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Equivalent Black Volatilities¤

Patrick S. Hagany Diana E. Woodwardz

March 30, 1998

Abstract

We consider European calls and puts on an asset whose forward price F (t) obeys

dF (t) = ®(t)A(F )dW (t)

under the forward m easure. By using singular perturbation techniques, we obtain explicitalgebraic formulas for the implied volatility ¾B in terms of today’s forward price F 0 ´ F (0),the strike K of the option, and the time to expiry tex. The price of any call or put can then becalculated simply by substituting this implied volatility into Black’s formula.

For example, for a power law (constant elasticity of variance) model

dF (t) = aF ¯dW (t)

we obtain

¾B = a

f 1¡¯av

n1 +

(1¡ ¯ )(2 + ¯ )

24

³F 0 ¡K

f av

´2

+ (1¡ ¯ )2

24

a2tex

f 2¡2¯av

+ : : :o

where f av = 1

2(F 0 + K ).

Our formula for the implied volatility is not exact. However, we show that the error isinsigni…cant, rarely approaching 1

1000 of the time value of the option. We also present more

accurate (albeit more complicated) formulas which can be used for the implied volatility.

¤The views presented in this report d o not necessarily re‡ect the views of NumeriX, The Bank of Tokyo-Mitsubishi,or any of their a¢liates.

yCurrent mailing address: NumeriX, 546 Fifth Avenue, 17th Floor, New York, NY 10036.zCurrent mailing address: The Bank of Tokyo-Mitsubishi, Ltd., 1251 Avenue of the Americas, New York, NY

10020.

1



Market prices of many European options are quoted in terms of their Black-Scholes volatility¾B . To obtain the cash price, let tex be the option’s exercise date, let ts be the option’s settlementdate, and de…ne F (t) to be the forward price of the a sset ( for a contract maturing at the settlement

date ts) as seen at date t . Then the cash price of calls and puts is given by Black’s formula [1, 6]

V call = D(0; ts)©

F 0(d1) ¡ K (d2)ª

(1.1a)

V put = D(0; ts)©

K (¡d2) ¡ F 0(¡d1)ª

(1.1b)

with

d1;2 = log(F 0=K ) § 1

2 ¾2Btex

¾B

p tex

:(1.1c)

Here F 0 ´ F (0) is today’s forward value of the asset, D(0; ts) is today’s discount factor to thesettlement date, and ¾B is the quoted volatility of the option.

The derivation of Black’s formula presumes that the forward prices F (t) are distributed log

normally about today’s forward price F 0,dF = ¾BF dW; F (0) = F 0:(1.2)

However, prices are r arely distributed log normally. Consequently market volatilities ¾B usuallyvary with the strike K and time-to-exercise tex .

Changing ¾B with the strike and exercise date essentially means that a di¤erent model is beingused for each strike and expiry. This presents several di¢culties when managing large books of options. First, it is not clear that the delta and vega risks calculated at a given strike are consistentwith the same risks calculated at other strikes, which interjects uncertainty into the consolidationand hedging of risks across strikes. Second, if ¾B varies with the strike K , it seems likely that ¾B

also varies systematically as the forward price F changes [3, 4]. Any vega risk arising from thesystematic change of ¾B with F could be hedged more properly (and inexpensively) as delta risk.Finally, it is di¢cult to know which volatility to use in pricing exotic options.

An alternative approach is to use a single model which correctly prices options at di¤erent strikesand exercise dates without “adjustment.” Commonly these models are of the form [3, 4]

dF = ® (t)A(F )dW; F (0) = F 0(1.3)

under the forward measure, with call and put prices given by the expected values

V call = D (0; ts)E ©

[F (tex) ¡ K ]+ j F (0) = F 0ª

(1.4a)

V put = D(0; ts)E ©

[K ¡ F (tex)]+ j F (0) = F 0ª

(1.4b)

in this measure.In Appendix A we use singular perturbation techniques [9] to analyze these models and …nd

explicit expressions for the values of European calls and puts. These formulas are then u sed to

obtain the implied volatilities ¾B of the options. For both calls and puts, we …nd that the impliedvolatility is

