a new formula for computing implied volatility

Applied Mathematics and Computation 170 (2005) 611–625

www.elsevier.com/locate/amc

A new formula for computingimplied volatility

Steven Li

School of Economics and Finance, Queensland University of Technology,

Brisbane QLD 4001, Australia

Abstract

This paper considers the explicit formulas for computing the implied volatility from

the Black–Scholes option pricing model. The existing formulas in the literature are sum-

marized and a uniform framework for deriving the formulas is given. A new explicit for-

mula for computing the implied volatility is provided. The new formula is valid for a

wide band of option moneyness and time to expiration. It is shown that the new formula

is more accurate than the existing ones. Moreover, the new formulas can be easily imple-

mented in spreadsheet applications. Thus the proposed formula is particularly impor-

tant for the calculation of intra-day implied volatility in real time.

� 2005 Elsevier Inc. All rights reserved.

Keywords: Options; Volatility; Implied volatility

0096-3003/$ - see front matter � 2005 Elsevier Inc. All rights reserved.

doi:10.1016/j.amc.2004.12.034

E-mail address: [email protected]

mailto:[email protected]

612 S. Li / Appl. Math. Comput. 170 (2005) 611–625

1. Introduction

In the classic option pricing framework developed by Black and Scholes [1]

and Merton [2], the value of a European call option on a non-dividend paying

stock 1 is stated as:

C ¼ SNðd1Þ � X e�rT Nðd2Þ; ð1Þwhere

d1 ¼lnðS=X Þ þ ðr þ r2=2ÞT

rffiffiffiffiT

p ; ð2Þ

d2 ¼lnðS=X Þ þ ðr � r2=2ÞT

rffiffiffiffiT

p ¼ d1 � rffiffiffiffiT

p; ð3Þ

NðxÞ ¼ 1ffiffiffiffiffiffi2p

pZ x

�1e�u2=2 du: ð4Þ

The stock price, strike price, interest rate, time to expiration and volatility

are denoted by S, X, r, T, r, respectively.A useful property of the Black–Scholes option pricing model is that all

model parameters except the volatility are directly observable from market

data. This allows a market-based estimate of a stock�s future volatility. Origi-nally suggested by Latane and Rendleman [3], implied volatilities are exten-

sively used in financial markets research. An implied volatility is the value of

the volatility that, when employed in the Black–Scholes formula, results in a

model price equal to the market price.

Unfortunately, a closed-form solution for an implied volatility from Eq. (1)

is not possible. Thus the implied volatility must be calculated numerically. In

general, this calculation is accomplished by feeding the value-price difference:

C(r) � CM into a root-finding program, where C(Æ) is an option pricing for-mula, r is the volatility parameter, and CM is the observed market price of

the option. Various algorithms can be used to find the value of r that makes

this expression equal to zero. Thus the calculation of implied volatility often

requires programming and numerical techniques. For example, Manaster

and Koehler [4] discuss the Newton–Raphson algorithm for calculating implied

volatilities.

As illustrated above, implied volatilities can be easily calculated by using

iterative techniques. However, to simplify some applications such as spread-sheets, it may be useful to have a closed-form approximation solution if that

1 Though the results presented in this paper can easily be extended to dividend paying stock

options and commodity options etc., the discussions here are restricted to non-dividend paying

stock options for the sake of simplicity.

S. Li / Appl. Math. Comput. 170 (2005) 611–625 613

solution is reasonable, simple, accurate and valid for a wide range of cases. The

cost and inconvenience of iterating also motivate the search for explicit formu-

las. For example, traders often need to plot intra-day implied volatility in real

time. In this case, the non-numerical approach such as using an explicit for-

mula is a must (cf. [4]). So far, a few approximation formulas have been

proposed.Brenner and Subrahmanyam [6] provide an elegant formula to compute an

implied volatility that is accurate when a stock price is exactly equal to a dis-

counted strike price. Their formula is as follows:

r �ffiffiffiffiffiffi2pT

rCS: ð5Þ

Feinstein [7] independently derived an essentially identical formula.

The accuracy of the well-known Brenner–Subrahmanyam [6] formula (5) de-

pends on the assumption that a stock-price is equal to a discounted exercise

price. Following Brenner and Subrahmanyam, an at-the-money option is

defined as one with S = Xe�rt. For convenience, the discounted strike price is

denoted by K (i.e. K = Xe�rT) throughout this paper.Corrado and Miller [8] provide an improved quadratic formula which is

valid when stock prices deviate from discounted strike prices. Their formula

is given as:


r1

S þ KC � S � K

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC � S � K

2

� �2

� ðS � KÞ2

p

s24

35: ð6Þ

This formula yields quite accurate estimates for some cases. Bharadia et al.

