The estimation of implied volatility from the Black-Scholes model: some new formulas and their applications

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DISCUSSION PAPERS IN ECONOMICS, FINANCE AND INTERNATIONAL COMPETITIVENESS

The estimation of implied volatility from the Black-Scholes model: some new formulas and

their applications

Steven Li

Discussion Paper No. 141, February 2003

Series edited by

The estimation of implied volatility from the Black-Scholes model: some new

formulas and their applications

Steven Li

School of Economics and Finance Queensland University of Technology

Brisbane, QLD 4001 Australia E-mail: [email protected]

Abstract

This paper provides a more accurate formula for estimating the implied volatilities for at-the-money calls than the existing formula as developed previously by Brenner and Subrahmanyam (1988). New formulas are also given for estimating the implied volatilities of in- or out-of-the-money calls. These formulas are derived mathematically and assessed by using numerical tests. All the new formulas are easy to use and accurate for a wide range of option moneyness and time to expiration.

JEL classification: G13

Keywords: options, implied volatility, implied standard deviation

1. Introduction and literature review

In the classic option pricing framework developed by Black and Scholes (1973) and Merton (1973), the value of a European call option on a non-dividend paying stock is stated as:

) ( X ) ( _{1 2} 0 N d e N d S C_{= −} −rT ₍₁₎ where ) 2 / 2 ( ) / ln( ₀ 1 T T r X S d σ σ + + = , (2) T d T T r X S d σ σ σ _{= −} − + = 0 2 2 ) 2 / 2 ( ) / ln( , (3)

∫

The stock price, strike price, interest rate, and time to expiration are denoted by S0,X,r,T, respectively. The annualized variance of the continuously compounded return on the underlying stock is denoted by_σ2_.

A useful property of the Black-Scholes-Merton option pricing model is that all model parameters except the log-price standard deviation are directly observable from market data. This allows a market-based estimate of a stock's future volatility. Originally suggested by Latane and Rendleman (1976), implied volatilities are extensively used in financial markets research.

An implied volatility is the value of the log-price standard deviation 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 standard deviation from Equation (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(σ)−C_M into a root-finding program, whereC(⋅) is an option pricing formula, σ is the volatility parameter, and C_M is the observed market price of the option. Various algorithms can be used to find the value of σ that makes this expression equal to zero (See e.g. Mayhew, 1995). Thus the calculation of implied volatility often requires programming and numerical techniques. For example, Manaster and Koehler (1982) discuss the Newton –Raphson algorithm for calculating implied volatilities. The implementation of such procedures is often cumbersome and manual calculation is error-prone. As the Black-Schoels model has become fundamental in modern finance and is being widely used in the financial industry, it is of great interest to find accurate estimates of implied volatility without resorting to numerical procedures. So far, a few approximation formulas have been proposed.

Brenner and Subrahmanyam (1988) provide an elegant formula to compute an implied stock return standard deviation that is accurate when a stock price is exactly equal to a discounted strike price. Their formula is as follows:

T S

C

π

σ = 2 . (5)

Feinstein (1989) independently derived an essentially identical formula.

The accuracy of the well-known Brenner-Subrahmanyam (1988) formula depends on the assumption that a stock-price is equal to a discounted stock price. Following Brenner and Subrahmanyam (1988), we define an at-the-money option as one in which_{S = Xe}−rt_{. For}

convenience, throughout this paper, we set

K

=

Xe

Corrado & Miller (1996) provide an improved quadratic formula which is valid when stock prices deviate from discounted strike prices. Their formula is given as:

] ) ( ) 2 ( 2 [ 2 1 2 2 π π σ C S K C S K S K K S T − − − − + − − + = . (6)

It is then shown that this formula yields quite accurate estimates for some cases. Bharadia, Christofides and Salkin (1996) derive a highly simplified volatility approximation as:

2 / ) ( 2 / ) ( 2 K S S K S C T − − − − = π σ (7)

As pointed out by Chambers and Nawalkha (2001), (7) is less accurate than (6) in general. Thus we shall not evaluate (7) in this paper. Note that both estimates (6) ad (7) are ad hoc in nature and lack rigorous mathematical derivations.

Chance (1996) provides a direct method of obtaining an accurate estimate of the implied volatility of a call option. His estimate is based on the formula for at-the-money options developed by Brenner and Subrahmanyam (1988). The adjusted formula by Chance (1996) is quite accurate for options no more than 20% in- or out-of-the-money and is simple to program and compute. But his formula contains many terms and requires an additional approximation. Thus it requires programming and is not very convenient to use. More recently, Chambers and Nawalkha (2001) develop a simplified extension of Chance’s model.