¾B = ajA(f av)j

f av

n1 + 1

24

£A00

A ¡ 2

¡A0

A

¢2+

2

f 2av

¤(F 0 ¡ K )2

+ 124

£2

A00

A ¡ ¡A0

A

¢2+

1

f 2av

¤a2A2(f av)tex + : : :

o:

(1.5a)

1



Here A and its derivatives are to be evaluated midway between today’s forward price and the strike,at

f av = 1

2 (F 0 + K );(1.5b)

and a is the “sum-of-squares average” of ®(t),

a =¡ 1

tex

Z tex0

®2(t0)dt0¢1=2

:(1.5c)

For example consider a power law model

dF = ®(t)F ¯dW:(1.6)

This model is just the constant elasticity of variance (CEV) model [2] written in terms of the forwardprice F (t). For this model the implied volatility is

¾B = af 1¡¯av

n1 + 1

24(1 ¡ ¯ )(2+ ¯ )

³F 0 ¡ K

f av

´2

+ 124

(1 ¡ ¯ )2 a2

texf 2¡2¯av

+ : : :o

;(1.7)

where f av and a are given by (1.5b) and (1.5c) as before.The “equivalent vol” formula (1.5) provides a very quick and very simple method for pricing

calls and puts. Instead of using lattice, PDE, or Monte Carlo methods to compute the value of theoption, one simply uses (1.5) to obtain the implied volatility ¾B , and then substitutes this ¾B intoBlack’s formula (1.1).

Option hedges can also be obtained by writing (1.5) as

¾B ´ ¾B(F 0; K ; tex)(1.8a)

and then writing the option price as

V = V Black(F 0; K ; tex ; ¾B(F 0; K ; tex));(1.8b)

where V Black is the call or put formula in (1.1). Di¤erentiating ( 1.8b) with respect to F 0 yields

@V

@F 0=

@V Black

@F 0+

@¾B

@F 0

@V Black

@¾B:(1.9)

Thus, the delta risk of the option is composed of two terms: the usual delta risk from Black’s formula,plus a term p roportional to the B lack vega. This last term arises from the s ystematic change in theimplied volatility ¾B of the option caused by changes in the forward price F .

The equivalent vol formula (1.5) is not exact. To test its accuracy, we compared the optionvalues obtained using the equivalent vol formula (1.5) against the “exact” option prices obtained bynumerical computation for di¤erent A(F ). We found that the equivalent vol formula is s urprisingly

accurate, usually as accurate as an excellent tree or PDE valuation, and much more accurate thanMonte Carlo solutions. To demonstrate its accuracy, consider the power law model (1.6). For powerlaw models, we found that the worst errors occur when ¯ = 0, which is to be expected since ¯ = 0 is“furthest” from the log normal Black model. Accordingly, in …gures 1-3 we show the error betweenthe exact option price and the price obtained using the equivalent vol formula (1.7) when ¯ = 0.Shown is the error as a function of the strike K for 1 year, 2 year, 5 year, and 10 year options.Figure 1 exhibits the error as the di¤erence between the exact implied vol and the equivalent vol

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obtained from (1.7); …gure 2 exhibits the error as a fraction of the forward price F 0; and …gure 3exhibits the error as a fraction of the option’s time value.

The singular perturbation analysis in Appendix A can be carried out to arbitrarily high order. In

Appendix B we state a more accurate equivalent vol formula obtained by carrying out the analysisthrough O(²4). Figure 4 shows the error made using this more a ccurate formula. Although thisformula is clearly much more accurate than (1.5), we believe that (1.5) is precise enough to makethis improvement super‡uous in most cases.

