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JournalofB A N K I N G &F I N A N C EELSEVIER Journal of Banking& Finance 20 (1996) 985 1001

V a l u i n g f u t u r e s a n d o p t i o n s o n v o l a t i l i t yA n d r e a s G r i i n b i c h l e r , F r a n c i s A . L o n g s t a f f *

The John E. Anderson Graduate School o['Management at UCLA, 105 Hilgard Acenue, Lo~ Angeles,CA 90024-1481, USA

Received 15 January 1995; accepted 15 May 1995

A b s t r a c t

This paper presents simple closed-form expressions for volatili ty futures and optionprices and examines their implications for the characteristics of these securities. We showthat the properties of these volatility derivatives are fundamentally different from those ofconventional option and futures contracts. This analysis also provides insights into the rolethat volatility derivatives may play in managing and hedging volatility risk in financialmarkets.JEL classification: G 13Kevwords." Options; Exotic options; Valuation;Stochastic volatility

1 . I n t r o d u c t i o n

Few proposed types of derivative securities have attracted as much attentionand interest as futures and option contracts on volatility. In April 1993, Reutersbegan reporting the VIX index which tracks the implied volatility of S & P 100index calls and puts. The Wall Street Journal recently reported that the ChicagoBoard Options Exchange plans to unveil options on the VIX index shortly. Arecent issue of Futures reports that the American Stock Exchange is also consider-

* Corresponding author. Tel.: 310-825-2218; fax: 310-206-5455.0378-4266/96/$15.00 Published by Elsevier Science B.V.SSDI 0378-4266(95)00034-8

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9 8 6 A. Griinbichler, F.A. Longstaff / Journal o f Banking & Finance 20 (1996) 985-1001i n g d e v e l o p i n g v o l a t i l i t y o p t i o n s o n t h e U . S . s t o c k m a r k e t a n d t h a t m a r k e tr e g u l a to r s h a v e p r i v a t e ly e n d o r s e d t h e c o n c e p t . A r ti c le s i n t he N o v e m b e r 1 99 3i s su e o f Fu t u r e s a n d Op t i o n s W o r l d a n d th e Ju l y 2 5 , 1 9 9 4 i s su e o f B a r r o n s e x p l a i nh o w v o l a t i l it y d e r i v a t i v e s c o u l d b e u s e d t o h e d g e t h e v o l a t i l i ty r isk o f o p t i o np o r t f o l i o s . Vo l a t i l i t y swa p s h a v e r e c e n t l y b e g u n t o t r a d e i n t h e o v e r - t h e - c o u n t e rm a r k e t . I n E u r o p e , t h e G e r m a n F u t u r e s a n d O p t i o n s E x c h a n g e ( D T B ) l a u n c h e d av o l a ti li ty in d e x c a l l e d V D A X o n D e c e m b e r 5 , 1 99 4. T h i s i n d e x t r ac k s th e i m p l i e dv o l a ti li ty o f D A X i n d e x c a ll s a n d p u ts . T h e D A X i n d e x i s a v a lu e - w e i g h t e d i n d e xo f t h e 3 0 l a r g e s t f i r m s t r a d ed o n t h e F r a n k f u r t S t o c k E x c h a n g e . T h e A u s t r i anF u t u r e s a n d O p t i o n s E x c h a n g e ( O T O B ) a n n o u n c e d a v o l a t i l i t y i n d e x o n i t sAu s t r i a n T r a d e d I n d e x ( AT X) c a l l s a n d p u t s f o r 1 9 9 5 . Vo l a t i l i t y f u t u r e s a n do p t i o n s o n v o l a t i l i t y i n d e x e s a r e c u r r e n t l y b e i n g d e v e l o p e d b y a n u m b e r o fi n v e s t m e n t b a n k i n g f i r m s in t h e U . S . a n d E u r o p e .

Fu t u r e s a n d o p t i o n s o n v o l a t i li t y a r e c le a r l y f u n d a m e n t a l t y p e s o f d e r i v a t i v ese c u r it i es . B y a l l o wi n g i n v e s t o r s t o h e d g e d i r e c t l y a g a i n s t sh i f ts i n v o l a t i li t y , t h e sese c u r i ti e s e n a b l e i n v e s t o r s t o a v o i d th e c o s t s o f d y n a m i c a l l y a d j u s t in g p o s i t i o n s f o rc h a n g e s i n v o l a ti l it y a n d s e r v e t o m a k e t h e m a r k e t m o r e c o m p l e t e . 1 O n e t r a d e rs t a t e d " b y b e i n g a b l e t o b u y a n d s e l l v o l a t i l i t y , i n v e s t o r s wo u l d n o w b e a b l e t om a n a g e t h e ir r i sk in t wo d i m e n s i o n s - p r i c e r i sk a n d v o l a t i l it y r i sk - a no p p o r t u n i t y p r e v i o u s l y o u t o f r e a c h f o r a ll b u t t h e l a r g e a n d so p h i s t i c a te d o p t i o n su s e r s . "

T h i s p a p e r d e r i v e s s i m p l e c l o s e d - f o r m v a l u a t i o n e x p r e s s i o n s f o r a v a r i e t y o fv o l a t i l it y d e r i v a t i v e s . W e t h e n e x a m i n e t h e i m p l i c a t i o n s o f th e v a l u a t i o n e x p r e s -s i o n s f o r th e p r o p e r t i e s o f th e se s e c u r it i es . T h e o b j e c t i v e o f t h i s a n a l y s i s i s tou n d e r s t a n d h o w t h e s e d e r i v a t i v e s e c u r i t i e s d i f f e r f r o m m o r e c o n v e n t i o n a l f u t u r e sa n d o p t i o n s a n d t o p r o v i d e so m e i n s i g h t s i n t o t h e e c o n o m i c r o l e t h a t v o l a t i l i t yd e r i v a t i v e s m a y p l a y . T h e v a l u a t i o n m o d e l w e u s e c a p t u r e s m a n y o f t h e o b s e r v e dp r o p e r t i e s o f v o l a t i l i t y . I n p a r t i c u l a r , t h e m o d e l a l l o ws v o l a t i l i t y t o b e m e a nr e v e r t i n g a n d c o n d i t i o n a l l y h e t e r o sk e d a s t i c .

W e f i r s t e x a m i n e t h e p r o p e r t i e s o f v o l a t i l i t y f u t u r e s p r i c e s . W e sh o w t h a t t h e sep r i c e s c a n d i f f e r f r o m t h o s e i m p l i e d b y t h e s t an d a r d c o s t - o f - c a r ry m o d e l i n an u m b e r o f i m p o r t a n t w a y s . F o r e x a m p l e , v o l a t i l i t y f u t u r e s p r i c e s a r e b o u n d e da b o v e z e r o a n d t h e b a s i s c a n b e e i t h e r p o s i t i v e o r n e g a t i v e . I n a d d i t i o n , we sh o wt h a t l o n g e r - h o r i z o n v o l a t i l i t y f u t u r e s c a n b e v i r t u a l l y u se l e s s a s h e d g i n g v e h i c l e s .

W e t h e n d e r i v e v a l u a t i o n e x p r e s s i o n s f o r v o l a t i l i t y o p t i o n s . T h e se o p t i o n s h a v em a n y s u r p r i s i n g p r o p e r t i e s a n d a r e f u n d a m e n t a l l y d i f f e r e n t f r o m c o n v e n t i o n a lo p t i o n s . Fo r e x a m p l e , t h e p r i c e o f a v o l a t i l i t y c a l l c a n b e b e l o w i t s i n t r i n s i c v a l u e ,t h e t r a d i t i o n a l p u t - c a l l p a r i t y r e l a t i o n d o e s n o t h o l d f o r t h e se o p t i o n s , a n d t h e

I B r e n n e r a n d G a l a i ( 1 9 8 9 ) a n d W h a l e y ( 1 9 9 3 ) p r o v i d e e x c e l l e n t d i s c u s s i o n s o f h o w v o l a t il i tyd e r i v a t i v e s c a n b e u s e d t o h e d g e t h e v o l a t i l i t y r i s k o f p o r t f o l i o s c o n t a i n i n g o p t i o n s o r s e c u r i t i e s w i t hop t ion - l i ke f e a tu r e s .

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A. Griinbichler, F.A. Longstaff / Journal o f Banking & Finance 20 (1996) 985-1001 987p a t t e rn o f t i m e d e c a y is p e r v e r s e . T h e u n d e r l y i n g r e a s o n w h y t h e s e o p t i o n s b e h a v es o d i f f e r e n t l y i s t h a t v o l a t i l i t y i s n o t t h e p r i c e o f a t r a d e d a s s e t .

A s w i t h f u tu r e s , t h e s e re s u l ts h a v e i m p o r t a n t i m p l i c a t io n s f o r t h e h e d g i n gb e h a v i o r o f v o l a ti l it y o p t i o n s . W e s h o w t h a t p r ic e s o f v o la t i li t y c a ll s a n d p u t sb e c o m e l e s s s e n s i t i v e t o t h e c u r r e n t l e v e l o f v o l a t i l i t y a s t h e i r t i m e t o e x p i r a t i o ni n c r e a se s . T h i s d a m p e n i n g e f f e c t i m p l i e s t h a t th e d e l ta s o f l o n g e r - t e r m o p t i o n st e n d t o w a r d z e r o . F u r t h e r m o r e , t h e a b s o l u t e v a l u e o f th e d e l ta f o r a v o l a ti l it yo p t i o n i s a l w a y s l e s s t h a n o n e , a n d i s g e n e r a l l y l e s s t h a n s o m e f i x e d n u m b e r .T h e s e f e a t u r e s d r a m a t i c a l l y a l t e r t h e w a y i n w h i c h t h e s e o p t i o n s c a n b e u s e d t oh e d g e t h e v o l a ti l it y r i sk o f o p t io n p o r t f o l i o s , a n d h a v e f u n d a m e n t a l i m p l i c a t i o n sf o r t h e w a y t h a t v o l a t i l i t y o p t i o n c o n t r a c t s s h o u l d b e d e s i g n e d .

