option pricing under multiscale stochastic volatility
TRANSCRIPT
Option Pricing under Multiscale Stochastic Volatility
Cristina Tessari∗1 and Caio Almeida†1
1EPGE-FGV
Ongoing Master’s ThesisFirst Draft: November 30, 2015
Abstract
The stochastic volatility model proposed by Fouque, Papanicolaou, and Sircar(2000) explores a fast and a slow time-scale fluctuation of the volatility processto end up with a parsimonious way of capturing the volatility smile implied byclose to the money options. In this paper, we test three different models of theseauthors using options on the S&P 500. First, we use model independent statisticaltools to demonstrate the presence of a short time-scale, on the order of days, and along time-scale, on the order of months, in the S&P 500 volatility. Our analysis ofmarket data shows that both time-scales are statistically significant. We also providea calibration method using observed option prices as represented by the so-calledterm structure of implied volatility. The resulting approximation is still independentof the particular details of the volatility model and gives more flexibility in theparametrization of the implied volatility surface. In addition, to test the model’sability to price options, we simulate options prices using four different specificationsfor the Data generating Process. As an illustration, we price an exotic option.
Keywords: Option pricing; Stochastic volatility; Mean-reversion.
∗[email protected]; Corresponding author.†[email protected]
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1 Introduction
Asset return volatility is a central concept in finance, whether in asset pricing, portfolioselection, or risk management. Although many well-known theoretical models assumedconstant volatility (see Merton, 1969; Black and Scholes, 1973), since Engle (1982) seminalpaper on ARCH models, the financial literature has widely acknowledge that volatility istime-varying, in a persistent fashion, and predictable (see Andersen and Bollerslev, 1997).
From the point of view of derivative pricing and hedging, continuous time stochasticvolatility models, which can be seen as continuous time versions of ARCH-type models,have become popular in the last twenty years. The idea of using stochastic volatilitymodels to describe the dynamics of an asset price comes from empirical evidence indicatingthat asset price dynamics is driven by processes with time-varying volatility. In fact, twophenomena are observed: (i) a non-flat implied volatility surface when the Black andScholes (1973) model is used to interpret financial data, and (ii) skewness and kurtosisare present in the asset price probability density function deduced from empirical data.These phenomena stands in empirical contradiction to the consistent use of a classicalBlack-Scholes approach to pricing options and similar securities, as the Black–Scholesmodel fails to correctly describe the market behavior.
Stochastic volatility models relax the constant volatility assumption of the Black-Scholes model by allowing volatility to follow a random process. In this context, themarket is incomplete because the volatility is not traded and the volatility risk cannot befully hedged using the basic instruments (stocks and bonds). To preclude arbitrage, themarket selects a unique risk neutral derivative pricing measure, from a family of possiblemeasures. As a result, in contrast to the Black-Scholes model, the stochastic volatil-ity models are able to capture some of the well-known features of the implied volatilitysurface, such as the volatility smile and skew.
The presence of volatility factors is well documented in the literature using underlyingreturns data (see Alizadeh et al., 2002; Andersen and Bollerslev, 1997; Chernov et al., 2003;Engle and Patton, 2001; Fouque et al., 2003b; Hillebrand, 2005; LeBaron, 2001; Gatheral,2006, for instance). While some single-factor diffusion stochastic volatility models such asHeston (1993) enjoy wide success, numerous empirical studies of real data have shown thatthe two-factor stochastic volatility models can produce the observed kurtosis, fat-tailedreturn distributions and long memory effect. For example, Alizadeh, Brandt, and Diebold(2002) used ranged-based estimation to indicate the existence of two volatility factorsincluding one highly persistent factor and one quickly mean-reverting factor. Chernov,Gallant, Ghysels, and Tauchen (2003) used the efficient method of moments (EMM)to calibrate multiple stochastic volatility factors and jump components. They showed
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that two factors are necessary for log-linear models. These evidences, among others, havemotivated the development of multiscale stochastic volatility models as an efficient way tocapture the principle effects on derivative pricing and portfolio optimization of randomlyvarying volatility.
