Sto chastic Volatility, Smile & Asymptotics
y
K. Ronnie Sircar George C. Papanicolaou
May 17, 1998
Abstract
We consider the pricing and hedging problem for options on sto cks whose volatility is a
random pro cess. Traditional approaches, such as that of Hull & White, have b een successful in
accounting for the much observed smile curve, and the success of a large class of such mo dels in
this resp ect is guaranteed by a theorem of Renault & Touzi, for whichwe present a simpli ed
pro of. We also present new asymptotic formulas that describ e the geometry of smile curves and
can b e used for interp olation of implied volatility data.
Motivated by the robustness of the smile e ect to sp eci c mo delling of the unobserved
volatility pro cess, we present a new approach to sto chastic volatility mo delling starting with the
Black-Scholes pricing PDE with a random volatility co ecient. We identify and exploit distinct
time scales of uctuation for the sto ck price and volatility pro cesses yielding an asymptotic
approximation that is a Black-Scholes typ e price or hedging ratio plus a Gaussian random
variable quantifying the risk from the uncertainty in the volatility. These lead us to translate
volatility risk into pricing and hedging bands for the derivative securities, without needing to
estimate the market's value of risk. For some sp ecial cases, we can give explicit formulas.
We outline how this metho d can b e used to save on the cost of hedging in a random volatility
environment, and run simulations demonstrating its e ectiveness. The theory needs estimation
of some statistics of the volatility pro cess, and we run exp eriments to obtain approximations to
these from simulated smile curve data.
Keywords: Option pricing, Volatility, Sto chastic Volatility mo dels, Hedging, Smile Curve.
Scienti c Computing and Computational Mathematics Program, Gates Building 2B, Stanford University, Stan-
ford CA 94305-9025; [email protected]; work supp orted by NSF grant DMS96-22854.
y
Department of Mathematics, Stanford University, Stanford CA 94305-2125; [email protected]; work
supp orted by NSF grant DMS96-22854. 1
Contents
1 Review 2
1.1 Intro duction ...... 2
1.2 Literature Survey ...... 3
1.3 Partial Di erential Equation for Option Price ...... 4
2 Smile Curve 6
2.1 Analytical Results ...... 7
2.2 Asymptotic Smile ...... 9
2.3 Uncorrelated Case ...... 9
2.4 Correlated Case ...... 10
3 Time Scales and Asymptotics 11
3.1 Mo del ...... 13
3.2 Results ...... 13
3.2.1 Pricing ...... 13
3.2.2 Hedging ...... 15
3.2.3 Numerical Simulations ...... 18
3.3 Connection with Smile Curve ...... 19
3.4 Estimation issues ...... 21
3.5 Numerical Simulation of Smile Fitting ...... 22
4 Conclusions 23
A Induction step 24
B Pro ofs of sto chastic averaging and CLT 24
C Classical Black-Scholes with sto chastic volatility co ecient 28
1 Review
1.1 Intro duction
Extensions of the Black-Scholes mo del for option pricing b egan app earing in the nance literature
not long after publication of the original pap er [7] in 1973. For example, Merton [24 ] generalised
the Black-Scholes formula to account for a deterministic time-dep endent rather than constant
volatility later the same year, and in 1976, he incorp orated jump di usion mo dels for the price of
the underlying asset [25 ]. However, the resulting deviations from the Black-Scholes derivative price
are typically quite small, at the cost of having to estimate more complex parameters recall that the
Black-Scholes mo del requires estimation only of a constant volatility parameter corresp onding to
a lognormal di usion mo del approximation of the underlying's price pro cess . For this reason, the
original form of the mo del has remained robust allowing for mo di cations for dividend payments
and transactions costs : although its limitations are recognised by traders, no simple `b etter' mo del
has b een universally accepted.
Indeed, there is an almost p erverse pressure by practitioners for real prices to conform to those
of the Black-Scholes world as demonstrated by prevalent use of the implied Black-Scholes volatility
measure on market prices. That is, traders are given to buying and selling in units of volatility 2
corresp onding to options prices through the Black-Scholes formula which is strictly increasing in
the volatility parameter . This is often termed `trading the skew' [35 ], where skew refers to a
trader's estimate of future volatility. Empirical studies of implied volatility date back to Latane &
Rendleman [22 ] 1976 , and include Beckers [4 ] 1981 , Rubinstein [30 ] 1985 and Canina & Figlewski
[8] 1993 . This measure has b een used to express a signi cant discrepancy between market and
Black-Scholes prices: the implied volatilities of market prices vary with strike price and time-to-
maturity of the options contract whereas, in the Black-Scholes mo del, the volatility is assumed
constant. In particular, the variation of implied volatility with strike price for options with the
same time-to-maturity is often a U-shap ed curve, the smile curve, with minimum at or near the
current asset price.
This particular shortcoming is remedied by sto chastic volatility mo dels rst studied by Hull
& White [19 ], Scott [33 ] and Wiggins [40 ] in 1987. The underlying asset price is mo delled as a
sto chastic pro cess whichisnow driven by a random volatility It^o pro cess that mayormay not b e
indep endent. As noted by Taylor [38 ], there is no economic intuition b ehind this generalisation,
although Nelson has shown sto chastic volatility arises as the continuous time limit of discrete
1
statistical mo dels suchasARCH , based on time-series analysis of nancial data [26 ]. As a result of
the increase in complexity, estimation of the unobserved volatility pro cess from market data is much
harder than estimation of the constant Black-Scholes volatility. Nonetheless, it is b elievable that
market volatility in the intuitive sense of wild price swings is random, and the fact that sto chastic
volatility prices pro duce the smile curve for any volatility pro cess instantaneously uncorrelated with
the price pro cess, as we shall see b elow, gives this extension of Black-Scholes a little more tractability
than earlier ones.
In the spirit of the smile result which is robust to di erent mo dels of the volatility, we shall
consider the scenario of the volatility pro cess uctuating on a di erent time scale to that of the
price pro cess: an asymptotic analysis will yield pricing and hedging bands which are not sensitive
to how the volatility is mo delled. In the sp ecial Black-Scholes typ e cases when volatilitydoesnot
dep end on sto ck price, we derive explicit formulas for the width of these bands, enabling a simple
mo di cation of the Black-Scholes formula.
We run simulations of sto ck prices with sto chastic volatility and test the e ectiveness of our
mo di ed hedging strategy against simple Black-Scholes delta hedging. Finally, we test the ease-
of-use of this mo del by using simulated smile data to infer estimates of the necessary parameters.
This is seen to work remarkably well.
Our results demonstrate the to ols of a separation-of-scales mo delling approach that can be
applied to a wide variety of incomplete market problems. This is discussed in the conclusions.
1.2 Literature Survey
We b egin with a brief review of the `traditional' approach to mo delling sto chastic volatility that
originates from 1987 see [19 , 33 , 40 ] . The asset price fX g is taken to b e a di usion pro cess that
t
is lognormal conditional on the path of a sto chastic volatility pro cess f g:
t
dX = X dt + X dW : 1.1
t t t t t
2
Thus X is driven by the pro cess , and is sub ordinate to it. We shall denote V = , and lo ok at
t t t
t
mo dels for V ; some authors mo del , but there are no real advantages of one or other approach.
t t
The price of a Europ ean option fC ; 0 t T g is a function of current price and volatility:
t
C = C X ;V ;t . This assumes that we can nd the presentvolatility although it is not directly ob-
t t t
servable. Then, the no-arbitrage price of a Europ ean call option with strike price K and expiration
1
Auto-regressive conditional heteroskedasticity 3
date T , in a market with constantspotinterest rate r , is given by
Q
r T t +
C = C X ;V ;t =E [e X K ]; 1.2
t t t T
t
2
where Q is an equivalent martingale measure under which the prices of the traded assets, b ond,
sto ck and option, are martingales dep ending on = X ; ;t , the volatility risk premium, or
t t
the market price of risk. The notation E means the conditional exp ectation given the past up
t
to time t: E = E jF , with F the -algebra of events generated by f X ; : s tg.
t t t s s
The presence of a second non-traded factor that the derivative price dep ends up on, the sto chastic
volatility, gives rise to an incomplete market in which the risk inherent in holding an option cannot
b e completely hedged by trading in just the underlying asset. Thus there must b e a parameter
which represents how the market values the unhedged risk, and this to o must be estimated from
data once the mo del is put into practice. The app earance of this parameter will b e more apparent
in the derivation of the next section.
The pricing formula 1.2 can be simpli ed under the following assumptions: if = ;t ,
t
indep endentof X , and is uncorrelated with X , then by iterated exp ectations,
t t t
p
Q
BS
C X ;V ;t =E [C V ]; 1.3
t t
t
where
Z
T
1
2
V = ds; 1.4
s
T t
t
BS
and C denotes the classical Black-Scholes formula written as a function of the volatility
p
V is the ro ot-mean-square time average of over the remaining tra jectory parameter . Thus
of each realisation. This is a generalisation of the Hull-White formula [19 ] which originally assumed
that 0, meaning that the risk from the volatility pro cess is non-comp ensated or can be
diversi ed away . In that case, the pricing measure is just that de ned by the volatility pro cess
p
BS
with no mo di cation, and the formula is C X ;V ;t =E fC V g.
t t t
1.3 Partial Di erential Equation for Option Price
An alternative characterisation of the option price is available when the sto chastic volatility is
also an It^o pro cess, namely as the solution of a parab olic partial di erential equation similar to
the Black-Scholes pricing PDE, but with an extra dimension representing the dep endence on the
volatility pro cess. It is often describ ed as a sp ecial case of the general asset pricing PDE presented
by Garman [15 ]. We give here a direct derivation which demonstrates how the market price of risk
function arises naturally,aswell as the breakdown of the hedged and unhedged p ortions of the risk.
