A Risk-Neutral Stochastic Volatility ModelYingzi Zhu1 and Marco Avellaneda2We construct a risk-neutral stochastic volatility model using no-arbitrage pric-ing principles. We then study the behavior of the implied volatility of optionsthat are deep in and out of the money according to this model. The motiva-tion of this study is to show the di�erence in the asymptotic behavior of thedistribution tails between the usual Black-Scholes log-normal distribution andthe risk-neutral stochastic volatility distribution.In the second part of the paper, we further explore this risk-neutral stochas-tic volatility model by a Monte-Carlo study on the implied volatility curve (im-plied volatility as a function of the option strikes) for near-the-money options.We study the behavior of this \smile" curve under di�erent choices of param-eter for the model, and determine how the shape and skewness of the \smile"curve is a�ected by the volatility of volatility (\V-vol") and the correlationbetween the underlying asset and its volatility.1Postal address: Citibank, N.A., 909 Third Avenue, 29th Floor, Zone 1, New York, NY100432Postal address: Courant Institute of Mathematical Sciences, 251 Mercer Street, NewYork, NY 10012

1 IntroductionThe Black-Scholes (BS) formula is widely used by traders because it is easy touse and understand. An important characteristic of the model is the assumptionthat the volatility of the underlying security is constant. However, practitionershave observed, especially after the crash of 1987, the so called volatility \smile"e�ect. Namely, options written on the same underlying asset usually trade,in Black-Scholes term, with di�erent implied volatilities3. Deep-in-the-moneyand deep-out-of-the-money options are traded at higher implied volatility thanat-the-money options. There is also a time e�ect. For example, in ForeignExchange market, options with longer maturities are traded at higher impliedvolatility than shorter maturities. This evidence is not consistent with theconstant volatility assumption made in Black-Scholes (Black & Scholes, 1973).This is due to the presence of \fat tails": extreme values for the priceare more likely in the real probability measure than in the lognormal model.There are several ways to address this empirical issue. Merton (R. Merton,1990) points out that a jump-di�usion process for the underlying asset couldcause such an e�ect. A more explored direction is the stochastic volatilityassumption. Hull and White (Hull and White, 1987) proposed a log-normalstochastic volatility model, namely, the volatility of the underlying asset followsanother Geometric Brownian Motion. However, these models have a drawback:since the volatility is not a traded asset, the option price in the stochasticvolatility context actually depends on investors risk preferences, that is, thepricing formula is not risk-neutral.In this paper, we study a risk-neutral pricing model in the context of log-normally distributed stochastic volatility. In order to �nd a risk-neutral prob-ability measure suitable for pricing options and OTC derivatives, we have toanalyze hedging strategies involving a traded asset which is perfectly correlatedwith volatility of the underlying security. For this purpose, we propose to useshort term options on the underlying asset to hedge the volatility risk. The keyassumption made here is that the maturities of these options are short enoughthat the options are reasonably marked-to-market (priced) by Black-Scholesformula.This paper is organized as follows. In Section 2 we derive a risk-neutralstochastic process for the underlying asset and its volatility. For this, we assumein particular a general correlation (negative, zero, positive) between the assetprice and its volatility. Section 3 is devoted to the asymptotic estimation of the3Implied volatility is the volatility value at which the option is traded if the Black-Scholesformula is used. Given an option price, there is a corresponding volatility value by invertingthe Black-Scholes formula. 1

implied volatility of the derived model, i.e., how the implied volatility behavesfor options that are deep-out-of-the-money. In Section 4 we study the impliedvolatility behavior of near-money options using Monte Carlo simulation. Oneof the interesting features of the model is the di�erent behavior among positive,zero, and negative correlations. Calls with positive correlations correspond toputs with negative correlations, and vice versa. The analysis therefore suggestsan asymmetry between puts and calls.4 The three appendices in the last sectioncontain mathematical details for the asymptotic analysis in section 3.2 The Risk-Neutral MeasureIn this section, the risk-neutral probability measure of the log-normal stochasticvolatility model is derived. Speci�cally, we consider an underlying security St,and its volatility �t, which follow the stochastic processes:dSt = �Stdt+ �tStdZtd�t = �tdt+ V �tdWtwhere Zt and Wt are two standard Brownian motions with correlation coe�-cient �. Formally, E(dZtdWt) = �dt. Let f be the price of a derivative securitycontingent on the price S of an underlying asset. Speci�cally,f = f(S; �; t)By Ito's lemma, the price process of the derivative security satis�es,df(S; �; t) = fSdS + f�d� + Lfdtwhere L is the in�nitesimal generator:L � @t + 12�2S2 @2@S2 + 12V 2�2 @2@�2 + ��2SV @2@S@� (1)In the same spirit of the original derivation of the Black-Scholes formula, wegive a hedging strategy using the underlying asset and a short-term call optionon it, along with a money market account. The riskless portfolio will includethe contingent claim with price f , underlying asset with price S and a shortterm call5 on asset S with price C(S;K; �;�t), where �t is the maturity of the4For example, for the options on Standard & Poor's 500 Index of Chicago Mercantile Ex-change, deep out-of-money puts trade at approximately � = 30%, and at-the-money optionstrade at 17%.5The strike of this short term option is to be determined.2

