Equivalent Black Volatilities

Equivalent Black Volatilities

¤

Patrick S. Hagan

_{Diana E. Woodward}

March 30, 1998

Abstract

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

dF(t) =®(t)A(F)dW(t)

under the forward measure. By using singular perturbation techniques, we obtain explicit algebraic formulas for the implied volatility¾B in terms of today’s forward price F0 ´F(0),

the strikeKof the option, and the time to expirytex. The price of any call or put can then be

calculated 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 f1¡¯

av

n

1 +(1¡¯)(2 +¯) 24

³_F0¡K fav

´2

+(1¡¯)

2

24 a2_t

ex f2¡2¯

av

+: : :o

where fav= 12(F0+K).

Our formula for the implied volatility is not exact. However, we show that the error is insigni…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 do not necessarily re‡ect the views of NumeriX, The Bank of Tokyo-Mitsubishi,} or any of their a¢liates.

y_{Current mailing address: NumeriX, 546 Fifth Avenue, 17th Floor, New York, NY 10036.}

Market prices of many European options are quoted in terms of their Black-Scholes volatility ¾B. To obtain the cash price, let tex be the option’s exercise date, letts be the option’s settlement

date, and de…neF(t)to be the forward price of the asset (for a contract maturing at the settlement datets) as seen at datet. Then the cash price of calls and puts is given by Black’s formula [1, 6]

Vcall =D(0; ts)

©

F0(d1)¡K(d2) ª

(1.1a)

Vput=D(0; ts)

©

K(¡d2)¡F0(¡d1) ª

(1.1b)

with

d1;2=

log(F0=K)§ 12¾2Btex

¾Bptex

: (1.1c)

Here F0 ´ F(0) is today’s forward value of the asset, D(0; ts) is today’s discount factor to the

settlement date, and ¾Bis 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 priceF0,

dF =¾BF dW; F(0) =F0:

(1.2)

However, prices are rarely distributed log normally. Consequently market volatilities ¾B usually

vary with the strikeK and time-to-exercisetex.

Changing¾B with the strike and exercise date essentially means that adi¤erent model is being

used 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 consistent with the same risks calculated at other strikes, which interjects uncertainty into the consolidation and 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 the systematic 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 strikes and exercise dates without “adjustment.” Commonly these models are of the form [3, 4]

dF =®(t)A(F)dW; F(0) =F0

(1.3)

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

Vcall=D(0; ts)E

©

[F(tex)¡K]+ jF(0) =F0 ª

(1.4a)

Vput =D(0; ts)E

©

[K¡F(tex)]+jF(0) =F0 ª

(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 used to obtain the implied volatilities¾B of the options. For both calls and puts, we …nd that the implied

volatility is

¾B =

ajA(fav)j

fav

n 1 + 1

24 £A00

A ¡2

¡A0 A ¢2 + 2 f2 av ¤

(F0¡K)2

+₂₄1£2A 00

A ¡

¡A0 A ¢2 + 1 f2 av ¤

a2A2(fav)tex+: : :

o

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

fav = 1₂(F0+K);

(1.5b)

andais the “sum-of-squares average” of®(t),

a=¡ 1

tex

Z tex

0

®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 forward priceF(t). For this model the implied volatility is

¾B = a

fav1¡¯

n 1 + 1

24(1¡¯)(2 +¯)

³_F0¡K

fav

´2 + 1

24(1¡¯) 2a2tex

fav2¡2¯

+: : :o; (1.7)

wherefavandaare 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 the option, one simply uses (1.5) to obtain the implied volatility ¾B, and then substitutes this ¾Binto

Black’s formula (1.1).

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

¾B´¾B(F0; K; tex)

(1.8a)

and then writing the option price as

V =VB lack(F0; K; tex; ¾B(F0; K; tex));

(1.8b)

whereVB lack is the call or put formula in (1.1). Di¤erentiating (1.8b) with respect toF0yields

@V @F0

= @VBlack

@F0

+ @¾B

@F0

@VB lack

@¾B

: (1.9)

Thus, the delta risk of the option is composed oftwoterms: the usual delta risk from Black’s formula, plus a term proportional to the Black vega. This last term arises from the systematic change in the implied volatility¾B of the option caused by changes in the forward priceF.

obtained from (1.7); …gure 2 exhibits the error as a fraction of the forward price F0; and …gure 3

exhibits 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 analysis through O(²4_{). Figure 4 shows the error made using this more accurate formula. Although this}

formula is clearly much more accurate than (1.5), we believe that (1.5) is precise enough to make this 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 to obtain quick and accurate pricing of many exotics.

