Asymptotics of a Two-Scale Sto chastic Volatility Mo del
y z
J.P. Fouque G. Papanicolaou K.R. Sircar
To Jacques-Louis Lions on the occasion of his seventieth birthday
Abstract
We present an asymptotic analysis of derivative prices arising from a sto chastic volatility
mo del of the underlying asset price that incorp orates a separation b etween the short tick-by-
tick time-scale of uctuation of the price and the longer less rapid time-scale of volatility
uctuations. The mo del includes leverage or skew e ects a negative correlation b etween price
and volatility sho cks , and a nonzero market price of volatility risk. The results can be used
to estimate the latter parameter, which is not observable, from at-the-money Europ ean option
prices. Detailed simulations and estimation of parameters are presented in [6].
1 Intro duction
Sto chastic volatility mo dels have b ecome p opular for derivative pricing and hedging in the last ten
years as the existence of a non at implied volatility surface or term-structure has b een noticed
and b ecome more pronounced, esp ecially since the 1987 crash. This phenomenon, which is well-
do cumented in, for example, [9 , 12 ], stands in empirical contradiction to the consistent use of
a classical Black-Scholes constant volatility approach to pricing options and similar securities.
However, it is clearly desirable to maintain as many of the features as p ossible that have contributed
to this mo del's p opularity and longevity, and the natural extension pursued in the literature and
in practice has b een to mo dify the sp eci cation of volatility in the sto chastic dynamics of the
underlying asset price mo del.
One approach, termed the implied deterministic volatility IDV approach [5, Chapter 8], is to
supp ose volatility is a deterministic function of the asset price X : volatility= t; X , so that the
t t
sto chastic di erential equation mo deling the asset price b ecomes
dX = X dt + t; X X dW :
t t t t t
The function C t; x giving the no-arbitrage price of a Europ ean derivative security at time t when
the asset price X = x then satis es the generalized Black-Scholes PDE
t
1
2 2
t; x x C + r xC C =0; C +
xx x t
2
with r the constant risk free interest rate and with terminal condition appropriate for the contract.
This has the nice feature that the market is complete which, in this context, means that the
CNRS-CMAP, Ecole Polytechnique, 91128 Palaiseau Cedex France, [email protected]. This work
was done while visiting the Department of Mathematics, Stanford University.
y
Department of Mathematics, Stanford University, Stanford CA 94305, [email protected]
z
Department of Mathematics, University of Michigan, Ann Arb or MI 48109, [email protected] 1
derivative's risk can theoretically b e p erfectly hedged by the underlying, and there is no volatility
risk premium to b e estimated.
However, while attempts to infer a lo cal volatility surface from market data by tree metho ds
[13 ] or relative-entropy minimization [2 ] or interp olation [15 ] have yielded interesting qualitative
prop erties of the risk-neutral probability distribution used by the market to price derivatives
such as excess skew and leptokurtosis in comparison to the lognormal distribution , this approach
has not pro duced a stable surface that can b e used consistently and with con dence over time.
For this reason, we concentrate on the \pure" sto chastic volatility approach of [8 ] reviewed in
[16 , Section 1.2] or [3 ], in which volatility is mo deled as an It^o pro cess driven by a Brownian
t
motion that has a comp onent indep endent of the Brownian motion W driving the asset price.
t
There has b een a lot of analysis of sp eci c It^o mo dels in the literature [14 , 17 , 7] bynumerical and
analytical metho ds, many of which have ignored skew e ects and/or the volatility risk premium
for tractability. Our goal [6 ] is to estimate these parameters from market data and to test their
stability over time and thus the p otential usefulness of sto chastic volatility mo dels for hedging
derivatives. What is to our knowledge new here in comparison with previous empirical work on
sto chastic volatility mo dels is our keeping of these two factors, use of high-frequency intraday
data, and an asymptotic simpli cation of option prices predicted by the mo del that allows for easy
estimation of the volatility risk premium from at-the-money market option prices.