short-term call option.6 We short7 1 unit of derivative security with price f , golong � units of the underlying asset with price S, and � units of the short termcall option on the asset S with price C. The major approximation we make inorder to derive a risk-neutral probability measure is to assume the short termcall price C(S;K; �;�t) to be the Black-Scholes price. This assumption is theessence of our model. If one doesn't make such an identi�cation, one can onlyachieve, in the general framework, a no-arbitrage pricing relationship betweenthe short-term call and the general derivative security f .8In addition to the underlying security and the short-term call, we considera money market account with riskless interest rate r For an in�nitesimal timeinterval dt, the value change of the portfolio is given as:df ��dS � �dC = (fSdS + f�d� + Lfdt) ��dS � �(CSdS + C�d� + LCdt)= (fS ��� �CS)dS + (f� � �C�)d� + (Lf � �LC)dtwhere L is the in�nitesimal generator de�ned in (1).A riskless portfolio is obtained by setting :�+ �CS = fS�C� = f�speci�cally, � = f�C�� = fS � CSC� f�Therefore, the value change of the riskless portfolio is:df ��dS � �dC = (Lf � �LC)dtAccording to no-arbitrage pricing principle, the return of the riskless portfoliomust be identi�ed with the riskless interest rate, i.e.,(Lf � �LC)dt = r(f ��S � �C)dt= r(f � fSS + �CSS � �C)dt (2)6�t is small compared with the maturity of the contingent claim in consideration, whilelarge compared with hedging period dt.7In �nancial terminology, short means sell, and long means buy.8A similar situation occurs in interest rate models. For instance, the Vascicek model(Vascicek, 1979) is a no-arbitrage model of interest rate derivatives. This is not a risk-neutral model because there is a non-determined parameter, the market price of risk, whichdepends on investor's risk preference. This risk premium, however, doesn't depend on theparticular choice of a derivative security. Therefore, it serves as a price relation betweendi�erent derivative securities. 3

where in the second equation, we substitute � = fS � �CS into the formula.Notice thatLC = (@t + 12�2S2 @2@S2 )C + ��2SV CS� + 12V 2�2C��= rSCS � rC + ��2SV CS� + 12V 2�2C�� (3)where for (3) we use the Black-Scholes PDE for function C.Substitute (3) into (2), we obtainLf � ��2V SCS�C� f� � 12�2V 2C��C� f� = rf � rfSS (4)Now use Black-Scholes formula for C,C(S;K; �;�t) = SN(d1)�Ke�r�tN(d2)where d1;2 = ln Ser�tK�p�t � 12�p�twe have the following Greeks:C� = SN 0(d1)p�tC�� = SN 00(d1)p�t(� ln Ser�tK�2p�t + 12p�t)CS = N(d1)CS� = N 0(d1)(� ln Ser�tK�2p�t + 12p�t)Now, in order to have a manageable drift term for �t, we set the strike of theshort term call to be ATMF (at the money forward), i.e.,K = S exp(r�t):Therefore we haveC� = SN 0(d1)p�tC�� = SN 00(d1)�t2 = 1p2�S exp(�d212 ) � 14(�t) 32CS� = N 0(d1) � 12p�t 4

Substitute the above formula into (4), and neglect the higher order term of �t,we have Lf � 12��2V f� = rf � rSfSL is de�ned as in (1). In terms of SDEs, the corresponding risk-neutral processcan be written as, dSS = rdt+ �tdZtd�t = �12�V �2t dt+ V �tdWt (5)This is the risk-neutral probability for the stochastic volatility model. Wesee that in the risk-neutral world, the drift term of the underlying securityis the short-term interest rate, while the drift of the stochastic volatility alsobecomes independent of that in \real" probability measure.Notice that the behavior of the volatility process is di�erent according tothe sign of the correlation coe�cient. In the case of positive correlation, thelocal volatility tends to zero when time goes to in�nity. In the case of negativecorrelation, the local volatility blows up in �nite time.9 This means that whenthe correlation is positive, the hedging procedure works rather nicely; while thecorrelation is negative, there is a contradiction between buying the underlyingand buying the ATM call option as a hedge. In other words, in the case ofnegative correlation, when market goes down, volatility goes up, one needs tobuy the short-term ATM call to \delta" hedge the volatility . But the callprice (therefore its sensitivity to volatility ) drops as the market goes down,the result is there is not enough \volatility" to buy.10 11The rest of the paper is devoted to classifying the asymptotic behavior ofthe implied volatility curve as a function of strike in this risk-neutral volatilitymodel. One possible application of this analysis is to �nd a suitable functionspace in which one could �t the \smile" curve observed in the market.9The blow-up time is typically very large compared with the maturity of options. Theblow-up time can be approximated by � 2�0�V . For a typical volatility �0 = 10% per annum,correlation � = �0:5 and \V-vol" V = 100% per annum, the blow-up time is 40 years.10At �rst sight, the reader may think that the situation could be resolved by using shortterm puts to hedge the volatility risk. However, the ATM puts' prices also drop when themarket goes down. So it doesn't make di�erence using puts or calls, as long as they areat-the-money.11Negative correlation is present in general in equity market.5