Singular perturbation techniques can be used to solve many related pricing problems. For exam-ple, these techniques can be used to solve intrinsically time-dependent models

dF = A(t; F )dW (1.10)

to obtain accurate equivalent vol formulas for calls and puts; to analyze one- and two-factor term-structure models to obtain accurate equivalent vol formulas for swaptions, ‡ oors, and caps; and toobtain quick and accurate pricing of many exotics.

A Singular Perturbation Expansion

Consider a European call with expiration date tex , settlement date ts , and strike K . As before, letF (t) be the stochastic process for the forward price as seen at date t. We are assuming that

dF = ®(t)A(F )dW (A.1)

under the forward measure. Under this measure, the value of the option at date t is V (t; F (t)),where the function V (t; f ) is given by the expected value [7, 8]

V (t; f ) = D(t; ts)E ©

[F (tex) ¡ K ]+ j F (t) = f ª

:(A.2)

Here D (t; ts) is the discount factor to the settlement date ts at date t.

To simplify the problem, let us strip the discount factor from V ,

V (t; f ) ´ D(t; ts)Q(t; f ):(A.3)

Then Q(t; f ) is de…ned by

Q(t; f ) = E ©

[F (tex) ¡ K ]+ j F (t) = f ª

;(A.4a)

and the expectation is over the probability distribution generated by the process

dF (t) = ®(t)A(F )dW (t):(A.4b)

Therefore Q (t; f ) satis…es the backward Kolmogorov equation [10]

Qt + 12 ®2(t)A2(f )Qff = 0; t < tex(A.5a)

with the …nal condition

Q = [f ¡ K ]+ at t = tex :(A.5b)

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A.1 Scaling and asymptotic solution

To obtain the option value, we …rst scale equtions (A.5a) and (A.5b) appropriately, and then usesingular perturbation methods to solve the scaled problem. To obtain the equivalent vol, we analyzeBlack’s model to determine the volatility ¾B which would yield the same value of the option.

To scale the equations appropriately, de…ne

² ´ A(K ) < < 1;(A.6a)

and the new variables

¿ (t) =

Z te

t

a2(t0)dt0; x = 1

²(f ¡ K ); ~Q(¿ ; x) =

1

²Q(t; f ):(A.6b)

In terms of the new variables, (A.5) becomes

~Q¿ ¡ 12

A2(K + ²x)

A2(K )~Qxx = 0 for ¿ > 0(A.7a)

with

~Q = x+ at ¿ = 0:(A.7b)

After solving (A.7) for ~Q(¿; x), the option value will be given by

V (t; f ) = D(t; ts)A(K ) ~Q¡

¿ (t) ; f ¡K A(K )

¢:(A.8)

By expanding

A(K + ²x) = A(K )©

1 + ²º 1x + 12

²2º 2x2 + : : :ª

(A.9a)

where

º 1 = A0(K )=A(K ); º 2 = A 00(K )=A(K ); : :: ;(A.9b)

problem (A.7) becomes

~Q¿ ¡ 12

~Qxx = ² º 1x ~Qxx + 12

²2(º 2 + º 21)x2 ~Qxx + : : : ; ¿ > 0(A.10a)

with

~Q = x+ at ¿ = 0:(A.10b)

We can solve (A.10) by expanding ~Q as

~Q(¿ ; x) = Q0(¿ ; x) + ²Q1(¿ ; x) + ²2Q2(¿; x) + : : : :(A.11)

Substituting (A.11) into (A.10) and equating like powers of ² yields a hierarchy of problems [9]. Atleading order, we obtain

Q0¿ ¡ 1

2Q0xx = 0 for ¿ > 0;(A.12a)

Q0 = x+ at ¿ = 0:(A.12b)

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At order ², we have

Q1¿ ¡ 1

2Q1xx = º 1xQ0

xx for ¿ > 0;(A.13a)

Q1 = 0 at ¿ = 0:(A.13b)

And at order ²2,

Q2¿ ¡ 1

2Q2

xx = º 1xQ1xx + 1

2(º 2 + º 21)x2Q0

xx for ¿ > 0;(A.14a)