In this paper, we first develop an explicit formula for estimating the volatility of an at-the-money call. It is shown mathematically that our estimate has a higher order of precision than (5). This is also verified numerically. In particular, our formula yields significantly better estimate for options with large volatilities and long time to maturities. Our formula is also easily applicable manually with a calculator.

For in- or out-of-the-money calls, we first develop a new quadratic formula which is shown to perform similarly as the quadratic formula (6) obtained by Corrado & Miller (1996). However, our new quadratic formula is based on a firm mathematical foundation, thus it is not ad hoc in nature. We also develop a new formula which is more accurate for closely near-the-money options or options with high volatility or long time to maturity. Both formulas are extensively tested numerically. Finally, we summarise and conclude in Section 4. Though our results can easily be extended to dividend paying stock options and commodity options etc., we restrict our discussions to non-dividend paying stock options for simplicity.

2. Implied volatility of at-the-money calls

In this section, we consider the implied volatilities of at-the-money calls. That is, we assume

rT

Xe K

S _{= =} − _{. In order to approximate the Black-Scholes equation (1), we expand the standard}

normal cumulative distribution function N(x) to the third order. That is, we use to the following Taylor expansion (see e.g. Abramowitz and Stegun,1965):

) ( 2 6 1 2 1 2 1 ) (_{x x x}3 _{O x}5 N = + − + π π (8)

Substituting the expansions for the cumulative distribution functions ) 2 1 ( T N σ and ) 2 1 ( T

N − σ into (1), we obtain the following equation:

3 3 1 2 2π _{= ξ− ξ} S C (9) where ξ σ T 2 1

= . Solving this equation (see Appendix I), we obtain

z z T z T 2 6 8 1 2 2 2 α σ = − − (10) where S C π α = 2 and )] 32 3 ( cos 3 1 cos[ −1 α = z .

Mathematically, our new formula (10) is of higher order of precision than Formula (5). Note that the precision of our formula is dependent on ξ σ T

2 1

= . Thus the two factors that impact the precision of Formula (10) are the volatility σand the time to maturity

T

Table 1 provides a numerical comparison of the accuracy of Formula (10) with that of the Brenner and Subrahmanyam formula (5). The estimation errors are given for both Formulas (5) and (10). The implied volatilities are estimated for at-the money calls with an exercise price of $100, a stock price of 95.123 (i.e. at-the-money) and a risk-free rate of 5% per annum. The prices of all calls used in the test are generated with the Black-Scholes model (1).

Table 1 reveals that Formula (10) yields more accurate estimate than Formula (5) in all the cases. This is consistent with the mathematical insight given in Appendix I. Moreover, the estimation error by Formula (10) is roughly one-tenth of that by Formula (5) for each case considered in Table 1. In conclusion, Formula (10) yields more accurate estimate than Formula (5) and it is also conveniently applicable with a calculator.

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

In this section, we develop some formulas that are valid for in- or out-of-the-money calls. For this purpose, we need to introduce the following variable:

S K =

η . (11)

Note that η measures the moneyness of an option: η=1represents at-the-money, η>1 represents out-of-the money and η<1represent in-the-money. Thus η is likely to be an important variable for obtaining an appropriate formula for in- or out-of-the-money options. Using the variables η andξ, the Black-Scholes equation (1) can be rephrased as:

) ( ) (d₁ N d₂ N S C _{= −η} (12) where ξ ξ η ξ ξ η_{+ =− −} − = 2 ln , 2 ln 2 1 d

d . Note that the right hand side of (12) is a function of ξ and η. Using Taylor expansions, we can obtain an algebraic equation forξ, which can be solved subsequently to produce new formulas forσ . The details of the derivation for the formulas given below are included in Appendix II. For in- or out-of-the-money calls, we are unable to derive a uniform formula which is accurate for all cases. We shall first consider two special cases and then consider the combination of them.