F i n a l l y , w e f o c u s o n t h e v a l u a t i o n o f o p ti o n s o n v o l a t il it y f u tu r e s . W e s h o wt h a t v o l a t i li t y f u t u r e s o p t i o n s c a n b e v a l u e d a s i f t h e y w e r e s i m p l e v o l a t i l it yo p t i o n s b y m a k i n g a t ra n s f o r m a t i o n o f t h e s tr ik e p r i c e . A s u r p r i s in g i m p l i c a t i o n o ft h is i s th a t f o r s o m e s t r i k e p r i c e s , t h e s e o p t i o n s a r e p r i c e d a s i f g u a r a n t e e d t o b e int h e m o n e y o r o u t o f th e m o n e y a t e x p ir a t io n .

T h e r e m a i n d e r o f th i s p a p e r is o r g a n i z e d a s f o ll o w s . S e c t i o n 2 p r e s e n ts t heb a s i c v a l u a t i o n f r a m e w o r k f o r v o l a t il i ty d e r i v a t iv e s . S e c t i o n 3 a p p l ie s t h e m o d e l t ov o l a t i l i t y f u t u r e s c o n t r a c t s a n d e x a m i n e s t h e i r p r o p e r t i e s . S e c t i o n 4 c o n s i d e r s t h ep r o p e r t i e s o f v o l a t i l i t y o p t i o n p r i c e s . S e c t i o n 5 a d d r e s s e s t h e v a l u a t i o n o f v o l a t i l it yf u t u r e s o p t i o n s . S e c t i o n 6 s u m m a r i z e s t h e r e s u l t s a n d m a k e s c o n c l u d i n g r e m a r k s .

2 . T h e v a l u a t io n f r a m e w o r k

I n th i s s e c ti o n , w e p r e s e n t t h e b a s i c v a l u a t io n f r a m e w o r k w i t h i n w h i c h s p e c i f i ce x p r e s s i o n s f o r v o l a ti l it y d e r i v a t i v e s e c u r i t ie s a r e d e r iv e d . A l t h o u g h t h e d i s c u s s i o nis c o u c h e d i n te r m s o f s t o c k i n d e x v o l a t i li ty , t h e f r a m e w o r k c o u l d a l s o b e a p p l i e dt o d e r i v a t i v e s o n o t h e r t y p e s o f v o l a t i l i t y s u c h a s c u r r e n c y o r i n t e r e s t - r a t ev o l a t i l it y . I n f a c t , t h e s e r e s u l ts c o u l d e a s i l y b e e x t e n d e d t o i n c l u d e d e r i v a t i v e s o ns u c h d i v e r s e n o n - p r i c e s t a t e v a r i a b l e s a s i n f l a t i o n r a t e s , c a s u a l t y i n s u r a n c e c l a i m s .o r h e a l t h c a r e c o s t i n d e x e s .

L e t V d e n o t e t h e c u r r e n t v a l u e o f th e s t a n d a r d d e v i a t i o n o f r e tu r n s f o r t h e s t o c ki n d e x . T h i s s t a n d a r d d e v i a t i o n c a n b e e i t h e r t h e i n s t a n t a n e o u s v o l a t i l i t y o r t h ev o l a t il i ty i m p l i e d f r o m s o m e o p t i o n p r i c in g m o d e l - t h e d i s t in c t i o n d o e s n o t a ff e c tt h e f o r m o f t h e re s u l ti n g v a l u a t i o n e x p r e s s i o n s . 2 L e t t h e d y n a m i c s o f V b e g i v e nb y

d V = ( a - K V ) d t + ~ r ~ d Z , 1)

For exam ple, Taylor and X u (19 94) show that w hen volatility is stochastic, the volatility estmmteimplied by inv erting the B lack-Sc holes mo del is nearly a linear function of the actual instantaneousvolatility.

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988 A. Griinbichler, F.A. Longstaff / Journal o f Banking & Finance 20 (1996) 985-1001w h e r e o r , K , a n d ~ r a r e c o n s t a n t s , a n d Z i s a s t a n d a r d W i e n e r p r o c e s s . T h i sf r a m e w o r k i s c o n c e p t u a l l y s i m i la r t o th a t u s e d b y W i g g i n s ( 1 9 8 7 ), H u l l a n d W h i t e( 1 9 8 7 ) , J o h n s o n a n d S h a n n o ( 1 9 8 7 ) , S c o t t ( 1 9 8 7 ) , S t e i n a n d S t e in ( 1 9 9 1 ), a n dH e s t o n ( 1 9 9 3 ) .

T h i s s p e c i f i c a t i o n o f th e v o l a t i l it y d y n a m i c s i s a ls o c o n s i s t e n t w i t h m a n y o f t h eo b s e r v e d p r o p e r t i e s o f s t o c k in d e x v o l a ti l it y . E m p i r i c a l e v i d e n c e b y F r e n c h e t a l.( 1 9 8 7 ) , H a r v e y a n d W h a l e y ( 1 9 9 2 ) , a n d S h e i k h ( 1 9 9 3 ) s u g g e s t s t h a t i n d e xv o l a ti li ty f o l lo w s a m e a n - r e v e r t i n g A R ( I ) p r o c e ss . R e g r e s s i o n s o f s q u a re d c h a n g e si n i m p l i e d v o l a t il i ty o n v o l a t il i ty l e v e l s i n d i c a t e t h a t t he v a r i a n c e o f c h a n g e s i ni m p l i e d v o l a t i l it y i s n o t c o n s t a n t , b u t i n c r e a s e s w i t h t h e l e v e l o f v o l a t i l i ty . T h ed y n a m i c s f o r V g i v e n i n ( 1 ) c a p t u r e b o t h o f t h e s e f e a tu r e s . I n a d d i t io n , t h e s ed y n a m i c s i m p l y t h a t V i s a l w a y s p o s i t i v e a n d h a s a l o n g - r u n s t a t i o n a r y g a m m a -d i s tr i b u ti o n . 3 T h e m e a n a n d v a r i a n c e o f th e s t a t io n a r y d i s tr i b u t io n a r e ~ / K a n dt ~ O ' 2 / 2 K 2 . S i m i l a r l y , t h e f i r s t - o r d e r s e r i a l c o r r e l a t i o n f o r V i s e - K ~ t . G i v e ne s t i m a t e s o f th e m e a n , v a r i a n c e , a n d s e r ia l c o r r e l a t io n o f V f r o m i n d e x e s s u c h a st h e V I X o r V D A X , t h e p a r a m e t e r s o r, K , a n d c r 2 c a n e a s i l y b e o b t a i n e d b yi n v e r t i n g t h e a n a l y t i c a l e x p r e s s i o n s f o r t h e m e a n , v a r i a n c e , a n d s e r ia l c o r r e l a t io n .

N o w c o n s i d e r t he v a l u a t i o n a t t im e z e r o o f a c o n t i n g e n t c la i m w i t h a p a y o f fB ( V r ) a t t i m e T d e p e n d i n g o n l y o n V r , t h e ti m e - T v a l u e o f V . S i n c e V i s n o t th ep r i c e o f a t r a d e d a s s e t , w e a l l o w f o r t h e p o s s i b i l i t y t h a t v o l a t i l it y r i s k is p r i c e d b yt h e m a r k e t . C o n s i s t e n t w i t h W i g g i n s ( 1 9 8 7 ) , S t e i n a n d S t e i n ( 1 9 9 1 ) , a n d o t h e r s ,w e m a k e t h e a s s u m p t i o n t h a t t h e e x p e c t e d p r e m i u m f o r v o l a t i l i t y r i s k i s p r o p o r -t i o n a l t o th e l e v e l o f v o l a t il i t y , { V . W e n o t e t h a t th i s a s s u m p t i o n i s s i m i l a r t o t h ei m p l i c a t i o n s o f g e n e r a l e q u i l i b r iu m m o d e l s s u c h a s C o x e t a l. ( 1 9 8 5 ) , H e m l e r a n dL o n g s t a f f ( 1 9 9 1) , a n d L o n g s t a f f a n d S c h w a r t z ( 19 9 2 ) , i n w h i c h r i sk p r e m i a i ns e c u r i t y r e t u rn s a r e p r o p o r t i o n a l t o th e l e v e l o f v o l a ti l it y . W e a l s o m a k e t h e u s u a la s s u m p t i o n s t h a t m a r k e t s f o r s e c u r i t i e s a r e p e r f e c t , f r i c t i o n l e s s , a n d a r e a v a i l a b l ef o r c o n t i n u o u s t r a d i n g . F u r t h e r m o r e , w e a s s u m e t h a t t h e r i s k le s s i n t e re s t r a te r i sc o n s t a n t .

I t i s im p o r t a n t t o n o t e t h a t t h e b a s i c n a t u r e o f o u r r e s u l ts i s u n a f f e c t e d b yw h e t h e r V c a n b e e x p r e s s e d a s a n o n - l i n e a r f u n c t i o n o f o t h e r s e c u r i t y p r i c e s a s int h e c a s e w h e r e V i s th e i m p l i e d v o l a t il i ty o f a n o p ti o n . T h i s i s b e c a u s e V c a n n o tb e r e p l i c a t e d b y a s e l f - f i n a n c i n g p o r t f o l i o o f th e o p t i o n a n d t h e i n d e x , e v e n t h o u g ha n o n - s e l f - f i n a n c i n g p o r t f o l i o c a n b e c o n s t r u c t e d w h i c h r e p l i c a t e s V e x a c t l y . T h ei n t u i t i o n f o r w h y t h i s i s t r u e i s r e l a t e d t o t h e f a c t t h a t a l l s e c u r i t i e s m u s t e a r n t h er i s k le s s r a te o f r e t u r n i n t he r i s k - n e u tr a l e c o n o m y . T h u s , a ll s e l f - f i n a n c i n gp o r t f o l i o s m u s t e a r n th e r is k l e s s r at e. A s i m p l i e d b y ( 1 ), h o w e v e r , t h e ' e x p e c t e d

3 The geometric Brownian motion process for V used by H ull and W hite (1987) implies that V isnon-stationary. Similarly with the constant elasticity of variance process assumed by Johnson andShanno (1987). The Ornstein-Uhlenback process use d by Sco tt (1987) and Stein and Stein (1991)implies that negative values of V are possible.