The latter exploits the separation of time-scales rst intro duced in this context in [16 ]. It is
often observed that while volatility might uctuate considerably over the many months comprising
the lifetime of an options contract, it do es not do so as rapidly as the sto ck price itself. That
is, there are p erio ds when the volatility is high, followed by p erio ds when it is low. Within these
p erio ds, there might be much uctuation of the sto ck price as usual , but the volatility can be
considered relatively constantuntil its next \ma jor" uctuation. The \minor" volatility uctuations
within these p erio ds are relatively insigni cant, esp ecially as far as option prices, which come from
an average of a functional of p ossible paths of the volatility, are concerned.
Many authors, for example [1], have prop osed nonparametric estimation of the pricing measure
for derivatives. The analysis in [16 ] is indep endent of sp eci c mo deling of the volatility pro cess, but
results in bands for option prices that describ e p otential volatility risk in relation to its historical
auto correlation decay structure, while obviating the need to estimate the risk premium. However,
the market in at-the-money Europ ean options is liquid and its historical data can b e used to estimate
1
this premium . For this reason, we shall attempt this with a mo del that is highly parametric, but
complex enough to re ect an imp ortantnumb er of observed volatility features:
1. volatility is p ositive;
2. volatility is rapidly mean-reverting see for example [10 ] ;
3. volatility sho cks are negatively correlated with asset price sho cks. That is, when volatility
go es up, sto ck prices tend to go down and vice-versa. This is often referred to as leverage [4 ],
and it at least partially accounts for a skewed distribution for the asset price that lognormal
or zero-correlation sto chastic volatility mo dels do not exhibit. The skew is do cumented in
empirical studies of historical sto ck data.
2 Mo del
The mo del wecho ose is that volatility is the exp onential of a mean-reverting Ornstein-Uhlenbeck
OU pro cess or, equivalently, log is mean-reverting OU . With a suitable initial distribution,
t
1
This was suggested to us by Darrell Due. 2
the volatility pro cess is stationary and ergo dic which allows us to use averaging principles to ap-
proximate the option price, separating the minor and ma jor uctuations. This mo del has b een
considered in [14 ] and it is related to EGARCH mo dels which, as shown in [11 ], are weak approx-
imations to the continuous-time di usion. Another mo del that is stationary and can be similarly
implemented and analyzed is when is a mean-reverting Feller or Cox-Ingersoll-Ross pro cess [3 ].
t
The nal ingredient is to mo del the two time scales describ ed previously. To this end, we
intro duce a small parameter " > 0 describing the discrepancy between the scales, and mo del the
"
volatilityas = , where is exp onential OU. Thus the volatilityisthe pro cess \sp eeded-
t t
t="
t
up" to re ect that there are many ma jor uctuations over the life of the options contract this time
scale is O 1 in the usual time-unit of years , but not as many as there are minor It^o uctuations.
We de ne Y := log and supp ose it satis es
t t
^
dY = m Y dt + dZ
t t t
" "
^
for constants >0; > 0;m and Z a Brownian motion. Then Y := log is describ ed by
t
t t
" "
^
p
= dY dt + m Y dZ
t
t t
" "
p
where now and have b een replaced by =" and = " to mo del rapid mean reversion and overall
^
variance of order one. Finally, to incorp orate the correlation skew e ect dhW; Z i = dt,we write
t
p
2
^
Z = W + 1 Z , where W and Z are indep endent Brownian motions, to arrive at the nal
t t t
"
sto chastic volatility mo del for the sto ck price X
t
"
" " Y "
t
dX = X dt + e X dW 1
t
t t t
q
" "
2
p
2 m Y dW + 1 dZ dt + dY =
t t
t t
" "
Then, by the usual no-arbitrage argument, as detailed for example in [16 , Section 1.3], the Europ ean
"
call option price C t; x; y satis es
y 2
1 xe
" 2y 2 " " " " "
p
C e x C C C + + + + r xC C
t xx xy yy x
2 " 2"
"
p
m y C = 0 3 +
y
" "
" +
C T ; x; y = x K
"
where is the market price of volatility risk whichwe assume constant. If C t; x; y satis es this
" " " "
equation then from It^o's formula C = C t; X ;Y satis es the sto chastic di erential equation
t t
"
" " " " " Y " " "
^
t
p p
dC =[rC + r X C + C ]dt + e X C dW + C dZ 4
t t
x y t x y
" "
p
From this expression we see that an in nitesimal change in the volatility risk = " changes the
in nitesimal rate of return of the option by times the change in volatility risk. This is why is
called the market price of volatility risk.