3 AsymptoticsThis section discusses the asymptotic behavior of the implied volatility of out-of-the-money calls for very large strikes. The main tool used is Large Deviationtheory. We classify the behaviors for positive, zero, and negative correlations.The main result is in Theorem 7 at the end of this section. Lemma 1 through6 are steps towards the derivation. Tedious and technical mathematical detailsare presented in appendix.To �x the notation, let BS(S; �;K; T ) be the Black-Scholes formula foran European call option price, with spot price S, strike K, volatility �, andmaturity T. Without loss of generality, we assume r = 0:Lemma 1. In the limit where the strike is large compared with spot price ofthe underlying asset, the Black-Scholes option price satis�es asymptoticallylimK!1BS(S; �;K; T ) = 1p2�X2 exp (� (X � 12�2T )22�2T ); (6)where X � ln (KS ).Proof. The key point is the following inequality, which can be found in, e.g.,Mckean (1969): x1 + x2 e�x22 � Z 1x e�u22 du � 1xe�x22or equivalently, Z 1x e�u22 du � 1xe�x22 for x largeWe also have Z 1x 1ue�u22 du � 1x2e�x22 for x largeThese approximations can be proved by using change the order of integration.By change the order of integration, we getE[(S �K)+]= 1p2� Z 1X+12 �2Tp�2T ( exp(p�2Ty � 12�2T )� exp(X)) exp(�y22 )dy= 1p2� Z 1X+12 �2Tp�2T exp(�y22 ) dy Z yX+12 �2Tp�2T exp (p�2Tz � 12�2T )�p�2T dz= p�2Tp2� Z 1X+12�2Tp�2T exp (p�2Tz � 12�2T ) dz Z 1z exp(�y22 ) dy6

� p�2Tp2� Z 1X+12�2Tp�2T exp (p�2Tz � 12�2T )(1z ) exp (� z22 )dz� 1p2�X2 exp (� (X � 12�2T )22�2T ):2QEDComment. It can be easily veri�ed that the deep out-of-the-money puts alsosatis�es this asymptotic formula. Therefore, the discussions that follow arealso true for puts.Lemma 2. For the above derived risk-neutral stochastic volatility model, thecall option price is EfBS(�;K; (1 � �2) Z T0 �2sds)gwhere E is the expectation with respect to the stochastic process �t, and� � exp[ �V (�t � �0)]BS(S, K, �2T ) is de�ned as above.Proof. From the last section, the risk-neutral measure of the stochastic volatil-ity model (5) can be written asdSS = ��tdWt +q1� �2�tdZt (7)d�t = �12�V �2t dt+ V �tdWt (8)Zt and Wt are two independent Wiener processes. Substitute (8) into (7), weget dSS = �V d�t + 12�2�2t dt+q1� �2�tdZtFormally integrating this SDE, we getS = expf �V (�t � �0) + 12�2 Z t0 �2sds� �22V 2 < �t; �t >g� expfZ t0 q1 � �2�sdZs � 1� �22 Z t0 �2sdsgwhere < �t; �t > is the quadratic variation of �t, < �t; �t >= V 2 R t0 �2sds:Therefore,S = expf �V (�t � �0)g � expfZ t0 q1 � �2�sdZs � 1� �22 Z t0 �2sdsg7