We supp ose that the asset price fX g satis es
t
p
V X dW ; 1.5 dX = X dt +
t t t t t
2
where the square of the volatility pro cess V = is a continuous solution of
t
t
dV = V ;t dt + V ;t dZ : 1.6
t t t t
The Brownian motions W and Z have constant correlation 2 [ 1; 1]: d
t t t
we lo ok for the option price as a smo oth function C X ;V ;t by trying to construct a hedged
t t
2
Details of the theory of pricing by equivalent martingale measures in the classical Black-Scholes complete markets
case can b e found in [16 ] or [11 , Chapter 6]. 4
p ortfolio of assets which can b e priced by the no-arbitrage principle. Unlike the Black-Scholes case,
it is not sucientto hedge solely with the underlying asset, since the dW term can b e balanced,
t
but the dZ term cannot. Thus we try and hedge with the underlying asset and another option
t
which has a di erent expiration date.
1
Let C x; v ; t be the price of an option with expiration date T , and try to nd pro cesses
1
fa ;b ;c g such that
t t t
1 2
^
C = a X + b + c C ; 1.7
T T T T T
1 1 1 1 1
T T
1 1
^
where is the price of a riskless b ond under the prevailing short-term constantinterest rate r , and
t
2
1
C is the price of a Europ ean option with the same strike price as C , but di erent expiration
t
date T >T >t. In other words, the right-hand side of 1.7 is a p ortfolio whose payo at time
2 1
1
T equals almost surely the payo of C . In addition, the p ortfolio is to b e self- nancing so that
1
2 1
^
: 1.8 = a dX + b d + c dC dC
t t t t t
t t
If such a p ortfolio can b e found, for there to b e no arbitrage opp ortunities, it must b e that
1 2
^
C = a X + b + c C ; 1.9
t t t t t
t t
for all t 1 ! 1 1 1 @C @C @C 1 + L C dt + dX + dV 1.10 1 t t @t @x @v ! 2 2 i h @C @C 2 ^ dt; a + c = dX + c dV + c L C + b r t t t t t t 1 t t @x @v where 2 2 2 p 1 @ @ 1 @ 2 2 L := vx v v; t + v; t + x ; 1 2 2 2 @x @x@v 2 @v 1 2 and C , C and its derivatives are evaluated at X ;V ;t . The risk from the dZ terms can b e t t t eliminated by balancing the dV terms, which gives t 1 @C =@ v ; 1.11 c = t 2 @C =@ v and to eliminate the dW terms asso ciated with dX ,wemust have t t 2 1 @C @C c : 1.12 a = t t @x @x 1 2 Substituting for a , c and b = C a X c C and comparing dt terms in 1.10 gives t t t t t t t t ! ! 1 1 2 1 @C @C 1 2 L C X ;V ;t = L C X ;V ;t ; 1.13 2 t t 2 t t @v @v where @ @ + L + r x : L := 1 2 @t @x p That is, L is the classical Black-Scholes di erential op erator with volatility parameter v , plus 2 second-order terms from the V di usion pro cess. Now, the left-hand side of 1.13 contains terms t 5 dep ending on T but not T and vice versa for the right-hand side. Thus b oth sides must b e equal to 1 2 a function that do es not dep end on expiration date. Denoting this function x; v ; t v; t v; t , the pricing function C x; v ; t , with the dep endence on expiry date supressed, must satisfy the PDE p @C 1 1 2 2 + vx C + x v v; t C + v; t C xx xv vv @t 2 2 + v; t x; v ; t v; t C + r xC C = 0: 1.14 v x + For a call option, the terminal condition is C x; v ; T = x K and C 0;v;t = 0, C x r T t Ke , as x ! 1. The v b oundary conditions dep end on the sp eci cs of the pro cess fV g. t The function is termed the market price of volatility risk b ecause dC X ;V ;t =[ r X C + V ;t C ] dt + X C dW + V ;t C dZ : t t t x t v t t x t t v t From this expression, we see that an in nitesimal fractional increase in the volatility risk increases the in nitesimal rate of return on the option by times that fraction. The equation 1.14 can b e considered as the Feynman-Kac PDE asso ciated with the exp ectation in 1.2 when the volatility is an It^o pro cess. In particular, the transformation of the measure is seen explicitly in the transformed drift co ecient in front of the C term. v A summary of sp eci c volatility mo dels prop osed in the literature is given in the following 3 table . Authors Correlation Risk Premium Volatility Pro cess Hull-White [19 ] =0 0 dV = V dt + V dZ , lognormal t t t t Scott [33 ] =0 0 log is mean-reverting OU t Stein-Stein [37 ] =0 =const. is mo dulus of a mean-reverting OU t p Ball-Roma [3] =0 0 dV = m V + V dZ t t t t p Heston [17 ] 6=0 = v dV = m V + V dZ t t t t There is little intrinsic intuition b ehind these volatility mo dels, beyond the need to enforce p ositivity and, sometimes, mean-reverting b ehaviour for the pro cess. This and the fact that a whole unknown and utterly unobservable function x; v ; t has to b e estimated in general in order to price consistently with this theory, has led us and other researchers [1, 13 , 20 ] to new approaches to translating uncertain volatility risk into price. We address this in Section 3. 2 Smile Curve De nition Given a price C K; for a Europ ean option with time-to-maturity = T t and strike price K from, say, market data or a di erent pricing mo del , we can calculate the implied BS Black-Scholes volatility I K ; x; , such that C x; ; I; K = C K; where x is the present time t price of the underlying asset. It can b e found uniquely b ecause of the monotonicityofthe BS Black-Scholes formula in the volatility parameter: @C =@ > 0. The most quoted phenomenon testifying to the limitations of the classical Black-Scholes mo del is the smile e ect: that implied volatilities of market prices are not constant across the range of options, but vary with strike price and time-to-maturity of the contract. In particular, the graph of 3 OU denotes an Ornstein-Uhlenbeck pro cess which is of the form dY = aY + b dt + kdZ . t t t 6 I K; against K for xed obtained from market options prices is often observed to b e U shap ed with minimum at or near the money K x . This is called the smile curve. Empirical evidence min for this observation can b e found, for example, in [8 , 10 ]. In this section, we show that under certain conditions, sto chastic volatility mo dels do give prices whose Black-Scholes implied volatilities plotted against strike price form a smile curve. The rst is a simpli ed pro of of a global result originally due to Renault & Touzi [28 ] which guarantees r T t existence of a smile lo cally centred around the current forward price of the sto ck xe . Then we derive new asymptotic results that are valid when the volatility pro cess has a small sto chastic comp onent. These give more detailed information ab out the geometry of the smile curve in cases where the volatility of the volatility is small, and this can be used for estimation purp oses when callibrating sp eci c mo dels from market data as we shall outline b elow. In addition, the more dicult correlated case 6= 0 , for which a global result ab out the shap e of implied volatility curves is not known, can b e tackled by asymptotic metho ds. 2.1 Analytical Results Theorem 1 Renault-Touzi In a stochastic volatility model de ned by 1.5 , where fV g is a stochas- t tic process uncorrelated with fX g, suppose the risk premium process is a function of X and t but t t 2 = X ;t . Then provided not of V : V de ned by 1.4 is an L random variable, the implied t t t volatility curve I K; for xed is a smile, that is, it is local ly convex around the minimum r K = xe , which is the forward price of the stock. min Proof Let us x t and T and start with V b eing a Bernoulli random variable V = with probability 1 with probability1 , 2 under the measure Q . Then provided the assumptions listed hold, the Hull-White formula 1.3 applies: BS BS BS C I ; K ; K = C ; K + 1 C ; K ; 2.15 1 2 BS where C denotes the classical Black-Scholes formula, and b elow we give only its volatility ar- gument. We have written I = I ; K to stress the dep endence on and K with x and xed. Di erentiating 2.15 with resp ect to K gives @I BS BS BS BS + C I ; K = C + 1 C ; C I ; K 1 2 K K K @K so that @I = sign f ; sign @K where, for xed K , BS BS BS f := C + 1 C C I ; K ; 1 2 K K K BS b ecause C > 0. Clearly if =0; 1 then I ; resp ectively, so f 0 = f 1 =0. The key 2 1 and somewhat surprising step is to now di erentiate f and 2.15 with resp ect to 0 BS BS BS f = C C C I ; K I ; 2.16 1 2 K K K BS BS BS C I ; K I = C C ; 2.17 1 2 7 from which we can obtain an expression for I . Di erentiating f again and substituting directly from the Black-Scholes formula then gives ~ LI 00 BS BS f =2[C C ] ; 1 2 3 I r BS ~ where L = log xe =K . Assuming, without loss of generality, that < , monotonicityofC 1 2 BS BS in the volatility parameter gives C < C and I < 0 from 2.16 . Since I > 0, we 1 2 00 r ~ ~ conclude that sign f = sign L . Thus for K min minimum in [0; 1], implying that f < 0 for 0 < <1. Hence @ I =@ K < 0. Similarly for K>K , min @ I =@ K > 0, and @ I =@ K =0 at K = K . The implied volatility curve is lo cally convex around min K . min V can take, and this This can b e extended by induction on the numb er of p ossible values that step is given in App endix A. The result then holds for all distributions of V that are the limit 2 of nite state distributions, as the number of states go es to in nity. This includes all L square integrable random variables. Numerical computations show that as expiration is approached, the region of concavity around r xe shrinks, and the curve b ecomes at outside this region - the smile b ecomes a very sharp downward spike. This is illustrated in Figure 1. To our knowledge, there has b een no discussion in the empirical literature as to how the smile curveevolves in time: it would b e interesting to see if it do es indeed b ecome convex nearer to the money as t ! T from real data . \tex{Smile Curves as $t\rightarrow T=0.5$} 0.3 \tex{\scriptsize $t=0.4994$} 0.295 \tex{\scriptsize $t=0.499$} 0.29 \tex{\scriptsize $t=0.495$} \tex{Implied Volatility} \tex{\scriptsize $t=0.49$} 0.285 0.28 0.94 0.96 0.98 1 1.02 1.04 1.06 \tex{Moneyness ratio $K/xe^{r(T−t)}$} Figure 1: The implied volatility smile curves b ecome spikier as t ! T . Here, V is Bernoulli with =0:2, =0:2 and =0:3. The shifting base of the smile curve with time has b een removed 1 2 r T t by the xe scaling. V the Bernoulli distribution ab ove, we can show that I ! maxf ; g Asymptotically, with 1 2 p r ~ ~ T t. At K L = 0 , I ! + 1 < , as t " T , provided jLj = j log xe =K j > min 1 2 max 8 so the narrow spike b ehaviour is con rmed analytically. Note that I is not de ned when t = T b ecause anyvalue I matches the terminal payo . 