the volatility is required to determine the vega, whereupon one would then

use the vega to determine if the volatility can be trusted. Moreover, as we show, when other

factors influence the smile, options with untrustworthy prices will tend to be retained, thereby

embedding this bias into all of the other factors that determine implied volatility. And even if

these concerns were not there at all, there is no evidence in the literature that anyone has ever

discarded observations for this reason. A decision to reject observations is such an important

one in research design that it surely would have been reported.

Can the problem be fixed? We come closest to eliminating these patterns by carrying

the option prices out to quadruple precision and demanding machine epsilon tolerance. Yet,

implementing the former requirement would require trading in prices with 32 decimals. It took

the exchange-listed options industry over 30 years to move from sixteenths to pennies, and even

with penny pricing, we can still observe volatility smiles with options priced by Black-Scholes-

Merton. And surely, trading in 32 decimal places would exact other extremely high costs.

We make no claim that the effect shown here is the entire source of the smile. Our

empirical estimates suggest that a lower bound is around 18%, but we find considerable evidence

that at times this effect can be extremely high, as shown in Table 1. Even under the most

conservative estimate possible, if approximately one-fifth of a result is driven by decisions made

by the researcher and there is no obvious fix, there are considerable grounds for concern in both

research and in trading. We are left with the disconcerting conclusion that the volatility smiles

observed in practice could be significantly biased, which can lead to false pricing signals, poor

trading strategies, and arbitrage and hedge transactions that should be nearly risk-free but in

fact contain a disturbingly wide zone of uncertainty.

Appendix - 1

Appendix A. Proof of Theorem 1

Using the second-order Taylor theorem for F(x), we have

2( )( ) ( ) ( )( ) ( ) ,

2!n

n n n n

FF x F x F x x x x x

where n is between x and xn and ˆx . Using the Newton-Raphson iteration formula in (6),

we have

21

( )ˆ ˆ( )

2 ( )n

n n

n

Fx x

F x

for every n = 0, 1, 2, …

Suppose that ˆrf

. If we choose the initial point 0

,rf

x then we have

01 0 0

0

( ),

( )

F xx x x

F x

since F(x0) > 0 and F´(x0) > 0. We also have

211 1

1

( )ˆ ˆ( ) 0,

2 ( )

Fx x

F x

since 1 satisfies 1ˆ

rf and F"(1) > 0. Therefore, 1 0ˆ

rfx x . Suppose this holds

for n, and we prove the n + 1 case. With ˆ n rfx , we have

1

21

( ),

( )( )

ˆ ˆ( ) 0,2 ( )

nn n n

n

nn n

n

F xx x x

F xF

x xF x

where n is between ˆand n rfx . Therefore, 1

ˆn n rfx x holds for all n. The Newton-

Raphson method is convergent since the sequence is strictly decreasing and bounded below.

The implied volatility is the unique solution for the root of F, so the algorithm must converge.

The errors are

2 21 1

2

1

2

1 1 2

12 ( ( ) ( ))/2

1 2

ˆ ( )ˆ

2 ( )

ˆ ( ( )) ( ) ( )

2 ( ( ))

ˆ ( ) ( ),

2

n n

n nn

n

n n n n

n n

d d x

n n n

n

x Fx

F x

x n d d d

n d x

x d de

where the second identity follows from the vega and vomma formulas. From the decreasing

property for the functions d1(n) and d1(n), we obtain the inequalities,

Appendix - 2

2 21 1

2 21 1

2( ( ) ( ))/2

1 21

ˆ( ( ) ( ))/2

1 2

ˆˆ ˆ( ) ( )ˆ

ˆ 2ˆˆ ˆ( ) ( )

ˆˆ 2

ˆ ,

n n

n

d d x

n

n

d x d

rf

n

n

xd dx e

d de x

x

Similarly for the case d2 < d1 < 0 with 1( ) 0rf

d , the algorithm is convergent if 2 21 1 1 ˆ( ( ) ( ))/2

1 2ˆˆ ˆ( ) ( )

1.ˆ 2

d x d

rfd de

The proof of the second case (ii) is the same, where d1 is positive and increasing and d2 is

decreasing and negative.