In this paper, motivated by both the popularity and appeal of stochastic volatilitymodels and by the difficulty associated with their estimation, we compare the performanceof three different specifications of the Fouque, Papanicolaou, and Sircar (2000) stochasticvolatility model for pricing European call options with the Black-Scholes model. Weassume that the underlying asset evolves according to a geometric Brownian motion, asin the Black and Scholes (1973) model, but with stochastic volatility. We use S&P 500data to demonstrate that the volatility of this index is driven by two diffusions: one fastmean-reverting and one slow-varying. In order to identify these scales, we analyze low-and high-frequency data using the empirical structure function, or variogram, of the logabsolute returns. To retrieve the scale on the order of months, we use daily closing pricesof the S&P 500 over several years. To extract the scale on the order of days, we followFouque et al. (2003a) and use high-frequency, intraday data.
In their work, Fouque, Papanicolaou, and Sircar (2000) show that multiscale stochas-tic volatility models lead to a first-order approximation of derivatives prices and of theimplied volatility surface. By using a combination of singular and regular perturbationsto approximate prices when volatility is driven by short and long time scale factors, theyderive a first-order approximation for European options prices and their induced impliedvolatilities. This first-order approximation is composed by the Black-Scholes price andthe first-order correction only involves Greeks. In terms of implied volatility, this pertur-bation analysis translates into an affine approximation in the log-moneyness to maturityratio (LMMR). This perturbation method allows a direct calibration using real data ofthe group market parameters, which are exactly those needed to price exotic contracts atthis level of approximation. The advantage of this approach is that the resulting formulafor the option price does not depend on the unobserved current value of the volatil-ity. However, calibration of this model requires information on near-the-money impliedvolatilities.
We have three different goals in this work. First, to demonstrate the existence of botha well-identified short time-scale (on the order of a few days) and a long time-scale (onthe order of months) in the S&P 500 volatility. In order to do that, we show that thefast time-scale and the slow time-scale can be simultaneously estimated using two dif-ferent datasets, making use of the empirical structure function, or variogram. To testthe statistical significance of the mean-reverting speeds, we use the circular block boot-strap of Politis and Romano (1992). We bootstrap pairs of residuals using the automatic
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block-length selection of Politis and White (2004). The second goal, conditional on themarket supporting evidence of a multiscale stochastic volatility process, is related to thecalibration of the model using option data on the underlying. At this stage, we comparethe observed prices with the corrected Black-Scholes prices by using three different first-order corrections: (i) a fast time scale correction; (ii) a slow time scale correction; and(iii) a multiscale (fast and slow) correction. Finally, the third goal of this paper is tosimulate both the trajectories of these three different corrected prices based on the modelof Fouque, Papanicolaou, and Sircar (2000) and the Black-Scholes prices, and comparethe prices of exotic derivatives based on these simulated prices.
The rest of this paper proceeds as follows. In Section 2, we describe the class ofmultiscale stochastic volatility models that we will work with. In Section 2.3, we presentthe empirical structure function, or variogram, which is used as an estimator of the speedof mean reversion. In Section 2.4, we present the first-order approximation derived inFouque, Papanicolaou, and Sircar (2000), which is valid for any European-style option.Additionally, we present an explicit formula for the implied volatility surface inducedby the option pricing approximation. In Section 3, we report the numerical results. InSection 4, we show how the corrected Black-Scholes prices can be used to price exoticoptions. Finally, Section 5 concludes.
2 A Model for the Stock Price Dynamics
2.1 The Model
Let (Ω,F ,P, Ftt≥0) be a complete probability space with a filtration satisfying theusual conditions. We consider a family of stochastic volatility models (Xt, Yt, Zt), whereXt is the underlying price, and Yt and Zt are two mean-reverting Ornstein–Uhlenbeck(OU) processes. Under the physical probability measure P, our model can be written asthe following system of stochastic differential equations (SDE):
dXt = µXtdt+ f(Yt, Zt)XtdW
0t ,
dYt = 1εα(Yt)dt+ 1√
εβ(Yt)dW 1
t ,
dZt = δc(Zt)dt+√δg(Zt)dW 2
t ,
(1)
where (W (0),W (1),W (2)) are correlated P-Brownian motions with
dW(0)t dW
(i)t = ρidt, i = 1, 2, dW
(1)t dW
(2)t = ρ12dt, (2)
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and |ρ1| < 1, |ρ2| < 1, |ρ12| < 1, and 1 + 2ρ1ρ2ρ12 − ρ21 − ρ2
2 − ρ212 > 0 in order to
ensure that the correlation matrix of the Brownian motions is positive-semidefinite. Theunderlying price Xt evolves as a diffusion with geometric growth rate µ and stochasticvolatility σt = f(Yt, Zt).1 The fast volatility factor, Yt, evolves with a mean reversiontime ε, while the slow volatility factor, Zt, evolves with a time scale 1/δ. In order to havea separation of scales, we must assume that δ 1/ε.
