Sto chastic Volatility Correction to Black-Scholes
y z
Jean-Pierre Fouque George Papanicolaou K. Ronnie Sircar
July 1999; revised August 1999. This version January 2000.
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 phe-
1
nomenon, whichiswell-do cumented , stands in empirical contradiction to the consistent use
of a classical Black-Scholes constant volatility approach to pricing options and similar se-
curities. 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
b oth 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.
There are many stories b ehind why we should mo del volatilityto be a random pro cess.
For example, it could simply represent estimation uncertainty, or it can arise as a friction from
transaction costs, or it could simulate non-Gaussian heavy-tailed returns distributions. In
other words, sto chastic volatilityis a far-reaching extension of the Black-Scholes lognormal
mo del, describing a much more complex market.
Any extended mo del must also sp ecify what data it is to be calibrated from. The pure
Black-Scholes pro cedure of estimating from historical sto ck data only is not p ossible in an
incomplete market if one takes the view as we shall that the market selects a unique
risk-neutral derivative pricing measure, from a family of p ossible measures, which re ects
its degree of "crash-o-phobia". This pricing measure is re ected in traded at-the-money
Europ ean options prices, so, as is common practice, this \smile data" is used for calibration.
Parameter estimation and stability of the estimates in time presents the ma jor mathe-
matical and practical challenge here. Without a formula for option prices under a particular
sto chastic volatility mo del, estimating the risk-neutral parameters is computationally inten-
sive we have to run a tree or simulations at each step in an iterative search pro cedure .
Often mo dels are chosen so that there is a closed-form solution, and this usually means tak-
ing the volatility to b e indep endent of the Brownian motion driving the sto ck price, whereas
Department of Mathematics, NC State University, Raleigh NC 27695-8205, [email protected].
y
Department of Mathematics, Stanford University, Stanford CA 94305, [email protected]
z
Department of Mathematics, University of Michigan, Ann Arb or MI 48109-1109, [email protected].
Work supp orted by NSF grant DMS-9803169.
1
See for example, Jackwerth, J. and Rubinstein, M, 1996, Recovering Probability Distributions from
Contemporaneous Security Prices, Journal of Finance 51 5 , pages 1611-1631, and Rubinstein, M., 1985,
Nonparametric Tests of Alternative Option Pricing Models, Journal of Finance 40 2 , pages 455-480. 1
common exp erience and empirical evidence suggests a negative correlation: when volatility
go es up, sto ck prices tend to go down.
Now supp ose that we have estimated the parameters. How go o d will these estimates b e
tomorrow? It is p ossible to haveavery tight tover a short time, but often these break down
2
signi cantly thereafter. This is certainly a problem with estimates of volatility surfaces ,
where unlike in sto chastic volatility mo deling , volatility is mo deled to be a function of
time and sto ck price with no indep endent randomness.
We present here a new approach to sto chastic volatility that has the following features:
It requires that volatility b e mean-reverting, but, other than that, do es not dep end in
an essential way on how the volatility is mo deled.
It translates the slop e and intercept of the implied volatility skew into information
ab out the correlation b etween volatility and sto ck price sho cks and the market's volatil-
ity risk premium.
It simpli es enormously the parameter estimation problem.
It gives a recip e for pricing and hedging other derivatives in a sto chastic volatility en-
vironmentby identifying their e ective approximating derivative security in a constant
volatilityenvironment. This includes barriers, Asians and Americans.
It pro duces parameter estimates from implied volatilityskews that are stable.
This is achieved by exploiting the mean-reverting b ehaviour of volatility and the much-
noted observation that volatility is p ersistent.
Framework
The sto ck price S satis es
t t0
dS = S dt + S dW ;
t t t t t
where is the volatility pro cess. To incorp orate the correlation with the Brownian
t t0
motion W which leads to the implied volatility skew in these mo dels , it is convenient
t
to take to be a di usion pro cess to o, although it can have jumps as well. To x
t
ideas, we shall write volatility as a p ositive function of a mean-reverting Gaussian Ornstein-
Uhlenbeck pro cess: = f Y , where
t t
q
2
dY = m Y dt + dW + ; 1 dZ
t t t t
with Z an indep endent Brownian motion the source of the additional randomness and
2
See Dumas, B., Fleming, J. and Whaley, R., 1998, ImpliedVolatility Functions: Empirical Tests, Journal
of Finance 53 6 , pages 2059-2106 for an empirical study of this and, for a mathematical analysis, Lee, R.,
1999, Local Volatilities under Stochastic Volatility, to app ear in International Journal of Theoretical and
Applied Finance. 2
= the rate of mean-reversion;
m = the long-run mean of Y;
= the \v-vol";
= the correlation co ecient :
Our ob jective is to analyze the e ect of sto chastic volatility in the basic Black-Scholes
mo del. Therefore, we assume that these parameters, as well as the rate of return are,
for simplicity, constant. For the same reason we assume that f Y do es not dep end on t
t
explicitly. In fact we do not need to sp ecify f in detail since the mean reversion asymptotics
give results that are insensitive to all but a few general features of f . The precise mo del
for the pro cess Y driving the volatilitydoes not matter either, so long as it is an ergo dic
t
pro cess like the Ornstein-Uhlenbeck pro cess ab ove.