Appendix B. Estimation Effects of the Normal Probability

In this appendix, we demonstrate that the estimation of the normal probability can lead

to errors that can generate a non-flat curve of the implied volatility against the strike. Let cM

be the observable and known current market price of a European call option. This value is then

taken as an input to calculate the implied volatility in the BSM option pricing model. The

implied volatility, denoted ˆ , is defined from the following nonlinear equation:

1 2

2

1

2 1

ˆ ˆ( , , , , ) ( , , , , ) ,

and

ˆln

2ˆ( , , , , )

ˆˆ ˆ ˆ( , , , , ) ( , , , , )

rMc SN d S K r Ke N d S K r

Sr

Kd S K r

d S K r d S K r

(B.1)

in which the unknown parameter ˆ is a function of the strike price K for the purpose of this

paper. The risk-free interest rate r is assumed to be known, and the underlying asset price St is

observed at the present time. Hence, the three parameters S, τ, and r are treated as constants

throughout this paper.

The nonlinear equation (B.1) can be solved for the implied volatility. The value of a call

option is strictly increasing with respect to volatility, which is to say that the vega is

monotonically increasing, 0c

. This condition guarantees that if a solution exists, it

will be unique. If options are priced in an arbitrage-free market, then the following bounds are

known for the value of the call option: (S – Ke-r)+ ≤ cM < S. Collectively, these conditions

guarantee the existence of a unique, nonnegative value for implied volatility, so the problem is

well defined.

Appendix - 3

The standard function to approximate the Gaussian cumulative distribution is given by

2

2 1/2

0

( 1)1 1 1( ) .

2 !2 (2 1)2 2

x k kz

kk

xN x e dz

k k (B.2)

The series in (B.2) is absolutely convergent everywhere. Thus, the nonlinear equation in (B.1)

can be rewritten in terms of the power series:

2 1 2 11 2

0

( 1)1ˆ ˆ( , , , , ) ( , , , , ) .

2 !2 (2 1)2

krk r k

M kk

S Kec Sd S K r Ke d S K r

k k (A.3)

The numerical method to solve the implied volatility is to truncate the power series into

a polynomial and to solve the implied volatility by the Newton-Raphson method for root

finding, as in Manaster and Koehler (1982). The method is illustrated below by varying the

truncation of the series.

B.1. The implied volatility 0ˆ for k = 0

Consider k = 0 in the power series (A.3) for the linear approximation ˆ of the implied

volatility ˆ . The following identity holds as a linear approximation of the market price of the

European call option from the linear approximation ˆt of the implied volatility ˆ in (B.1),

1 0 2 0

0 01 1 1 2

0 00

02 1

00

1ˆ ˆ( , , , , ) ( , , , , )

2 2

ˆ ˆ( ) 1

ˆ ˆˆ

ˆ( ) 1.

ˆˆ

rr

M

S Kec Sd S K r Ke d S K r

d d d d d

dK K K KK

d d d

dK KK

(B.4)

Taking the partial derivative with respect to K on both sides of equation (B.4), we obtain25

2

0 0

0 2 1

2

0

1 1 0

0

2 2 0 0

2 1( )

2ˆ ˆ1

ˆ

where

ˆln

2ˆ, , , ,

ˆ

ˆ ˆ, , , ,

r r

r

d e S KeK

K Sd Ke d

Sr

Kd d S K r

d d S K r

(B.5)

Examination of the expression for 0ˆ

Kin (B.5) shows that it need not be zero, so it can produce

non-constant implied volatilities across strikes.

25Unlike Corrado and Miller (1996), we do not intend to solve the implied volatility directly as a function of the strike price, since our purpose it to understand the implied volatility smile. Hence, we are only interested in understanding the sign behavior of the slope of the implied volatility (i.e., the fact that the slope has a minimum strike price that creates the upward curve of the volatility smile).

Appendix - 4

B.2. The implied volatility 1̂ for k = 1

Consider the power series (A.3) for the second term approximation 1̂ of the implied

volatility ˆ . The following identity holds as the second term (cubic) approximation of the

market price of the European call option from the cubic approximation of the implied volatility.