Since Zt and �t are independent, one can think that, for each realization of�t, the distribution of the underlying asset price at maturity is equivalent to ageometrical Brownian Motion, starting from� � exp[ �V (�t � �0)];with total variance (1 � �2) R t0 �sds. The lemma is proved. 2 QEDUsing the asymptotic Black-Scholes formula (6), as derived in Lemma 1, wecan formally write down the asymptotic formula for the call price under therisk-neutral stochastic volatility measure, namely,1p2� Z Z 1X2 exp[�(X � 1��22 At � �V (�t � �0))22(1 � �2)At ]f(�t; At)d�tdAt (9)while X !1where At � R t0 �2sds. We suppose S0 = 1, X is de�ned as in Lemma 1. f(�t; At)is the joint probability density of �t and At.f(�t; At) = g(�tjAt)h(At)g(�tjAt) is the probability density of �t conditional on At.Next we are going to characterize the probability distribution g(�tjAt).Observe that �t = �12�V Z t0 �2sds + V Z t0 �sdZsLet �1 be the �rst time �s = 0. LetA1 = Z �10 �2sdsUse the random time change formula, we have�t = �12�V At + V B(At) (At < A1)= 0 (At � A1)where Bt is another Brownian motion. Therefore, the above formula is equiva-lent to saying that the distribution of �t conditional on At is equivalent in lawto the distribution of a drifting Brownian motion conditional on not hittingupon 0.Based on the above observation, we can derive an approximate formula forthe conditional distribution g(�tjAt).In what follows, we treat the cases of � > 0 and � < 0 di�erently.8

Lemma 3. When � > 0 and At � 1, the conditional distribution g(�tjAt)satis�es g(�tV 2 d�jAt) � �02p2�� exp (� (� + �0At)22At ) exp(12�02At)d�:where we let �0 = 12�.Proof. Let � � �tV = ��0At + BAt. The distribution of ��0t + Bt conditionalon not hitting 0 is(P (��0)(t; �; �0V )� P (��0)(t; �;��0V ))= �N(t) (Mckean, 1969)where P (b)(t;x; y) � 1p2�t exp (� (x� y)22t ) exp(bx� 12b2t); (10)�N(t) is the normalization factor given by�N(t) � 2�0�02A 32t exp (� �02At2 ): (see Appendix 1)When At � 1, we can make the following approximationg(�tV 2 d�jAt) = (P (��0)(At; �; �0V )� P (��0)(At; �;��0V ))= �N(At)� dP (��0)(At; �; 0)d� � 2�0VSubstituting the formula of P (b) given by (10) into the above approximation,we get the formula as stated in the Lemma.2 QED.Lemma 4. When � > 0, the call price satis�es�28� Z 1X2 �(At)h(At)dAt; X � 1 (11)where �(At) � (1� �2)At exp (� 12 (X � (12 � �2)At)2�2(1� �2)At ) exp(18�2At) (12)as At � X1��22 (1 + 1�)�(At) � (1 � �2)At exp (� 12 (X � 1��22 At)2(1� �2)=�2At ) (13)as At � X1��22 (1 + 1�)9

Proof. From the asymptotic call price (9) discussed after Lemma 2, under thecondition � > 0 and X !1, the call price isZ Z 1p2� 1X2 exp (� (X � 1��22 At � �V (�t � �0)22(1 � �2)At )g(�tjAt)h(At)d�tdAtSubstituting the expression (3) of the conditional distribution g(�tjAt) fromLemma 3, after some calculations (details are in Appendix 2), we prove theLemma.2 QED.The following two lemmas deal with the case � < 0. The analysis is mostlythe same as the case of positive �, so we only emphasize the part which isdi�erent between positive and negative �, omitting the similar parts.Lemma 5. When � < 0 and At � 1,g(�tV 2 d�jAt) ' �(At) exp (� (� + �2At)22At )where �(At) = 2�0�p2�AtV At= �N(t) (14)with the normalization factor being�N(t) = 2 sinh(��0�2V ):Proof. This result follows from the same calculation as in Lemma 3 but withdi�erent normalization factor �N(t), The reader should refer to Appendix 1 forthe calculation of the normalization factor �N(t): 2 QED.Lemma 6. When � < 0, the call price satis�es1p2� Z 1X2 �(At)h(At)dAt; X � 1 (15)where �(At) � (1 � �2)�(At) exp (� 12 (X � (12 � �2)At)2�2(1 � �2)At ) (16)�(At) is de�ned in (14).Proof. According to Lemmas 1,2,5, the call price isC Z Z 1X2 exp(�(X � 1��22 At � �(� � �0))22(1� �2)At )�(At) exp(�(� + �2At2At )2)d�dAt10