A Singular Perturbation Expansion

Consider a European call with expiration datetex, settlement datets, and strikeK. As before, let

F(t)be the stochastic process for the forward price as seen at datet. 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 functionV(t; f)is given by the expected value [7, 8]

V(t; f) =D(t; ts)E

©

[F(tex)¡K]+jF(t) =f

ª

: (A.2)

HereD(t; ts)is the discount factor to the settlement datets at date t.

To simplify the problem, let us strip the discount factor fromV,

V(t; f)´D(t; ts)Q(t; f):

(A.3)

ThenQ(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)

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

Qt+ 12®

2_(t)A2_(f)Q

f f = 0; t < tex

(A.5a)

with the …nal condition

Q= [f ¡K]+ at t=tex:

A.1 Scaling and asymptotic solution

To obtain the option value, we …rst scale equtions (A.5a) and (A.5b) appropriately, and then use singular perturbation methods to solve the scaled problem. To obtain the equivalent vol, we analyze Black’s model to determine the volatility¾B 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) Q~xx = 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);_Af¡_(KK₎¢: (A.8)

By expanding

A(K+²x) =A(K)©1 +²º1x+ 1₂²2º2x2+: : : ª

(A.9a)

where

º1=A0(K)=A(K); º2 =A00(K)=A(K); : : : ;

(A.9b)

problem (A.7) becomes

~

Q¿¡ 1₂Q~xx=²º1xQ~xx+1₂²2(º2+º21)x2Q~xx+: : : ; ¿ >0

(A.10a)

with

~

Q=x+ _{at¿ = 0:}

(A.10b)

We can solve (A.10) by expanding_Q~ _as

~

Q(¿ ; x) =Q0_{(¿ ; x) +²Q}1_{(¿ ; x) +²}2_Q2_{(¿ ; x) +: : : :}

(A.11)

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

Q0¿ ¡12Q 0

xx= 0 for¿ >0;

(A.12a)

At order ², we have

Q1

¿ ¡ 12Q 1

xx =º1xQ0xx for¿ >0;

(A.13a)

Q1= 0 at¿ = 0: (A.13b)

And at order ²2_,

Q2

¿ ¡ 12Q 2

xx=º1xQ1xx+ 12(º2+º 2

1)x2Q0xx for ¿ >0;

(A.14a)

Q2_{= 0 at¿ = 0;}

(A.14b)

and so on.

The solution of (A.12a) yields

Q0(¿ ; x)´G(¿ ; x) =x¡_px ¿

¢ +q¿

2¼e¡ x2=2¿_;

(A.15a)

as can be veri…ed by direct substitution. Note thatG(¿ ; x)is essentially the value of a call option using a normal (as opposed to a log normal) model for the forward price. For later convenience, we note that

G¿ = 1₂e

¡x2=2¿

p

2¼¿ ; Gx= (

x

p

¿);

(A.15b)

G¿ ¿ =

x2_¡¿ 4¿2

e¡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¿4

e¡x2=2¿

p 2¼¿ : (A.15d)

AtO(²), we substitute Q0 _{´ G(¿ ; x)into (A.13) and use G}

xx = 2G¿ and (A.15c). This yields

Q1¿ ¡ 12Q 1

xx=º1xGxx=¡2º1¿ Gx¿ for ¿ >0

(A.16a)

Q1= 0 at¿ = 0: (A.16b)

The solution of (A.16) is

Q1_=¡º

1¿2Gx¿ ´º1¿ xG¿;

(A.17)

as can be veri…ed by direct substitution.

AtO(²2), we substituteQ0=G and Q1=º1¿ xG¿ into (A.14). This yields

Q2¿ ¡12Q 2

xx=

³

º21

x4¡2¿ x2

2¿ +

1 2º2x

2´e¡x

2_=2¿

p

Q2_{= 0 at¿ = 0:}

(A.18b)

The solution of (A.18) is

Q2=º2₁£¿4G¿ ¿ ¿ + 8₃¿3G¿ ¿ + 1₂¿2G¿¤+º2£2₃¿3G¿ ¿+ 1₂¿2G¿¤:

(A.19)

To understand the solution, let us use (A.15) to writeQ2 _{in a convenient way,}

Q2= 1₂º21¿2x2G¿ ¿+ ₁₂1º21(x2¡¿)¿ G¿+ 1₆º2(2x2+¿)¿ G¿:

(A.20)

Then through O(²2_{), our solution is}

~

Q=Q0+²Q1+²2Q2+: : :

=G+²º1¿ xG¿ + 12² 2_º2

1¿2x2G¿ ¿+²2

¡4º2+º21

12 x

2₊2º2¡º21

12 ¿

¢

¿ G¿;

(A.21)

which we can re-write as

~

Q(¿ ; x) =G(~¿ ; x); (A.22a)

with

~

¿ =¿£1 +²º1x+²2

¡4º2+º21

12 x

2₊ 2º2¡º21

12 ¿

¢ +: : :¤: (A.22b)