1 1 2 1

3 31 1 2 1

1ˆ ˆ( , , , , ) ( , , , , )

2 21 1

ˆ ˆ( , , , , ) ( , , , , )62

rr

M

rt

S Kec Sd S K r Ke d S K r

Sd S K r Ke d S K r (B.6)

Similar to the previous case, we take the partial derivative with respect to K on both sides of

equation (B.6) and obtain

3 2 22 1 2

2

1 1 1

1 1 22 1 1 2

21

1 1 1

1

2 1 1

2 1 1( )

2 6 2 2ˆ ˆ ˆ1

ˆ

2where

ˆln

2ˆ, , , ,

ˆ

ˆ, , , ,

r r r

r r

d d dd e S Ke S Ke

K K

K d dSd Ke d Sd Ke d

Sr

Kd d S K r

d d S K r1̂.

(B.7)

The expression for 1̂

Kin (B.7) reveals that the second term (cubic) approximation from

the standard Gaussian cumulative distribution function does cause a potential smile effect in the

computed implied volatility, because the numerator in (B.7) can change sign from positive to

negative. Consider the numerator of (B.7), reformulated as follows:

3 2 22 1 2

2

1 1

2 1 1.

2 6 2 2ˆ ˆ

r r r rd d dd e S Ke e S Ke

K K

From this, we have the explicit equation for the strike price K1 that is necessary to reach the

minimum of the implied volatility. That is, 11

ˆ0K

Kin which K1 is a solution of the

following equation:

3 2 22 1 2

2

1 1

2 1 1.

2 6 2 2ˆ ˆ

r r r rd d dd e S Ke e S Ke

K K (B.8)

Examination of this expression shows that the numerical approximation of the Gaussian

cumulative distribution function can produce non-constant implied volatilities across strikes.

B.3. The implied volatility ̂1 for k = n

Appendix - 5

The power series (A.3) for the n-th term approximation n̂ of the implied volatility ˆ .

The following identity holds as the n-th term ((2n + 1)-polynomial) approximation of the

market price of the European call option from the (2n + 1)-polynomial approximation n of the

implied volatility ˆ in (A.3):

1 2

2 1 2 1

0

1 ( 1)ˆ ˆ( , , , , ) ( , , , , )

2 !2 (2 1)2

r kk r k

M kk

S Kec Sd S K r Ke d S K r

k k

(B.9)

By taking the partial derivative with respect to K, we obtain

2 1 2 21 22 1 2

0 0

2

1 1

1

2 1

( 1) ( 1)2

2 !2 (2 1) !2

where

ˆln

2ˆ, , , ,

ˆ

ˆ ˆ, , , , .

k kn nk r k r k

k kk k

n

n

n n

dd ddd e Sd Ke d

dK dKk k k

Sr

Kd d S K r

d d S K r

Combining 1dd

dKand 2dd

dKwith simplifications, we get

2 2

1 2 2 1

0

2 1 2 2

2 1 2

0 0

ˆ( 1) 1

ˆ!2

2 ( 1) 1 ( 1).

2 !2 (2 1) !2ˆ

knk r k n

kk n

k kn nk r k r k

k kk kn

Sd d Ke d dk K

d e Sd Ke dk k kK

(B.10)

In order to understand the smile effect, we need to understand when there is a minimal K* such

that the K* is a solution of the following equation:

* *

* * *

* *

*

2 1 2 2

2 1 2* *0 0

2 ( 1) 1 ( 1)0

2 ˆ!2 (2 1) !2

k kn nk r k r k

tk kk kn

d e S d Ke dKk k k

(B.11)

This depends solely upon the level of the order of approximation, n. Note that as n , the

left-hand side of the above equation approaches

* *

* * *

* ** *

2 1 2 22 1 2* * *

0 0

2

( 1) ( 1)2 1

2 !2 (2 1) !2ˆ

2 ( ) 0.

k kn nk r k r k

tk kk kn

r

d e S d Ke dk k kK

N d e

Therefore, there exists a critical level n0 such that when n > n0, equation (B.11) has no solution

for K*. Hence, the problem disappears when the approximation becomes increasingly accurate.

The theoretical results are summarized in the following theorem.