Use the formula in Appendix 2 to integrate d� part, we obtain the result.2QED.Now, we are in the position to prove the following result:Theorem 7 The implied volatility of deep out-of-the-money calls has the fol-lowing asymptotic properties:a)When � > 0; and X !1; �imppT ' p2X .b)When � < 0; and �2 � 12; X !1; �imppT ' p2X .c)When � < 0; and �2 > 12 ; X ! 1; �imp ' Const � pX , where Const =p2C + 2 �p2C with C = 4�2 (�2� 121� �2 ): Notice that Const < p2.Proof.a) It is shown in Appendix 3 that the tail distribution ofAt � Z t0 �2sdsis log-normal, i.e.,h(At) � exp(�C � (ln(At))2) as At !1From lemma 4 we know that the call price asymptotically satis�es�28� Z 1X2 �(At)h(At)dAtwhere � is de�ned in (12) and (13). Use \steepest descent" technique (Ap-pendix 2) to integrate the above call price. Notice that there is a total squarein the integrand � de�ned in (13), and when calculating the \saddle point", thecontribution from h(At) is small as At is large, (note the exponent of h(At) is�(ln(At))2.) Therefore, the saddle point is At = 2(1��2)X, and the resulted callprice is (to leading order of X)Call price � C2 exp(�C1 � (lnX)2)where C1; C2 are constant depending on T and �; V . Compare it with theBlack-Scholes asymptotic formula 6 (to leading order of X)exp(�(X � 12�2impT )22�2impT ) � exp(�C1 � (lnX)2)we get �imp � p2X:11

b) When � < 0 , we have E[At] = 1. This is because, from Appendix 3, wehave the formula for �t: �t = Mt1�0 + �2V R t0 MsdsWhen � < 0, and for �nite t, the probability thatZ t0 Msds � 2�0(��)Vis always positive, i.e., �t goes to in�nity with positive (but small) probability.Therefore, we have E[At] =1.When we use \steepest descent" technique to integrate (15) the factor h(At)doesn't contribute (refer to Appendix 2).Moreover, when �2 < 12 , At = X12��2 is the \saddle point", and the exponen-tial part of X disappears. When �2 = 12 , the exponential part disappears too.Therefore, we have (X � 12�2impT )22�2impT � ConstTherefore, �imppT ' p2Xc) Refer to Lemma 6, when � < 0; and �2 > 12there is not a complete square, the \steepest descent" technique applied to theintegral of the call price (15) results in an exponential of the form ofexp(�C �X)where C = 4�2 (�2� 121� �2 ). Therefore,(X � 12�2impT )22�2impT � C �XWe conclude that �imppT ' (p2C + 2�p2C) � pX:2QED 12

Comment: Notice that the asymptotic behaviors are di�erent for the case� > 0 and � < 0. This actually suggests the skewness of the \smile" curvewhen � is not equal to 0, i.e., the \smile" curve goes to in�nity with di�erentexponents. The argument is the following. WhenX � ln(KS )! �1the same large deviation results we get above is valid for put options. Whileputs are equivalent to calls with � becomes ��. In fact, in Foreign Exchangemarket, a put on currency 1/currency 2 is a call on currency 2/currency 1.Therefore, the behavior of \smile" curve when X ! �1 is the same as thatwhen X !1 with opposite sign of �. The main result is therefore:Corollary: Asymptotic behaviors for the implied volatility smile:1. If � > 0,for deep-out-of-money calls, �imppT ! p2X .for deep-out-of-money puts, �imppT ! p2X . when �2 � 12and �imppT ! C � pX (C < p2) when �2 > 12 .2. If � < 0,for deep-out-of-money calls, �imppT ! p2X . when �2 � 12and �imppT ! C � pX (C < p2) when �2 > 12 .for deep-out-of-money puts, �imppT ! p2X:13

4 Monte Carlo Study for Near-Money OptionsIn this section, we use the same risk-neutral stochastic volatility model derivedbefore to study the implied volatility \smile" curve for the near-the-moneyoptions. We show how the shape of the \smile" curve changes in terms of themodel parameters such as correlation � and volatility of volatility V . We alsopresent some results on the term structure of implied volatility.12The methodology we use in this section is Monte Carlo simulation, dueto the fact that there is no close-form solution for the risk-neutral stochasticvolatility model. The results show that, for zero correlation between the un-derlying asset and its volatility, one obtains a symmetric \smile" curve, i.e.,approximately a centered parabola13. For positive correlation, the center ofthe parabola moves to the left; for negative correlation, the center moves tothe right. With other parameters �xed, the bigger the absolute value of �, thefurther the center is moved. Since the observable options are those near-the-money, when � is close to 1 or -1, the center of the parabola is further awayfrom at-the-money region, e�ectively the \smile" curve resembles a line morethan a parabola, but is actually the part of a parabola that is far away fromthe center.On the other hand, the volatility of volatility , V , has a \centering" e�ecton the \smile" curve. In other words, with other parameters held �xed, thelarger the V , the larger the curvature of the implied volatility, and the center ofthe parabola returns to the near-the-money region. So for the same correlation�, the implied volatility curve with larger V looks like a \smile", while withsmaller V the \smile" curve is degenerated to a line.We use two techniques to reduce the standard deviation of the simulation.One is the antithetic variate, the other is control variate (Hammersley, J., 1964).The antithetic variate is to use one Brownian path along with its mirror pathin simulation. The resulting estimate is still unbiased, but with their perfectcorrelation, the standard deviation is largely reduced.Control variate is another widely used error deduction technique in MonteCarlo simulations. The idea is to �nd a variable with which the unknown vari-able is highly correlated, and which has explicit evaluation formula. One cansimulate this variable using the same sample paths as those used for simulat-ing the unknown variable. E�ectively, one simulates the di�erence betweentwo positively correlated random variables. The di�erence is smaller than the12A more detailed study of the term structure of implied volatility is presented in thesecond essay.13From the analysis of last section, the \smile" curve is not strictly a parabola out-of-the-money or in-the-money. Nevertheless, in the near-money region, the curve can be wellapproximated by a parabola. 14