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A. Griinbichler, F.A. Longstaff Jo urn al of Banking & Finance 20 (1996) 985-1001 98 9r e t u r n ' o n V , c a n b e e i t h e r p o s i t i v e o r n e g a t i v e a n d g e n e r a l l y w i l l n o t e q u a l t h er is k l e ss r a te . T h i s m e a n s V c a n n o t b e th e v a l u e o f a s e l f - f in a n c i n g p o r t f o l i o o fs e c u r i t i e s a n d t h a t s t a n d a r d h e d g i n g a r g u m e n t s a r e n o t a p p l i c a b l e . N o t e t h a t t h i si s su e i s d i f f e r e n t f r o m t h e i s s u e o f w h e t h e r w e c a n s u b s t i t u te o u t th e v a l u e o f V i nt h e p r i c in g e x p r e s s i o n s u s i n g t h e n o n - l in e a r m a p p i n g b e t w e e n V a n d o b s e r v e dm a r k e t p r i c e s . T h i s l a t t e r i s s u e i s s i m p l y w h e t h e r i t i s p o s s i b l e t o m a k e a c h a n g e o fv a r i a b le s . I f s o , t h e c h a n g e o f v a r i a b l e s i s m a t h e m a t i c a l l y n e u t r a l a n d d o e s n o ta f f e c t a n y o f t h e p r o p e r t ie s o f t h e p r i c i n g e x p r e s s i o n s .

G i v e n t h i s f r a m e w o r k , t h e c u r r e n t v a l u e o f t h is c l a im , A ( V , T ) , sa t i s f i e s t hef u n d a m e n t a l v a l u a t i o n e q u a t i o n

o- 2- - V A v v + ( e t - 1 3 V ) A v - r A = A T , ( 2 )2

w h e r e 13 = K + 4 , s u b j e c t t o t h e e x p i r a t i o n - d a t e c o n d i t i o nA ( V , 0 ) = B ( V r ) . ( 3 )

L e t D ( T ) d e n o t e t h e c u r r e n t p r i c e o f a T - m a t u r i t y r i s k le s s u n i t d i s c o u n t b o n d . T h es o l u t i o n t o t h i s p a r t i a l d i f f e r e n t i a l e q u a t i o n c a n b e e x p r e s s e d a s

A ( V , T ) = D ( T ) E [ B ( V r ) ] , ( 4 )w h e r e t h e e x p e c t a t i o n i s t a k e n w i t h r e s p e c t t o t h e r i s k - a d j u s t e d p r o c e s s f o r V

d V = ( , , - 1 3 V )d t + ~ d Z . ( 5 )F rom F e l le r ( 1951) and Cox e t a ] . ( 1985) , t h i s r i sk -ad jus t ed p rocess imp l ies t ha ty V T i s d i s t r i bu t ed as a non -cen t ra l ch i - squared va r ia t e w i t h v deg rees o f f reedomand non-ce n t ra l i t y pa ramete r h , where

4 1 33 ' = r2(1 _ exp ( - 13T) ) '

40~v - ~ r 2 , ( 6 )

h = "y ex p( - 13T) V.T he non-cen t ra l ch i - squared d i s t r i bu t ion i s desc r ibed in Ch . 28 o f Johnson andKo t z ( 1970 ) . W i t h t h i s rep resen t a ti on o f so lu t i ons to t he pa r t ia l d i f f e ren t ia le q u a t i o n i n ( 2 ) , v a l u a t i o n e x p r e s s i o n s f o r v o l a t i l i t y d e r i v a t i v e s c a n b e o b t a i n e d b yd i r e c t l y e v a l u a t i n g t h e e x p e c t a t i o n i n ( 4 ) .

3 . V o l a t i l i t y f u t u r e sI n t h is s e c t i o n , w e d e r i v e f u t u r e s p r i c e s f o r fu t u r e s c o n t r a c t s o n v o l a t il i ty a n d

e x a m i n e s o m e o f t h e p r o p e r ti e s o f t h e se p r i c es . 4

4 Since the riskless interest rate is not stochastic, volatility futures and lbrw ard prices are identical.

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L e t F ( V , T ) d e n o t e t h e f u t u r e s p r i c e f o r a f u t u r e s c o n t r a c t o n V w i t h m a t u r i t yT . F o l l o w i n g ( 4 ) a n d E q . ( 4 6 ) o f C o x e t a l. ( 1 9 8 1 ) , t h e f u t u re s p r i c e c a n b ee x p r e s s e d a s t h e e x p e c t e d v a l u e o f V a t t i m e T

F ( V , T ) = E [ V T ] , ( 7 )w h e r e t h e e x p e c t a t i o n i s t a k e n w i t h r e s p e c t t o t h e r i s k - a d j u s t e d p r o c e s s f o r V .E v a l u a t i n g t h i s e x p e c t a t i o n g i v e s t h e f o l l o w i n g e x p r e s s i o n f o r t h e v o l a t il i ty f u tu r e sp r i c e

F ( V , T ) = ( a / [ 3 ) ( l - e x p ( - [ 3 T ) ) + e x p ( - f S T ) V . ( 8 )I n t h is m o d e l , v o l a t i l it y f u tu r e s p r i c e s a r e e x p o n e n t i a l l y w e i g h t e d a v e r a g e s o f

t h e c u r r e n t v a l u e o f V a n d t h e l o n g - r u n m e a n a / J 3 o f t h e r i s k - a d j u s te d p r o c e s s .A s T ~ 0 , t h e f u t u r e s p r i c e c o n v e r g e s t o t h e c u r r e n t v a l u e o f V . A s T ~ oo, t h ef u t u r e s p r i c e c o n v e r g e s t o ~ x / [ 3 . N o t e t h a t s i n c e t h e f u t u r e s p r i c e i s t h e e x p e c t e dv a l u e o f V t a k e n w i t h r e s p e c t t o t h e r is k - a d j u s t e d p r o c e s s f o r V , t h e fu t u r e s p r ic eg e n e r a l l y w i ll b e a b i a s e d e s t i m a t e o f t h e a c tu a l e x p e c t e d f u t u r e s p o t v a l u e o f V .3 .2 . P ro p e r t i e s o f v o la t il i ty f u tu re s p r i c e s

V o l a t i li t y f u tu r e s p r i c e s h a v e a n u m b e r o f in t e r e s ti n g p r o p e r t ie s . F o r e x a m p l e ,a s V ~ 0 , t h e v o l a t il i ty f u t u r e s p r i c e d o e s n o t c o n v e r g e t o z e r o . T h u s , v o l a t i l it yf u t u r e s p r i c e s a r e b o u n d e d a b o v e z e r o . T h e i n t u i t i o n f o r t h i s i s r e l a t e d t o t h e m e a nr e v e r s i o n o f t h e v o la t i li t y p r o c e s s . W h e n V r e a c h e s z e r o , V i m m e d i a t e l y r e t u r n s top o s i t i v e v a l u e s . T h u s , t h e e x p e c t e d v a l u e o f V r i s s t r i c t l y g r e a t e r t h a n z e r o e v e nw h e n t h e c u r r e n t v a l u e o f V i s z e r o . T h i s f e a t u r e o f v o la t i li t y fu t u r e s p r i c e sc o n t r a s t s w i t h t h o s e o f f u t u r e s p r ic e s o n t r a d e d a s s e t s. T h e l o w e r b o u n d o n f u t u r e sp r i c e s a l s o h a s i m p o r t a n t i m p l i c a t i o n s f o r p r i c i n g f u t u r e s o p t i o n s w h i c h i s d is -c u s s e d l a t e r .

A n o t h e r i m p o r t a n t p r o p e r t y o f v o la t i l it y f u tu r e s p r i c e s i s th a t t h e ir h e d g i n ge f f e c t i v e n e s s i s a f u n c t i o n o f t h e i r m a t u r i t y . I n p a r t i c u l a r , t h e p a r ti a l d e r i v a t i v e o fF ( V , T ) w i t h r e s p e c t to V i s e x p ( - [ 3 T ) . T h i s m e a n s t h a t v o l a t i li t y f u t u r e s p r i c e sd o n o t m o v e i n a o n e - t o - o n e r a t io w i t h c h a n g e s i n V . F u r t h e r m o r e , a s T i n c r e a se s ,a c h a n g e i n V h a s l e s s o f a n e f f e c t o n t h e f u t u r e s p r i c e . I n t h e li m i t a s T ~ oo,f u tu r e s p ri c e s a p p r o a c h t h e l o n g - r u n m e a n a / [ 3 o f th e v o l at il it y p ro c e s s a n d a r eu n a f f e c t e d b y t h e c u r r e n t v a l u e o f V . F o r t h i s r e a s o n , l o n g e r - t e r m f u t u r e s c o n t r a c t sm a y n o t b e e f f e c t i v e i n s t r u m e n t s f o r h e d g i n g v o l a t i l i t y r i s k . T h e i n t u i t i o n f o r t h i si s a g a i n r e l a t e d t o t h e m e a n r e v e r s i o n o f V . A n y c h a n g e i n t h e c u r r e n t v a l u e o f Vi s e x p e c t e d t o b e p a r t i a l ly r e v e r s e d p r i o r t o t h e e x p i r a t i o n o f t h e c o n t r a c t . H e n c e ,v o l a t i l i t y f u t u r e s p r i c e s m o v e s l u g g i s h l y i n r e s p o n s e t o v o l a t i l i t y s h o c k s . 5

5 Brenner and G alai (1989) and W haley (1993) describe how v olatility futures can be used to hedgevolatility risk, but do not provide explicit solutions for volatility futures prices.