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

For deep in- or out-of-the-money options, we have the following formula:

T 2 1 ) 1 ( 4 ~ ~ 2 2 η η α α σ + − − + = (13) where ] 1 2 [ 1 2 ~ _{+ −} + = η η π α S C . (14)

Note that Formula (13) reduces to (5) when η=1. This formula is particularly accurate when_ξ2 _<<η−1_{, where “<<” means “far less than”. This condition is equivalent} to

T 1 − << η

expiration or deep in- or out-of –the-money. In order to compare Formula (13) with the quadratic formula (6) proposed by Corrado and Miller (1996), we rewrite (13) as:

] 2 / 1 ) ( ) 2 ( 2 [ 2 1 2 2 S K K S K S C K S C K S T + − − − − + − − + = π π σ . (15)

This is slightly different from (6) only in the last term under the square root. Thus the difference between (13) and (6) is ignorable in most cases1.

Table 2 presents a numerical comparison of the accuracy of Formula (13) with that of Corrado and Miller formula (6). 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 calls with an exercise price of $100, a stock price of 104.738 (i.e. η=0.95), and risk free-rate of 5% per annum. The prices of all calls used in Table 2 are generated with the Black-Scholes model using a given standard deviation ranging from 15% to 135% and maturity ranging from 0.1 to 1.5 years. Note a higher range of true volatilities has been chosen to avoid the square root of a negative number in (13) and (6) for small time to expiration. This problem exists for both (13) and (6) in some extreme cases when the time to expiration is small and true volatility is low.

Table 2 reveals that the estimation errors for (13) and (6) are very close to each other for all cases. The same result can be obtained for other values of η. This is also confirmed in Table 3 by using options with the real-life data, where the time to expiration is very small.

It should be observed from Table 2 that the estimation errors for both (13) and (6) are quite large for options with long time to expiration and large true volatility. Thus an alternative formula needs to be developed for these cases. This is considered in the next subsection.

3.2. Nearly at-the-money option

As mentioned above, both (13) and (6) lead to relatively large errors when

T 1 − >> η

σ where “>>” means “far bigger than”. In this case, we need the following formula:

z z T z T 2~ 6 ~ 8 1 ~ 2 2 2 α σ = − − (16) where )] 32 ~ 3 ( cos 3 1 cos[

~_{z =} −1 α _{and α}~ is defined by (14). Note that this formula reduces to (10)

when η=1. According to our derivation in Appendix II, (16) can yield 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.

Table 4 presents a comparison of (13) and (16). 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 calls with an exercise price of $100, a stock price of 104.738 (i.e. η=0.95), and risk free-rate of 5% per annum. The prices of all calls used in Table 4 are generated with the Black-Scholes model using a given standard deviation ranging from 15% to 135% and maturity ranging from 0.1 to 1.5 years. The estimation errors in Table 4 clearly demonstrate that (13) is more accurate than (16) in some cases, and less accurate than (16) in the other cases. Next, we consider how to combine (13) and (16) to obtain a higher accuracy.

3.3. The combination

As shown above, neither (13) nor (16) dominates the other. A natural question is: how can one choose between these two formulas in practice? For this purpose, we introduce a new determinant variable: 1 − = η σ ρ T . (17)

A rough numerical test shows that ifρ<2, (13) should be chosen and otherwise (16). In practice, if one has some rough idea of the volatility scale, we can use (17) to pick up the right formula. If not, one can apply either of the two formulas and then use (17) to check if the right one has been used. Thus the condition (17) is applicable in practice. As an example, we apply the above decision rule and present the estimation errors using the combination of (13) and (16) in Table 5. Comparing to Table 4, the estimation errors in Table 5 have been reduced on average.

4. Summary

This paper provides some accurate formulas to estimate the implied volatility from the Black-Scholes model. One formula is developed for at-the-money options. It has higher order of precision than the famous Brener-Subrahmanyam formula.

We also give two formulas for in- or out-of-the-money options. One is very similar to the Corrado-Miller formula, and the other is an extension of the new formula we obtained for at-the-money options. The validity range for the two formulas is analysed and identified. We also give a decision rule for making the choice between the two formulas.

Besides, our formulas are tested numerically and assessed against the existing formulas in the literature.

In this appendix, we show how the formula (10) is derived. Set S C π α = 2 . Then Equation (9) is equivalent to 0 3 6 3 _{− ξ + α =} ξ . (A1)

Using the cubic formula (see e.g. Stilwell, 1989), we know one solution is given by

3 3 2 3 2 1 2 32 9 3 32 9 3 + − + − − − − = α α α α ξ . (A2)

Let _∆=9_α2 ₋32_{. Then the above can be further simplified as:}

3 3 3 1 2 3 3 + ∆ + − ∆ − = α α ξ . (A3)

Assuming that ∆<02, this formula gives a negative ξ, henceforth a negative standard deviation. Therefore this solution is not the appropriate one. Using the above solution, we can lower the order of the cubic equation of (A1) by 1 and find the other two solutions. The other solutions are given by

1 1 4 1 2 1 2 2 12 ξ αξ ξ ξ ξ =− + + , (A4) 1 1 4 1 2 1 3 2 12 ξ αξ ξ ξ ξ =− − + (A5)

Using some real examples, we can see that (A5) should be chosen. Simplifying (A5), we can obtain the formula (10).