[9] derive a highly simplified volatility approximation as:


rC � ðS � KÞ=2S � ðS � KÞ=2 : ð7Þ

As pointed out by Chambers and Nawalkha [10], Formula (7) is less accu-

rate than Formula (6) in general. Thus it is natural to take Formula (6) as abenchmark for any new formulas to be developed.

Most of the above explicit formulas are based on the Taylor series expan-

sions, but some do involve guesswork to a certain degree. This paper aims to

offer a uniform approach in deriving the approximation formulas for calculat-

ing the implied volatility and to derive some formulas, which are more accurate

and applicable, by using higher order Taylor series expansions.

It should be mentioned that Chance [11] provides a direct method of obtain-

ing an accurate estimate of the implied volatility of a call option. His estimate isbased on the formula for at-the-money options developed by Brenner and

Subrahmanyam [7]. The adjusted formula by Chance [11] is quite accurate

for options no more than 20% in- or out-of-the-money and is simple to


program and compute. More recently, Chambers and Nawalkha [10] develop a

simplified extension of the Chance [11] model. The approach taken in these two

papers uses the first and second derivatives of the call price with respect to vol-

atility. In addition, they need a reasonable estimate of volatility to serve as a

starting point to the approximation.

Since the publication of the seminal works of Black and Scholes [1] andMerton [2], a large amount of research has been devoted to the option pricing

models. It has been noticed for some time the existence of the so-called volatil-

ity smile, which implies some deficiencies of the Black–Scholes formula. Con-

sequently, stochastic volatility models including Hull and White [12] and

Heston [13] have been proposed. Nevertheless, many traders still use the im-

plied volatility from the Black–Scholes model to forecast future volatility

(Van Den Brink [5]). Hence an improved explicit formula for calculating im-

plied volatility from the Black–Scholes model is still of some practical value.The rest of the paper is organized as follows. In Section 2, an explicit for-

mula for calculating the volatility of an at-the-money call is developed. This

formula is assessed and tested against Formula (5). In Sections 3, a new

quadratic formula similar to the quadratic formula (6) is first derived for deep

in- or out-of-the-money calls. Then a new formula for nearly at-the-money op-

tions or options with high volatility or long time to maturity is developed. By

combining these two formulas, a generic explicit formula is proposed. This gen-

eric formula is assessed against Formula (6). A real-world application of thegeneric formula is also given. In Section 4, the major conclusions are presented.

The technical derivations are provided in Appendix A.

2. At-the-money calls

In this section, at-the-money calls are considered. That is, S = K = Xe�rT is

assumed throughout this section. In order to approximate the Black–ScholesEq. (1). The Taylor series expansion to the third order for the standard normal

cumulative distribution function N(x) can be used [15]:

NðxÞ ¼ 1

2þ 1ffiffiffiffiffiffi

2pp x� 1

6ffiffiffiffiffiffi2p

p x3 þ Oðx5Þ: ð8Þ

Substituting the expansions for the cumulative distribution functions

Nð12r

ffiffiffiffiT

pÞ and Nð� 1

2r

ffiffiffiffiT

pÞ into Eq. (1) results in the following equation:ffiffiffiffiffiffi

2pp

CS

� 2n � 1

3n3; ð9Þ

where n ¼ 12r

ffiffiffiffiT

p. This equation can be solved by using the cubic formula [14].

After some tedious derivation and simplification, the following formula is

obtained:

Table 1

Comparison of Formula (10) and the Brenner–Subrahmnyam formula (5) for at-the-money calls

Time to

expiration

True volatility

15% 35% 55% 75% 95% 135%

Formula

(5)

Formula

(10)

Formula

(5)

Formula

(10)

Formula

(5)

Formula

(10)

Formula

(5)

Formula

(10)

Formula

(5)

Formula

(10)

Formula

(5)

Formula

(10)

0.1 �0.000013 0.000001 �0.000177 0.000001 �0.000691 0.000002 �0.001753 0.000004 �0.003560 0.000012 �0.006306 0.000032

0.2 �0.000027 0.000001 �0.000356 0.000001 �0.001383 0.000004 �0.003501 0.000015 �0.007097 0.000049 �0.012550 0.000128

0.3 �0.000042 0.000001 �0.000535 0.000001 �0.002072 0.000007 �0.005240 0.000034 �0.010610 0.000111 �0.018732 0.000293

0.4 �0.000056 0.000001 �0.000713 0.000002 �0.002760 0.000013 �0.006973 0.000060 �0.014099 0.000200 �0.024853 0.000530

0.5 �0.000070 0.000001 �0.000891 0.000002 �0.003447 0.000020 �0.008698 0.000095 �0.017564 0.000316 �0.030915 0.000840