Theorem B1. Let the implied volatility ˆ at time t be defined as in the nonlinear equation (B.1)

,

1 2

ˆ ˆ( , , , , ) ( , , , , ) ,rMc SN d S K r Ke N d S K r

Appendix - 6

With 2 1

0

( 1)1( )

!2 (2 1)2

k k

kk

xN x

k k. The n-th term numerical approximation ˆ

n of the implied

volatility ˆ in (B.1) from the market price of the European call option is given by equation (B.9),

1 2

2 1 2 1

0

2

1 1

1

2 1

1 ( 1)ˆ ˆ( , , , , ) ( , , , , ) ,

2 !2 (2 1)2

ˆln

2ˆ, , , ,

ˆ ˆ, , , , .

r knk r k

M n nkk

n

n

n n

S Kec Sd S K r Ke d S K r

k k

Sr

Kd d S K r

d d S K r

Then the following three propositions hold:

(1) The rate of change of the n-th level implied volatility ˆn with respect to the strike K is

given by the formula (B.10)

(2) There is a critical level n0 for the accuracy of the approximation such that for every n ≥

n0 and n = 0, no smile effect occurs. That is to say,ˆ

0n

K (there is no solution for

equation(B.7)).

(3) There are possible solutions for 1 ≤ n < n0 such that the small n-th level approximation

of the implied volatilityn̂has a potential smile effect. That is to say,

ˆ0n

K

for some

strike K*.

This theorem explains how computational approximations of the cumulative normal density can

induce a smile effect in implied volatilities. At a minimum, they can lead to non-constant

implied volatilities across strikes and in some cases those volatilities can turn down and then

upward, in the pattern of a smile. Alternatively, they can appear like skews and smirks. The

important point is that the slope of the implied volatility with respect to strike may not be zero

due to the inaccuracy of the approximation of the normal distribution.

We have also developed these relationships with dividends on the underlying, but the

results are not appreciably altered.

Appendix C. Example of the Implied Volatility Solution for Newton-Raphson

Consider a 30-day call struck at 80 in which the underlying is at 100, the true volatility

is 20%, and the risk-free rate is 4%. The procedure we follow is to price the option using the

BSM formula. We then insert the BSM price back into the formula and using an iterative

search method, we attempt to arrive back at the 20% volatility. For this illustration we use the

Newton-Raphson method and stop iterating when the difference between the BSM price

obtained using 20% and the price produced by the iteration is less than 0.01. We illustrate the

Appendix - 7

steps below and show the results using 12 decimal places, though except where noted, we retain

full precision.

Step 1: Obtain the BSM price using volatility of 20%:

Result: c = 20.262627603558

For the illustration we round this to the nearest penny, 20.26. We will also show the results

with this value not rounded.

Step 2: Determine the starting point of the Newton-Raphson method using the Manaster-

Koehler formula

The Manaster-Koehler starting point guarantees convergence. Here the starting point is

234.7301943788%. We retain precision of this value by not rounding it.

Step 3: Iterate until the price of the ith iteration is within the absolute value of the tolerance of

0.01. The results are illustrated in the table below. The numbers below are rounded for display

purposes but are not rounded in the computations. In Newton-Raphson, the increment is

obtained as the difference divided by the derivative.

Iteration

#

Volatility Guess Price Difference Derivative Increment

1 2.347301943788 35.082353410446 -14.8223534104464 9.119823440372 -1.62528951

2 0.722012429429 21.541814035481 -1.2818140354811 5.584540218349 -0.22952902

3 0.492483410859 20.552511188393 -0.2925111883927 2.815679594479 -0.10388653

4 0.388596878824 20.339883967690 -0.0798839676898 1.292605777194 -0.06180072

5 0.326796158837 20.284193999601 -0.0241939996008 0.549936602275 -0.04399416

6 0.282801997509 20.268222705577 -0.0082227055769 0.206590213306 -0.03980201

As shown, the process converges in six iterations. The result is 28.28%, which is about 41%

higher than the volatility used to price the option. Had the original option price not been

rounded, the solution would differ only slightly at 28.96%.

2ˆ ln

2 100ln 0.04(0.082192)

0.082192 80

2.347301943788

i

Manaster Koehlor

SrT

T K

Appendix - 8

Although not shown here, the results are further off at 14.67% for the bisection method

and 234.72% for Dekker-Brent. The regula falsi and secant methods do not converge at all.