original variable that is being calculated, accordingly, the standard deviation issmaller. The original simulation is obtained by adding the simulated di�erenceand the theoretical evaluation of the auxiliary variable.In our simulation, we use the Black-Scholes option price as the controlvariate. Namely, we simulate the Black-Scholes price with a constant volatilitywhich can be chosen as the initial value of stochastic volatility , using the sameBrownian sample path. This reduces the standard deviation greatly.Figure 1 and 2 exhibit the near-money smile curves corresponding to dif-ferent model parameters.Figure 1 shows how the curve changes for di�erent correlation �. We ob-serve that, in general, negative correlations correspond to negative skewness,ie, out-of-money puts are more expensive than out-of-money calls; and positivecorrelations correspond to positive skewness, ie, out-of-money calls are moreexpensive than out-of-money puts. In particular, strong correlations (positiveor negative) corresponds to strong skewness. This can be seen in Figure 1,which compares small correlation coe�cient � = �0:2 with that as large as� = �0:9.Figure 2, however, exhibits the di�erent V e�ect. One can see that thevolatility of volatility has the e�ect of changing the convexity as well as thecenter of the \smile" curve. In the left panel, where V is 1, what we see ismostly a line (skewness) rather than a \smile" for non-zero correlations. In theright panel, with V equals 3, we see \smiles" with di�erent center rather thana line (skewness).In Figure 3 and 4, we show the \smile" e�ect across di�erent time horizons.One can see that for � = �0:2, the longer the maturity, the higher level and themore convexity for the implied volatility curve. This e�ect is less for � = �0:9.This is because for � = �0:9, the volatility goes to equilibrium or blows upfast.15

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rho=0.9Figure 1: Simulated Smile Curves for Options with Maturity of 120 days. TheVolatility of Volatility, V , is Fixed at 1 per annum; the Level of Correlations,�, varies from -0.9 to 0.9, as Marked in Each Plot.16

0.9 1 1.10.1

0.15

0.2

0.25

Low Vol of Vol, V=1 per annum

Impl

ied

Vol

rho=−0.5

0.9 1 1.10.1

0.15

0.2

0.25

Impl

ied

Vol

rho=0

0.9 1 1.10.1

0.15

0.2

0.25

Strike/Spot

Impl

ied

Vol

rho=0.5

0.9 1 1.10.1

0.15

0.2

0.25

High Vol of Vol, V=3 per annum

Impl

ied

Vol

rho=−0.5

0.9 1 1.10.1

0.15

0.2

0.25Im

plie

d V

olrho=0

0.9 1 1.10.1

0.15

0.2

0.25

Strike/Spot

Impl

ied

Vol

rho=0.5Figure 2: Simulated Smile Curves for Options with Maturity of 120 days. TheVolatility of Volatility, V , is Fixed at 1 per annum for the Left Panels, and 3per annum for the Right Panels; the Level of Correlations, �, varies from -0.5to 0.5, as Marked in Each Plot.17

0.9 0.95 1 1.05 1.1

0.15

0.16

0.17

0.18

Impl

ied

Vol rho=−0.2

Maturity=60 days

0.9 0.95 1 1.05 1.1

0.15

0.16

0.17

0.18

Impl

ied

Vol rho=0

0.9 0.95 1 1.05 1.1

0.15

0.16

0.17

0.18

K/S

Impl

ied

Vol rho=0.2

0.9 0.95 1 1.05 1.1

0.15

0.16

0.17

0.18

Impl

ied

Vol rho=−0.2

Maturity=180days

0.9 0.95 1 1.05 1.1

0.15

0.16

0.17

0.18Im

plie

d V

ol rho=0

0.9 0.95 1 1.05 1.1

0.15

0.16

0.17

0.18

K/S

Impl

ied

Vol rho=0.2Figure 3: Simulated Smile Curves for Options with Maturity of 60 days and180 days. The Volatility of Volatility, V , is Fixed at 1 per annum; the Level ofCorrelations, � takes value of �0:2 as Marked in Each Plot.