0.6 �0.000084 0.000000 �0.001069 0.000003 �0.004131 0.000029 �0.010415 0.000138 �0.021006 0.000459 �0.036917 0.001227

0.7 �0.000098 0.000000 �0.001246 0.000004 �0.004815 0.000039 �0.012126 0.000188 �0.024426 0.000631 �0.042860 0.001695

0.8 �0.000112 0.000000 �0.001424 0.000005 �0.005496 0.000051 �0.013829 0.000248 �0.027822 0.000832 �0.048746 0.002247

0.9 �0.000126 0.000000 �0.001601 0.000007 �0.006176 0.000065 �0.015525 0.000316 �0.031196 0.001064 �0.054575 0.002889

1 �0.000140 0.000000 �0.001778 0.000008 �0.006855 0.000081 �0.017214 0.000392 �0.034547 0.001327 �0.060347 0.003624

1.1 �0.000154 0.000000 �0.001955 0.000010 �0.007532 0.000098 �0.018896 0.000477 �0.037876 0.001622 �0.066063 0.004458

1.2 �0.000168 0.000000 �0.002132 0.000012 �0.008207 0.000117 �0.020571 0.000571 �0.041182 0.001951 �0.071724 0.005395

1.3 �0.000182 0.000000 �0.002309 0.000014 �0.008881 0.000138 �0.022238 0.000675 �0.044467 0.002314 �0.077330 0.006443

1.4 �0.000196 0.000001 �0.002485 0.000016 �0.009553 0.000160 �0.023899 0.000788 �0.047730 0.002713 �0.082882 0.007608

1.5 �0.000210 0.000001 �0.002661 0.000019 �0.010224 0.000185 �0.025554 0.000910 �0.050972 0.003149 �0.088381 0.008896

Each entry represents the estimation error: the difference between the true volatility and estimated volatility (estimated volatility–true volatility). The implied volatilities are

estimated for at-the-money calls with an exercise price of $100, and risk free-rate of 5% p.a. The prices of all calls used in this table are generated with the Black–Scholes

model for a given volatility ranging from 15% to 135% and maturity ranging from 0.1 to 1.5 years. The average error by using Formula (10) is 0.023010 less than that by

Formula (5).

S.Li/Appl.Math.Comput.170(2005)611–625

615


r ¼ 2ffiffiffi2

pffiffiffiffiT

p z� 1ffiffiffiffiT

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8z2 � 6affiffiffi

2p

z

s; ð10Þ

where a ¼ffiffiffiffi2p

pC

S and z ¼ cos½13cos�1ð 3affiffiffiffi

32p Þ.

A few remarks are in order. First, one can show that 8z2 � 6affiffiffi2z

p > 0 holds. 2

Further, 0 < 3affiffiffiffi32

p < 1 holds as long as 0 < CS <

ffiffiffiffi32

p

3�ffiffiffiffi2p

p ¼ 0:7522, which is valid for

nearly all options in reality. Thus, Formula (10) is valid for nearly all at-the-

money calls.

It follows from the derivation that Formula (10) is of higher order of accu-

racy than Formula (5). This can also be verified by a numerical test. The two

factors that impact the accuracy of Formula (10) are the volatility r and the

time to maturity T. Thus Formula (10) can be tested by using options with a

range of different maturities and volatilities.Table 1 provides a numerical comparison of the accuracy of Formula (10)

with that of the Brenner–Subrahmanyam formula (5). The estimation errors

are given for both Formulas (5) and (10). The implied volatilities are estimated

for at-themoney calls with an exercise price of $100, and a risk-free rate of 5%per

annum. The prices of all calls used in the test are generated with the Black–Scho-

les model (1) for a given volatility (true volatility) ranging from 15% to 135%.

Table 1 reveals that Formula (10) yields consistently more accurate estimate

than Formula (5) in all cases. Moreover, the estimation error by Formula (10)is roughly one-tenth of that by Formula (5) for each case considered in Table 1.

A more detailed calculation shows that the estimation error by Formula (10)

on average is 0.023 less than that by Formula (5). Note that average true

volatility is about 68%. Thus Formula (10) is on average about 3.4% more

accurate than Formula (5).

It should also be noted that Formula (10) can be easily implemented on

spreadsheet applications or calculators. Further more, due to its higher accu-

racy, replacing Formula (5) by Formula (10) in the Chance [11] model leadsto higher accuracy virtually with barely any extra effort.

3. In- or out-of-the-money calls

In this section, in- or out-of-the-money calls are considered. For this pur-

pose, it is necessary to introduce the following variable:

g ¼ KS: ð11Þ

Note that g measures the moneyness of an option: g = 1 represents at-the-money, g > 1 represents out-of-the money and g < 1 represents in-the-money.