2.2 Asymptotic Smile We utilise two approaches to demonstrate the smile e ect when the randomness of the volatility pro cess is small. For the uncorrelated case, a direct asymptotic expansion of the Hull-White formula is simplest. Similar Taylor series expansions were presented in [19 , 3 ], but these are not generally valid without an asymptotic approximation. When 6= 0, where the result is far more complex, we construct an asymptotic solution of the PDE satis ed by the sto chastic volatility option pricing function. As describ ed by Black [6 ], one often nds that <0 rising volatility pushes sto ck prices down , so the latter is an imp ortant case. 2.3 Uncorrelated Case " Supp ose t = + "z t for "<<1 where z t is some pro cess with rst and second moments. 0 In this subsection, we shall assume for simplicity that volatility risk is non-comp ensated so that 0 and the measure Q is denoted just Q. Then we can expand pathwise 1 " Z p 2 T 1 " 2 2 " = + "H t; T +" G t; T + ; 2.18 V = s ds 0 T t t where Z T 1 H t; T = z s ds; T t t " Z T 1 1 2 2 : z s ds H t G t; T = 2 T t t 0 h i p " Q " BS " BS We lo ok for an asymptotic expansion of the implied volatility I de ned by C I =E C V t " 2 of the form I = + "I + " I + : Expanding and comparing p owers of " reveals 0 1 2 Q I = E [H t; T ] ; 1 t BS C 1 0 Q + E [G t; T ] var [H t; T ] I = t 2 t BS 2 C 0 ! 2 ~ 1 1 L Q 2 = E [G t; T ] var [H t; T ] ; t t 0 2 2 4 0 0 " 2 ~ and so the plot of I against K deviates from a at line at O " with a minimum at L = 0 that r is, K = xe , and the smile b ecomes steep er as t " T i.e. 0 . Provided the expansion 2.18 is " valid, which dep ends on and the pro cess z , the expansion for I as a function of K is asymptotic in the region p p r r " 0 0 2 2 ~ that is, when L = is smaller than ". This is shown in Figure 2. It remains convex, in this 0 p r +1= approximation, for K Thus the asymptotic analysis gives detailed information ab out the geometry of the smile curve when the sto chastic comp onent of the volatility is small, which is not available in the global result. A similar calculation will b e presented in section 3.3 when the volatility is oscillating on a time 9 \tex{Asymptotic Smile and Region of Validity, $\eps=0.1$} 0.3862 0.386 0.3858 0.3856 0.3854 0.3852 \tex{Implied Volatility to order $\eps^{2}$} 0.385 0.3848 94 96 98 100 102 104 106 108 110 \tex{Strike Price $K$} Figure 2: The dashed lines enclose the region of validity of the asymptotic approximation to the smile curve when the sto chastic comp onent of the volatility is small. Here = 0:3, time to 0 expiration =0:5, and the exp ectations in the expressions for I and I are taken to b e of order 1. 1 2 scale that is short compared to the length of the derivative contract. This case will b e imp ortantin connecting the traditional approach to sto chastic volatility with the approach presented in Section 3. In addition, the expansion shows that such sto chastic volatility mo dels do not all display the same time-to-maturity e ect as they do the smile e ect. To see this, we consider the variation of " I with T for xed t, and as 1 @I 1 Q = E f z T H t; T g ; t @T T t it is clear that the variation of implied volatility with time-to-maturity will dep end on the statistics of the volatility pro cess. This is consistent with Rubinstein [30 ] who found di erent time-to- maturity e ects in di erent time p erio ds and for di erent moneyness ratios. 2.4 Correlated Case When fV g is an It^o pro cess as in section 1.3 , we mo del the small randomness of volatility by t supp osing that p " " " " V ;t dZ dV = " V ;t dt + t t t t " for some "<<1, so that the option pricing function C x; v ; t satis es p p 1 1 2 " " 2 " " vx C "x v v; t C " v; t C + + 2.19 C + xx xv vv t 2 2 p " " " C = 0; 2.20 + " v; t + + r xC "d v; t C x v 10 with appropriate terminal and b oundary conditions. Wehave assumed that is indep endentof x and de ned d v; t = v; t v; t . p When " = 0, 2.20 reduces to the Black-Scholes PDE with volatility parameter v ; the solution p BS is C v . Constructing an asymptotic solution p p " BS 1 2 C = C v + "C + "C + ; we nd p p p 1 d v; t 1 BS BS p v C = x v; t C v C v ; L BS x 2 2 v 1 2 2 1 where L u := u + x u + r xu u , and C has zero terminal and b oundary conditions. BS t xx x 2 Because v is essentially a parameter in this equation, we can construct a solution of the form p p p p 1 BS BS C = A v; t xC v +B v; t C v : x For simplicity, supp ose ; and do not explicitly dep end on t. Then 2 T t ; 2.21 A ;t = 2 +1 " 3 2 2 2 2 d 2 2 1+ + T t T t : B ;t = 4 +1 2 2 +1 p p " 1 2 In any case, the rst-order correction to the implied volatility I = v + "I + "I + is given by ! ~ p p 1 L 1 : 2.22 v; t +A v; t I = B 2 v T t 2 Since A vanishes and B ;t = T t d when = 0, the implied volatility is still at to p " O " in the uncorrelated case, as was shown in the previous section. When 6=0, I is linear to ~ rst-order in L, so there is no apparent lo cal minimum. In the case of t-indep endent co ecients, 1 " if <0, then A>0 and @I =@ K > 0, so I increases slowly upwards as K increases, within the ~ region of validity L = O 1 . This ties in with the empirical observations of Rubinstein [30 ] and MacBeth & Merville [23 ] where monotonic implied volatility curves were rep orted: sometimes I was seen to increase with K and in other time p erio ds decrease with K . Hull & White and Wiggins [40 ], from numerical exp eriments, related the slop e of @ I =@ K to the sign of the correlation , and this is con rmed by the analytical formulas 2.21,2.22 . However, as they noted, it is not clear why the correlation should change sign in di erent time p erio ds. This b ehaviour holds indep endently of sp eci c volatility mo delling and choice of v; t , and the formulas 2.21-2.22 can be used to t smile data and suggest tractable volatility mo dels. 2 ~ We can further construct C , and much calculation reveals that it is a quartic in L if 6= 0 and ~ ~ a quadratic in L with minimum at L =0 if =0, whichis exp ected from the previous section's result. It is worth noting that if , or dep end on x, it will not in general b e p ossible to construct asymptotic solutions from the Black-Scholes formula and its derivatives. 3 Time Scales and Asymptotics Wenow take a somewhat di erent approach to the traditional view of sto chastic volatility consid- ered ab ove. The various mo dels of the volatility pro cess in the literature exhibit di ering degrees 11 of success that might seem to commend or condemn them. For example, with Hull-White's log- normal mo del, the volatility stays p ositive, but it can grow unb oundedly with time which do es not re ect true b ehaviour. The square-ro ot pro cess of Heston [17 ] and Ball-Roma [3 ] also remains p ositive and re ects an intuitively app ealing idea of volatility b eing mean-reverting; in addition, the option price has a closed-form solution, up to calculation of Fourier integrals. The same is also true for Stein's mo del [37 ]. What they all have in common is that they give prices that are slightly 4 5 di erent from Black-Scholes prices , and that these prices always show the smile e ect when the correlation is zero . The mo dels can then be callibrated from market prices for example by the metho d of moments outlined by Scott at some exp ense. At the very least, two mo del-describing parameters plus the correlation ,today's value of the volatility and the market price of risk haveto b e estimated, so the complexityover Black-Scholes is considerably greater while the pricing biases are small. This at least suggests that we might take the random volatility to b e something simpler than a continuous It^o pro cess: if atwo-state Markovchain gives the smile, why mo del it with a t far more complicated pro cess? New approaches to mo delling uncertain volatilityhave b een initiated byAvellaneda et al. [1 , 2] who construct worst case hedging strategies for complex p ortfolios of options assuming volatility lies in a known band, and Fournie et al.