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T h e b a s i s f o r v o l a t i l i t y f u t u r e s i s g i v e n b y( e ~ /1 3 - V ) ( 1 - e x p ( - 1 3 T ) ) . ( 9 )

T h e b a s i s c o n v e r g e s t o z e r o a s T ~ 0 , a n d c o n v e r g e s t o th e d i f f e r e n c e b e t w e e n t h ec u r re n t v a l u e o f V a n d t h e l o n g - ru n m e a n a / 1 3 a s T ~ ~ . A n i m p l i c a t io n o f t h i si s t h a t t h e v o l a t i l i t y f u t u r e s b a s i s c a n b e e i t h e r n e g a t i v e o r p o s i t i v e , d e p e n d i n g o nw h e t h e r t h e c u r r e n t v a l u e o f V i s a b o v e o r b e l o w i ts l o n g - r u n m e a n c~ /1 3. T h i sa g a i n c o n t r a s t s w i t h t he p r o p e r t i e s o f f u tu r e s o n t r a d e d c o m m o d i t i e s o r a ss e ts .

F i n a l l y , i t i s a l s o u s e f u l t o d e r i v e t h e d y n a m i c s o f th e f u t u r e s p r i c e . T h i s i si m p o r t a n t s i n c e if v o l a t il i ty f u t u re s p r i c e s w e r e t o b e c o m e o b s e r v a b l e , o n e m i g h ts u s p e c t t h a t i t w o u l d b e e a s i e r t o v a l u e d e r i v a t i v e s o n v o l a t i l i t y i n t e r m s o f t h ef u t u r es p r i c e r a t h e r t h a n V . T h e d y n a m i c s f o r t h e fu t u r e s p r i c e a re g i v e n b y as i m p l e a p p l ic a t io n o f l to ' s L e m m a t o ( 8) ,

d F = cr e x p ( - [ 3 ( T - t ) ) V/V d Z . ( 1 0 )N o t s u r p r i s i n g l y , s i n c e ( 7 ) i m p l i e s t h a t F ( V , T ) i s a n e x p e c t a t i o n , t h e f u t u r e s p r i c ei s a m a r t i n g a l e . I n v e r t i n g ( 8 ) a n d s u b s t i t u t i n g i n f o r V g i v e s

d F = c r e x p ( - 1 3 (T - t ) ) v / F e x p ( ~ 3 ( T - t ) ) - - ( e L / ~ ) ( e x p ( 1 3 ( T - - t ) ) -- 1 ) d Z . ( 1 1 )

S i n c e F i s a M a r k o v p r o c e s s , d e r i v a t i v e s o n V c a n b e v a l u e d u s i n g t h e f u t u re sp r i c e a s t h e u n d e r l y i n g s t at e v a r ia b l e . N o t e , h o w e v e r , t h a t t h e m a r k e t p r i c e o f ri sks t il l a p p e a r s i n t h e d y n a m i c s f o r F v i a t h e 13 p a r a m e t e r . T h u s , u s i n g t h e f u t u r e sp r i c e ra t h e r th a n V d o e s n o t e l im i n a t e t h e n e e d t o u s e r is k - a d j u s t e d p a r a m e t e r s int h e m o d e l . I n a d d i ti o n , t h e d y n a m i c s f o r F a r e n o w t i m e d e p e n d e n t a n d m o r ec o m p l e x t h a n t h e d y n a m i c s o f V g i v e n i n ( 5 ). F u r t h e r m o r e , s i n c e F i s n o tl o g n o r m a l l y d i s t r i b u t e d , t h e B l a c k ( 1 9 7 6 ) f r a m e w o r k i s n o t a p p l i c a b l e i n p r i c i n gd e r i v a t i v e s . T h e s e c o n s i d e r a t i o n s s u g g e s t t h a t t h e r e m a y b e l i m i t e d b e n e f i t s t ou s i n g t h e f u t u r e s p r i c e r a t h e r t h a n t h e v o l a t i l i t y i n d e x i n v a l u i n g v o l a t i l i t yd e r i v a t i v e s .

4 . V o l a t i l i t y o p t i o n sI n t h i s s e c t i o n , w e d e r i v e v a l u a t i o n e x p r e s s i o n s f o r v o l a t i l i t y o p t i o n s a n d

e x a m i n e t h e i r i m p l i c a t i o n s f o r t h e p r o p e r t i e s o f t h e s e s e cu r i ti e s . W e f o c u s f i r st o nE u r o p e a n o p t i o n s s i n c e c u r r e n t p r o p o s a l s f o r v o l a ti l it y o p t i o n p r i c e s h a v e E u r o -p e a n e x e r c i s e f e a t u r e s .4.1. Valuat ion express ions

L e t C ( V , K , T ) d e n o t e t h e c u r re n t v a l u e o f a c a l l o p t i o n o n V , w h e r e K i s t h es t r i k e p r i c e o f t h e o p t i o n a n d T i s t h e t i m e u n t i l e x p i r a t i o n . F r o m ( 4 ) , t h e v a l u e o f

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C ( V , K , T ) = D ( T ) E [ m a x ( 0 , V r - K ) ] . ( 1 2 )Eva lua t ing th i s e xpe c ta t ion g ive s the f o l lowing c lose d - f o r m e xpr e ss ion f o r theva lue o f a vo la t i l i t y c a l l op t ion

C ( V , K , T ) = D ( T ) e x p ( - f ST ) V Q ( ' , / K I v + 4 , h )+ D ( T ) ( o L / f 3 ) ( 1 - e x p ( - [ 3 T ) ) Q ( ' y K I v + 2 , h )- D ( T ) g o ( ' v g l v , h ) , ( 1 3 )

w h e r e Q ( . Iv + i , h ) i s t he c om ple m e n ta r y d i s t r ibu t ion f unc t ion f o r the non- c e n t r a lc h i - squa r e d de n s i ty w i th v + i de g r e e s o f f r e e do m a nd n on- c e n t r a l i ty pa r a m e te rh . 6 The vo la t i li t y c a ll p r i c e is a n e xp l i c i t f unc t ion o f V a nd T , a nd de pe nds on theexer c ise pr ice K , the r i sk less in te res t ra te r throu gh D ( T ) , a nd the pa r a m e te r s o fthe r isk-adjusted volat i l i ty process c~, 13, and or .

I n som e wa ys , t h i s e xpr e s s ion f o r the va lue o f a vo la t i l i t y c a l l r e se m ble s they i e l d o p t io n f o r m u l a d e r i v e d in L o n g s t a f f ( 1 9 9 0) . T h i s i s b e c a u s e L o n g s t a f fa s sum e s tha t t he sho r t - t e r m in te r e s t r a t e f o l lows a squa r e - r oo t p r oc e ss s im i l a r to( 1 ) . De sp i t e th i s , how e ve r , t he r e a re m a ny d i f f e r e nc e s be tw e e n the two m ode l s . I npa r t ic u la r , t he d i sc oun t f a c to r is un c or r e l a t e d wi th the e xp e c te d pa y of f o f theop t ion in th i s m od e l . I n c on t r a s t, t he d i sc oun t fa c to r a nd the e xpe c te d pa y of f o f theop t ion a r e pe r f e c t ly ne ga t ive ly c o r r e l a t e d in the L on gs ta f f y i e ld op t ion m od e l . Th i sd i s t inc t ion l e a ds to s ign i f i c a n t d i f f e r e nc e s be tw e e n the p r i c ing be h a v io r o f vo la ti l -i t y op t ions a nd y ie ld op t ions . F o r e xa m ple , t he de l t a o f a vo la t i l i t y c a l l i s a lwa yspos i t ive wh i l e the de l t a o f a y i e ld c a l l c a n be ne ga t ive . Thus , t he s im i l a r i tybe tw e e n th i s m od e l a nd the L on gs ta f f y i e ld op t ion m od e l i s l a r ge ly supe rf i c ia l .

T h e p r i c e o f a E u r o p e a n p u t P ( V , K , T ) i s g i v e n b y t h e f o l l o w i n g p u t - c a l lpa r i ty r e l a t ion f o r vo la t i l i ty op t ions

P ( V , K , T ) = C ( V , K , T ) - O ( T ) F ( V , T ) + O ( T ) K . ( 1 4 )Th i s pu t - c a l l pa r i ty r e l a t ion d i f f e r s f r om the pu t - c a l l pa r i ty r e l a t ion f o r op t ions ont r a de d a s se ts . T he r e a so n f o r th i s i s tha t t he p r e se n t va lue o f a po r t f o l io tha t pa ys Va t tim e T i s no t e qua l to the c u r r e n t va lue o f V . R a the r , i t e qua l s the p r e se n t va lueof the f u tu r e s p r i c e . Thus , t he pu t - c a l l pa r i ty r e l a t ion f o r vo la t i l i t y op t ions d i f f e r sf r om the usua l pu t - c a l l pa r i ty r e l a t ion by the subs t i tu t ion o f D ( T ) F ( V , T ) f o r Von the r igh t - ha nd s ide o f ( 14 ) .

C o m p ut ing c a l l a nd pu t p r ic e s r e qu i r e s c al c u la t ing the c om ple m e n ta r y non-c e n t r a l c h i - squa r e d d i s t r ibu t ion . W hi le th is f unc t ion i s s t r a igh t f o r wa r d to e va lua te ,i t i s o f t e n m or e c onve n ie n t to c a l c u la t e i t s va lue us ing the h igh ly a c c u r a te no r m a la ppr ox im a t ion sugge s te d by S a nka r a n ( 1963) . Th i s a ppr ox im a t ion i s

Q ( ~ K I v , h ) = 1 - U ( d ) , ( 1 5 )6 The derivation s available upo n request from the authors.

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- t ) ,

h = 1 - + + 3 X ) ( v + 2 X ) - 2k = { h 2 2 ( v + 2 k ) [ 1 - ( 1 - h ) ( 1 - 3 h )( v+ 2 h ) ( v + h) - 2 ]} - ' /2

x ) 2v + 2 X ( v W 2 X ) 2l = l + h ( h - 1 ) ( v + k ) 2 h (h 1) (2 - h ) (1 3 h) 2 ( v + h ) 4

An advantage of this approximation is that call and put prices can be calculatedusing the same types of programming routines used to compute Black and Scholes(1973) option prices. Examples of the accuracy of this algorithm are presented inCh. 28 of Johnson and Kotz (1970).4.2. Pro perties of volatility calls

Taki ng the limit of the volatility call valuatio n expression in (13) shows that thesolution satisfies the expiration date condition C (Vr, K,0) = max(0,V - K). Thisfollows since Q('yKlv,h) converges to zero when K < V, and conver ges to onewhen K > V.