Here, it is worth mentioning that this model characterizes an incomplete market, incontrast to the Black and Scholes (1973) model, since the volatility presents its own in-dependent sources of uncertainty, Yt and Zt. The introduction of these two new sourcesof randomness give rise to a family of equivalent martingale measures that will be pa-rameterized by the market price of risk, λ(y, z) = (µ− r)/f(y, z), and two market pricesof volatility risk, which we denote by ξ(y, z) and ζ(y, z), associated respectively with Ytand Zt. All these market prices are not determined within the model, but are fixed ex-ogenously by the market. To preclude arbitrage, we assume that the market chooses onemeasure P∗ through the combined market price of volatility risk (ξ, ζ).
Under the risk neutral measure P∗, an application of the Girsanov’s theorem providesthat our model can be written as
dXt = rXtdt+ f(Yt, Zt)XtdW0∗t ,
dYt =(
1εα(Yt)− 1√
εβ(Yt)Λ1(Yt, Zt)
)dt+ 1√
εβ(Yt)dW 1∗
t ,
dZt =(δc(Zt)−
√δg(Zt)Λ2(Yt, Zt)
)dt+
√δg(Zt)dW 2∗
t ,
(3)
where f is a positive function, bounded above and away from zero, r ≥ 0 is the risk-freerate of interest, and the combined market prices of volatility risk associated with Yt andZt are
Λ1(y, z) = ρ1λ(y, z) + ξ(y, z)√
1− ρ21, (4)
Λ2(y, z) = ρ2λ(y, z) + ξ(y, z) ˜ρ1,2 + ζ(y, z)√
1− ρ22 − ρ2
12. (5)
In our model, the coefficients α(y) = mY − y and β(y) = νY√
2 describe the dynamicsof Y under the physical measure P and the coefficients c(z) = mZ − z and g(z) = νZ
√2
describe the dynamics of Z under P. Their particular form does not play a role inthe perturbation analysis provided that they are defined so that the processes Y andZ are mean-reverting and have a unique invariant distribution denoted by ΦY and ΦZ ,respectively. Since Y and Z are OU processes, it can be shown that ΦY is the density of thenormal distribution N(mY , ν
2Y ), where ν2
Y = β2/2α and ΦZ is the density of the normal1If σt is constant, then Xt is a geometric Brownian motion and corresponds to the classical model
used in the Black and Scholes (1973) theory.
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distribution N(mZ , νZ).2 Finally, the P∗-standard Brownian motions (W 0∗t ,W
1∗t ,W
2∗t )
present the same correlation structure as between their P-counterparts in Equation (2).
2.2 Pricing Equation
Consider a European option with smooth and bounded payoff function h(x) and expira-tion date T . The fact that the discounted price Pt = e−rtPt is a P∗-martingale guaranteesthat the no-arbitrage pricing function of this option at time t < T is given by
P ε,δ(t,Xt, Yt, Zt) = E∗e−r(T−t)h(XT )
∣∣∣Xt, Yt, Zt, (6)
where the expectation E∗· is taken under the risk-neutral pricing measure P∗, and wehave used the Markov property of (Xt, Yt, Zt). By an application of the Feynman-Kacformula, we obtain a characterization of P ε,δ as the solution of the parabolic partialdifferential equation (PDE)
∂P ε,δ
∂t+ L(X,Y,Z)P
ε,δ − rP ε,δ = 0 (7)
with terminal condition P ε,δ(T, x, y, z) = h(x), where L(X,Y,Z) is the infinitesimal generatorof the Markov processes (Xt, Yt, Zt). Define the infinitesimal generator Lε,δ as
Lε,δ = ∂
∂t+ L(X,Y,Z) − r, (8)
so that (7) can be written asLε,δP ε,δ = 0, (9)
with terminal condition P ε,δ(T, x, y, z) = h(x). It is convenient to re-write the operatorLε,δ as a sum of components that are scaled by the different powers of the small parameters(ε, δ) that appear in the infinitesimal generator of (X, Y, Z) as
Lε,δ = 1εL0 + 1√
εL1 + L2 +
√δM1 + δM2 +
√δ
εM3, (10)
where the operators are defined as
L0 = 12β
2(y) ∂2
∂y2 + α(y) ∂∂y, (11)
L1 = β(y)(ρ1f(y, z)x ∂2
∂x∂y− Λ1(y, z) ∂
∂y
), (12)
2For additional details about the derivation of the invariant distributions, see Fouque, Papanicolaou,Sircar, and Solna (2011).