3 Asymptotic Analysis
Now, as " 0, the distinction b etween the time scales disapp ears and the ma jor uctuations o ccur
in nitely often. In this limit, volatility can be approximated by a constant as far as averages of 3
functionals of its path are concerned that is, weakly , and we return to the classical Black-Scholes
"
setting. What is of interest is the next term in the asymptotic approximation of C t; x; y valid
for small ", that describ es the in uence of ; and the randomness >0 of the volatility.
" "
To obtain this, let us write 3 as L C = 0, where
1 1
"
p
L + L + L ; L :=
0 1 2
" "
and
2
1 @ @
2
L := + m y ;
0
2
2 @y @y
2
@ @
y
L := xe ;
1
@x@y @y
2
1 @ @ @
2y 2
+ e x + r x : L :=
2
2
@t 2 @x @x
Then, constructing an expansion
p
"
C t; x; y =C t; x; y + "C t; x; y +"C t; x; y + ;
0 1 2
we nd, comparing p owers of "<<1,
L C =0
0 0
1
at the O " level. Since L involves only y -derivatives and is the generator of the OU pro cess
0
Y , its null space is spanned byany nontrivial constant function, and it must b e that C do es not
t 0
dep end on y : C = C t; x .
0 0
1=2
At the next order, O " , wehave
L C + L C =0; 5
1 0 0 1
and since L takes y -derivatives, L C =0. By the same reasoning, 5 implies that C = C t; x .
1 1 0 1 1
Thus, up till O " , the option price do es not dep end on the currentvolatility.
Comparing O 1 terms,
L C + L C =0:
0 2 2 0
Given C t; x , this is a Poisson equation for C t; x; y and there will b e no solution unless L C
0 2 2 0
is in the orthogonal complement of the null space of L Fredholm Alternative . This is equivalent
0
to saying that L C has mean zero with resp ect to the invariant measure of the OU pro cess. We
2 0
denote this
hL C i =0;
2 0 ou
2
where hi denotes the exp ectation with resp ect to this invariant measure whichis N m; , where
ou
2 2
:= =2 :
Z
1
2
2 1
y m =2
p
e f y dy : hf i =
ou
2
1
2
2y
Since C is indep endent of y and L only dep ends on y through the e co ecient, hL C i =
0 2 2 0 ou
hL i C , and
2 ou 0
2
@ @ 1 @
2 2
+ r x hL i = L ^ := + ^ x ;
2 ou BS
2
@t 2 @x @x
2
2 2y 2m+2
. where ^ := he i = e
ou 4
Thus C t; x = C t; x;^ , and the rst term in the expansion is the Black-Scholes pricing
0 BS
formula with the averaged volatility constant ^ . The and havethus far played no role, and we
pro ceed to nd the next term in the approximation, C t; x .
1
Comparing terms of O " , we nd
L C = L C + L C ; 6
0 3 1 2 2 1
whichwe lo ok at as a Poisson equation for C t; x; y . Just as the Fredholm solvability condition
3
for C determined the equation for C , the solvability for 6 will give us the equation for C t; x .