18

0.9 0.95 1 1.05 1.10.1

0.15

0.2

Impl

ied

Vol rho=−0.9

Maturity=60 days

0.9 0.95 1 1.05 1.10.1

0.15

0.2

Impl

ied

Vol rho=0

0.9 0.95 1 1.05 1.10.1

0.15

0.2

K/S

Impl

ied

Vol rho=0.9

0.9 0.95 1 1.05 1.10.1

0.15

0.2

Impl

ied

Vol rho=−0.9

Maturity=180days

0.9 0.95 1 1.05 1.10.1

0.15

0.2Im

plie

d V

ol rho=0

0.9 0.95 1 1.05 1.10.1

0.15

0.2

K/S

Impl

ied

Vol

rho=0.9Figure 4: Simulated Smile Curves for Options with Maturity of 60 days and180 days. The Volatility of Volatility, V , is Fixed at 1 per annum; the Level ofCorrelations, � takes value of �0:9, as Marked in Each Plot.19

References[1] Avellaneda, M., and A. Par�as (1996). \Managing the volatility risk ofportfolios of derivative securities: The Lagrangian Uncertain Volatilitymodel." Applied Mathematical Finance 3, 21-52.[2] Black, F., Scholes, M. (1973). \The pricing of options and corporate lia-bilities". Jour. Pol. Ec., 81, 637-653.[3] Derman, E., and I. Kani (1994). \Riding on the Smile."Risk 7 (February):32-39.[4] Dupire, B. (1994). \Pricing with a Smile." Risk (January) 7:18-20.[5] Hammersley, J. (1964) Monte Carlo Methods. Wiley (1964).[6] Mckean, H. P., Jr. (1969) Stochastic Integrals. Academic Press, New York.[7] Merton, R. C. (1990). \Continuouis-Time Finance". Blackwell Publishers,inc.[8] Hull, J., and A. White (1987) \The Pricing of Options on Asset withStochastic Volatilities." Journal of Finance (June):281-300.[9] Judge,G.G., W.E. Gri�ths, R.C. Hill, H. Lutkepohl and T.C.Lee (1988)Introduction to the Theory and Practice of Econometrics, 2nd ed. NewYork: John Wiley.[10] Karatzas, I., and S. Shreve (1991) Brownian Motion and Stochastic Cal-culus. Springer-Verlag.[11] Rubinstein, M. (1994). \Implied Binomial Trees." Journal of Finance49:771-818.[12] Vasicek, O. (1977)\An Equilibrium Characterization of the Term Struc-ture." Journal of Financial Economics 5:177-88.20

A Appendix 1In this appendix, we calculate the normalization factor �N for the conditionalprobability density of g(�tV 2 �jAt).Throughout this appendix, let �0 � �2 .�N(At)= Z 10 d� 1p2�At ( exp (� (� � �0V )22At )� exp (� (� + �0V )22At )) � exp(��0� � 12�02At)= Z 10 d� 1p2�At exp (� (� � �0V + �0At)22At ) � exp(��0�0V )� Z 10 d� 1p2�At exp (� (� + �0V + �0At)22At ) � exp(�0�0V )= exp(��0�0V ) �N( � �0pAtV + �0qAt)� exp(�0�0V ) �N( �0pAtV + �0qAt)From here, it is di�erent for �0 positive or negative. For �0 > 0,�N (At)� exp (� �02At2 ) � ( 1�0pAt � �0pAtV � 1�0pAt + �0pAtV )= exp (� �02At2 ) � 2 �0pAtV�02At � �20AtV= 2�0�02A 32t exp (� �02At2 )For �0 < 0, we have �N (At)� exp(��0�0V )� exp(�0�0V )= 2 sinh(��0�0V )B Appendix 2The main tool of our calculation is \steepest descent".In general, to evaluate integrals of the formI(�) = Z e�f(z)dz (� large and positive)21

observe that for large value of �, the main contribution to the integral comesfrom the small neighborhood of the maximum points of f(z).Suppose z0 is a maximal point of f(z), i.e., f 0(z0) = 0. Near the point z0,f(z) � f(z0) + 12f 00(z0)(z � z0)2The integral is approxmated asI(�) = e�f(z0) Z e�2 f 00(z0)(z�z0)2dz= C � e�f(z0)C is a constant depending on � and f 00(z0).In our calculation, X is the large parameter. the result is obtained by takingthe leading order of X.When calculating � of Lemmas 4 and 6, we encountered the integral of theform Z 10 exp (� (� �m1)22�21 ) � exp (� (� �m2)22�22 )d�In this case, steepest descent method actually gives the exact result. Observethat the maximal point of the integrand is �0 = m1�21 +m2�221�21+ 1�22 . The integral can bewritten as: exp (� (m1 �m2)22(�21 + �22) ) � 1q 1�21 + 1�22 �N1(� m1�21 + m2�22q 1�21 + 1�22 )� 1q 1�21 + 1�22 � exp (� (m1 �m2)22(�21 + �22) ) as m1�21 + m2�22 > 0� 1q 1�21 + 1�22 � exp (� 12(m21�21 + m22�22 )) as m1�21 + m2�22 < 0while in our context,m1 = X � 1��22 At� �21 = 1 � �2�2 Atm2 = ��2At �22 = AtIn general, for integral of the form:Z 10 exp (� (M � �)22�2 ) � exp(�(ln �)�)d�22