Three observable quantities emerge from the asymptotics and they are the only ones that
must b e calibrated from historical data and the term structure of volatility. These quantities
are complicated functions of the primitive mo del parameters, the function f and the market
price of volatility risk intro duced b elow, which need not be calibrated separately. This is
a new approach to the study of sto chastic volatility mo dels.
Volatility Persistence
It is often noted in empirical studies of sto ck prices that volatility is p ersistent or bursty - for
days at a time it is high and then, for a similar length of time, it is low. However, over the
lifetime of a derivative contract a few months , there are many such p erio ds, and lo oked at
on this timescale, volatility is uctuating fast, but not as fast as the rapidly changing sto ck
price.
In terms of our mo del, wesay that the volatility pro cess is fast mean-reverting relativeto
the yearly timescale, but slow mean-reverting by the tick-tick timescale. Since the derivative
pricing and hedging problems we study are p osed over the former p erio d, we shall say that
volatility exhibits fast mean-reversion without explicitly mentioning the longer timescale of
reference.
1
The rate of mean-reversion is governed by the parameter , in annualized units of years .
2
Fast mean-reversion means that is in fact large and that = 2 , the variance of the
invariant distribution of the OU pro cess, is a stable O 1 constant. As an illustration,
Figure 1 shows simulated volatility paths for the mo del ab ove in which = 1 on the top
and = 200 b elow. In practice the volatility pro cess is not directly observable. In fact the
true observable is the de-meaned returns pro cess
dS
t
dt = dW ;
t t
S
t
at discrete times. In Figure 2 we show simulated tra jectories of this returns pro cess corre-
sp onding to these volatility tra jectories in the rst two graphs. We observe that the size
of the uctuation in the returns pro cess of the top picture is relatively constant over time,
while for the large picture, in the middle, it is changing a lot. This is exactly what we call
fast mean-reverting or p ersistent sto chastic volatility. 3
=1 0.5
0.4
0.3
0.2
0.1
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
= 200 0.5
0.4
0.3
0.2
0.1
0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Figure 1: Simulated volatility for smal l and large rates of mean-reversion for the OU model, with
y
the choice f y =e . Note how volatility \clusters" in the latter case.
In the b ottom graph of Figure 2, we show the returns pro cess for the S&P 500 over the
rst ve months of 1996. By comparing with the top two paths, we note how the structure
of the returns uctuation size resembles more the one from a fast mean-reverting sto chastic
volatility mo del middle , than one from a mo del with slow mean-reversion top . A detailed
3
analysis of the mean reverting structure of the S&P 500 is presented in a forthcoming pap er .
In the metho dology we describ e next we will not need the precise value of , only that
it is large.
Derivative Pricing
We start with Europ ean derivatives with terminal payo h S . The no arbitrage price P
T t
dep ends on the present sto ck price S and the present level of the volatility-driving pro cess
t
Y . It is given by the risk-neutral exp ected discounted payo
t
Q
r T t
P = IE fe h S jS ;Y g;
t T t t
where Q represents probabilities in the risk-neutral world and is the volatility risk
4
premium . In this world,
?
dS = rS dt + S dW ;
t t t t
t
3
Fouque, Papanicolaou, Sircar and Solna, 1999, Mean-Reversion of S&P500 Volatility.
4
A sto chastic volatility mo del is an incomplete market mo del, and as such there is a whole family of
risk-neutral pricing measures, unlike in the constantvolatility case when there is only one. It can b e shown
from Girsanov's theorem, that each p ossible measure in this It^o framework corresp onds to a choice of the
volatility risk premium, also known as the market price of volatility risk. See, for example, Hull, J. and 4 2
1
0 =1 −1
−2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
2
1
0
=−1 200
−2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
2
1 500 0
−1 S&P
−2
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4
Time
Figure 2: The top and midd le graphs show simulated returns for smal l and large rates of mean-
y
reversion for the OU model, with the choice f y = e . The bottom graph shows 1996 S&P 500
returns computed from half-hourly data.
= f Y ;
t t
" !
q q
r
? ?