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L2 = ∂
∂t+ 1
2f2(y, z)x2 ∂
2
∂x2 + r
(x∂
∂x− ·), (13)
M1 = g(z)(ρ2f(y, z)x ∂2
∂x∂z− Λ2(y, z) ∂
∂z
), (14)
M2 = 12g
2(z) ∂2
∂z2 + c(z) ∂∂z, (15)
M3 = β(y)ρ1,2g(z) ∂2
∂y∂z. (16)
Note that L2 is the Black-Scholes operator, corresponding to a constant volatility levelf(y, z). The Black-Scholes price CBS(t, x;σ), the price of a European claim with payoffh at the volatility σ, is given as the solution of the following PDE
LBSCBS = 0, (17)
with terminal condition CBS(T, x;σ) = h(x). If we are able to calculate the solutionof the PDE given by Equation (9), then we know how to price derivatives under theproposed model. However, for general coefficients (f, α, β, c, g,Λ1,Λ2), we do not havean explicit solution to this equation. In this context, Fouque, Papanicolaou, and Sircar(2000) developed an asymptotic approximation for the option price in the neighborhoodof the Black-Scholes price that made the calibration problem computationally tractable.In order to obtain the first-order approximation, they expand P ε,δ in powers of
√ε and√
δ as follows:P ε,δ(t, x, y, z) =
∑j≥0
∑i≥0
√εi√δjPi,j(t, x, y, z), (18)
where P0,0 = PBS is the Black-Scholes price, P ε1,0 =
√εP1,0 is the first-order fast scale
correction, and P δ0,1 =
√δP0,1 is the first-order slow scale correction. In the Appendix, we
present the derivation of the this first-order correction to the Black-Scholes solution.
If we define Dk asDk = xk
∂k
∂xk, (19)
then, for nice payoff functions h, Fouque, Papanicolaou, and Sircar (2000) show that thecorrected call price P ∗ for European options using a combination of singular and regularperturbation is given by
P ∗ = P ∗BS + (T − t)(V δ
0 (z)∂P∗BS
∂σ+ V δ
1 (z)D1∂P ∗BS∂σ
+ V ε3 (z)D1D2P
∗BS
), (20)
where P ε,δ = P ∗ + O(ε + δ). Note that this formula involves only Greeks of the Black-Scholes price P ∗BS evaluated at σ∗(z). In addition, observe that only the group marketparameters (σ∗, V δ
0 , Vδ
1 , Vε
3 ) are needed to compute the approximate price P ∗, where V δ0
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and V δ1 are of order
√δ and V ε
3 is of order√ε. Thus, the main advantages of this
price approximation are the ease of implementation and the parsimony in the number ofparameters. In addition, if δ = 0, then only the group market parameters σ∗(z) and V ε
3 (z)are needed to compute the contribution due to the fast time scale rather than the fullspecification of the stochastic volatility model. Similarly, if ε = 0, then the slow time scalecontribution to P ∗ is contained in only two parameters, V δ
0 (z) and V δ1 (z). Accordingly, the
corrected price (20) obtained when we consider a two-factor stochastic volatility modelsincludes the correction due to the fast time scale and the correction due to the slow timescale as its particular cases.
2.3 The Variogram and Time Scales in Market Data
In order to identify the time scales in the volatility, this section introduces the vari-ogram, or empirical structure function, as a tool for estimating the mean-reversion times.
Consider a discrete version of time, with ∆t = tn− tn−1, 0 ≤ n ≤ N , and let Xn denotethe S&P 500 data recorded at time tn. Adopting a discrete version of Equation (1), wedefine the normalized fluctuation of the data, Dn, as
Dn = 1√∆t
(∆Xt
Xt
− µ∆t)
= σt∆Wt
∆t , (21)
where Xn, Yn, Zn andWn represent the processes Xt, Yt, Zt andWt sampled at time n∆t,σt = f(Yt, Zt), t ≥ 0, is the (positive) volatility process, µ is a constant, and ∆Wt is theincrement of a Brownian motion. From the basic properties of a Brownian motion, weare able to model Dn by
Dn = f(Yn, Zn)εn, (22)
where εn is a sequence of i.i.d. N(0, 1) random variables. In order to obtain an expres-sion similar to that found in Fouque, Papanicolaou, and Sircar (2000), we assume thatthe stochastic volatility can be written as f(y, z) = f1(y)f2(z). This functional form wasproposed by Fouque, Papanicolaou, Sircar, and Solna (2003c) for the case in which thecorrelation between the instantaneous volatility components is zero.3 In this case, in orderto obtain an additive noise process, we define the log absolute value of the normalizedfluctuations as
Fn = log |Dn| = log |f1(Yn)|+ log |f2(Zn)|+ log |εn|. (23)
3Machuca (2010) demonstrated that this equation remains valid even when the correlation betweenY and Z is different from zero.