2 The proof is available upon request.


Thus g is likely to be an important variable for obtaining an appropriate for-

mula for in- or out-of-the-money options. Using g and n, the Black–ScholesEq. (1) can be rephrased as:

CS¼ Nðd1Þ � gNðd2Þ; ð12Þ

where d1 ¼ � ln g2n þ n, d2 ¼ � ln g

2n � n. Note that the right hand side of (12) is afunction of n and g. Using Taylor series expansions, an algebraic equation for ncan be obtained similarly as before. This equation can be solved subsequently

to produce new formulas for r. The details of the derivation for the formulas

given below are included in Appendix A. Two special cases are first considered.

Then the formulas for the two special cases are combined to yield a generic ex-

plicit formula.

3.1. Deep in- or out-of-the-money calls

For deep in- or out-of-the-money options, the following formula is

obtained:

r �~a þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~a2 � 4ðg�1Þ2

1þg

q2

ffiffiffiffiT

p ; ð13Þ

where

~a ¼ffiffiffiffiffiffi2p

p

1þ g2CS

þ g � 1

� : ð14Þ

Note that Formula (13) reduces to Formula (5) when g = 1. This formula isparticularly accurate when n2 � jg � 1j, where ‘‘�’’ means ‘‘far less than’’.

This condition is equivalent to r �ffiffiffiffiffiffiffiffijg�1jT

q. Thus Formula (13) is more accurate

for options with small volatility, short time to expiration or deep in- or out-of -

the-money. Note that Formula (13) is equivalent to:

r � 1ffiffiffiffiT

pffiffiffiffiffiffi2p

p

S þ KC � S � K

2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiC � S � K

2

� �2

� ðS � KÞ2

p1þ K=S

2

s24

35:ð15Þ

This is slightly different from Formula (6) only in the last term under the

square root. Thus the difference between Formulas (13) and (6) can be ignored

in most cases. 3

3 For example, for g = 1.10 or 0.9, the difference between the two estimates is only about S/

10,000.


3.2. Nearly at-the-money calls

Another special case is when r �ffiffiffiffiffiffiffiffijg�1jT

q, where ‘‘�’’ means ‘‘far bigger

than’’. In this case, the following formula is obtained:

r � 2ffiffiffi2

pffiffiffiffiT

p ~z� 1ffiffiffiffiT

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8~z2 � 6~affiffiffi

2p

~z

s; ð16Þ

where ~z ¼ cos½13cos�1ð 3~affiffiffiffi

32p Þ and ~a is defined by (14). It should be noted that For-

mula (16) is very similar to Formula (10) and it reduces to Formula (10) when

g = 1. Note that ~a > 0 is equivalent to C > S�K2, which holds true as the call

price is bounded below by max(0,S � K). Further, it can be shown that3~affiffiffiffi32

p < 1 holds as long as C < ð12þ 2

3ffiffip

p ÞS ¼ 0:88S. Therefore, Formula (16) isvalid for almost all (nearly) at-the-money options.

Note that Formula (16) yields more accurate volatility estimate when the

option under consideration is nearly at-the-money, or when it has a long time

to maturity or high volatility.

3.3. The combination

In this subsection, the combination of Formulas (13) and (16) is considered.

Note that (16) holds when r �ffiffiffiffiffiffiffiffijg�1jT

q. Using Formula (5), this condition is

roughly equivalent to 2pðCS Þ2 � jg � 1j. This suggests that one can use the fol-

lowing variable for combining Formulas (13) and (14):

q ¼ jg � 1jðC=SÞ2

¼ jK � SjSC2

: ð17Þ

A rough numerical test shows that if q 6 1.4, Formula (16) should be chosenand otherwise Formula (13) should be chosen. In sum, the following generic

formula is proposed:

r �

2ffiffi2

pffiffiffiT

p ~z� 1ffiffiffiT

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi8~z2 � 6~affiffi

2p

~z

qif q 6 1:4;

~aþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~a2�4ðg�1Þ2

1þg

q2ffiffiffiT

p if q > 1:4;

8>><>>: ð18Þ

where ~z ¼ cos½13cos�1ð 3~affiffiffiffi

32p Þ and ~a is defined by (14). Note that Formula (10) can

be treated as a special case of Formula (18). Thus Formula (18) is generic and

valid regardless of the option moneyness. It should also be noted that Formula

(18) can be easily implemented in spreadsheet applications such as Microsoft

Excel.