[14 ], who suggest simple, rapidly uctuating volatility pro cesses. The approach we prop ose is to supp ose that uncertainty in the volatility leads to uncertainty in the option price. Since the volatility cannot b e directly observed and mo delled, we shall restrict ourselves to nding pricing bands rather than exact prices, and hedging strategies that account for the risk inherent in a randomly uctuating volatilityenvironment, without assuming knowledge of the exact distribution. Furthermore, we shall exploit the fact that the asset price and its volatility may uctuate on di erent time scales. The price is mo delled as an It^o pro cess as usual, whose driving Brownian motion oscillates on a scale that is \in nitely fast". Now, if we supp ose the volatility is sto chastic, to account for the smile e ect, it is intuitive to exp ect that its time scale of variation is small compared with the lifetime of the derivative contract we are interested in typically many months , but not as small as the in nitesimal scale of the Brownian motion. Thus the fastest random uctuations of price are picked up by the Wiener pro cess, and intermediate uctuations by the volatility pro cess. For example, it is reasonable that asset price variations are on the scale of minutes whilst volatility uctuations are daily, which is small compared to a nine-month options contract. Over the longer O T time scale, the fast uctuations of the volatility give the app earance of a constant in time volatility function which would corresp ond to a Black-Scholes price; the randomness of the volatility would manifest itself as con dence bands around this price whose width can be calculated. Thus the fact that sto ck price and volatility change on di erent time scales allows us to develop an asymptotic pricing theory around Black-Scholes. The law of large numb ers and the central limit theorem remove the dep endence on the sp eci cs of the volatility pro cess, re ecting the robustness of this metho d. 4 It is not obvious what the Black-Scholes price is in the context of a sto chastic volatility mo del: Hull-White use E V as the volatility squared parameter in the Black-Scholes formula, and Stein-Stein use the mean that reverts t to in their mo del. But b oth of these, which are used to describ e pricing biases of the new mo dels from Black-Scholes are really measuring errors in approximating sto chastic volatility options prices with Black-Scholes prices by adjusting the volatility parameter - it is like trying to guess the implied volatility curve and then inferring biases from the errors of the guess. A more logical measure would b e to assume volatility is sto chastic according to the mo dels, simulate the asset price, and then calculate a historical Black-Scholes volatility parameter by tting the generated data to a lognormal pro cess. 5 We shall lo osely refer to the smile e ect as meaning any non-constantvariation of Black-Scholes implied volatility with strike price. 12 3.1 Mo del The generalised Black-Scholes analysis tells us that the price C ; x of a Europ ean option on a sto ck whose price is mo delled as the It^o pro cess ^ dX = t; X dt + t; X dW t t t t satis es the PDE 1 2 C = C + r xC C ; 3.23 ; x xx x 2 6 ^ where = T t is the time to expiration of the contract , and ; x := T ; x is a deterministic function called the volatility. If wenow supp ose the volatility is random and rapidly " ;x , where ; x = ^ T ; x for some varying in time, so that it is given by ; x = " 7 sto chastic pro cess f^ t; x ;t 0g that stays p ositive, taking values in a function space, and some " small " > 0, then we de ne the asso ciated stochastic option price C ; x as the solution of the sto chastic PDE 1 " 2 " " " C = ;x C + r xC C ; 3.24 xx x 2 " " + with initial condition C 0;x = x K for a call option. We have allowed the volatility to dep end on the sto ck price, but our results will not dep end on sp eci c mo delling of such a function space valued pro cess. " We are interested in the asymptotic b ehaviour of C as " 0: this approximation will tell us how to deal with the risk from the randomness of the volatility. 3.2 Results 3.2.1 Pricing 2 Supp ose that ; x is an ergo dic Markov pro cess and that ; x is stationary. For the moment, we shall restrict admissible pro cesses to be b ounded and continuous almost surely. Then, under 8 further technical assumptions given in App endix B , and with kk denoting the space-time sup-norm de ned there in B.48 , wehave that as " 0, " C k > ! 0; P kC p 2 C ; x; x denotes the solution to the generalised for any > 0 and each 0 T , where 9 2 2 Black-Scholes PDE 3.23 with di usion co ecient x = E f ; x g which do es not dep end on by the stationarity assumption. This is an averaging principle result and will b e derived in App endix B bymultiple scales and the ergo dic theorem. It is analogous to the law of large numb ers, " and says that C converges in probability to the Black-Scholes price with an averaged volatility function. In the sp ecial case when the volatility pro cess is separable in and x, ; x =q x, where 2 fq g is an ergo dic Markov real-valued pro cess with q stationary, the ergo dic theorem implies that Z s 1 " 2 2 2 q V = ds ! q = E fq g 3.25 " 0 6 Wehavechanged time variable from t to so as to deal with forward problems for which `forward-p ointing' It^o integrals are well-de ned. 7 The pro cess ^ t; x de nes an increasing ltration F F for t t s 8 The necessary smo othing of the initial data for options is dealt with there. 9 The di usion co ecient is the co ecient in front of the second-derivative term. 13 as " 0 with probability 1, and so, bychange of time variable, q p " " BS 2 C x; v ; =C ; x; V ! C ; x; x q 3.26 in probability. Thus only the statistical average of the stationary pro cess is needed, rather than the path average. " " C by the following expression: We de ne the scaled error Z in approximating C by q p " " 2 C ; x =C ; x; x + "Z ; x ; 3.27 and consider the statistics of this uctuation around the Black-Scholes average: the quanti cation of the sto chastic volatility risk will be in this term. From the sto chastic PDE 3.24 and the p " 2 C ; x; x , Z satis es Black-Scholes PDE 3.23 satis ed by ! 2 2 ;x x 1 1 " 2 " " " " p Z = ;x Z + r xZ Z + C : 3.28 xx xx x 2 " 2 " Writing this in time-integrated form, we can use the central limit theorem for Markov pro cesses: Z Z D E 1 s 2 2 ~ p ; ;g s; ds ! g s; ;dW s; ; 3.29 " " 0 0 2 + as " 0 where h; i denotes the L R inner pro duct and convergence is in the sense of distributions + 1 + for all test functions g ; 2S R , the Schwarz space of rapidly decreasing C R functions. 10 2 ~ Here W ; x is a Brownian motion with covariance function x; y de ned by Z 1 2 R s; x; y ds; 3.30 x; y = 2 0 o n 2 2 2 2 x s ;y y : 3.31 R js s j;x;y = E s ;x 2 1 2 1 " Applying this to 3.28 tells us that the uctuation Z converges weakly to a pro cess Z ; x which satis es 1 1 2 ~ x Z + r xZ Z ]d + C dW; 3.32 dZ =[ xx x xx 2 2 and Z 0;x 0. Further details are given in App endix B. This is a linear sto chastic PDE, and so Z ; x is Gaussian with mean zero and covariance U ; x; y :=E fZ ; x Z ; y g that satis es the following Lyapunov partial di erential equation: 1 1 1 2 2 2 U = x U + y U + r xU + yU 2U + C ; x x; y C ; y ; 3.33 xx yy x y xx xx 2 2 4 with U 0;x;y 0. It gives the pricing bands or con dence intervals accounting for the unhedged volatility risk: for example, q q 2 C ; x; x 2 " ; x ; 2 where ; x =E fZ ; x g = U ; x; x . Pricing within two standard deviations of the appropriate Black-Scholes mean provides roughly 95 protection against volatility uctuations. These bands could, for example, b e utilised to quote a bid-ask spread. 10 ~ Again, we mean that W evolves forward with natural time and is evaluated at . 14 When ; x =xq , corresp onding to the `traditional' separable sto chastic volatility situation of X lognormal driven by the pro cess fq g, t ! 2 q a+1 [log x=K ] K ^ q exp ; x = ; 3.34 a 2 x 2q 2 2 2 q 2 2 2 2 2 q 1=2, = r=2+q =8+r =2q , and ^ is the integral of the correlation time or where a = r= 2 power sp ectral density for q : Z 1 2 2 2 2 2 ^ =2 q q 0 q gds: E f q s 0 The formula 3.34 is calculated using the Green's function for the classical Black-Scholes PDE see App endix C . A plot of the pricing bands around the Black-Scholes average when q isatwo-state Markov chain is shown in Figure 3. The bandwidth is widest in the neighb ourho o d of the strike price x K and tends to zero as x ! 0, 1. Thus one can b e con dent of the Black-Scholes price with the correct averaged volatility deep in- and out-of-the-money, but the greatest uncertainty ab out the option price arising from uncertainty ab out the volatility is at-the-money. 