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Thus, the empirical structure function or variogram of Fn, which is a measure of thecorrelation structure of the sampled version of the process log (f(Yt, Zt)), is given by
V Nj = 1
Nj
∑n
(Fn+j − Fn)2 , (24)
where j is the lag for which we are measuring the correlation and Nj is the total numberof points for each lag.
In this paper, we consider a model with two components driving the volatility, one fastand one slow. In this case, the stochastic volatility model is given by
f(Yt, Zt) = eYt+Zt , (25)
where Yt and Zt are independent OU processes. Fouque, Papanicolaou, and Sircar(2000) show that, for each j = 1, 2, . . . , J , the empirical variogram V N
j is an unbiasedestimator of
Vj = 2γ2 + 2ν2Y
(1− e−αj∆t
)+ 2ν2
Z
(1− e−δj∆t
), (26)
where α ≡ 1/ε, γ2 = Var(log |ε|), ν2Y is the variance of the stationary distribution of
the process log f1(Yt) and ν2Z is the variance of the stationary distribution of the process
log f2(Zt).
Note that, for the range of lags we are looking at, that is j∆t up to a week, δj∆t issmall and the last term is negligible. Hence, we can use the following approximation inorder to obtain the component of fast mean-reversion:
V Nj ≈ 2γ2 + 2ν2
Y
(1− e−αj∆t
). (27)
At the opposite extreme, if we had looked at a longer data sample and included largerlags, then the term related to α in (26) is sufficiently large. In this case, we can considerthe following approximation to obtain the component of slow mean-reversion:
V Nj ≈ 2(γ2 + ν2
Y ) + 2ν2Z
(1− e−δj∆t
). (28)
Using this result, we are able to estimate by non linear regression methods the ratesof mean reversion of the volatility processes, 1/ε and δ. Note that Equations (27) and(28) share a parameter, γ. In order to take this fact into account, we fit two differentdatasets to different equations with a shared parameter. For the high frequency dataset,we use the empirical variogram from Equation (22) as the dependent variable, and adjustit to the functional form given in Equation (27). At the same time, for the slow frequencydataset, we adjust the empirical variogram to the functional form given by Equation (28).
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The resulting estimates are reported in Section 3.
2.4 Implied Volatility Asymptotic and Calibration
It is common practice to quote option prices in units of implied volatility, by invertingthe Black-Scholes formula for the European option with respect to the volatility parame-ter. This quantity is a convenient change of unit through which to view the departure ofmarket data from the Black-Scholes theory.
The multiscale asymptotic theory developed in Fouque, Papanicolaou, and Sircar(2000) is designed to capture some of the important effects that fluctuations in the volatil-ity have on derivative prices. In their work, they propose a first order approximationfor the implied volatility function obtaining a very nice linear relation between impliedvolatility and the ratio log moneyness to time to maturity, from which we can estimatethe parameters from Equation (20). This procedure is robust and no specific model ofstochastic volatility is actually needed.
In order to obtain the implied volatilities, we consider a European call option withstrike K and maturity T . In this case, the corresponding payoff function is given byh(x) = (x−K)+, and the Black-Scholes price at the corrected effective volatility σ∗ is
P ∗BS = xN(d∗1)−Ke−rτN(d∗2), (29)
where τ = T −t is the time to maturity, N is the cumulative standard normal distributionand
d∗1,2 =log(x/K) +
(r ± 1
2σ∗2)τ
σ∗√τ
. (30)
Given an observed European call option price Cobs, the implied volatility I is definedto be the value of the volatility parameter that must go into the Black-Scholes formula(29) in order to have CBS(t, x;K,T ; I) = Cobs. Thus, as ∂P ∗
BS
∂σ= τσ∗D2P
∗BS for European
vanilla options, we can rewrite the corrected price (20) as
P ∗ = P ∗BS +(τV δ
0 + τV δ1 D1 + V ε
2σ∗D1
)∂P ∗BS∂σ
, (31)
where ∂PBS∂σ
= x√τe−d2
1/2√
2π .