The first major difference between the properties of volatility calls and calls ontraded assets is that C(V,K ,T) does not converge to zero as V ~ 0. This is shownin Fig. 1, which graphs the values of volati lity calls as a function of V for variousvalues of T. The parameters used in Fig. 1 imply a long-r un mea n and standarddeviation for V of 0.15 and 0.05, respectively. 7 The reason why the call pricedoes not converge to zero as V --* 0 is related to the me an- rev ert ing behavio r of V.If the value of V ever reaches zero, the process for V immediat ely returns tonon-zero values. Consequently, the value of a volatility call is not zero whenV = 0 since the call could still be in the money on its expirat ion date. In contrast,if the price of a traded asset equals zero, there is no possibility that the price of theasset will eventually become non-zero (otherwise the present value of the assetcould not be zero). This is also the reason why there are three terms in (13) ratherthan the usual two terms in model s such as the Black and Scholes (1973) equation.

7 The parameters used in the figures are consistent with findings in the literature. We assume along-term volatility of 0.15, which is similar to estimates reported by Harvey and Whaley (1992) forimplied volatilities on the OEX. The same holds for o 2 which is equal to 0.133 in the figures. Weassume a half-life of 90 trading days for volatility shocks. This is consistent with many recent paperswhich find that shocks in volatility are highly transitory.

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9 9 4 A. Gri inbichler , F.A. Lon gsta f f Jo ur na l o f Banking & Finance 20 (1996) 98 5-1 0010 . 2 5

0 .2

0 . 1 5

-4,~ 0.10.05-

o . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 0 8 . 1 6 2 4 . 3 2 .4 0V O L A T I L I T Y

- - T = . 1 0 - - T = . 3 0 . . . . .. . T = . 50

F i g . 1 . T h e v a l u e o f a v o l a t i l i t y c a l l g r a p h e d a s a f u n c t i o n o f t h e u n d e r l y i n g v o l a t i l i t y . T d e n o t e s t h em a t u r i t y o f th e o p t i o n i n y e a r s . T h e p a r a m e t e r s u s e d a r e a = 0 . 6 0 , 1 3 = 4 . 0 0 , ~ 2 = 0 . 1 3 3 , r = 0 . 0 5 , a n dK = 0 . 1 5 .

T h e e x t r a t e r m r e f l e c t s t h a t t h e o p t i o n s ti ll h a s v a l u e e v e n w h e n V = 0 ( w h e nV = 0 , t h e f i r s t t e r m i n ( 1 3 ) d i s a p p e a r s ) .

B e c a u s e t h e v a l u e o f a v o l a t i l i ty c a ll i s g r e a t e r t h a n z e r o w h e n V = 0 , t h ev o l a t i l i t y c a l l d o e s n o t s a t i s f y t h e u p p e r b o u n d a r y r e s t r i c t i o n d e r i v e d b y M e r t o n( 1 9 7 3 ) f o r c a ll s o n t r a d e d a s se t s. I n p a r ti c u l a r , t h e v a l u e o f a v o l a t i li t y c a ll e x c e e d st h e n u m e r i c a l v a l u e o f t h e u n d e r l y i n g v a r i a b l e V f o r s u f fi c i e n t l y s m a l l V .I n t u i t iv e l y , t h e r e a s o n f o r t h is p r o p e r t y i s th a t t h e e x p e c t e d p a y o f f f o r a v o l a t il i tyc a l l i s h i g h e r t h a n t h e c u r r e n t v a l u e o f V w h e n V i s s m a l l b e c a u s e o f th em e a n - r e v e r t i n g b e h a v i o r o f v o l a t i li t y . N o t e t h a t s i n c e V i s n o t t h e p r i c e o f a t r a d e da s se t, t h e v i o l a ti o n o f t h e M e r t o n ( 1 9 7 3 ) u p p e r b o u n d a r y d o e s n o t i m p l y t h ee x i s t e n c e o f a r b it r a g e o p p o r t u n i t ie s .

D i f f e r e n t i a t i o n s h o w s t h a t v o l a t i l i t y c a l l s a r e i n c r e a s i n g f u n c t i o n s o f V . I n t h el im i t , t h e v a l u e o f a v o l a t i l it y c a l l a p p r o a c h e s D(T)exp(- 3 T ) V , w h i c h b e c o m e si n f in i t e a s V - -* ~ . A n i m p o r t a n t p r o p e r t y o f t h is l im i t , h o w e v e r , i s t h a t th e v a l u eo f a v o l a ti l it y c a l l b e c o m e s l e s s t h a n i ts i n tr i n s ic v a l u e f o r s o m e v a l u e o f V . T h i si s a l s o i ll u s tr a t e d i n F i g . 1 , w h i c h s h o w s t h a t th e p r i c e o f a v o l a t i li t y c a ll c a n b el e s s t h a t i t s i n t r i n s i c v a l u e w h e n t h e c a l l i s o n l y s l i g h t l y i n t h e m o n e y . T h e r e a s o nf o r th is p r o p e r t y a g a i n f o l l o w s f r o m t h e m e a n r e v e r s io n o f v o la t il it y . W h e n V i sa b o v e i t s l o n g - r u n m e a n , m e a n r e v e r s i o n i m p l i e s t h a t t h e e x p e c t e d f u t u r e v a l u e o fV w i ll b e l o w e r t h a n i ts c u r r e n t v a l u e . T h i s i m p l i e s t h a t th e e x p e c t e d p a y o f f f o r av o l a t i l it y c a ll c a n b e l e ss t h a n i ts c u r r e n t i n tr i n s ic v a l u e - t h e e x p e c t e d c h a n g e i nV i s n e g a t i v e . T h i s c o u l d n o t o c c u r i f V w e r e t h e p r ic e o f a t r a d e d a s s e t s i n c en e g a t i v e r e t u rn s w o u l d n o t b e c o n s i s t e n t w i t h t h e a b s e n c e o f a r b it ra g e . A s b e f o r e ,t h e v i o l a t io n o f th e M e r t o n ( 1 9 7 3 ) l o w e r b o u n d a r y r e s tr ic t io n b y v o l a ti li ty c a llo p t i o n s d o e s n o t i m p l y th e e x i s t e n c e o f a r b i t r a g e o p p o r t u n i t i e s .

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0"9f~"0 . 8 \o.z ',.o.s ',,0 . 5 4 " -

0 . 3 - " . .0 . 2 - " ' - ~ ' > , ,0 . 1 -

0 r , , . . , , , , , , , , ,, 2 . 4 . 6 . 8 1 . 0T I M E T O E X P IR A T IO NV = . 1 0 - - V = .1 5 . . . .. . . V = . 2 0

F i g . 2 . T h e d e l t a o f a v o l a t i l i t y ca l l g r a p h e d a s a f u n c t i o n o f it s t i m e u n t il e x p i r a t i o n m e a s u r e d i n y e a r s .T h e p a r a m e t e r s u s e d a r e ~x = 0 . 6 0 , 13 = 4 . 0 ( I, 0 2 = 0 . 1 3 3 . r = 0 . 0 5 , a n d K - 0 . 1 5 .

A s w i t h vo l a t i l i t y f u t u r e s p r i c e s , t he p r i c i ng e xp r e s s i on f o r vo l a t i l i t y c a l l s ha si m p l i c a t i ons f o r t he he dg i ng b e ha v i o r o f t he se op t ions . B y i n spe c t i on o f ( 13 ) , t hevo l a t i l i ty ca l l p r i c e de pe n ds o n V on l y t h r ough t he t e r m e xp ( - [3T ) V . T h i s m e a nst h at w h e n e x p ( - [ 3 T ) i s sm a l l , V h a s l it tl e in f l u e n c e o f th e c u r r en t v a l u e o f t h ec a l l op t i on . I n o t he r w or ds , a s T i nc r e a se s , t he de l ta o f t he c a l l a pp r oa c h e s z e r oa nd t he g r a ph o f the c a l l va l ue i s e s se n t ia l l y f l a t f o r r e l e va n t r a nge s o f V . T h i s c a nbe seen in F ig . 1 , where the ca l l p r i ce f l a t t ens ou t a s T inc reases . S imi la r ly , F ig . 2s h o w s t h a t t h e d e l t a s o f a t - t h e - m o n e y a n d i n - t h e - m o n e y c a l l s c a n d e c r e a s e a s Ti nc r e a se s . F o r l a r ge e nough T , t he de lt a s o f a ll c a l ls a pp r oa c h z e r o .

A n i m m e d i a t e i m p l i c a t i on o f t he se r e su l t s i s t ha t l onge r - m a t u r i t y c a l l op t i onsha ve l i t t l e o r no va l ue a s he dg i ng i n s t r um e n t s s i nc e t he i r p r i c e s a r e no t a f f e c t e d byc ha n ge s i n V . T h e i n t u i t ion f o r w hy t h is p r o pe r t y ho l d s i s re l a t e d to t he ha l f - l if e o fde v i a t i ons in V f r o m i ts lon g - t e r m m e a n . A ss um e tha t a sudde n i nc r e a se in Vo c c u r s . T h e d y n a m i c s o f V i m p l y th a t r o u g h l y o n e - h a l f o f th e d e v i a t io n w i ll b ee l i m i na t e d i n 1 / [ 3 pe r i ods . I f t he ho r i z on o f the op t i on T i s m a n y t im e s t ha t o f t heha l f - li f e , t he n a n i nc r e a se i n V w i l l ha ve l i t tl e e f f e c t o f the e x pe c t e d p a y o f f o f t heop t i ons a nd , t he r e f o r e , on t he c u r r e n t p r i c e o f the c a l l. T hus , t he s e ns i t i v it y o f c a l lp r i c es t o c h a n g e s i n t h e c u rr e n t v a l u e o f V i s d a m p e n e d b y t h e e f f e c ts o f m e a nr e ve r s i on .