13 12 11 10 9 8 7 Option Pricing Confidence Interval 6 5 4 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time Figure 3: Pricing bands for separable ; x = xq with q a two-state Markov chain taking values 0:2or0:3. The option strike price is K = 100, expiration T =0:5 and " =0:05. The option price is shown along a particular realisation of the asset price. 3.2.2 Hedging Wenow address the problem of hedging the risk of having written an option on the sto ck, whichis an imp ortant issue for large trading institutions. In the classical Black-Scholes case, a p erfect hedge 15 BS BS can be achieved by holding = @C =@ x shares of the underlying to eliminate the risk from the Brownian uctuations. The feedback e ect of a signi cant amount of such hedging strategies on market volatilitywas studied in [36 ]. In the traditional sto chastic volatility case, the pro cedure is less clear. For a p erfect hedge, another derivative security on the same sto ck is needed and hedging ratios are given by formulas like 1.11,1.12 when, for example, an option with a later expiration date is used. This is known as - hedging b ecause the original notation had of the underlying and of the other derivative de ning the strategy , but is often regarded as unsatisfactory b ecause of the transaction costs involved in purchasing the other derivative. Romano & Touzi [29 ] lo oked for an optimal second option strike price to hedge with to lower such costs, and Renault & Touzi [28 ] studied the e ects of just hedging with the underlying sto ck. Another line of researchbyFollmer & Sondermann [13 ] and Schweizer [32 ] tackles the implicit uncertainty of how to hedge represented by the dep endence of the pricing measure Q on the volatility risk parameter see expression 1.2 by cho osing the measure that minimises the exp ected variance of the cost of hedging with just the underlying. Solving this sto chastic least- squares problem de nes a so-called minimal equivalent martingale measure that characterises the risk preference of the investor, and yields a hedging strategy that is self- nancing on the average. The analysis for a more general market mo del in which, for example, the volatility pro cess can also dep end on the sto ck price as wehave allowed in this section is presented in Hofmann et al.[18 ]. In this vein, we are interested in hedging strategies that can be adapted to the amount of volatility risk each investor will p ermit him or herself to be exp osed to. This is characterised by hedging con dence intervals. In the generalised Black-Scholes case, the hedging ratio = @ C=@x gives the number of shares of the underlying sto ck that an option writer should buy to entirely hedge away the risk from the randomness in the sto ck price. It satis es the PDE h i 1 1 2 2 ; x = ; x + rx + ; xx x x 2 2 11 with initial condition 0;x =H x K for a call option . When the volatility is sto chastic, we " de ne the stochastic hedging ratio ; x as the solution of the corresp onding sto chastic PDE 1 1 " 2 " 2 " = + rx + ; 3.35 ;x ;x xx x 2 " 2 " x where wehave assumed that ; x is restricted to a space of admissible pro cesses that are smo oth enough that ; x exists almost surely. As " 0, x " k > ! 0; P k for any >0 and each0 T , where ; x satis es the averaged hedging PDE 1 1 2 = x + rx + a x ; xx x 2 2 2 a x := E fa ; x g. We assume the technical restrictions of Ap- with a ; x := and ; x x p endix B on the pro cess a ; . " Thus the uctuation Y ; x de ned by p " " ; x = ; x + "Y ; x ; 11 Here, H z denotes the Heaviside function H z =0 if z<0 and H z =1 if z>0 16 satis es 1 1 " 2 " " Y = ;x Y + rx + a ;x Y xx x 2 " 2 " " " 2 2 x a a x ;x ;x 1 1 " " p p + + : xx x 2 " 2 " Under our regularity assumptions, a central limit theorem of the form 3.29 holds for a ; x Z Z D E 1 s ~ p ; a a ;g s; ds ! g s; ;dB s; ; 3.36 " " 0 0 2 ~ where B ; x is a Brownian motion with covariance x; y de ned by Z 1 2 ^ R s; x; y ds; 3.37 x; y = 2 0 ^ a x a s ;y a y g: 3.38 R js s j;x;y = E f a s ;x 2 1 2 1 ~ ~ The Brownian motion B ; x is related to W ; x , the driving Brownian in the limit pro cess for 2 ; x in 3.29 , by ZZ Z Z Z ~ ~ s; x x; y x; y s; y dxdy ds: h s; ;dB s; i = h s; ;dW s; i E 0 0 0 " Hence Y converges weakly to a pro cess fY ; x g satisfying the linear sto chastic PDE 1 1 1 2 ~ ~ dY = x Y + rx + a x Y d + dW + dB : 3.39 xx x xx x 2 2 2 As we did with the sto chastic price, wecharacterise the con dence bands for the hedging strategy taking into account the randomness of the volatility by the variance of the limiting uctuation 2 pro cess, ; x :=E fY ; x g: In general, the covariance function V ; x; y :=E fY ; x Y ; y g will satisfy the following Lyapunov PDE, 1 1 1 1 2 2 V = x V + y V +[rx + a x ]V +[ry + a y ]V + M ; x; y ; xx yy x y 2 2 2 2 where h 1 2 M ; x; y := ; x x; y ; y xx xx 4 i 2 + x; y ; x x; y x; y ; y + ; x ; y ; xx x x x with V 0;x;y 0. In the `traditional' separable case, we nd, using the Green's function for the classical Black- Scholes hedging PDE see App endix C , that ! ! 2 q a+1 2 ^ K x [log x=K ] 2 q ; x = log + 2r +3q exp ; 3.40 2 x K 2q 2 2 6 q where a and are as de ned after 3.34 . Typical risk bands q ; x 2 " ; x for the two-state Markovchain example are shown in Figure 4. 2 x ; a x ; x; y ; x; y of the pro cess The results of this section dep end only on the statistics ; x in the most general case, and so are not sensitive to sp eci c mo delling of the volatility. This will b e imp ortant for estimation. 17 0.65 0.6 0.55 0.5 Hedging Confidence Interval 0.45 0.4 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Time Figure 4: Hedging bands for the example describ ed in Figure 3. 3.2.3 Numerical Simulations We simulate a sto ck price tra jectory that is conditionally lognormal, driven by a rapidly uctuat- ing two-state Markov chain volatility pro cess, and compare two hedging strategies, one with the averaged volatility, corresp onding approximately to the traditional approach through the ergo dic theorem 3.25 , and one at the edge of the hedging band describ ed ab ove which oughttoprovide similar protection against catastrophic losses, but for a lower cost. Thus the asset price follows t " " " dX = X dt + q X dW ; t " and our hedging strategies against the risk of having sold a call option are q BS @C SV 2 = q ; 3.41 @x q BS p @C " 2 = q 2 ": 3.42 @x The second stategy costs less since it involves buying less of the underlying, but it should pro- vide the same protection. The writer using the pure Black-Scholes strategy sells the option for q q BS BS 2 2 C 0;X 0 ; q , and the writer employing the edge-of-band hedge sells it for C 0;X 0 ; q + p 2 " T; X 0 . In Figure 5 we show the sto ck, volatility and hedging pro cesses along a typical realisation, and in Figure 6 we show histograms of 1000 runs implementing the strategies over the length of a twelve 18 month contract T = 1 with 200 equally spaced re-hedgings and the asset price b eginning at 90 of the strike price. Clearly greater pro ts are obtained with the second strategy. Typical Realization 100 90 80 Stock Price 70 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.4 0.2 Volatility 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 0.5 Hedges 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Time Figure 5: Sto ck price, Volatility and hedging strategies when " =0:0005;K = 100;T =1. In the SV " last, the solid curve is , the dotted curve is up to rst-order, and " is exaggerated in the plot to show the di erence. 3.3 Connection with Smile Curve We detail here the connection b etween rapidly uctuating volatility pro cesses, traditional sto chastic volatility mo dels and the smile curve. In particular, this will show that the sto chastic option price " C for the sp ecial separable volatility case gives the smile on average, and that we can obtain expressions for the geometry of the implied volatility curveby asymptotic metho ds combining the averaging techniques of section 3.2.1 and the calculations of section 2.3 . 2 Thus we lo ok at ; x = xq where q is a stationary ergo dic sto chastic pro cess. The " sto ck price X satis es T t " " " ^ X t dW t : dX t =X t dt + q " SV " Then recall that the `traditional' sto chastic volatility price C X ;q ; of a Europ ean call option is de ned to b e q " SV BS C = E fC V g; t " where we assume the market price of volatility 0, and V is de ned in 3.25 . Wenow use the following well-known result that is the averaging and central limit theorem for nite-dimensional SV pro cesses to construct an asymptotic expansion for C see, for example, [27 ] . 