Table 2 presents a numerical comparison of Formulas (18) and (6) for in-

the-money calls. Each entry represents the estimation error: the difference

Table 2

Comparison of Formula (18) and the Corrado–Miller formula (6) for in-the-money calls

Time to

expiration

True

volatility

15% 35% 55% 75% 95% 115% 135%

Formula

(6)

Formula

(18)

Formula

(6)

Formula

(18)

Formula

(6)

Formula

(18)

Formula

(6)

Formula

(18)

Formula

(6)

Formula

(18)

Formula

(6)

Formula

(18)

Formula

(6)

Formula

(18)

0.1 �0.026300 �0.014890 �0.000994 0.000059 �0.000993 �0.000368 �0.001989 �0.001539 �0.003803 �0.003451 �0.006576 �0.006285 �0.010487 �0.0102400.2 �0.002937 �0.001356 �0.000610 �0.000113 �0.001549 �0.001243 �0.003682 �0.003459 �0.007312 �0.007136 �0.012803 0.005846 �0.020520 0.005164

0.3 �0.001213 �0.000301 �0.000689 �0.000364 �0.002213 �0.002010 �0.005411 �0.005263 �0.010819 0.004726 �0.018982 0.004109 �0.030425 0.003920

0.4 �0.000698 �0.000055 �0.000833 �0.000592 �0.002891 �0.002740 �0.007140 0.004443 �0.014306 0.003663 �0.025101 0.003395 �0.040196 0.003653

0.5 �0.000482 0.000015 �0.000996 �0.000803 �0.003573 �0.003452 �0.008862 0.003602 �0.017770 0.003088 �0.031160 0.003135 �0.049837 0.003887

0.6 �0.000375 0.000029 �0.001165 �0.001005 �0.004256 �0.004155 �0.010579 0.003062 �0.021211 0.002772 �0.037161 0.003143 �0.059349 0.004472

0.7 �0.000319 0.000023 �0.001337 �0.001201 �0.004937 0.003453 �0.012288 0.002696 �0.024629 0.002615 �0.043103 0.003341 �0.068734 0.005350

0.8 �0.000287 0.000008 �0.001511 �0.001392 �0.005618 0.003040 �0.013991 0.002443 �0.028024 0.002571 �0.048987 0.003692 �0.077994 0.006502

0.9 �0.000271 �0.000010 �0.001686 �0.001580 �0.006297 0.002722 �0.015686 0.002268 �0.031397 0.002611 �0.054814 0.004177 �0.087130 0.007925

1 �0.000263 �0.000030 �0.001862 �0.001766 �0.006975 0.002472 �0.017375 0.002150 �0.034747 0.002721 �0.060584 0.004787 �0.096146 0.009627

1.1 �0.000261 �0.000051 �0.002037 �0.001950 �0.007652 0.002273 �0.019056 0.002076 �0.038075 0.002892 �0.066299 0.005519 �0.105042 0.011623

1.2 �0.000263 �0.000071 �0.002213 �0.002134 �0.008327 0.002111 �0.020730 0.002038 �0.041381 0.003117 �0.071959 0.006373 �0.113821 0.013932

1.3 �0.000268 �0.000092 �0.002389 �0.002315 �0.009000 0.001979 �0.022398 0.002030 �0.044665 0.003393 �0.077564 0.007350 �0.122485 0.016580

1.4 �0.000275 �0.000111 �0.002565 �0.002497 �0.009672 0.001870 �0.024058 0.002047 �0.047927 0.003717 �0.083115 0.008454 �0.131035 0.019599

1.5 �0.000283 �0.000131 �0.002740 �0.002677 �0.010343 0.001781 �0.025712 0.002086 �0.051168 0.004088 �0.088612 0.009692 �0.139474 0.023028


estimated for calls with an exercise price of $100, moneyness of g = 0.95, and risk free-rate of 5% p.a. The prices of all calls used in this table are generated with the Black–

Scholes model using a given volatility ranging from 15% to 135% and maturity ranging from 0.1 to 1.5 years. The average error by using Formula (18) is 0.021497 less than

that by Formula (6).


619

Table 3

Comparison of Formula (18) and the Corrado–Miller formula (6) for out-of-the-money calls

Time to

expiration

True volatility

15% 35% 55% 75% 95% 115% 135%

Formula

(6)

Formula

(18)

Formula

(6)

Formula

(18)

Formula

(6)

Formula

(18)

Formula

(6)

Formula

(18)

Formula

(6)

Formula

(18)

Formula

(6)

Formula

(18)

Formula

(6)

Formula

(18)

0.1 �0.015448 �0.023199 �0.000847 �0.001794 �0.000949 �0.001513 �0.001961 �0.002367 �0.003777 �0.004096 �0.006548 �0.006811 �0.010457 �0.0106810.2 �0.002329 �0.003733 �0.000570 �0.001018 �0.001529 �0.001806 �0.003663 �0.003865 �0.007291 �0.007450 �0.012779 �0.012910 �0.020492 0.004700