A l t hough l onge r - m a t u r i t y c a l l s ha ve de l t a s ne a r z e r o , sho r t e r - m a t u r i t y c a l l s c a nbe u se d t o he dge . A n i m por t a n t f e a t u r e o f vo l a t i l i t y c a l l s , how e ve r , i s t ha t t he i rd e lt as ar e b o u n d e d a b o v e b y D ( T ) e x p ( - [ 3 T ) . T h i s u p p e r b o u n d o n th e d el ta o ft he c a l l c a n be a s i gn i f i c a n t r e s t r i c t i on on t he he dg i ng p r ope r t i e s o f e ve nsho r t - t e rm o pt ions . T hi s i s i llus t ra t ed in F ig . 3 , wh ich p lo t s the de l t a o f a ca l l a s af unc t i on o f V f o r va r i ous va l ue s o f T . F o r e xa m pl e , t he sho r t - t e r m ca l l w i t hT = 0 . 10 ha s a de l ta tha t i s ne v e r g r e a t e r t ha n 0 .70 . T h e de l t a s f o r t he l onge r - m a -

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9 9 6 A. G riinbichler , F.A. Long sta f f Journa l o f Banking & Finance 20 (1996) 98 5-1 0010.7-

0 .6-

0.5-

0 .4

0 . 3 -

0.2- ~ ~"

. . . . . . . . . . . . . . . . . . . . . . . .0.1-

. 0 8 . 1 6 , 2 4 , , 3 2 .4 .0V O L A T I U T Y

- - T = . 1 0 - - - - T = . 3 0 . .. .. .. T = . 5 0F i g . 3 . T h e d e l t a o f a v o ] a t i l i t y c a l l g r a p h e d a s a f u n c t i o n o f t h e u n d e r l y i n g v o l a t i l i t y . T d e n o t e s t h em a t u r i t y o f th e o p t i o n i n y e a r s . T h e p a r a m e t e r s u s e d a r e ~ = 0 . 6 0 , 13 = 4 . 0 0 , ~ 2 = 0 . 1 3 3 , r = 0 . 0 5 , a n dK = 0 . 1 5 .

t u r i ty c a l l s ne ve r e xc e e d 0 . 30 . N o t e t ha t the de l t a s f o r t he l onge r - m a t u r i t y c a ll s a r ea l so b o u n d e d a b o v e ze r o a s V ~ 0 .

T h e s e c o n d d e r i v a t i v e o f C w i t h r e s p e c t t o V , th e g a m m a o f th e c a ll , ispos i t i ve . A s w i t h op t i ons on t r a de d a s se t s , t he ga m m a o f a vo l a t i l i t y c a l l i s h i ghe s tf o r n e a r - e x p i r a ti o n a t - t h e - m o n e y o p t i o n s . C o n s e q u e n t l y , t h i s f e a t u re m a k e s i t c l e a rt h a t h e d g e r s n e e d t o m o n i t o r t h e d e l ta o f t h e se t y p e s o f o p t io n p o s i ti o n s c a r e f u l l ys i nc e t he de l t a o f t he pos i t i on c a n c ha nge s i gn i f i c a n t l y i n r e sponse t o sm a l lc ha nge s i n V . I n c on t r a s t , t he ga m m a f o r a l onge r - m a t u r i t y vo l a t i l i t y c a l l i s ne a rz e r o f o r a l l va l ue s o f V .

I n c o n t r a s t to t h e B l a c k - S c h o l e s f o r m u l a , t h e v a l u e o f a v o l a ti li t y c a ll o p t i o n i sno t a l w a ys a n i nc r e a s i ng f unc t i on o f T . I n f a c t , t he l i m i t o f C(V,K,T) a s T ~e qua l s z e r o . T h i s c a n be se e n i n F i g . 4 , w h i c h g r a phs vo l a t i l i t y c a l l p r i c e s a s af unc t i on o f T . I n t u i t i ve l y , t he r e a son f o r t h i s p r ope r t y i s t ha t V ha s a l ong - r uns t a t i ona r y d i s t ri bu t ion . Co nse qu e n t l y , a s T i nc r e a se s , t he e xpe c t e d pa y o f f f o r t hec a l l op t i on i s bounde d . H ow e ve r , a s T i nc r e a se s , t he va l ue o f D(T) use d t od i s c o u n t t h e e x p e c t e d p a y o f f a p p r o a c h e s z e r o . A s a re s u lt , t h e p r o d u c t o f D(T)a n d t h e e x p e c t e d p a y o f f c o n v e r g e s t o z e r o . B e c a u s e o f t h is p r o p e r t y , th e s ig n o ft he de r i va t i ve CT, t he t he t a o f the c a l l, i s a m b i gu ous . F o r sm a l l T , t he t a c a n beg r e a t e r t ha n z e r o . A s T i nc r e a se s , t he ta u l t im a t e l y be c om e s ne ga t i ve . T he t he t a o fvo l a t i l i t y c a l l s i s g r e a t e s t i n a b so l u t e t e r m s f o r sho r t - t e r m i n - t he - m one y op t i ons .

T he e f f e c t o f a n i nc r e a se i n t he s t r i ke p r i c e o f a vo l a t i l i t y c a l l i s a l w a ysne ga t i ve . I n t e r e s t i ng l y , a n i nc r e a se in K doe s no t ha ve a n e f f e c t sym m e t r i c t o ade c r e a se i n V . A n i nc r e a se in K ha s a s i gn i f i c a n t e f f e c t on t he p r i c e s o f bo t hsho r t - t e r m a nd l ong - t e r m c a l l s . I n c on t r a s t , a de c r e a se i n V ha s l i t t l e e f f e c t on t hev a l u e o f a l o n g - t e r m c a l l . T h u s , t h e n o t i o n o f ' m o n e y n e s s ' i s s u b t l y d i f f e r e n t f o r

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A. Gr~nbichler , F .A. Lo ngsta f f Journa l o f Banking & Finance 20 (1996) 9 85-1 001 9 9 70 , 0 5 -

0 " 0 4 5 " , . .0 . 0 4 -

0 .035- " , ,0 . 0 3 - " " - ~n-O. 0.0"25-0. 02 - " ..................................

0.015-0.01-

0.005-

.2 .4 .6 .8 1.0TIME TO EXPIRATION

~ V =.1 0 -- V=. 15 . . . .. . . V=. 20F i g . 4 . T h e p r i c e o f a v o l a ti l i t y c a l l g r a p h e d a s a fu n c t i o n o f i ts t i m e t o e x p i r a t i o n m e a s u r e d i n y e a r s .T h e p a r a m e t e r s u s e d a r e a = 0 . 6 0 , 13 = 4 . 0 0 , o r : = 0 . 1 3 3 , r = 0 . 0 5 , a n d K = 0 . 1 5 .

these types of calls. The 'moneyness' of a call is a function not only of thedifference between the current value of V and K, but also of the differencebetween the long-run mean of V and K.

In the Black-Scholes model, call options are increasing functions of theriskless rate r. The intuition for this is that an increase in r increases the upwarddrift of the risk-neutral process for the underlying asset. In contrast, volatility callsare decreasing functions of the riskless rate, through the discount factor D(T).This is because an increase in r has no effect on the drift of the process for V.Consequently, the only effect of an increase in r is to reduce the discount factorD(T) , which in turn, reduces the value of the volatility call.

Finally, volatil ity call prices also depend on the values of the c~, 13, and er 2parameters. To examine the sensitivity to changes in these parameters, wecompute the elasticity of the price of an at-the-money volatility call with strikeprice K = 0.25 and T = 0.25. A one percent increase in the value of a increasesthe call price by 2.7 percent; a one percent increase in 13 decreases the call priceby 2.8 percent; and a one percent increase in or: increase the call price by 4.0percent.

Brenner and Galai (1989) and Whaley (1993) also present models for volatilityoption prices. Although innovative, these models do not consider the effects ofmean reversion on the option prices. Hence, the properties of volatility optionsimplied by these models are similar to those implied by the Cox et al. (1979) orBlack (1976) models.4.3. Prop erties o f uolatility puts

As for volatility calls, the price of a volatility put converges to its payoff valuemax(0, K - VT). Differentiation shows that the deltas of volatility puts are nega-

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t iv e . I n a d d i t io n , t h e v a l u e o f a p u t o p t i o n c a n a g a i n b e l e s s t h a n i t s i n t ri n s ic v a l u e .A s i n t h e c a se o f v o l a t i l i t y c a l l s , t h e d e l t a o f a p u t i s a d e c r e a s i n g f u n c t i o n o f T .A s T i n c r e a se s , t h e p r i c e f u n c t io n f o r th e p u t f l a t te n s o u t a n d t h e d e l t a o f t h eo p t i o n c o n v e r g e s t o z e r o .

T h e r e l a ti o n s b e t w e e n p u t d e l t a s a n d T a n d p u t d e lt a s a n d V m i r r o r t h e p a t te r n sf o r v o l a t i l i t y c a l l s . T h e se r e su l t s a g a i n i m p l y t h a t l o n g e r - m a t u r i t y v o l a t i l i t y p u t sw i l l b e o f l i m i t e d u se t o h e d g e r s s i n c e p u t d e l t as a p p r o a c h z e r o a s T i n c r e a se s .T h e s e c o n d d e r i v a t i v e o f v o l a t i l i t y p u t p r i c e s w i t h r e sp e c t t o V i s a g a i n p o s i t i v e .