19 SIMPLE HEDGE 80 60 40 FREQUENCY 20 0 −8 −6 −4 −2 0 2 4 6 8 10 Profit/Loss EDGE OF BAND HEDGE 100 80 60 40 FREQUENCY 20 0 −8 −6 −4 −2 0 2 4 6 8 10 Profit/Loss Figure 6: Pro t-loss statistics after 1000 runs of hedging against having sold a call option struck at K = 100 with T =1. " Theorem 2 Finite-dimensional averaging Under the conditions on q stated above, V con- 2 verges strongly to the average q in the sense that " 2 lim P sup jV q j > =0; "0 ^ 0 T ^ for any >0 and T < 1. In addition, the appropriately scaled error process in the approximation p " " " 2 2 V by q , also referred to as the uctuation and de nedbyR := V q = ",converges of 12 . weakly to a Gaussian random variable R with mean zero t;T To derive the asymptotic smile arising from the rapidly uctuating volatility pro cess, we write p " " 2 V =q + "R , and expand 1 0 q q q q BS BS 2 2 " C B C q q C p " E fR g t " 2 SV BS BS 2 2 C B q + E f R g: C = C q + "C q t 3 A @ 2 8 2 q 2 2 2 q q q p 3=2 BS " " 2 q + "I + "I + O " , we nd Equating this with C I where I = 1 2 E [R ] t t;T q I = =0; 3.43 1 2 2 q R T ^ 12 dB s , where fB g is a standard Brownian motion adapted to the Sp eci cally,we could write R = t;T t ltration F fromq ^ t . t 20 q BS 2 C q 1 1 2 q I = var [R ] E [R ] 2 t t;T t 3 t;T 2 2 8q BS 2 2 C q 8 q ! 2 ~ var [R ] 1 L t t;T 2 = q 1 ; 3.44 3=2 2 4 q 2 8 q " where we have replaced Z by its weak Gaussian mean zero limit in the exp ectations and variance. As in section 2.3 , these expressions give the asymptotic geometry of the smile curve arising from a rapidly uctuating volatility pro cess. The connection with the sto chastic volatility mo del for derivative prices intro duced in section 3.1 is clear. When the pro cess ; x is separable as q x, the sto chastic option price is q " " BS C ; x =C V ; as can be shown by the usual change of time variable in the sto chastic PDE 3.24 . Thus SV " C x; q ; = E fC ; x jq T = q g and so, the averaged sto chastic option price gives the t asymptotic smile curve. 3.4 Estimation issues Traditionally, the approach to the estimation asp ect of implementing sto chastic volatility mo dels has b een to write down a sp eci c mo del of the volatility pro cess as in the table at the end of section 1.3 , and then use historical sto ck price data to identify the relevant parameters using the metho d of moments. Furthermore, it is necessary to assume a priori a sp eci c functional form for the volatility risk premium as explained in Wiggins [40 ], who derives his using a market equilibrium argument for investors with log utility functions. Scott [34 ] assumes is a constant and estimates it and to day's value of volatility as minimizers of the squared discrepancy b etween observed and mo del options prices. Renault & Touzi [28 ] suggest using implied volatilities to obtain the sto chastic volatility parameters from a maximum likeliho o d estimator. In this manner, we plan to use observed smile curves to estimate the parameters needed to calculate the pricing and hedging bands from our rapidly uctuating volatility theory. Since traditional mo dels coupled with estimation themselves lead to con dence intervals for predicted prices b ecause of the uncertainty in the parameter identi cation pro cess , our seeking of mo del- indep endent bands in the rst place might not b e to o great a weakening of the goals of sto chastic volatility theory in practice. Use of implied volatility or option price data from Europ ean options for market analysis purp oses has b een rep orted recently by Shimko [35 ], Dupire [12 ], Derman & Kani [10 ] and Rubinstein [31 ], who all sought a deterministic term-structure of volatility analogous to ; x in 3.23 and a corresp onding risk-neutral probability density for the asset price. The rst three exploit the Breeden-Litzenb erger formula relating the density function to the option price convexity in obs strike price, C , using interp olation or binomial trees. Rubinstein uses an optimization scheme KK to nd the probability density values at the no des of a binomial tree that are least-squares closest to lognormal risk-neutral probabilities, and allow exp ected sto ck and option prices to lie in the observed bid-ask spreads. His results indicate greater skew and kurtosis than lognormal distributions from data after the 1987 crash. However, the extraction of such detailed information ab out the market's view of volatility risk from to day's data has p o or predictive ability - tomorrow's estimate of the term-structure will likely be di erent from to day's. This problem of consistency over time is further discussed in Wilmott et al. [42 ]. Of course, since we take the view that volatility 21 is genuinely sto chastic rather than just as a function of the sto ck price t; X t , we do not seek the whole term-structure, but a few mo del-indep endent statistics. We outline a simple pro cedure for the separable volatility case for which we have explicit asymptotic formulas. Supp ose that at discrete times t we observe the sto ck price x and a set n n obs of implied volatilities I x ;t ; K ;T from options with expiration dates T and strike prices n n ij i i K , and that for each T , the interp olated p oints seem to form a shallow smile as a function of K . ij i 2 Then, for our theory,we need to estimate "; q and ^ for use in 3.34 or 3.40 . " We could view each observed smile curve as an approximation to the average smile curve I " BS " coming from E fC g = C I . By a suitable b est- t of the implied volatility data to curves de- q " 13 2 2 scrib ed explicitly by I = q + "I and 3.44 ,we can obtain estimates for "; q and var [R ]. 2 t t ;T n n i It remains to relate var [R ] to ^ . This is trivial in the separable case by the representation t t ;T n n i R T ^ 2 dB s , and var [R ]= ^ = T t . Thus in fact we shall end up estimating R = t t ;T i n t;T n n i t T t 2 the pro duct " ^ , but it is exactly the square ro ot of this that is needed in the con dence band formulas. What wehave assumed is that the market through its trading of options comes to some `average' view of volatility risk, and that this is re ected in market prices. Thus we want to infer the market's quanti cation of this risk contained in the observed smile curves, and from this obtain a rst-pass approximation to our parameters. More precise information will come from sp ectral analysis of ltered sto ck price data. Our taking actual market prices to pro duce bands is consistent with the philosophy given by Wilmott [41 ] who writes: \If we drop the assumption that markets are always correct and that nancial model ling is at best an approximate science, then there is no such thing as risk-free hedging . [An] alternative approach assumes uncertain volatility: if volatility is so dicult to measure, why not accept this situation and instead work on volatility ranges? A consequence is that there is no such thing as a single fair value for an option, al l prices within a given range are possible." 3.5 Numerical Simulation of Smile Fitting 2 2 To demonstrate how smile data can b e used to infer q and " ^ , we ran the following exp eriment using generated sto ck and options prices. We assumed that prices for options with 10 evenly distributed strike prices between $90 and $110 all with the same expiration date T = 1 were observed at times 0 t t =0:025 n n f " 1. Simulate a realisation of the sto ck price X t at discrete times t , n = 0; ;N, using a n realisation of q t=" , a rapidly uctuating two-state Markovchain. Wechose " =0:0005 and avolatility q taking values 0:1or0:4 with p ower sp ectral density ^ =1. obs 2. At each time-step n, generate C corresp onding to strike price K i =1; ;I by running i n;i P " " 2 " V = [q t ] and then averaging 5000 realizations of fq t ;t t T g, calculating n k k k k p " BS obs C V ; K over the 5000 paths. The C pro duced this way displaya shallow smile as i p " 5000 required for the exp eriment, b ecause they are an approximation to E fC g to O 1= 2 10 . obs 3. Calculate the observed implied volatility curves I x ;t ; K ;T by numerically inverting n n i the Black-Scholes formula. obs " 4. Set-up I N +1 least-squares equations to t I to the formula 3.44 for I . In our case, I = 10 and N =5. This is e ectively an interp olation of the smile data simultaneously in K t space to functions that are quadratic in log K and linear in t. Such basis functions 13 ~ The formula tells us we should t each strike time's data to a quadratic curveinL . 22 arise naturally from the theory and suggest an improvement to the aribitrary quadratic-in-K basis used by Shimko [35 ] to nd the term-structure of volatility. 