0.3 �0.000985 �0.001801 �0.000668 �0.000962 �0.002198 �0.002381 �0.005394 �0.005528 �0.010799 �0.010905 �0.018958 0.003746 �0.030397 0.003610

0.4 �0.000577 �0.001154 �0.000818 �0.001037 �0.002878 �0.003015 �0.007123 �0.007224 �0.014286 0.003334 �0.025077 0.003122 �0.040169 0.003420

0.5 �0.000407 �0.000853 �0.000983 �0.001157 �0.003561 �0.003670 �0.008847 0.003269 �0.017750 0.002824 �0.031137 0.002917 �0.049810 0.003700

0.6 �0.000324 �0.000688 �0.001154 �0.001299 �0.004243 �0.004335 �0.010563 0.002783 �0.021191 0.002551 �0.037137 0.002961 �0.059322 0.004315

0.7 �0.000280 �0.000588 �0.001327 �0.001451 �0.004925 �0.005004 �0.012273 0.002457 �0.024609 0.002426 �0.043080 0.003184 �0.068707 0.005215

0.8 �0.000258 �0.000524 �0.001502 �0.001610 �0.005606 �0.005675 �0.013975 0.002234 �0.028005 0.002405 �0.048964 0.003554 �0.077967 0.006383

0.9 �0.000247 �0.000482 �0.001677 �0.001773 �0.006286 �0.006347 �0.015671 0.002082 �0.031378 0.002464 �0.054791 0.004054 �0.087104 0.007819

1 �0.000244 �0.000454 �0.001853 �0.001939 �0.006964 0.002245 �0.017359 0.001983 �0.034728 0.002589 �0.060562 0.004676 �0.096120 0.009531

1.1 �0.000245 �0.000435 �0.002029 �0.002107 �0.007640 0.002066 �0.019041 0.001924 �0.038056 0.002771 �0.066277 0.005418 �0.105017 0.011535

1.2 �0.000249 �0.000422 �0.002205 �0.002277 �0.008315 0.001921 �0.020715 0.001899 �0.041362 0.003006 �0.071936 0.006280 �0.113796 0.013850

1.3 �0.000255 �0.000415 �0.002381 �0.002447 �0.008989 0.001804 �0.022383 0.001901 �0.044646 0.003290 �0.077541 0.007264 �0.122460 0.016504

1.4 �0.000263 �0.000411 �0.002557 �0.002618 �0.009661 0.001708 �0.024043 0.001927 �0.047908 0.003621 �0.083093 0.008374 �0.131010 0.019527


estimated for calls with an exercise price of $100, moneyness of g = 1.05, and risk free-rate of 5% p.a. The prices of all calls used in this table are generated with the Black–

Scholes model using a given volatility ranging from 15% to 135% and maturity ranging from 0.1 to 1.5 years. The average error by using Formula (18) is 0.020928 less than

that by Formula (6).

620



between the true volatility and estimated volatility. The implied volatilities are

estimated for calls with an exercise price of $100, moneyness of g = 0.95, and

risk free-rate of 5% per annum. The prices of all calls used in Table 2 are gen-

erated with the Black–Scholes model using a given volatility (true volatility)

ranging from 15% to 135% and maturity ranging from 0.1 to 1.5 years. Table

2 reveals that Formula (18) yields consistently more accurate estimate thanFormula (6) in all cases considered. A more detailed calculation shows that

the estimation error by Formula (18) on average is 0.021497 less than that

by Formula (6). Note that average true volatility is about 68%. Thus Formula

(18) is on average about 3.2% more accurate than Formula (6).

Similar to Table 2, Table 3 presents a numerical comparison of Formulas

(18) and (6) for out-of-the money calls (g = 1.05). In this case, Formula (18)

yields more accurate estimates in most cases only. However, the estimation er-

ror by Formula (18) on average is still 0.020928 less than that by Formula (6).Thus Formula (18) is still on average about 3.1% more accurate than Formula

(6). In sum, the estimation errors in Tables 2 and 3 reveal that Formula (18)

yields significantly more accurate estimates than Formula (6).

As an application of Formula (18), we consider call options written on QQQ

(The NASDAQ 100 trust shares). The Nasdaq-100 Trust is a unit investment

trust designed to correspond generally to the performance, before fees and ex-

penses, of the Nasdaq-100 Index. The fund holds all the stocks in the Nasdaq-

100 Index, which consists of the largest non-financial securities listed on theNASDAQ Stock Market. The fund issues and redeems shares of Nasdaq-100

0.18

0.19

0.2

0.21

0.22

0.23

0.24

0 50 100 150 200 250

Bid Ask

Fig. 1. The implied volatilities from QQQCK contracts on February 18, 2004. The bid and ask

implied volatilities are calculated by using new formula (18) based on some intra-day data of

QQQCK contracts on 18 February 2004. The QQQCK contracts are call options with 30 days to

expire and an exercise price of 37.