Q2 = 0 at ¿ = 0;(A.14b)

and so on.The solution of (A.12a) yields

Q0

(¿ ; x) ´ G(¿ ; x) = x¡ xp ¿

¢ +

q ¿ 2¼ e¡

x2=2¿

;(A.15a)

as can b e veri…ed by direct substitution. Note that G(¿ ; x) is essentially the value of a call optionusing a normal (as opp osed to a log normal) model for the forward price. For later convenience, wenote that

G¿ = 12

e¡x2=2¿

p 2¼¿

; Gx = ( xp ¿ );(A.15b)

G¿¿ = x2 ¡ ¿

4¿ 2e¡x2=2¿

p 2¼¿

; Gx¿ = ¡ x

2¿

e¡x2=2¿

p 2¼ ¿

; Gxx = e¡x2=2¿

p 2¼¿

;(A.15c)

G¿¿¿ = x4 ¡ 6x2¿ + 3¿ 2

8¿ 4e¡x2=2¿

p 2¼¿

:(A.15d)

At O(²), we substitute Q0 ´ G(¿ ; x) into (A.13) and use Gxx = 2G¿ and (A.15c). This yields

Q1¿ ¡ 1

2 Q1xx = º 1xGxx = ¡2º 1¿ Gx¿ for ¿ > 0(A.16a)

Q1 = 0 at ¿ = 0:(A.16b)

The solution of (A.16) is

Q1

= ¡º 1¿ 2

Gx¿ ´ º 1¿ xG¿ ;(A.17)

as can be veri…ed by direct substitution.At O (²2), we substitute Q 0 = G and Q1 = º 1¿ xG¿ into (A.14). This yields

Q2¿ ¡ 1

2Q2xx =

³º 21

x4 ¡ 2¿ x2

2¿ + 1

2º 2x2´ e¡x

2=2¿

p 2¼¿

for ¿ > 0(A.18a)

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Q2 = 0 at ¿ = 0:(A.18b)

The solution of (A.18) is

Q2 = º 21£¿ 4G¿¿¿ + 8

3¿ 3G¿¿ + 12 ¿ 2G¿

¤+ º 2

£23 ¿ 3G¿¿ + 1

2 ¿ 2G¿ ¤

:(A.19)

To understand the solution, let us use (A.15) to write Q2 in a convenient way,

Q2 = 12

º 21¿ 2x2G¿¿ + 112

º 21(x2 ¡ ¿ )¿G¿ + 16

º 2(2x2 + ¿ )¿ G¿ :(A.20)

Then through O(²2), our solution is

~Q = Q0 + ²Q1 + ²2Q2 + : : :

= G + ²º 1¿ xG¿ + 12 ²2º 21¿ 2x2G¿¿ + ²2

¡4º 2 + º 2112

x2 + 2º 2 ¡ º 21

12 ¿

¢¿G¿ ;

(A.21)

which we can re-write as

~Q(¿ ; x) = G(~¿ ; x);(A.22a)

with

~¿ = ¿ £

1 + ²º 1x + ²2¡ 4º 2 + º 21

12 x2 +

2º 2 ¡ º 2112

¿ ¢

+ : : :¤

:(A.22b)

To obtain the option value, recall that

V (t; f ) = D(t; ts)² ~Q(¿; x) = D(t; ts)²G(~¿ ; x)(A.23a)

and note that

²G(~¿ ; x) = G(²2~¿ ; ² x) = G (A2(K )~¿; f ¡ K ):(A.23b)

Thus the value of the option is

V (t; f ) = D(t; ts)G(¿ ¤; f ¡ K );(A.24a)

where

¿ ¤ = A2(K )¿ £1 + º 1(f ¡ K ) +

4º 2 + º 2112

(f ¡ K )2 + 2º 2 ¡ º 21

12 A2(K )¿ + : : :

¤:(A.24b)