5. The rst-order equations for the least-squares problem give a 2 2 nonlinear system of q 2 2 equations for q and " ^ that is easily solved by Newton-Raphson since the Jacobian can b e calculated explicitly. We ran this simulation and inversion algorithm 100 times so that each time, the realised asset q 2 q within 3 of price path is di erent . Most of the time 94 , we obtained a go o d estimate of 14 2 the theoretical value of 0:2915 and an estimate of " ^ that was of the right order of magnitude 4 2 10 , but inaccurate by up to 26. On other runs, the estimate of " ^ was negative 4 , or the algorithm did not converge 2 . These preliminary ndings from a simple tting pro cedure indicate that a rst-pass extraction of the necessary parameters can be obtained from smile data alone, b efore even lo oking at historical sto ck price data for further accuracy. In particular, the 2 p otential problem of " ^ b eing estimated as negative can b e xed by constraining it to b e p ositive. We defer further improvements of the algorithm until we use real market data, which will b e tackled in future work. Furthermore, empirical work [8] suggests that implied volatility from options prices is not a good forecaster of future realised volatility, and that historical volatility estimates from sto ck price data are sup erior in this sense. Thus it is crucial to employ a detailed analysis of past 2 2 sto ck prices to extract a go o d estimate of " ^ and q . 4 Conclusions Wehave studied the pricing and hedging problem for options and by extension, other Europ ean- style derivative securities on sto cks whose price volatility is sto chastic, motivated by the success of such mo dels in accounting for the frequently observed smile curve. New asymptotic formulas, valid when volatility has a small sto chastic comp onent or uctuates on a distinct time scale give detailed description of the geometry of implied volatility curves and are e ective for interp olation of observed implied volatilities, or as a to ol for parameter identi cation. The usefulness of separating the time scales on which the separate price and volatility pro cesses uctuate enables us to present a new approach to sto chastic volatility mo delling starting with the Black-Scholes option pricing PDE with a random volatility co ecient. This leads to an approxi- mation that is the Black-Scholes price plus a Gaussian random variable quantifying the risk from the uncertain volatility. In particular, wecharacterise the risk by pricing and hedging bands which satisfy a PDE and dep end only on the average and p ower sp ectral density of the volatility pro cess. This robustness to di erent mo dels of the volatility is in the spirit of the global smile result. In a sp ecial separable case, we have explicit formulas for the bands that can be combined with the Black-Scholes formula. The explicit representation of the average asymptotic smile curve provides a simple metho d for obtaining a rst-pass approximation to the statistics of the volatility pro cess that are needed. We plan to address more accurate estimation pro cedures using sto ck price data in future work, as well as extension of this approach to American securities for which there is little sto chastic volatility literature at present. In broad terms, wehave presented a set of to ols that can b e used to quantify risk in an incomplete markets setting where the mo delling identi es distinct time-scales of uctuation b etween sto chastic state variables. The metho dology, which is easily extendable to derivative securities on n > 2 14 obs This is the theoretical value if C were computed byaveraging over all volatility paths, whereas we generate our data from an average over only 5000 paths 23 underlying assets and factors such as volatility , is an alternative to the general pricing PDE of Garman [15 ] which needs estimation of an unseen market-price-of-risk-typ e function for each untradeable state factor, and whose solution will b e sp eci c to mo dels of these factors. The separation-of-scales technique values the contribution to the derivative price from the hedge- able factors classically no arbitrage and quanti es the risk from the untradeable factors as pricing bands. Thus there is no longer one numb er known as the price, but there is no dep endence on unob- servable market risk premium factors or sp eci c mo dels of the non-traded factors. In addition, this approachevaluates the risk for the investor of hedging with just the tradeable underlyings, which is preferable to hedging with other derivatives in view of transaction costs. The Garman approach would give a theoretically p erfect hedge, but with hedging strategies involving other derivatives. The new metho dology presents a wide varietyofchallenging mo delling and estimation problems for future work on pricing and hedging in incomplete markets. A Induction step Supp ose V is an n-state random variable taking values < < < with probabilities 1 2 n n ; ; resp ectively. Analogous to the case n =2,we are interested in sign f where 1 n n X BS BS n n C C f := I ;K ; i i K i=1 n n is the vector of probabilities and I ;K is de ned by n X BS BS n C : C I ;K = i i i=1 Di erentiating these with resp ect to for j =1; ;n 1 and using =1 + + , j n 1 n 1 we nd n BS BS BS I ;K = C C < 0; I C j n j and ~ LI BS BS k f =2[C C ] ; j n j k 3 I n ~ so that I < 0 and sign f = sign L for j = 1; ;n 1. Finally, we write f = j j k n 1 n 1 f ; with denoting the vector of the rst n 1 probabilities, and supp ose for n n 1 ~ induction that sign f ; 0 = sign L on the hyp erplane + + = 1. This is the 1 n 1 n 1 result. This implies that f 1; 0 = 0, and we also know that f 0; 1 =0. Since f can only ~ achieve one typ e of extremum given the sign of L, it follows that f is of one sign in the region 0 < + + < 1, provided = 1 + + , which is the desired result for 1 n 1 n 1 n 1 n-states. B Pro ofs of sto chastic averaging and CLT Averaging The averaging problem for deterministic parab olic PDEs with rapidly varying co ecients is consid- ered in Bensoussan et al.[5 ], and Watanab e [39 ] presents results for the case when the co ecients are sto chastic, under assumptions of stationarity and strong regularity of co ecients. Similar results are given by Dawson and Kouritzin [9 ] under less stringent regularity. The co ecients also need 24 not be stationary, but must satisfy a weak law of large numb ers and a weak invariance principle p ointwise in x and t. The basic metho d of the pro ofs, in which an `extra' term of the asymptotic series representation of the series is taken and the error shown to b e small, is due to Khas'minskii [21 ]. " We rst use a multiple scales analysis to motivate the approximation of C ; x in 3.24 by p 2 C ; x; x . Considering the fast time scale as an indep endentvariable s := =",we lo ok for a solution of the form " 0 1 C ; x =C ; s; x +"C ; s; x + : Substituting into 3.24 with the formal replacement @ 1 @ @ 7 ! + ; @ @ " @s 0 0 0 and comparing p owers of " gives C =0, from the O 1=" term, so that C = C ; x . The s O 1 terms then give Z h i s 1 0 2 0 0 1 0 0 0 C s ;x ds : B.45 C = s C r xC C + xx x 2 0 2 Now as " 0, s ! 1, and the integral in the second term b ehaves like s x by the ergo dic 1 theorem. Hence this expression for C is non-asymptotic on average unless the right-hand side has mean zero: 1 0 0 0 0 2 x C + r xC C =0: B.46 C + xx x 2 p 0 2 Thus C ; x =C ; x; x . Wenow make this approximation concrete by showing the error is small. We shall need the following result and then explain how it can b e used in the problem at hand. " n Theorem 3 In nite-dimensional averaging Consider the PDE for u ; y :[0;T] R !R, n n 2 " " " X X @ u @u @u " " ; y ; y = a + b ; B.47 ij j @ @y @y @y i j j i;j =1 j =1 " u 0; = h y ; " " where the coecients a = a ;y , b = b ;y are random on a probability space ; F ; P , ij j ij j " " ">0 and T < 1. Assume the fol lowing conditions hold n 1. a ; y; ! , b ; y; ! are measurable functions of ; y; ! 2 [0;T] R . ij j 2. The coecients fa g are uniformly el liptic: ij n n n X X X 2 2 c a ; y; ! c ; 1 ij i j 2 i i i=1 i;j =1 i=1 for some 0 1 2 3. The fa ; g and fb ; g are stationary ergodic processes with at least four uniformly ij j bounded and continuous spatial derivatives: max D D a ; b c < 1; ij j 3 y y 25 where j j @ D := ; j j = + ; 1 n y n 1 @y @y n 1 for j j4, with probability 1. The norm is de ned by kf k := sup sup f ; y : B.48 n 0 T y 2R 4. The initial data has at least four bounded and continuous derivatives: sup jD h y jc for 4 y j j4. Then " uk > ! 0; P ku as " 0 for each >0, where u ; y satis es n n 2 X X u @ u @ u @ = a y + b y ; B.49 ij j @ @y @y @y i j j i;j =1 j =1 u 0;y = h y ; and a y =E fa ; y g, b y =E fb ; y g. ij ij j j Proof We present the calculation for n = 1 for simplicity of notation; the higher dimensional case go es through analogously. Let Z s 0 0 s; y = [a s ;y a y ]ds ; 1 0 Z s 0 0 s; y = [b s ;y b y ]ds ; 2 0 " and de ne H ; y by " " u ; y = ;y ;y u ; y +" u + " u + H ; y : B.50 1 yy 2 y " " " Then from B.47 and B.49 , H satis es " " " " " " H = a ; y H + b ; y H + R ; y ; B.51 yy y with zero initial data where ! 2 @ @ " " " u + u : + b ;y ;y R = " a yy 2 y 1 2 @y @y " " We have used the assumed smo othness of the co ecients and initial data which guarantees four " derivatives in space on the solution u ; y of B.49 , so that R is well-de ned. Now, by the b oundedness of the co ecients and the ergo dic theorem: " D ! 0; w.p.1 as " 0; ;y 1;2 y " " for j j 2, we have that kR k ! 