2 2
dY = m Y 1 1 dZ ; + Y dt + dW +
t t t
t t
f Y
t
? ?
where W ;Z are indep endent Brownian motions under the Equivalent Martingale Measure
Q . The interest rate r is assumed to be constant and known. As for the other mo del
parameters, we assume a market price of volatility risk Y driven only by the volatility-
t
driving pro cess Y . This can b e validated aposteriori when the formula presented b elowis
t
tted to implied volatility data. The new function y is included in the asymptotics and
not directly estimated using other derivatives for instance .
5
The main result of an asymptotic analysis of this problem says that when volatility p er-
sists, we can approximate the derivative price P in the sto chastic volatilityenvironmentby
t
pricing a more complicated path-dep endent security in the Black-Scholes constantvolatil-
ity environment. The payo structure of the new security dep ends on the Black-Scholes
pricing formula for the original one and accounts appropriately for volatility risk.
The pro cedure is as follows:
1. Let S be the Black-Scholes lognormal mo del:
t t0
?
dS = r S dt + S dW ;
t t t
t
White, A., 1987, The Pricing of Options on Assets with Stochastic Volatilities, Journal of Finance 42 2 ,
pages 281-300.
5
See Fouque, Papanicolaou and Sircar, 1999, Mean-Reverting Stochastic Volatility, to app ear in Interna-
tional Journal of Theoretical & Applied Finance, for details of the calculation. 5
with the usual constant historical volatility. Price the derivative with this mo del.
That is, nd
? r T t
P t; S =IE fe h S jS = S g:
BS T t
2. The sto chastic volatility price P is well-approximated by the price in the Black-Scholes
t
~
model P of the path-dep endent contract with the same terminal payo h S and a
t T
payout rate H t; S between times t and T , given by
t
2 3
@ P @ P
BS BS
2 3
H t; S =V S t; S +V S t; S ;
2 3
2 3
@S @S
where V and V are constants related to the original parameters ; ; m; and the
2 3
functions f and . They contain information ab out the market, but are not sp eci c to
~
any derivative contract. That is, P is given by the mo di ed Black-Scholes formula
Z
T
? r T t r u t
~
P = IE e h S e H u; S dujS :
t T u t
t
The path-dep endent payment stream H t; S is computed from the second and third
order derivatives of the classical Black-Scholes price P . It may b e p ositive or negative
BS
and accounts dynamically for volatility randomness in a robust mo del-indep endentway.
It also accounts for the market price of volatility risk e ectively selected by the market.
3. There is a simple explicit formula whenever there is a formula for the Black-Scholes
price of the contract P t; S , for example calls, puts, binaries. It is given by
BS
~
P = P t; S T t H t; S :
t BS t t
The Black-Scholes price is corrected by a term containing the Gamma of the contract
and a term containing its third derivative which we name Epsilon since it is related
to a small correction.
p
The error is of order less than 1= which is small when mean-reversion is fast. Notice
that this approximation do es not dep end on the present level of volatility, which is not
directly observable and usually dicult to estimate.
Calibration
Where this simpli cation is extremely useful is in parameter estimation. We rst estimate
in the usual way the historical volatility which can be related to the mo del parameters:
2
it is the square ro ot of the average of f with resp ect to the invariant distribution of the
OU pro cess Y . We are not using this explicit relation since we are not aiming at estimating
t
the mo del parameters. The quantities V and V are easily calibrated by using the full term
2 3
+
structure of the volatility smile across strikes and maturities. If we take h S = S K ,
T T
a call option, compute the approximation to the sto chastic volatility price describ ed ab ove
and then work out the implied volatility I , we obtain the simple formula
log K=S
I = a + b:
T t 6
The parameters V and V are related to the smile parameters a; b and the mean volatility
2 3
by
3
2
; V = b a r +
2
2
3
V = a :
3
It turns out that a has the same sign as , so that a downward sloping skew indicates a
negative correlation. In the particular uncorrelated case, =0,we deduce that, at this order
of approximation, the smile gives only a constant correction to the historical volatility . In
that case, one can verify that the corrected price itself satis es a Black-Scholes equation with
q
2
the corrected volatility +2 b used e ectively by the market b is close to for
fast mean-reverting sto chastic volatility markets . This can b e thought of as a pure kurtosis
e ect, as p ointed out to us by a referee. The general case 6=0 is more interesting since it
re ects the skew e ect. The formulas ab ove are derived in the pap er cited in fo otnote 5.
So, the calibration pro cedure is as follows:
1. Fit near-the-money implied volatilities for several maturities, to a straight line in the
comp osite variable called the log-moneyness-to-maturity-ratio LMMR
Strike Price
log
Sto ck Price
LMMR := :
Time to Maturity
Estimate the slop e a and the intercept b. Since LMMR = 0 when sto ck price = strike
price, b is exactly the at-the-money implied volatility.