when M is large. To �nd the saddle point, set the derivative of the exponentto zero, we get �M22�2 + 12 + � ln �� = 0:In this case, the saddle point is approximately � =M , because whenM is large,the third term on the left hand side is about zero. Therefore, the integral, toleading order, is C � exp(�(lnM)�)More generally, if we have an integral of the formZ 10 exp (� (M � �)22�2 ) � exp(�f(�))d�when M is large, and f(�) is of the formf(�) = ��with � < 1, then the saddle point is � =M . And the integral isC � exp(�f(M)):If � = 1, the integral is of the formC � exp(C 0 �M):All the integral we used in Section (1.3) can be transformed into the one ofthe forms of the above.C Appendix 3In this appendix, we show that the tail distribution ofAt � Z t0 �2sds;for positive �0 � �2 ; is log-normal.Lemma 1. Let �t be the risk-neutral stochastic volatility, i.e., �t satis�es theSDE: d�t = ��0V �2t dt+ V �tdZt;then �t = Mt1�0 + �0V R t0 Msds (17)23

Proof.Di�erentiate (17), and notice thatdMt = V �MtdZt;we get d�t = �Mt( 1�0 + �0V R t0 Msds)2 � �0V Mtdt+ dMt1�0 + �0V R t0 Msds= ��0V �2t + V �tdZti.e., �t satis�es the original SDE.Lemma 2. Tail distribution of R t0 Msds is a log-normal distribution.Proof.Let M be a large number, we have the following estimations:P [Z t0 exp(Zs)ds > M ] � P [C1 exp(Z t0 Zsds) > M ]according to Jensen's inequality, and we haveP [Z t0 exp(Zs)ds > M ] � P [C2 exp(Zmax)(t) > M ]where Zmax(t) is the maximum of Brownian motion between 0 to t. SinceR t0 Zsds is normally distributed, the �rst inequality tells us that the tail ofR t0 exp(Zs)ds is no fatter than a log-normal distribution. Moreover, since dis-tribution of the maximum of Brownian Motion is the same as the absoluteBrownian Motion, the tail of which is log-normally distributed. Therefore,the second inequality shows that the the tail of R t0 exp(Zs)ds is no \thinner"than a log-normal distribution. We conclude that the tail of R t0 exp(Zs)ds is alog-normal distribution, and so is R t0 Msds, because one can always �nd twoconstants D1 and D2 (depending on t) such that:D1 � Z t0 exp(Zs)ds � Z t0 Msds � D2 � Z t0 exp(Zs)ds:From Lemmas 1 and 2, we now show that the tail distribution of�t = Mt1�0 + �0V R t0 Msdsis log-normal. The argument is the following.24

a) Because �0 and V are positive, andMt is the positvie exponential Martingale,we have �t � �0Mt;so the tail of �t is no fatter than a log-normal.b) Since the tail of R t0 Msds is also log-normal, (from lemma 1), one can con-struct a random variable � such that� =M; when Z t0 Msds �M;� = Z t0 Msds when Z t0 Msds �M:where M is a large number. Apparently,�t � Mt1�0 + �0V � ;and the latter has tail distribution of log-normal. Hence, the tail of �t is nothinner than a log-normal. So it is log-normal.Finally, we claim that the tail of R t0 �2t is also a log-normal. Since the tailof �t is log-normal, so is that of �2t . If we can conclude that a tail of thesummation of two log-normal r.v.'s is log-normal, by using induction, we canprove that the tail of R t0 �2t is log-normal. The next lemma is to show that thesum of log-normal has tail of log-normal.Lemma 3. Let X and Y be two log-normal r.v.'s. Then the tail of Z = X + Yis log-normal.Proof. Let f(Z) be the probability density function of Z. Thenf(Z) � Z exp(�(lnX)2) � exp(�(ln(Z �X))2)dX= Z exp(�[(lnX)2 + (ln(Z �X))2]dXWhen Z is large, we use the \steepest descent" technique to integrate it. Bydi�erentiate the exponent (lnX)2 + (ln(Z �X))2we get 2 lnXX � 2 ln(Z �X)Z �XSet it to zero, we �nd the saddle point X = Z. Therefore, the tail distributionof Z behaves as exp(�2( ln(Z2 ))2)which is log-normal. 25