With no price rounding, the results are 29.33% for the bisection method and 0.0014% for the

Dekker-Brent method. There is no convergence for the regula falsi and secant methods.

Figure 1. Volatility Smile, eBay Call Options, January 18, 2013, expiring February 16, 2013.

The stock price is $54.21 and the risk-free rate is 0.05%. The option prices are taken from the

CBOE’s web site at the close of business.

0.10

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B. Option Prices in Pennies

C. Continuous Option Prices

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Figure 3. Volatility Smiles, Secant Method

A. Option Prices in Sixteenths

B. Option Prices in Pennies

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Figure 4. Volatility Smiles, Regula Falsi Method

A. Option Prices in Sixteenths

B. Option Prices in Pennies

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Figure 5. Volatility Smiles, Bisection Method

A. Option Prices in Sixteenths

B. Option Prices in Pennies

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C. Continuous Option Prices

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Figure 6. Volatility Smiles, Dekker-Brent Method

A. Option Prices in Sixteenths

B. Option Prices in Pennies

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Figure 7. Implied Volatilities for Calls

Stock price = $100, strike ranges from $75 - $125 in $2.50 intervals, risk-free rate = 4%, true

volatility = 20%, and expirations of 180 days (0.4986), 90 days (0.2466), 30 days (0.0822), and 7

days (0.0192) as shown; solution is obtained using Newton-Raphson with tolerance of $0.01.

Option prices are not rounded.

A. Implied Volatilities

B. Vegas

Figure 8. Difference between True Vegas and Proxy Vegas

Proxy vegas are obtained by solving for the implied volatilities and using the computed implied

volatilities to obtain the vegas.

Figure 9. Implied Volatilities for Calls with 1% Price Premium

Stock price = $100, strike ranges from $75 - $125 in $2.50 intervals, risk-free rate = 4%, true

volatility = 20%, and expirations of 180 days (0.4986), 90 days (0.2466), 30 days (0.0822), and 7

days (0.0192) as shown; solution is obtained using Newton-Raphson with tolerance of $0.01.

Option prices are calculated using the Black-Scholes-Merton model and amplified by 1% and not

rounded.

A. Implied Volatilities

B. Vegas

Figure 10. Volatility Smile, Market and Model Prices, of eBay Call Options, January 18, 2013,

expiring February 16, 2013. The stock price is $54.21 and the risk-free rate is 0.05%. The

option market prices are taken from the CBOE’s web site at the close of business.

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Market prices Model prices

Table 1. Means, Standard Deviations, and Maximum Values of the Relative Proportion of the

Smile Accounted for by Computational Error

Estimates of the statistic for the Dow Jones 30 stocks and the S&P 500 and Nasdaq over

2004-2010. Data are taken at quarterly intervals with approximately one month to maturity.

The implied volatility of the at-the-money option is used as the true volatility.

(table continues on next page)

Stock Mean Standard Deviation Maximum

AA 18.99 24.08 87.69

AXP 11.14 13.72 40.49

BA 8.35 13.23 61.57

BAC 27.00 29.11 94.19

CAT 7.98 15.77 74.04

CSCO 9.76 15.74 78.45

CVX 18.55 20.04 75.14

DD 17.68 16.87 44.16

DIS 25.09 23.15 77.61

GE 30.76 25.12 77.61

HD 24.53 23.49 82.72

HPQ 11.99 20.12 85.56

IBM 17.23 18.95 79.97

INTC 17.21 21.91 91.98

JNJ 27.92 19.38 63.94

JPM 19.16 24.01 88.85

KO 24.47 17.84 50.81

MCD 17.67 16.82 64.52

MMM 16.12 16.26 50.40

MRK 13.78 19.67 90.74

MSFT 28.61 26.42 89.16

NDX 6.38 8.67 24.90

PFE 30.29 24.41 83.39

PG 15.84 24.56 53.44

SPX 14.98 16.17 47.52

T 18.91 20.86 95.72

TRV 5.30 9.50 33.66

UNH 10.52 16.11 67.11

UTX 14.11 13.83 59.28

VZ 28.65 25.65 96.34

WMT 25.53 21.98 90.56

XOM 10.32 14.12 62.00

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