A.2 Implied Volatility

Although (A.24) gives a closed form expression for the price of the option, it is not very convenient.So we compute the equivalent Black volatility implied by this price. To simplify the calculation, wetake the square root of (A.24b) and obtain

p ¿ ¤ = A(K )

p ¿ h

1 + 12

º 1(f ¡ K ) + 2º 2 ¡ º 21

12 (f ¡ K )2 +

2º 2 ¡ º 2124

A2(K )¿ + : : :i

:(A.25)

Recall that º 1 = A0(K )=A(K ), º 2 = A00(K )=A(K ), so the …rst two terms of (A.25) arep ¿ [A(K ) + 1

2A0(K )(f ¡ K ) + : : : ]. This suggests expanding A around the average

f av = 12

(f + K )(A.26)

6



instead of K . Accordingly, we de…ne

° 1 = A0(f av )=A(f av); ° 2 = A 00(f av)=A(f av); : : : ;(A.27)

and re-write (A.25) in terms of ° 1 and ° 2 instead of º 1 and º 2. This then shows that the optionprice is

V (t; f ) = D(t; ts)G(¿ ¤; f ¡ K );(A.28a)

with

p ¿ ¤ = A(f av)

p ¿ h

1 + ° 2 ¡ 2° 21

24 (f ¡ K )2 +

2° 2 ¡ ° 2124

A2(f av)¿ + : : :i

:(A.28b)

Now suppose we had started with Black’s model

dF (t) = ¾BF (t)dW (A.29)

instead of dF = ® (t)A(F )dW . Repeating the preceding analysis for the special case ®(t) = ¾B ; A(F ) = F ,

shows that the option price is

V (t; f ) = D(t; ts)G(¿ B; f ¡ K );(A.30a)

with

p ¿ B = ¾Bf av

p tex ¡ t

h1 ¡ (f ¡ K )2

12f 2av¡ ¾2

B(tex ¡ t)

24 + : : :

i:(A.30b)

Since G (¿ B ; f ¡ K ) is an increasing function of ¿ B, the Black price (A.30) matches the correct price(A.28) if and only if

p ¿ B =

p ¿ ¤:(A.31)

Equating yields the implied volatility

¾B = aA(f av)

f av

n1 + (° 2 ¡ 2° 21 +

2

f 2av)

(f ¡ K )2

24 + (2° 2¡ ° 21 +

1

f 2av)

a2A2(f av)(tex ¡ t)

24 + : : :

o;

(A.32a)

where

f av = 12

(f + K ); ° 1 = A0(f av )

A(f av); ° 2 =

A00(f av)

A(f av) ;(A.32b)

and

a = 1

tex ¡ t

Z text

®2(t0)dt0:(A.32c)

Setting t = 0 at f = F 0 yields (1.5).Equation (A.32) gives the implied volatility for the European call option for the model

dF (t) = ®(t)A(F )dW (t):(A.33)

A similar analysis shows that the implied volatility for a European put option is given by the sameformula.

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B Higher order results

The above analysis can be carried out to arbitrarily high order. Carrying it out through O (²4) yields

the more accurate (but more complicated) equivalent vol formula

¾B = aA(f av)

f av

n1 +

¢

24

¡2° 2¡ ° 21 + 1

f 2av

¢ +

µ2

24

¡° 2 ¡ 2° 21 + 2

f 2av

¢

+ ¢2

480

¡2° 4 + 4 ° 1° 3 + 3° 22 ¡ 3° 21° 2 + 3

4° 41 ¡ 3

4f 4av

+ 10° 2 ¡ 5° 21 + 5=f 2av

2f 2av

¢

+ ¢µ2

2880

¡6° 4 ¡ 18° 1° 3 + 14° 22 ¡ 29° 21° 2 + 1 1° 41 ¡ 11

f 4av

+ 35° 2 ¡ 40° 21 + 40=f 2av

f 2av

¢

+ µ4

1440

¡34

° 4 ¡ 6° 1° 3 ¡ 2° 22 + 17° 21° 2 ¡ 8° 41 + 8f 4av

+ 5° 2 ¡ 10° 21 + 10=f 2av

f 2av

¢ + : : :

o:

(B.1)

Here

f av = 12

(f + K ); µ = f ¡ K;(B.2a)

a =³ 1

tex ¡ t

Z text

®2(t0)dt0´1=2

; ¢ = a2A2(f av)(tex ¡ t);(B.2b)

and

° 1 = A0(f av)

A(f av) ; ° 2 =

A00(f av)

A(f av) ; ° 3 =

A000(f av)

A(f av) ; ° 4 =

A0000(f av )

A(f av) :(B.2c)

For the special case of a power-law model, dF = ®(t)F ¯

dW , the equivalent Black vol reduces to

¾B = a

f 1¡¯av

n1 +

(1¡ ¯ )2¢

24f 2av+ (1 ¡ ¯ )(2 + ¯ )

µ2

24f 2av+

(1¡ ¯ )2¢2

1920f 4av(7¡ 54¯ + 2 7¯ 2)

+ (1¡ ¯ )2¢µ2

2880f 4av(29 ¡ 13¯ ¡ 16¯ 2) +

(1 ¡ ¯ )µ4

5760f 4av(72 + 34¯ ¡ 9¯ 2 ¡ 7¯ 3) + : : :

o:

(B.3)

References

[1] F. Black, “The Pricing of Commodity Contracts," J. Financial Economics 3 (1976), pp. 167-79

[2] J. Cox, “Notes on Option Pricing I: Constant Elasticity of Variance Di¤usion,” (Working paper,Stamford University, Stanford Calif., 1975)

[3] B. Dupire, “Pricing with a Smile," RISK 7 (1994), pp. 18-20

[4] B. Dupire, “Pricing and Hedging with Smiles," Working Paper, Paribas Capital Markets (1993)

[5] P. S . Hagan and D. E. Woodward “Markov Interest Rate Models,” p reprint (1997)

8



[6] J. C. Hull, Options, Futures, and Other Derivatives (Prentice-Hall, Upper Saddle River, 1997)

[7] J. M. Harrison and D. Kreps, “Martingales and arbitrage in multiperiod securities markets,” J.

Econ. Theory 20 (1979), pp. 381-408

[8] J. M. Harrison and S. R. Pliska, “Martingales and stochastic integrals in the theory of continuoustrading,” Stoch. Proc. Applications 11 ( 1981), p p. 215-260

[9] J. Kevorkian and J. D. Cole, Perturbation Methods in Applied Mathematics (Springer-Verlag,New York, 1981)

[10] P. Wilmott, J. Dewynne, and S. Howison, Option Pricing (Oxford Financial Press, Oxford,1993)

9



Figure 1: Error in the equivalent vol for a normal model (¯ = 0). Shown is the di¤erence betweenthe equivalent vol given by (1.5) and the exact implied volatility as a function of the strike K for 1year, 2 year, 5 year, and 10 year options. The case shown has F 0 = 100 and an at-the-money Blackvol of 20%. The graph is scaled so that a 1 bp error would change 20% to 20.01%.

Figure 2: The error in the option price caused by using the equivalent vol formula (1.5). Shown isthe error as a function of the strike K for 1 year, 2 year, 5 year, and 10 year options. Here 1 bpcorresponds to an error of 10¡4F 0 for the same case as Figure 1.

10



Figure 3: The error in the option price caused by using the equivalent vol formula (1.5). Shown isthe ratio of the error to the time value of the option as a function of the strike K for 1 year, 2 year,5 year, and 10 year options. Same case as Figure 1. Note that when the error is an appreciablepercent of the time value, the time value of the option is near zero.

Figure 4: Error in the equivalent vol for a normal model (¯ = 0). Shown is the di¤erence betweenthe more accurate equivalent vol given by (B.3) and the exact implied volatility as a function of thestrike K for 1 year, 2 year, 5 year, and 10 year options. Same case as Figure 1. Note that the graphis scaled so that a 2% error would change 20% to 20.0002%.

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