2 0 1
Substituting for C t; x; y with
2
1
C = L L hL i C ;
2 2 2 ou 0
0
this condition is
E D
1
=0; L C L L L hL i C
2 1 1 2 2 ou 0
0
ou
where
hL C i = hL i C = L ^ C
2 1 ou 2 ou 1 BS 1
since C do es not dep end on y .
1
De ning
D E
1
A := L L L hL i ;
1 2 2 ou
0
ou
the equation determining C is
1
L ^ C = AC ;
BS 1 0
as C do es not dep end on y .
0
Again, using that L acts only on y -dep endent functions, we can write
0
* ! !+
2 2
@ 1 @ @
y 2
A = xe ; y x
2
@x@y @y 2 @x
ou
where
1
2 00 0 2y 2y
L y = y + m y y =e he i ;
0 ou
2
and so
2 3
@ @
2 3
+ Bx ; A = Ax
3 2
@x @x
with
1
y 0
A := he i
ou
2
y 0 0
B := he i h i :
ou ou
2
Thus wemust solve
3 2
@ C @ C
BS BS
3 2
L ^ C = Ax ^ +Bx ^
BS 1
3 2
@x @x
2
=2 d
1
d xe
1
p
p
B A 1+ ; =
^ 2 T t
^ T t
where
1
2
log x=K + r + ^ T t
2
p
d = ;
1
^ T t 5
and where we have used the explicit expression for the Black-Scholes price C ^ . Using the
BS
Green's function for the Black-Scholes PDE, as given for example in [16 , App endix C], and loga-
rithmic transformations to reduce to Gaussian integrals, we nd that
2
d =2
p
1
xe d
1
p
C = A + A B T t :
1
^
^ 2
Finally,we compute
3m
2e
2 2
9 =2 5 =2 y 0
e e he i =
ou
2
2
2^
0
h i = ;
ou
and so
2
d =2
p p
1
xe d
1
2
p
C = T t T t ; 7 + ^
1
^
^ 2
p
2 2
3m 9 =2 5 =2 "
where := e e e = is a p ositive constant. Note that to order ", C is decreasing
in .
" " "
We can now calculate the implied volatility I de ned by C = C I . Constructing an
BS
p
"
expansion I =^ + "I + ,we nd that
1
1
@C t; x;^
BS
I = C t; x
1 1
@ ^
d ^
1
p
; 8 = 1
^
^ T t
which shows that
"
2 2
2 3 =2 =2
p
K
^ e e
" 1=2 1
p
I =^ + log + O r T t 2^ ; 9
" x 2 "
2 T t
2
m+ 2 2
where we have used the expression for ab ove and where ^ = e and = =2 . We have
1 "
. also expressed the expansion for I in terms of the inverse of the fast mean reversion rate
"
For < 0, which is the usual case, this gives a decreasing implied volatility curve when plotted
against strike price K , that is, a decreasing smirk.
The analysis gives rise to an explicit formula describing the geometry of the implied volatility
surface across strike prices and expiration dates. In particular, the relationship to the risk premium
parameter in 9 considerably simpli es the pro cedure for its estimation, which otherwise would
b e a computationally-intensiveinverse problem for the PDE 3 .
4 Conclusions
We have shown that an incomplete market asset mo deled by a fast mean reverting sto chastic
volatility leads to an asymptotic formula for options pricing and asso ciated implied volatility 9 .
This formula involves in a direct way the otherwise unobservable market price of volatility risk
, which can then be estimated by tting it to observed smirks observed implied volatility as a
function of strike price K . The other parameters in the mo del, the mean and variance of the log
2
volatility m and and the fast mean reversion rate =", can be estimated from historical asset
price data. The remaining parameter of the mo del, the skew , can in principle also b e estimated
from historical asset price data but it is b etter in practice to estimate it by tting formula 9 to
option pricing data, as is done for the market price of volatility risk . This is done in [6 ]. 6
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