Appendix II. Derivation of the formulas (13) and (16) for in- or out-of-the-money calls

Set

S K =

η (A6)

Using η and ξ, the Black-Scholes equation can be rephrased as: ) ( ) (d₁ N d₂ N S C _{= −η} (A7) where

ξ ξ η ξ ξ η_{+ =− −} − = 2 ln , 2 ln 2 1 d d . (A8 )

Note that for at-the-money calls, we have η=1. Here our aim is to derive an approximate formula for ξ (and thus forσ ) from (A7) when η≠1. For this purpose, we treat the right hand side of (A7) as a function of η and expand it at η=1 using the Taylor expansion. Let

) ( ) ( ) ( N d₁ N d₂ f η = −η . (A9) Then we have ) ( ) ( ) 1 ( =N ξ −N −ξ f , (A10) ) ( )] ( ' ) ( ' [ 2 1 ) 1 ( ' ξ ξ ξ ξ − − − − − = N N N f , (A1 ) )] ( ' ) ( ' [ 2 1 )] ( '' ) ( '' [ 4 1 ) 1 ( '' ₂ ξ ξ ξ ξ ξ ξ − − + + − = N N N N f (A12)

where the primes denote differentiation. Note that we haveN'(ξ)−N'(−ξ)=0. Using the Taylor expansion for f(η) and (A10)-(A12), we obtain the following:

) ) 1 (( )] ( ' ) ( ' [ 4 ) 1 ( )] ( '' ) ( '' [ 8 ) 1 ( ) ( ) ( ) ) 1 (( ) 1 )( 1 ( '' 2 1 ) 1 )( 1 ( ' ) 1 ( ) ( 3 2 2 2 3 2 − + − + − + − − − + − − = − + − + − + = η ξ ξ ξ η ξ ξ ξ η ξ η ξ η η η η O N N N N N N O f f f f (A13)

Using the Taylor expansion for N(x), we know that:

) ( 6 1 2 1 2 1 2 1 ) ( ) ( η_ξ 3 _ξ4 π ξ π η η ξ η ξ N O N − − = − + + − + + , (A14) ) ) 1 (( 2 1 4 ) 1 ( )] ( '' ) ( '' [ 8 ) 1 ( 2 2 2 2 η π ξ η ξ ξ ξ η _{− − =−} − _{+ −} − O N N , (A15) ) ) 1 (( 2 1 2 ) 1 ( )] ( ' ) ( ' [ 4 ) 1 ( 2 2 _η 2 π ξ η ξ ξ ξ η _{+ − =} − _{+ −} − O N N (A16)

) ) 1 (( ) ( 2 1 4 ) 1 ( 6 1 2 1 2 1 2 1 ) ( ₌ − ₊ + ₋ + 3 ₊ − 2 _{+ ξ}4 _{+ η−} 2 π ξ η ξ η π ξ π η η η O O f . (A17)

Now two cases can be distinguished as follows.

Case 1: Suppose T 1 1 ) 1 ( 2 2 3 _<< − _{⇔ξ << −η ⇔σ <<} η− ξ η ξ . (A18)

We obtain the following approximation equation from (A13):

π ξ η ξ π η η 2 1 4 ) 1 ( 2 1 2 1 ₋ 2 + + = − − S C . (A19) This is equivalent to 0 ) 1 ( 2 ) 1 ( ~ 2 2 2 ₌ + − + − η η ξ α ξ (A20) where [2 1] 1 2 ~ _{+ −} + = η η π α S C

. Note that (A20) is a quadratic equation forξ, thus it can be easily solved. Further, (13) can be obtained by using the definition ofξ.

Case 2: Suppose T 1 1 ) 1 ( 2 2 3 _>> − _{⇔ξ >> −η ⇔σ >>} η− ξ η ξ . (A21)

We obtain the following equation:

3 6 1 2 1 2 1 2 1 η_ξ π ξ π η η₊ + ₋ + − = S C . (A22)

This equation can be simplified as:

0 ~ 3 6 3_{− ξ+ α =} ξ (A23) where

] 1 2 [ 1 2 ~ _{+ −} + = η η π α S C . (A24)

Note that equation (A23) is the same as (A3) except that the constant α is replaced byα~. Hence the solution can be obtained similarly as before.