2. Estimate , the historical volatility from sto ck price returns, and compute V and V
2 3
using the formulas ab ove.
3. Price any other Europ ean by pricing the adjusted claim with the payout rate H for
that contract in a Black-Scholes world using the explicit formula given ab ove. A
similar pro cedure that needs only the market-describing parameters V and V holds
2 3
6 7
for Asian, American and barrier options.
We stress again that this is not mo del sp eci c: it do es not dep end on a particular choice
of the functions f or or a particular ergo dic driving di usion Y , in the sense that many
such choices will lead to the three observable quantities ; a; b or ; V ;V with no need
2 3
to estimate the parameters ; ; m; separately: only the V 's, which contain these, are
needed. Nor is the present value Y required.
t
6
See Fouque, Papanicolaou and Sircar, 1999, From the ImpliedVolatility Skew to a Robust Correction to
Black-Scholes American Option Prices, preprint.
7
See Fouque, Papanicolaou and Sircar, 1999, Financial Modeling in a Fast Mean-Reverting Stochastic
Volatility Environment, Asia-Paci c Financial Markets 6 1 , pages 37-48. 7
Stability of Parameter Estimates
We have undertaken, in the reference in fo otnote 3, a detailed empirical study of high-
frequency S&P 500 index data to establish that volatility reverts slowly to its mean compared
to the tick-by-tick scale uctuations, but it reverts fast when lo oked at over the longer time
scale of months. The key conclusion of this study is that while the rate of mean-reversion in
1
units years is large, it is a dicult parameter to estimate precisely, b eing the recipro cal
of the correlation time of a hidden Markov pro cess. However, the asymptotic derivatives
theory do es not need the value of , only that it b e large.
We have also tested a posteriori the feasibility of the theory-predicted LMMR line t
for actual implied volatility data. We show in Figure 3 daily estimates of the slop e and
^
intercept co ecients a^ and b from tting Black-Scholes implied volatilities from observed
S&P 500 Europ ean call option prices:
!
log K=S
obs
^
I t; S ; K; T =a ^ + b;
T t
across strikes K and maturities T .
Liquid Slop e Estimates: Mean= 0:154, Std= 0:032
−0.05
−0.1
−0.15
−0.2
−0.25
0 10 20 30 40 50 60
Liquid Intercept Estimates: Mean= 0:149, Std= 0:007 0.2
0.15
0.1
0.05
0
0 10 20 30 40 50 60
Trading Day Numb er: 9/20/94 - 12/19/94
Figure 3: Daily ts of S&P 500 European cal l option implied volatilties to a straight line in LMMR,
excluding days when there is insucient liquidity 16 days out of 60 .
We observe from the results that the slop e co ecientsa ^ are small. This strongly supp orts
the fast mean-reverting hyp othesis and validates use of the asymptotic formula as the full
p
skew formula shows that a is a term of order 1= . We also nd that the estimates a^ and 8
^
b over a sixty-day p erio d are stable as attested to by the standard deviations of the daily
estimates rep orted in the Figure.
Hedging
8
There is a related theory for hedging which we do not describ e here in detail . It relies
on the parameter V estimated from the smile to account for the e ect of correlation or
3
leverage. The amount of sto ck to hold is given by a correction to the Black-Scholes Delta
by a combination of the Gamma, Epsilon and the fourth derivative which we call Kappa:
!
2 3 4
@P @ P @ P @ P V T t
BS BS BS BS 3
2 3 4
4S +5S + S t; S : t; S
t t
t t t
2 3 4
@S S @S @S @S
t
This strategy removes a bias in the shortfall of a Black-Scholes hedge used in a random
volatility market. The new average hedging error measured, for example, as the exp ected
shortfall is of order 1= .
Conclusion
Wehave presented a mo di ed Black-Scholes pricing and hedging metho dology to account for
p ersistent sto chastic volatility. It is e ective in identifying the imp ortant comp onents of the
implied volatilityskew, from which we calibrate, and gives a recip e for pricing and hedging
more exotic securities. As the implied volatility ts show, the calibrated group parameters
are quite stable over time.
It dynamically captures the main skewness of the observed implied volatility surface,
indep endent of sp eci c mo deling of volatility. This, plus the simplicity of the pro cedure,
makes the results extremely suitable for practice.
8
The hedging problem and a presentation from rst principles of the work discussed here plus application
to interest-rate derivatives app ears in the forthcoming b o ok Derivatives in Financial Markets with Stochastic
Volatility by the authors of this article. 9