Index Tracking Stock in multiples of 50,000 in exchange for the stocks in the

Nasdaq-100 and cash. The options on QQQ are one of the most heavily traded

options in the world.

The bid and ask implied volatilities based on some intra-day data of

QQQCK contracts on 18 February 2004 are presented in Fig. 1. 4 The

QQQCK contracts are call options with 30 days to expire and an exercise priceof 37. This application clearly demonstrates that Formula (18) can be useful in

practice.

4. Conclusions

Implied volatility can be easily calculated by using an iterative searching

technique. However, to simplify some applications such as spreadsheet, itmay be useful to have an approximation formula if that formula is reasonably

simple, accurate and valid for a wide range of cases. The cost and inconve-

nience of iterating also motivate the search for explicit formulas.

This paper summarizes the existing explicit formulas for computing the im-

plied volatility from the Black–Scholes model. Moreover, a uniform approach

for deriving possible approximation formulas is given. It is shown that some

accurate explicit formulas can be obtained by solving some cubic algebra equa-

tions. More accurate formulas would require solving some quartic or higherorder algebra equations, for which no simple closed-form solutions can be

obtained.

An accurate explicit formula for computing the implied volatility from the

Black–Scholes model is obtained. The new formula is of a simple form and

can be easily applied by using calculators or implemented in spreadsheet appli-

cations. The new formula is a generic explicit formula that is valid regardless of

option moneyness. This formula is accurate for a wide range of moneyness and

time to expiration.In the case of at-the-money calls, it is significantly more accurate than the

Brenner–Subrahmanyam [6] formula. Due to its higher accuracy, replacing

the Brenner–Subrahmanyam [6] formula by the new formula in the Chance

[11] model leads to higher accuracy with barely any extra effort. For the cases

of in- or out-of-money calls, the generic formula can yield significantly better

estimates than the Corrado–Miller formula [8].

4 I am grateful to Dr Van Den Brink for providing the data.


Acknowledgement

The author is grateful to Dr. Ning Gong from University of Melbourne and

Professor Dave Allen from Edith Cowan University for some useful comments

and suggestions. Of course, any possible remaining errors are my own

responsibility.

Appendix A

This appendix provides the derivations for Formulas (13) and (16). Using gand n, the Black–Scholes Eq. (1) can be rephrased as:

CS¼ Nðd1Þ � gNðd2Þ; ðA:1Þ

where

d1 ¼ � ln g2n

þ n; d2 ¼ � ln g2n

� n: ðA:2Þ

Note that for at-the-money calls, g = 1. Here the aim is to derive an approx-

imate formula for n (and thus for r) from (A.1) for g 5 1. To this end, the right

hand side of (A.1) is treated as a function of g. Set

f ðgÞ ¼ Nðd1Þ � gNðd2Þ: ðA:3Þ

Then the following results can be easily obtained:

f ð1Þ ¼ NðnÞ � Nð�nÞ; ðA:4Þ

f 0ð1Þ ¼ � 1

2n½N 0ðnÞ � N 0ð�nÞ � Nð�nÞ; ðA:5Þ

f 00ð1Þ ¼ 1

4n2½N 00ðnÞ � N 00ð�nÞ þ 1

2n½N 0ðnÞ þ N 0ð�nÞ; ðA:6Þ

where the primes denote differentiation. Note that there holds N 0(n) �N 0(�n) = 0. Using the Taylor expansion for f(g) and (A.3)–(A.6) yields

f ðgÞ ¼ f ð1Þ þ f 0ð1Þðg � 1Þ þ 12f 00ð1Þðg � 1Þ2 þ Oððg � 1Þ3Þ

¼ NðnÞ � gNð�nÞ þ ð1� gÞ2

8n2½N 00ðnÞ � N 00ð�nÞ þ ð1� gÞ2

4n

� ½N 0ðnÞ þ N 0ð�nÞ þ Oððg � 1Þ3Þ ðA:7Þ


Using the Taylor expansion for N(x), the following equations can be obtained:

NðnÞ � gNð�nÞ ¼ 1� g2

þ 1þ gffiffiffiffiffiffi2p

p n � 1ffiffiffiffiffiffi2p

p 1þ g6

n3 þ Oðn4Þ; ðA:8Þ

ð1� gÞ2

8n2½N 00ðnÞ � N 00ð�nÞ ¼ � ð1� gÞ2

4n1ffiffiffiffiffiffi2p

p þ Oðð1� gÞ2Þ; ðA:9Þ

ð1� gÞ2

4n½N 0ðnÞ þ N 0ð�nÞ ¼ ð1� gÞ2


p þ Oðð1� gÞ2Þ: ðA:10Þ

Combining (A.7) with (A.8)–(A.10) yields:

f ðgÞ ¼ 1� g2

þ 1þ gffiffiffiffiffiffi2p

p n � 1ffiffiffiffiffiffi2p

p 1þ g6

n3 þ ð1� gÞ2


p þ Oðn4Þ

þ Oððg � 1Þ2Þ: ðA:11Þ

Now two cases can be distinguished as follows.

Case 1: Suppose

n3 � ð1� gÞ2

n() n2 � j1� gj () r �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffijg � 1j

T

r: ðA:12Þ

The following approximation equation from (A.11) can be obtained:

CS� 1� g

2¼ 1þ gffiffiffiffiffiffi

2pp n þ ð1� gÞ2


p : ðA:13Þ

This is equivalent to

2n2 � ~an þ ð1� gÞ2

2ð1þ gÞ ¼ 0; ðA:14Þ

where ~a ¼ffiffiffiffi2p

p

1þg ½2CS þ g � 1. Note that (A.14) is a quadratic equation for n, thus

it can be easily solved. Further, (A.13) can be obtained by using the definition

of n.Case 2: Suppose

n3 � ð1� gÞ2

n() n2 � j1� gj () r �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffijg � 1j

T

r: ðA:15Þ

The following equation can be obtained:

CS¼ 1� g

2þ 1þ gffiffiffiffiffiffi

2pp n � 1ffiffiffiffiffiffi

2pp 1þ g

6n3: ðA:16Þ

This equation can be simplified as:

n3 � 6n þ 3~a ¼ 0; ðA:17Þ


where

~a ¼ffiffiffiffiffiffi2p

p

1þ g2CS

þ g � 1

� : ðA:18Þ

Note that (A.17) can be similarly solved as (9). Hence (16) can be obtainedsimilarly.

Final remark: It should be noted that a more accurate n (and thus r) can be

obtained from the following quartic equation:

n4 � 6n2 þ 3~an � 3ð1� gÞ2

2ð1þ gÞ ¼ 0; ðA:19Þ

which results from combining (A.1) and (A.11). Unfortunately, the radical

solutions for (A.19) contain too many terms. Therefore it is necessary to con-

sider the two special cases above.

References

[1] F. Black, M.S. Scholes, The pricing of options and corporate liabilities, Journal of Political

Economy 81 (3) (1973) 637–654.

[2] R.C. Merton, The theory of rational option pricing, Bell Journal of Economics and

Management Science 4 (1973) 141–183.

[3] H.A Latane, R.J. Rendleman, Standard deviation of stock price ratios implied by option

premia, Journal of Finance 31 (1976) 369–382.

[4] S. Manaster, G. Koehler, The calculation of implied variances from the Black-Scholes model:

A note, Journal of Finance 37 (1982) 227–230.

[5] R.Van Den Brink, Personal communication (2004).

[6] M. Brenner, M.G. Subrahmanyam, A simple formula to compute the implied standard

deviation, Financial Analysts Journal 5 (1988) 80–83.

[7] S. Feinstein,, A source of unbiased implied volatility, Working paper 88–9, Federal Reserve

Bank of Atlanta, December (1988).

[8] C.J. Corrado, T.W. Miller, A note on a simple, accurate formula to compute implied standard

deviations, Journal of Banking and Finance 20 (1996) 595–603.

[9] M.A. Bharadia, N. Christofides, G.R. Salkin, Computing the black Scholes implied

volatilityAdvances in Futures and Options Research, vol. 8, JAI Press, London, 1996, pp.

15–29.

[10] D.R. Chambers, S.K. Nawalkha, An improved approach to computing implied volatility, The

Financial Review 38 (2001) 89–100.

[11] D.M. Chance, A generalized simple formula to compute the implied volatility, The Financial

Review 31 (4) (1996) 859–867.

[12] J.C. Hull, A. White, An analysis of the bias in option pricing caused by by a stochastic

volatility, Advances in Futures and Options Research 3 (1988) 29–61.

[13] S.L. Heston, A closed-form solution for options with stochastic volatility with applications to

bond and currency options, Review of Financial Studies 6 (2) (1993) 327–343.

[14] A. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions, Dover, 1965.

[15] J. Stilwell, Mathematics and Its History, Springer-Verlag, 1989.

a new formula for computing implied volatility

Documents