Final remark: We note that a more accurate ξ (and thusσ ) can be obtained from the following quartic equation:

0 ) 1 ( 2 ) 1 ( 3 ~ 3 6 2 2 4 ₌ + − − + − η η ξ α ξ ξ (A25)

which results from combining (A7) and (A17). Unfortunately, the radical solutions for (A25) are too messy though they are obtainable. Therefore it is necessary to consider the two special cases as above.

References

Abramowitz, A. and Stegun, I.A., 1965. Handbook of Mathematical Functions. Dover.

Bharadia, M.A., Christofides N. and Salkin, G.R., 1996. Computing the black Scholes implied volatility. Advances in Futures and Options Research, Vol. 8, 15-29, JAI Press, London.

Black, F. and Scholes, M.S., 1973. The pricing of options and corporate liabilities. Journal of Political Economy, May-June, 81(3), 637-54.

Brenner, M. and Subrahmanyam, M.G., 1988. A simple formula to compute the implied standard deviation. Financial Analysts Journal 5 (July/August).

Chambers, D.R. and Nawalkha, S.K., 2001. An improved approach to computing implied volatility. The Financial Review, 38, 89-100.

Chance, D,M., 1996. A generalized simple formula to compute the implied volatility. The Financial Review, Vol .31 No. 4, 859-867.

Corrado, C.J. and Miller, T.W., 1996. A note on a simple, accurate formula to compute implied standard deviations. Journal of Banking and Finance, 20, 595-603.

Feinstein, S.,1989. The Balck-Scholes formula is nearly linear in σ for at-the-money options; therefore implied volatilities from at-the-money options are virtually unbiased. Working paper, Federal Reserve Bank of Atlanta and Boston University.

Latame, H.A. and Rendleman, R.J., 1976. Standard deviation of stock price ratios implied by option premia. Journal of Finance 31, May, 369-382.

Manaster, S. and Koehler, G., 1982. The calculation of implied variances from the Black-Scholes model: a note. Journal of Finance 37, March, 227-230.

Mayhew, S., 1995. Implied volatility. Financial Analysts Journal, July/August, 8-20.

Merton, R.C., 1973. The theory of rational option pricing. Bell Journal of Economics and Management Science 4, 141-183.

Table 1. Comparison of the Brenner-Subrahmnyam (5) and the improved formula (10)

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, a stock price of 95.122942 (i.e. at-the-money), 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 standard deviation ranging from 5% to 125% and maturity ranging from 0.1 to 1.5 years.