0 almost surely. Thus by the maximum principle for B.51 , " kH k! 0 almost surely, and the result follows from B.50 . Remarks 26 1. For our application, we shall assume that we can write ; x ~ ; x x with ~ ; having four continuous b ounded derivatives almost surely. A logarithmic transformation y = log x reduces 3.24 to the case ab ove. 2. The payo function must b e four times continuously di erentiable. For call and put options, this means that some slight arti cial smo othing will b e required at the kink at x = K , but this can b e designed to have relatively small consequences for the pricing and hedging bands some time away from expiration >> 0 . CLT for Fluctuation The statistics of the uctuation of the sto chastic option price and hedging ratio ab out the Black- Scholes average are deduced from the following result, similar versions of which can be found in [9, 39 ]. Theorem 4 In nite-dimensional uctuation theory We assume that the coecients satisfy a weak invariance principle Z Z D E 1 " p s; g s; ;a a ds hg s; ;dW s; i ! 0 as " 0; ij ij ij " 0 0 with convergence in the sense of distributions on the probability space. Here the Brownian motions fW s; g have covariance functions ij Z i h 1 2 a E f[a s; x x; y := 2 a x ][a 0;y a y ]gds; ij ij ij ij ij 0 n n with x; y 2 R , g s; 2 S R , the space of Schwarz distributions, and h; i denotes integration n b x; y and fW s; g. Then over R . Similar is assumed to hold for the b 's with corresponding j j j the uctuation p " " u ; y = " S ; y := u ; y converges weakly to a process S ; y satisfying 3 2 n n n 2 2 X X X @ S @S u @ 5 4 a y + b y dW ; y d + dS = ij j ij @y @y @y @y @y i j j i j j =1 i;j =1 i;j =1 n X @ u + dW ; y : B.52 j @y j j =1 " Proof Again wework with n = 1 for simplicity. By B.47 and B.49 , S ; y satis es ! " " a b b a " " " " " p p u + u : S = a S + b S + yy y yy y " " " ~ Let S ; y satisfy ! " " a a b b " " " ~ ~ ~ p p + + = a y S b y S u + u : S yy y yy y " " " Then we can write S ; y =I ; y +I ; y where 1 2 Z Z " 1 a s; a p G I = y; ; ; s u s; d ds; 1 yy a;b " 0 1 27 and similarly I . Here G is the Green's function asso ciated with the averaged co ecients. Under 2 a;b the prevailing assumptions, the invariance principles of the co ecients apply and we see that " ~ S ; y converges weakly to S ; y de ned by B.52 . Finally,we can write the equation satis ed " " ~ by S S and conclude " " ~ P kS S k > ! 0as " 0, by the maximum principle. C Classical Black-Scholes with sto chastic volatility co ecient Here, we brie y give details of the sp ecial case at the end of section 3.2.1 where the covariance ; x is calculated for ; x =xq . The Green's function for the classical Black-Scholes PDE with constantvolatility is given by 0 ! 2 1 [log x= ] x p exp s ; C.53 G x; ; ; s = 2 2 s 2 s 0 0 where 3 r = ; 2 2 0 2 r r 1 2 = + + : 0 2 2 8 2 0 Thus the price of a put option P ; x , for example, can b e written Z 1 + P ; x = G x; ; ; 0 K d : 0 Then the solution Z ; x of 3.32 is Z D E 1 2 ~ G x; ; ; s C s; ;dW s; : Z ; x = xx 2 0 Finally, ; x is calculated from the expression Z Z 2 1 1 2 2 G x; ; ; s ds; ^ C s; d ; x = xx 4 0 0 2 2 2 2 b ecause the covariance function is x; y = ^ x y , which gives 3.34 . The Green's function for the classical Black-Scholes hedging PDE is identical to C.53 with replaced by 1. This is used in the calculation in section 3.2.2 . References [1] M. Avellaneda, A. Levy, and A. Paras. Pricing and hedging derivative securities in markets with uncertain volatilities. Applied Mathematical Finance, 1 2 :73{88, 1995. [2] M. Avellaneda and A. Paras. Managing the volatility risk of p ortfolios of derivative securities: the Langrangian uncertain volatility mo del. Applied Mathematical Finance, 1 3 :21{52, 1996. [3] C. Ball and A. Roma. Sto chastic Volatility Option Pricing. J. Financial and Quantitative Analysis, 29 4 :589{607, Decemb er 1994. 28 [4] S. Beckers. Standard Deviations Implied in Option Pro cess as Predictors of Future Sto ck Price Vari- ability. J. Banking and Finance, 5:363{382, 1981. [5] A. Bensoussan, J.L. Lions, and G. Papanicolaou. Asymptotic Analysis for Periodic Structures. North- Holland, 1978. [6] F. Black. Living up to the mo del. RISK, March 1990. [7] F. Black and M. Scholes. The Pricing of Options and Corp orate Liabilities. J. Political Econ., 81:637{ 659, 1973. [8] L. Canina and S. Figlewski. The informational content of implied volatilty. Review of Financial Studies, 6 3 :659{681, 1993. [9] D.A. Dawson and M.A. Kouritzin. Invariance principles for parab olic equations with random co ecients. Technical Rep ort 1433, IMA Preprint Series, Septemb er 1996. [10] E. Derman and I. Kani. Riding on a Smile. RISK, 7 :32{39, 1994. [11] D. Due. Dynamic Asset Pricing Theory, 2nd ed. Princeton University Press, 1996. [12] B. Dupire. Pricing with a smile. RISK, 7 :18{20, January 1994. [13] H. Fo ellmer and D. Sondermann. Hedging of non-redundant contingent claims. In W. Hildenbrand and A. Mas-Colel l eds. , Contributions to Mathematical Economics, pages 205{223, 1986. [14] E. Fournie, J. Lasry, and P.L. Lions. Some nonlinear metho ds to study far-from-the-money contingent claims. Numerical Methods in Finance, Cambridge University Press, 1996. [15] M. Garman. A general theory of asset valuation under di usion state pro cesses. Technical Rep ort 50, Center for Research in Management Science, Berkeley, July 1976. [16] J.M. Harrison and D.M. Kreps. Martingales and arbitrage in multip erio d securities markets. J. Eco- nomic Theory, 2 3 :381{408, June 1979. [17] S. Heston. A closed-form solution for options with Sto chastic Volatility with applications to b ond and currency options. Review of Financial Studies, 6 2 :327{343, 1993. [18] N. Hofmann, E. Platen, and M. Schweizer. Option pricing under incompleteness and sto chastic volatility. Math. Finance, 2 3 :153{187, 1992. [19] J. Hull and A. White. The Pricing of Options on Assets with Sto chastic Volatilities. J. Finance, XLI I 2 :281{300, June 1987. [20] N. El Karoui and M. Quenez. Dynamic programming and pricing of contingent claims in an incomplete market. SIAM J. Control and Optimization, 33 :29{66, 1995. [21] R.Z. Khas'minskii. On sto chastic pro cesses de ned by di erential equations with a small parameter. Theor. Probability Appl., 11 2 :211{228, 1966. [22] H. Latane and R. Rendleman. Standard deviations of sto ck price ratios implied in option prices. J. Finance, 31:369{381, 1976. [23] J. MacBeth and L. Merville. An empirical examination of the Black-Scholes call option pricing mo del. J. Finance, 34:1173{86, 1979. [24] R.C. Merton. Theory of rational option pricing. Bel l Journal of Economics, 4 1 :141{183, Spring 1973. [25] R.C. Merton. Option pricing when underlying sto ck returns are discontinuous. J. Financial Economics, 3 1/2 :125{144, Jan/March 1976. [26] D.B. Nelson. Arch mo dels as di usion approximations. J. Econometrics, 45 1-2 :7{38, July/Aug 1990. [27] G. C. Papanicolaou. Asymptotic analysis of sto chastic equations. In Murray Rosenblatt ed. , Studies in Probability Theory, Mathematical Association of America, 18:111{179, 1978. 29 [28] E. Renault and N. Touzi. Option Hedging and Implied Volatilities in a Sto chastic Volatility Mo del. Mathematical Finance, 6 3 :279{302, July 1996. [29] M. Romano and N.Touzi. Contingent claims and market completeness in a sto chastic volatility mo del. To appear in Math. Finance. [30] M. Rubinstein. Nonparametric Tests of Alternative Option Pricing Mo dels. J. Finance, XL 2 :455{480, June 1985. [31] M. Rubinstein. Implied binomial trees. J. Finance, LXIX 3 :771{818, July 1994. [32] M. Schweizer. Option hedging. Stoch. Process Appl., 37:339{363, 1991. [33] L. Scott. Option Pricing when the Variance changes randomly: Theory, Estimation, and an Application. J. Financial and Quantitative Analysis, 22 4 :419{438, Decemb er 1987. [34] L. Scott. Random-Variance Option Pricing: Empirical tests of the mo del and delta-sigma hedging. Advances in Futures and Options Research, 5:113{135, 1988. [35] D. Shimko. Bounds of probability. RISK, 6 4 :33{37, April 1993. [36] K.R. Sircar and G.C. Papanicolaou. General Black-Scholes mo dels accounting for increased market volatility from hedging strategies. Submitted to Applied Mathematical Finance, June 1996. [37] E. Stein and J. Stein. Sto ck Price Distributions with Sto chastic Volatility: An Analytic Approach. Review of Financial Studies, 4 4 :727{752, 1991. [38] S.J. Taylor. Mo deling sto chastic volatility: A review and comparitive study. Mathematical Finance, 4 2 :183{204, April 1994. [39] H. Watanab e. Averaging and uctuations for parab olic equations with rapidly oscillating random co ecients. Prob. Th. Rel. Fields, 77:358{378, 1988. [40] J. Wiggins. Option Values under Sto chastic Volatility. J. Financial Economics, 19 2 :351{372, 1987. [41] P. Wilmott. Volatility smiles revisited. Derivatives Week, 4 38 :8, Sept. 1995. [42] P. Wilmott, J. Dewynne, and S. Howison. Option Pricing, 2nd Ed. Oxford Financial Press, forthcoming, 1997. 30