T h e r e l a t i o n b e t w e e n p u t p r i c e s a n d T i s s i m i l a r t o t h a t f o r ca l ls . A s T --* ~ , t h ep u t p r i c e c o n v e r g e s t o z e r o . T h e r a t i o n a le f o r t h is f o l l o w s f r o m t h e b o u n d e d n e s s o ft h e p a y o f f f u n c t i o n a n d f r o m t h e f a c t t h a t t h e d i s c o u n t f a c t o r f o r th e e x p e c t e dp a y o f f d e c r e a se s t o z e r o a s T - - * ~ . T h i s f e a t u r e i m p l i e s t h a t t h e t h e t a o f av o l a t i l i t y p u t c a n b e e i t h e r p o s i t i v e o r n e g a t i v e . F o r so m e v a l u e s o f V , t h ei n - t h e - m o n e y p u t s a r e d e c r e a s i n g f u n c t i o n s o f T , w h i l e t h e o p p o s i t e i s t r u e f o r t h eo u t - o f - th e - m o n e y p u ts .

T h e r e m a i n i n g c o m p a r a t i v e s t a t i c s p a r a l l e l t h o se o f v o l a t i l i t y c a l l s . A n i n c r e a sei n t h e r i sk l e s s i n t e r e s t r a t e h a s t h e e f f e c t o f r e d u c i n g t h e d i s c o u n t f a c t o r a n dd e c r e a s i n g t h e v a l u e o f t h e p u t . I n t h i s r e sp e c t , t h e p r i c e s o f v o l a t i l i t y p u t s a r es i m i l a r t o t h o s e i m p l i e d b y t h e B l a c k - S c h o l e s f o r m u l a . A n i n c r e a s e i n t h e s t r i k ep r i c e t e n d s t o m a k e t h e p u t o p t i o n f u r t h e r i n t h e m o n e y f o r a l l v a l u e s o f V a n d T .H e n c e , t h e p u t i s a n i n c r e a s i n g f u n c t i o n o f i t s s t r i k e p r i c e .

5 . V o l a t i l i t y f u t u r e s o p t i o n sI n a d d i t i o n t o o p t i o n s o n v o l a t i l i t y , i t i s i m p o r t a n t t o c o n s i d e r t h e v a l u a t i o n o f

v o l a t i l i t y f u t u r e s o p t i o n s . L e t C F ( V , T , t + T , K ) a n d P F ( V , T , t + T , K ) d e n o t e t h er e sp e c t i v e p r i c e s f o r a c a l l a n d p u t o p t i o n w i t h m a t u r i t y T o n a f u t u r e s c o n t r a c te x p i r i n g a t t i m e t + T . T h e se o p t i o n s c a n b e p r i c e d d i r e c t l y g i v e n t h e r e su lt s in t h ep r e v i o u s s e c t i o n s . I n p a r t ic u l a r , r e c a l l th a t t h e p a y o f f f u n c t i o n f o r a T - m a t u r i t y c a llo p t i o n o n a f u t u r e s c o n t r a c t e x p i r i n g a t ti m e t + T c a n b e w r i t t e n a s

m a x ( O , F ( V T , t ) - K ) . ( 1 6 )Subs t i tu t ing in fo r F ( V T , t ) f r o m t h e f u t u r e s p r i c e e x p r e s s i o n i n ( 8 ) g i v e s

m a x ( 0 , e x p ( - f3 t ) V r + ( c t / [ 3 ) ( 1 - e x p ( - [ 3 t) ) - K ) . ( 1 7 )R e a r r a n g i n g t e r m s a n d f a c t o r i n g o u t e x p ( - [ 3 t ) g i v e s

e x p ( - [ 3 t ) m a x ( 0 , V r + ( a / 1 3 ) ( e x p ( 1 3 t ) - 1) - K e x p ( [ 3 t ) ) . ( 1 8 )H o w e v e r , t h is i s eq u a l t o e x p ( - 1 3 t ) t i m e s t h e p a y o f f f o r a s i m p l e v o l a ti li t y c a ll

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A. Gri inbichler , F.A. Lon gsta f f Journa l Of Banking & Finance 20 (1996) 9 85-1 001 999

o p t i o n w h e r e t h e s t ri k e p r i c e d i f f e rs f r o m K . T h u s , t h e c u r r e n t v a l u e o f a v o l a t i l it yf u t u r e s c a l l o p t i o n i s

CF( v, r , t + T , K ) = e x p ( - 1 3t) C ( V , K e x p ( 1 3 t )- ( e ~ / [ 3 ) ( e x p ( 1 3 t ) - l ) , r ) . ( 1 9 )

B e c a u s e o f th e f u n c t i o n a l f o r m o f t h e v o l a t i l i t y f u tu r e s p r i c e , th e v a l u e o f av o l a t i l i t y f u t u r e s c a ll e q u a l s a s c a l e f a c t o r e x p ( - 1 3 t ) t i m e s t h e v a l u e o f a s im p l ev o l a t i l i t y c a ll w h e r e t h e s t r ik e p r i c e o f th e o p t i o n i s tr a n s f o r m e d . B e c a u s e o f th i s ,t h e p r o p e r t i e s o f t h e s e o p t i o n s a r e b a s i c a l l y s i m i l a r to t h o s e d e s c r i b e d in th ep r e v i o u s s e c t io n f o r v o l a t i l i ty o p t io n s . N o t e , h o w e v e r , t h a t b e c a u s e e x p ( - [3 t) < I ,v o l a t i l i t y fu t u r e s c a l ls m a y b e e v e n l e s s s e n s i t iv e t o c h a n g e s in V t h a n s i m p l ev o l a t i l i t y c a l l s .

O n e i m p o r t a n t d i f f e r e n c e b e t w e e n v o l a t i l i t y f u t u r e s c a l l s a n d s i m p l e v o l a t i l i t yc a l l s a ri s e s b e c a u s e o f th e t r a n s f o r m a t i o n o f t h e s t r ik e p r i c e i n ( 1 9 ) . R e c a l l t h a te v e n t h o u g h v o l a t i l i t y c a n t a k e o n a n y p o s i t i v e v a l u e , v o l a t i l i t y f u t u r e s p r i c e s a r eh o u n d e d b e l o w . T h i s i m p l i e s t h a t if t h e s t r ik e p r i c e o f t h e v o l a t il i ty f u t u r e s c a l l i ss m a l l e n o u g h , t h e n t h e ca l l o p t i o n i s g u a r a n t e e d t o e x p i r e i n t h e m o n e y . T h u s .w h e n

K < ( ~ x / 6 ) ( 1 - e x p ( - [ 3 t ) ) , ( 2 0 )t h e c a l l o p t i o n e x p i r e s i n t h e m o n e y a n d t h e v a l u a t i o n e x p r e s s i o n i n ( 1 9 ) b e c o m e s

D ( T ) e x p ( - [ 3 t ) F ( V , T ) + D ( T ) ( o ~ / [3 ) ( 1 - e x p ( - f 3 T ) ) - K D ( T ) . ( 2 1 )T h i s r e p r e s e n t a t i o n o f th e v a l u e o f a d e e p - i n - t h e - m o n e y f u tu r e s c a l l m a k e s i t c l e a rt h a t th e d e l t a o f t h is o p t i o n i s a g a i n l e s s th a n o n e s i n c e t h e V t e r m in ( 2 1 ) i sm u l t i p l i e d b y a n e x p o n e n t i a l t e r m w i t h v a l u e l e s s t h a n o n e .

S i m i l a r a r g u m e n t s c a n b e a p p l i e d t o d e r i v e t h e v a l u e o f a v o l a t i l i t y fu t u r e s p u t .T h e v a l u e o f th is o p t i o n c a n b e s h o w n t o b e

PF (V,T, t + T,K) = e x p ( - [ 3 t ) P(V , Xexp (p t )- ( c ~ / 1 3 ) ( e x p ( 1 3 t ) - 1 ) , r ) . ( 2 2 )

T h i s t im e , th e l o w e r b o u n d o n t h e v a l u e o f t h e f u t u r e s p r i c e i m p l i e s t h a t if Ks a t is f i e s t h e i n e q u a l i t y i n ( 2 0 ) , t h e f u tu r e s p u t o p t i o n i s g u a r a n t e e d t o b e o u t o f th em o n e y a t e x p i r a t i o n . I n t h is c a s e , th e v a l u e o f t h e p u t o p t i o n i s z e r o e v e n t h o u g h ith a s a p o s i t i v e s t r i k e p r i c e . T h i s a g a i n d e m o n s t r a t e s h o w d i f f e r e n t t h e p r o p e r t i e s o fv o l a t i l i ty d e r i v a t iv e s a r e f r o m t h o s e o f m o r e c o n v e n t io n a l d e r i v a t iv e s e c u r i ti e s.

6 . C o n c l u s i o n

V o l a t i li ty d e r i v a t i v e s h a v e t h e p o t e n t ia l t o b e o n e o f th e m o s t i m p o r t a n t n e wf i n a n c i a l i n n o v a t i o n s o f t h is d e c a d e . I n l ig h t o f t h is , th i s p a p e r d e v e l o p s s i m p l e

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c lose d - f o r m va lua t ion m ode l s f o r f u tu r e s a nd op t ions c on t r a c t s on s toc k inde xvo la t i l i t y a nd e xa m ine s the i r im p l i c a t ions f o r the p r i c ing a nd he dg ing be ha v io r o fthe se se c u r i t i e s . We show tha t t he p r ope r t i e s o f vo la t i l i t y de r iva t ive s c a n be ve r yd i f f e r e n t f r om those f o r f u tu r e s a nd op t ions on t r a de d a s se t s .

A m a jo r im p l i c a t ion o f ou r a na lys i s i s t ha t l onge r - t e r m vo la t i l i t y f u tu r e s a ndop t ions m a y be l e s s e f f e c t ive in he dg ing a nd m a na g ing vo la t i l it y r i sk tha nc o m m o n l y b e l i e v e d . T h e u n d e r l y i n g r e a s o n f o r t h i s i s t h a t m e a n r e v e r s i o nda m pe ns the e f f e c t o f c u r r e n t shoc ks in vo la t i l it y on the op t ion o r f u tu r e s p r i c e.The se r e su l t s ha ve m a ny im por ta n t im p l i c a t ions f o r the de s ign o f vo la t i l i t yde r iva t ive p r oduc t s .

7 . F o r f u r t h e r r e a d i n gF o r f u r t h e r r e a d in g , s e e A b r a m o w i t z a n d S t e g u n ( 1 9 7 0 ) a n d C o x a n d R o s s

( 1976) .