True volatility 5% 25% 45% 65% 85% 105% 125% Time to Expiration Brenner-Subrahmanya m (5) New Formula (10) Brenner-Subrahman yam (5) New Formula (10) Brenner-Subrahman yam (5) New Formula (10) Brenner-Subrahman yam (5) New Formula (10) Brenner-Subrahman yam (5) New Formula (10) Brenner-Subrahman yam (5) New Formula (10) Brenner-Subrahma nyam (5) New Formula (10) 0.1 _{0.000000 0.000000 -0.000064 0.000001 -0.000378 0.000001 -0.001142 0.000003 -0.002551 0.000007 -0.004803 0.000020 -0.008091 0.000048} 0.2 _{-0.000001 0.000000 -0.000129 0.000001 -0.000758 0.000002 -0.002281 0.000008 -0.005090 0.000028 -0.009568 0.000081 -0.016088 0.000196} 0.3 _{-0.000001 0.000000 -0.000195 0.000001 -0.001136 0.000003 -0.003417 0.000016 -0.007615 0.000063 -0.014293 0.000185 -0.023991 0.000449} 0.4 _{-0.000002 0.000000 -0.000260 0.000001 -0.001514 0.000005 -0.004548 0.000029 -0.010126 0.000114 -0.018979 0.000333 -0.031804 0.000812} 0.5 _{-0.000002 0.000000 -0.000325 0.000001 -0.001891 0.000007 -0.005677 0.000046 -0.012623 0.000179 -0.023627 0.000526 -0.039525 0.001291} 0.6 _{-0.000003 0.000000 -0.000390 0.000001 -0.002268 0.000010 -0.006801 0.000067 -0.015107 0.000260 -0.028237 0.000767 -0.047158 0.001892} 0.7 _{-0.000003 0.000000 -0.000455 0.000001 -0.002644 0.000014 -0.007922 0.000091 -0.017578 0.000357 -0.032809 0.001057 -0.054704 0.002622} 0.8 _{-0.000004 0.000000 -0.000520 0.000001 -0.003019 0.000019 -0.009040 0.000120 -0.020035 0.000470 -0.037344 0.001398 -0.062162 0.003488} 0.9 _{-0.000004 0.000000 -0.000584 0.000001 -0.003394 0.000024 -0.010154 0.000152 -0.022479 0.000599 -0.041842 0.001791 -0.069536 0.004500} 1 _{-0.000005 0.000000 -0.000649 0.000002 -0.003768 0.000029 -0.011264 0.000189 -0.024910 0.000746 -0.046304 0.002240 -0.076825 0.005667} 1.1 _{-0.000005 0.000000 -0.000714 0.000002 -0.004142 0.000035 -0.012371 0.000229 -0.027328 0.000910 -0.050729 0.002746 -0.084031 0.006999} 1.2 _{-0.000006 0.000000 -0.000779 0.000002 -0.004515 0.000042 -0.013474 0.000274 -0.029733 0.001091 -0.055119 0.003312 -0.091156 0.008508} 1.3 _{-0.000007 0.000000 -0.000844 0.000003 -0.004888 0.000050 -0.014574 0.000323 -0.032125 0.001291 -0.059473 0.003940 -0.098199 0.010208} 1.4 _{-0.000007 0.000000 -0.000908 0.000003 -0.005260 0.000058 -0.015671 0.000377 -0.034505 0.001510 -0.063792 0.004634 -0.105163 0.012113} 1.5 _{-0.000008 0.000000 -0.000973 0.000004 -0.005631 0.000067 -0.016764 0.000435 -0.036872 0.001748 -0.068077 0.005396 -0.112049 0.014241}

Table 2. Comparison of the Corrado-Miller formula (6) and the new formula (13)

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 calls with an exercise price of $100, a stock price of 104.738 (i.e η=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 standard deviation ranging from 15% to 135% and maturity ranging from 0.1 to 1.5 years. Note a higher range of true volatility has been chosen to avoid the square root of a negative number in (6) and (13) for small time to expiration.

True volatility 15% 35% 55% 75% 95% 115% 135% Time to Expiration Corrado-Miller Formula (6) New Formula (13) Corrado-Miller Formula (6) New Formula (13) Corrado-Miller Formula (6) New Formula (13) Corrado-Miller Formula (6) New Formula (13) Corrado-Miller Formula (6) New Formula (13) Corrado-Miller Formula (6) New Formula (13) Corrado-Miller Formula (6) New Formula (13) 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 -0.010240} 0.2 _{-0.002937 -0.001356 -0.000610 -0.000113 -0.001549 -0.001243 -0.003682 -0.003459 -0.007312 -0.007136 -0.012803 -0.012658 -0.020520 -0.020396} 0.3 _{-0.001213 -0.000301 -0.000689 -0.000364 -0.002213 -0.002010 -0.005411 -0.005263 -0.010819 -0.010702 -0.018982 -0.018884 -0.030425 -0.030341} 0.4 _{-0.000698 -0.000055 -0.000833 -0.000592 -0.002891 -0.002740 -0.007140 -0.007028 -0.014306 -0.014217 -0.025101 -0.025027 -0.040196 -0.040134} 0.5 _{-0.000482 0.000015 -0.000996 -0.000803 -0.003573 -0.003452 -0.008862 -0.008773 -0.017770 -0.017699 -0.031160 -0.031101 -0.049837 -0.049787} 0.6 _{-0.000375 0.000029 -0.001165 -0.001005 -0.004256 -0.004155 -0.010579 -0.010504 -0.021211 -0.021151 -0.037161 -0.037111 -0.059349 -0.059307} 0.7 _{-0.000319 0.000023 -0.001337 -0.001201 -0.004937 -0.004851 -0.012288 -0.012224 -0.024629 -0.024578 -0.043103 -0.043060 -0.068734 -0.068697} 0.8 _{-0.000287 0.000008 -0.001511 -0.001392 -0.005618 -0.005542 -0.013991 -0.013935 -0.028024 -0.027979 -0.048987 -0.048949 -0.077994 -0.077961} 0.9 _{-0.000271 -0.000010 -0.001686 -0.001580 -0.006297 -0.006230 -0.015686 -0.015636 -0.031397 -0.031357 -0.054814 -0.054781 -0.087130 -0.087101} 1 _{-0.000263 -0.000030 -0.001862 -0.001766 -0.006975 -0.006914 -0.017375 -0.017330 -0.034747 -0.034711 -0.060584 -0.060554 -0.096146 -0.096120} 1.1 _{-0.000261 -0.000051 -0.002037 -0.001950 -0.007652 -0.007596 -0.019056 -0.019015 -0.038075 -0.038042 -0.066299 -0.066272 -0.105042 -0.105018} 1.2 _{-0.000263 -0.000071 -0.002213 -0.002134 -0.008327 -0.008276 -0.020730 -0.020693 -0.041381 -0.041351 -0.071959 -0.071933 -0.113821 -0.113799} 1.3 _{-0.000268 -0.000092 -0.002389 -0.002315 -0.009000 -0.008953 -0.022398 -0.022363 -0.044665 -0.044637 -0.077564 -0.077540 -0.122485 -0.122465} 1.4 _{-0.000275 -0.000111 -0.002565 -0.002497 -0.009672 -0.009629 -0.024058 -0.024026 -0.047927 -0.047901 -0.083115 -0.083093 -0.131035 -0.131016} 1.5 _{-0.000283 -0.000131 -0.002740 -0.002677 -0.010343 -0.010302 -0.025712 -0.025682 -0.051168 -0.051143 -0.088612 -0.088592 -0.139474 -0.139456}