To obtain the option value, recall that

V(t; f) =D(t; ts)²Q~(¿ ; x) =D(t; ts)²G(~¿ ; x)

(A.23a)

and note that

²G(~¿ ; x) =G(²2_{~¿ ; ²x) =G(A}2_{(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 +º1(f¡K) + 4º2+º 2 1

12 (f¡K)

2₊ 2º2¡º21

12 A

2_{(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, we take the square root of (A.24b) and obtain

p

¿¤_=A(K)p_¿h_{1 +}1

2º1(f¡K) +

2º2¡º21

12 (f¡K)

2₊ 2º2¡º21

24 A

2_{(K)¿ +: : :}i_:

(A.25)

Recall that º1=A0(K)=A(K),º2=A00(K)=A(K), so the …rst two terms of (A.25) are p

¿[A(K) +1

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

fav= 1₂(f+K)

instead of K. Accordingly, we de…ne

°1=A0(fav)=A(fav); °2 =A00(fav)=A(fav); : : : ;

(A.27)

and re-write (A.25) in terms of °1 and°2 instead ofº1 andº2. This then shows that the option

price is

V(t; f) =D(t; ts)G(¿¤; f¡K);

(A.28a)

with

p

¿¤_{= A(fav)}p_¿h_{1 +}°2¡2°21

24 (f ¡K)

2₊2°2¡°21

24 A

2_(f

av)¿ +: : :

i

: (A.28b)

Now suppose we had started with Black’s model

dF(t) =¾BF(t)dW

(A.29)

instead ofdF =®(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 =¾Bfav

p

tex¡t

h

1¡ (f ¡K) 2

12f2

av ¡

¾2

B(tex¡t)

24 +: : : i

: (A.30b)

SinceG(¿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 =a

A(fav)

fav

n

1 + (°2¡2°21+ 2

f2

av

)(f¡K) 2

24 + (2°2¡° 2 1+ 1 f2 av )a 2_A2_(f

av)(tex¡t)

24 +: : : o

; (A.32a)

where

fav=1₂(f+K); °₁= A

0_(fav)

A(fav); °2=

A00_(fav)

A(fav) ;

(A.32b)

and

a= 1

tex¡t

Z tex

t

®2_(t0_)dt0_:

(A.32c)

Settingt= 0atf=F0 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)

B Higher order results

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

the more accurate (but more complicated) equivalent vol formula

¾B =

aA(fav)

fav

n 1 + ¢

24 ¡

2°2¡°21+ f12 av ¢ + µ 2 24 ¡

°2¡2°21+ f22 av ¢ + ¢ 2 480 ¡

2°4+ 4°1°3+ 3°22¡3°21°2+3₄°41¡ _4f34 av +

10°2¡5°21+ 5=fav2

2f2

av

¢

+ ¢µ 2

2880 ¡

6°4¡18°1°3+ 14°22¡29°21°2+ 11°41¡ f114 av +

35°2¡40°21+ 40=fav2

f2 av ¢ + µ 4 1440 ¡3

4°4¡6°1°3¡2°22+ 17°21°2¡8°41+f84 av +

5°₂¡10°2

1+ 10=fav2

f2

av

¢

+: : :o: (B.1)

Here

fav= 1₂(f +K); µ=f¡K;

(B.2a)

a=³ 1

tex¡t

Z tex

t

®2(t0)dt0´1=2; ¢ =a2A2(fav)(tex¡t);

(B.2b)

and

°1=

A0(fav)

A(fav)

; °2=

A00(fav)

A(fav)

; °3=

A000(fav)

A(fav)

; °4=

A0000(fav)

A(fav)

: (B.2c)

For the special case of a power-law model,dF =®(t)F¯dW, the equivalent Black vol reduces to

¾B= a

fav1¡¯

n

1 +(1¡¯) 2_¢

24f2

av

+ (1¡¯)(2 +¯) µ 2

24f2

av

+ (1¡¯) 2_¢2

1920f4

av

(7¡54¯+ 27¯2)

+ (1¡¯) 2_¢µ2

2880f4

av

(29¡13¯¡16¯2) + (1¡¯)µ 4

5760f4

av

(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)

[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 continuous trading,” Stoch. Proc. Applications 11(1981), pp. 215-260

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

Figure 1: Error in the equivalent vol for a normal model (¯= 0). Shown is the di¤erence between the equivalent vol given by (1.5) and the exact implied volatility as a function of the strikeK for 1 year, 2 year, 5 year, and 10 year options. The case shown hasF0 = 100and an at-the-money Black

vol of 20%. The graph is scaled so that a 1 bp error would change 20% to 20.01%.

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