A c k n o w l e d g e m e n t sWe a r e g r a t e f u l f o r the c om m e nts a nd sugge s t ions o f Ya c ine Ai t - S a ha l i a ,

H e n d r i k B e s s e m b i n d e r , M e n a c h e m B r e n n e r , J e f f F l e m i n g , D a n G a l a i, A y m a nH i n d y , N a r a s i m h a n J e g a d e e s h , D a v i d S h i m k o , E d u a r d o S c h w a r t z, P e t e r S w o b o d a ,S t e p h e n T a y l o r , B r u c e T u c k m a n , R o b e r t W h a l e y , J o s e f Z e c h n e r , a n d w o r k s h o ppa r t i c ipa n t s a t t he AM EX Opt ions C o l loqu ium , Ar iz ona S ta t e Un ive r s i ty , C h ic a goB o a r d o f T r a d e , t h e U n i v e r s i t y o f G r a z , t h e L o n d o n B u s i n e s s S c h o o l , S a l o m o nB r o the r s I nc . , t he Un ive r s i ty o f S ou th e r n C a l i f o r n ia , t he Un ive r s i ty o f S t r a thc lyde ,t h e A m e r i c a n F i n a n c e A s s o c i a t i o n m e e t i n g s , t h e W e s t e r n F i n a n c e A s s o c i a t i o nm e e t ings , a nd the Eur ope a n F ina nc e Assoc ia t ion m e e t ings . We a r e pa r t i c u la r lyg r a te f u l f o r the c om m e nts a nd sugge s t ions o f the e d i to r G io r g io S z eg~5 a nd th r e ea nonym ous r e f e r e e s . A l l e r r o r s a r e the r e spons ib i l i t y o f the a u tho r s .

R e f e r e n c e sA b r a m o w i t z , M . a n d I . A . S t e g u n , 1 97 0, H a n d b o o k o f m a t h e m a t i c a l f u n c t io n s ( D o v e r P u b l i c a ti o n s ,

N e w Y o r k , N Y ) .B l a c k , F . , 1 9 7 6, T h e p r i c i n g o f c o m m o d i t y c o n tr a c t s , J o u r n a l o f F i n a n c ia l E c o n o m i c s 3 , 1 6 7 - 1 7 9 .B l ack , F . an d M . S c h o l es , 1 9 73 , Th e p r i c i n g o f o p t i o n s an d c o rp o ra t e l i ab i l it i e s , J o u rn a l o f P o l i t ica l

E c o n o m y 8 1 , 6 3 7 - 6 5 4 .B ren n e r , M . an d D . Ga l a i , 1 9 89 , New f i n an c i a l i n s tru m en t s fo r h ed g i n g ch an g es in v o l a t il i ty , F i n an c i a lA n a l y s t s J o u r n a l ( J u l y / A u g u s t ) , 6 1 - 6 5 .C o x , J . C . an d S . A . R o s s , 1 9 76 , Th e v a l u a t i o n o f o p t i o n s fo r a l t e rn a t iv e s t o ch as t i c p ro ce s s es , J o u rn a l o f

F i n a n c i a l E c o n o m i c s 3 , 1 4 5 - 1 6 6 .

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A. Gri~nbichler , F.A. Lo ngs ta f f Jo ur na l o f Banking & Finance 20 (1996) 985 -10 01 lit01C o x , J . C . , S . A . R o s s a n d M . R u b i n s t e i n , 1 9 7 9, O p t i o n p r i c in g : A s i m p l i f i e d a p p r o a c h , J o u r n a l o f

F i n a n ci a l E c o n o m i c s 7 , 2 2 9 - 2 6 4 .Co x , J . C . , J . E . I n g e r s o l l a n d S . A . Ro s s , 1 9 8 1 , T h e r e l a t i o n b e tw e e n f o r w a r d p r i c e s a n d f u tu r e s p r i c e s ~

J o u r n al o f F in a n c ia l E c o n o m i c s 9, 3 2 1 - 3 4 6 .Co x , J . C . , J . E . I n g e r s o l l a n d S . A . Ro s s . 1 9 8 5 , A th e o r y o f t h e t e r m s t r u c tu r e o f i n t e r e s t r a t e s .E c o n o m e t r i c a 5 3 , 3 8 5 - 4 0 8 .F e ll e r, W . , 1 95 1, T w o s i n g u l a r d i f f u s i o n p r o b l e m s , A n n a l s o f M a t h e m a t i c s 5 4 , 1 7 3 - 1 8 2 .F r e n c h , K . R . , G . W . S c h w e r t a n d R . F . S t a m b a u g h , 1 98 7, E x p e c t e d s t o c k r e t u r n s a n d v o l a ti l it y , J o u rn a l

o f F i n a n c i a l E c o n o m i c s 1 9 , 3 - 3 0 .H a r v e y , C . R . a n d R . E . W h a le y , 1 9 92 , M a r k e t v o l a t i li t y p r e d i c t i o n a n d t h e e f f i c i e n c y o f t h e S &P 1 0 l)

i n d e x o p t i o n m a r k e t , J o u r n a l o f F i n a n c i a l E c o n o m i c s 3 I , 4 3 - 7 4 .H e m l e r , M . L . a n d F . A . L o n g s t a f f , 1 9 9 1 , G e n e r a l e q u i l i b r i u m s t o c k i n d e x f u t u r e s p r i c e s : T h e o r y a n d

e m p i r i c a l e v i d e n c e , J o u r n a l o f F i n a n c i a l a n d Q u a n t i t a t iv e A n a l y s i s 2 6 , 2 8 7 - 3 0 8 .H e s to n , S . A , , 1 9 9 3 , A c lo s e d - f o r m s o lu t i o n f o r o p t i o n s w i th s t o c h a s t i c v o l a t i l i t y w i th a p p l i c a t i o n s t o

b o n d a n d c u r r e n c y o p t i o n s , T h e R e v i e w o f F i n a n c i a l S tu d i e s 6 , 3 2 7 3 4 3 .H u l l , J. a n d A . W h i t e , 1 9 87 , T h e p r i c in g o f o p t i o n s o n a s s e t s w i th s t o c h a s t i c v o l a t il i t ie s , J o u r n a l o fF i n a n ce 4 2 , 2 8 1 - 3 0 0 .J o h n s o n , N . L . a n d S . K o t z , 1 97 0, C o n t i n u o u s u n i v a r i a te d i s t ri b u t i o n s , V o l s . I a n d 2 ( J o h n W i l e y . N e w

Y o r k . N Y ) .J o h n s o n , H . a n d D . S h a n n o , 1 9 8 7 , O p t i o n p r i c i n g w h e n t h e v a r i a n c e is c h a n g i n g , Jo u r n a l o f F i n a n c i al

a n d Q u a n t i t a t i v e A n a l y s i s 2 2 , 1 4 3 - 1 5 2 .L o n g s t a f f , F , A . , 1 99 0, T h e v a l u a t i o n o f o p t i o n s o n y i e l d s , Jo u r n a l o f F i n a n c i a l E c o n o m i c s 2 6 , 9 7 - 1 2 I.L o n g s t a f f , F . A . a n d E . S . S c h w a r t z , 1 9 92 , I n t e r e s t r a t e v o l a t i l i ty a n d t h e te r m s t r u c tu r e : A tw o - f a c to r

g e n e r a l e q u i l i b r i u m m o d e l , T h e J o u r n a l o f F i n a n c e 4 7 , 1 2 5 9 - 1 2 8 2 .M e r t o n , R . C . , 1 97 3, T h e t h e o r y o f r at i o n al o p t i o n p r i ci n g . B e ll J o u r n a l o f E c o n o m i c s a n d M a n a g e m e n t

S c i e n c e 4 , 1 4 1 - 1 8 3 .S a n k a r a n , M . , 1 96 3, A p p r o x i m a t i o n s to t h e n o n - c e n t r a l c h i - s q u a r e d is t r ib u t i o n , B i o m e t r i k a 5 0 ,

199 2(14.S c o t t . L . O . , 1 9 8 7 , O p t i o n p r i c i n g w h e n t h e v a r i a n c e c h a n g e s r a n d o m l y : T h e o r y , e s t i m a t i o n , a n d a n

a p p l i c a t i o n , J o u r n a l o f F i n a n c i a l a n d Q u a n t i t a t i v e A n a l y s i s 2 2 , 4 1 9 - 4 3 8 .S h e ik h , A . M . . 1 9 9 3 , T h e b e h a v io r o f v o l a t i l i t y e x p e c t a t i o n s a n d t h e i r e f f e c t s o n e x p e c t e d r e tu r n s , T h e

J o u r na l o f B u s i n e s s 6 6 , 9 3 - 1 1 6 .S t e in , E . M . a n d J . C . S t e in , 1 9 9 1 , S to c k p r i c e d i s t r i b u t i o n s w i th s t o c h a s t i c v o l a t i l i t y : A n a n a ly t i c

a p p r o a c h . T h e R e v i e w o f F i n a n c i a l S t u d i e s 4 , 7 2 7 - 7 5 2 .T a y lo r , S .J . a n d X . X u , 1 9 94 , I m p l i e d v o l a t i l i t y s h a p e s w h e n p r i c e a n d v o l a t i l it y s h o c k s a r e c o r r el a t e d .

W o r k i n g p a p e r ( L a n c a s t e r U n i v e r s i t y ) .W h a le y , R . E , 1 9 93 , D e r iv a t i v e s o n m a r k e t v o l a ti l it y : H e d g in g to o l s l o n g o v e r d u e , J o u r n a l o f D e r iv a -

t i v e s ( F a l l ) , 7 1 - 8 4 .W i g g i n s , J . B ., 1 98 7, O p t i o n v a l u e s u n d e r s t o c h a s t i c v o la t il it y : T h e o r y a n d e m p i r i c a l e s t i m a t e s , J o u rn a l

o f F i n a n ci a l E c o n o m i c s 1 9, 3 5 1 - 3 7 2 .