Table 3. Comparison of the Corrado-Miller formula (6) and the new formula (13) using real life data.

Examples are taken from Corradon and Miller (1996). They are based on closing prices reported in the financial press for options expiring in 29 days. The interest rate used is 3 percent. No ex-dividend dates occurred during this period.

Implied volatility Stock Price Strike Call

Corrado-Miller (6) New formula (13) Actual volatility Borland 22.25 20 3.375 0.850465 0.855483 0.8505 22.5 1.5 0.635195 0.635192 0.6352 Ford 52.125 50 2.75 0.231148 0.232605 0.2336 55 0.5 0.243068 0.240540 0.2471 Gen Elec 88.5 85 4.125 0.162780 0.165317 0.1699 90 1.0625 0.162046 0.161984 0.1621 IBM 54 50 4.75 0.346381 0.352711 0.3548 55 1.4375 0.300489 0.300445 0.3006 Microsoft 84.25 80 5.75 0.326112 0.327903 0.3282 85 2.625 0.304117 0.304114 0.3042 Tel Mex 52.75 50 3.625 0.317679 0.319754 0.3203 55 0.8125 0.273724 0.272898 0.2766 Unisys 13.50 12.5 1.5625 0.657239 0.659757 0.6595 15 0.4375 0.624098 0.617018 0.629

Table 4. Comparison of the formulas (13) and (16)

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, a stock price of 104.738 (i.e. η=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 standard deviation ranging from 15% to 135% and maturity ranging from 0.1 to 1.5 years. Note a higher range of true volatility has been chosen to avoid the square root of a negative number in (13) for small time to expiration.

True volatility 15% 35% 55% 75% 95% 115% 135% Time to Expiration New Formula (16) New Formula (13) New Formula (16) New Formula (13) New Formula (16) New Formula (13) New Formula (16) New Formula (13) New Formula (16) New Formula (13) New Formula (16) New Formula (13) New Formula (16) New Formula (13) 0.1 _{0.080050 -0.014890 0.036924 0.000059 0.023745 -0.000368 0.017473 -0.001539 0.013825 -0.003451 0.011452 -0.006285 0.009806 -0.010240} 0.2 _{0.041828 -0.001356 0.018625 -0.000113 0.011917 -0.001243 0.008768 -0.003459 0.006966 -0.007136 0.005846 -0.012658 0.005164 -0.020396} 0.3 _{0.028317 -0.000301 0.012454 -0.000364 0.007960 -0.002010 0.005874 -0.005263 0.004726 -0.010702 0.004109 -0.018884 0.003920 -0.030341} 0.4 _{0.021404 -0.000055 0.009355 -0.000592 0.005981 -0.002740 0.004443 -0.007028 0.003663 -0.014217 0.003395 -0.025027 0.003653 -0.040134} 0.5 _{0.017204 0.000015 0.007491 -0.000803 0.004797 -0.003452 0.003602 -0.008773 0.003088 -0.017699 0.003135 -0.031101 0.003887 -0.049787} 0.6 _{0.014383 0.000029 0.006248 -0.001005 0.004011 -0.004155 0.003062 -0.010504 0.002772 -0.021151 0.003143 -0.037111 0.004472 -0.059307} 0.7 _{0.012356 0.000023 0.005359 -0.001201 0.003453 -0.004851 0.002696 -0.012224 0.002615 -0.024578 0.003341 -0.043060 0.005350 -0.068697} 0.8 _{0.010830 0.000008 0.004693 -0.001392 0.003040 -0.005542 0.002443 -0.013935 0.002571 -0.027979 0.003692 -0.048949 0.006502 -0.077961} 0.9 _{0.009639 -0.000010 0.004174 -0.001580 0.002722 -0.006230 0.002268 -0.015636 0.002611 -0.031357 0.004177 -0.054781 0.007925 -0.087101} 1 _{0.008685 -0.000030 0.003760 -0.001766 0.002472 -0.006914 0.002150 -0.017330 0.002721 -0.034711 0.004787 -0.060554 0.009627 -0.096120} 1.1 _{0.007902 -0.000051 0.003422 -0.001950 0.002273 -0.007596 0.002076 -0.019015 0.002892 -0.038042 0.005519 -0.066272 0.011623 -0.105018} 1.2 _{0.007249 -0.000071 0.003140 -0.002134 0.002111 -0.008276 0.002038 -0.020693 0.003117 -0.041351 0.006373 -0.071933 0.013932 -0.113799} 1.3 _{0.006696 -0.000092 0.002902 -0.002315 0.001979 -0.008953 0.002030 -0.022363 0.003393 -0.044637 0.007350 -0.077540 0.016580 -0.122465} 1.4 _{0.006221 -0.000111 0.002698 -0.002497 0.001870 -0.009629 0.002047 -0.024026 0.003717 -0.047901 0.008454 -0.083093 0.019599 -0.131016} 1.5 _{0.005809 -0.000131 0.002522 -0.002677 0.001781 -0.010302 0.002086 -0.025682 0.004088 -0.051143 0.009692 -0.088592 0.023028 -0.139456}

Table 5. Estimation errors by using the combination of (13) and (16).

Each entry represents the estimation error: the difference between the true volatility and estimated volatility (estimated volatility-true volatility). The estimated volatility is obtained via (13) whenρ<2, otherwise (16). The implied volatilities are estimated for at-the-money calls with an exercise price of $100, a stock price of 104.738 (i.e. η=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 standard deviation ranging from 15% to 135% and maturity ranging from 0.1 to 1.5 years. Note a higher range of true volatility has been chosen to avoid the square root of a negative number in (13) for small time to expiration.

True volatility Time to Expiration 15% 35% 55% 75% 95% 115% 135% 0.1 -0.014890 0.000059 -0.000368 -0.001539 -0.003451 -0.006285 -0.014890 0.2 -0.001356 -0.000113 -0.001243 -0.003459 -0.007136 0.005846 -0.001356 0.3 -0.000301 -0.000364 -0.002010 -0.005263 0.004726 0.004109 -0.000301 0.4 -0.000055 -0.000592 -0.002740 0.004443 0.003663 0.003395 -0.000055 0.5 0.000015 -0.000803 -0.003452 0.003602 0.003088 0.003135 0.000015 0.6 0.000029 -0.001005 -0.004155 0.003062 0.002772 0.003143 0.000029 0.7 0.000023 -0.001201 0.003453 0.002696 0.002615 0.003341 0.000023 0.8 0.000008 -0.001392 0.003040 0.002443 0.002571 0.003692 0.000008 0.9 -0.000010 -0.001580 0.002722 0.002268 0.002611 0.004177 -0.000010 1 -0.000030 -0.001766 0.002472 0.002150 0.002721 0.004787 -0.000030 1.1 -0.000051 -0.001950 0.002273 0.002076 0.002892 0.005519 -0.000051 1.2 -0.000071 -0.002134 0.002111 0.002038 0.003117 0.006373 -0.000071 1.3 -0.000092 -0.002315 0.001979 0.002030 0.003393 0.007350 -0.000092 1.4 -0.000111 -0.002497 0.001870 0.002047 0.003717 0.008454 -0.000111 1.5 -0.000131 -0.002677 0.001781 0.002086 0.004088 0.009692 -0.000131