Remember that σ∗ solves P ∗BS = CBS(σ∗). Thus, by expanding the difference I − σ∗
between the implied volatility I and the volatility used to compute P ∗BS in powers of√ε
and√δ, Fouque, Papanicolaou, and Sircar (2000) show that the first-order approximation
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for the implied volatility, I ≈ σ∗ +√εI1,0 +
√δI0,1, takes the simple form
I ≈ b∗ + τbδ +(aε + τaδ
)LMMR, (32)
where LMMR, the log-moneyness to maturity ratio, is defined by
LMMR = log (K/x)T − t
= log (K/x)τ
. (33)
The parameters (b∗, bδ, aε, aδ) depend on z and are related to the group parameters(σ∗, V δ
0 , Vδ
1 , Vε
3 ) by
b∗ = σ∗ + V ε3
2σ∗(
1− 2rσ∗2
), (34)
bδ = V δ0 + V δ
12
(1− 2r
σ∗2
), (35)
aε = V ε3
σ∗3, (36)
aδ = V δ1
σ∗2. (37)
The coefficients b∗ and aε are due to the fast volatility factor, while the coefficientsbδ and aδ are due to the slow volatility factor, which becomes more important for largematurities. If we ignore the slow scale, then the implied volatility can be approximatedby
I ≈ b∗ + aε(LMMR), (38)
which corresponds to assuming δ = 0. On the other hand, if we assume ε = 0, then
I ≈ σ∗ + bδτ + aδ(LM), (39)
where we denote LM = τ(LMMR), the log-moneyness. The estimated parameters forEquations (32), (38) and (39) will be presented in Section 3. In Section 4, we show thatthese same parameters are exactly those needed to price exotic derivatives. Thus, in theregime where ε and δ are small, corresponding to stochastic volatility models with fastand slow volatility factors, we can handle a wide class of pricing problems which wouldbe very complicated without the perturbation theory.
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3 Numerical Results
In this section, we present the dataset that will be used in this paper and the numericalresults of our analysis. We divide our empirical study in two parts: the results related tothe dynamics of the volatility and price processes of the S&P 500 index, and the resultsregarding the calibration of the model using option prices on this index.
3.1 Data
We are interested in estimating the fast and slow mean-reverting times of the S&P500. In order to find the scale on the order of days, we use high frequency, 1-min data,from October 23, 2014, to October 23, 2015. To extract the fast scale, we average thedata over 5-min intervals so that we have 72 data points per day. We collapse the timeby eliminating overnights, weekends and holidays, so that we have 252 trading days with18,072 data points per year. To estimate the time scale on the order of months, we usedaily closing prices from January 02, 1980, to October 23, 2015. In this case, we averagethe data over 5-day intervals so that we have 50 data points per year.
To perform the fitting procedure described in Section 2.4, we use S&P 500 options dataon November 03, 2015, obtained from the Bloomberg database. Our dataset containsS&P 500 index option quoted bid-ask prices, implied volatilities, and contract detailssuch as strikes and expiration dates. We shall present results based on European calloptions, taking the average of bid and ask quotes to be the current price. Throughout theempirical analysis, we will restrict ourselves to options between 0.95 and 1.05 moneynessso as to be on the safe side of liquidity issues.
In order to remove inconsistencies and unreliable entries, we roughly follow the data-cleaning procedure described in Fouque, Papanicolaou, Sircar, and Solna (2011). We filterbid quotes less than $0.50 and options with no implied volatility value. We also removethe implied volatilities with the shortest maturity from our dataset. To smooth the jumpbetween the implied volatility curves coming from calls or puts with the same maturity, wefollow the general procedure described in Figlewski (2009), blending the implied volatilityvalues for options with strikes between certain cutoffs L and H, with L < H. We chooseL = 0.95 and H = 1.05. For strikes less than L, we use only puts; for strikes greater thanH, we use only calls; and for strikes between L and H, we blend the two.
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3.2 Mean-Reversion Rates and Block Bootstrap
In a previous study, Fouque, Papanicolaou, Sircar, and Solna (2003a) identify a fastmean-reversion time for the volatility of the returns of the S&P 500 index. Empiricalevidences also show that the S&P 500 exhibits long memory, for example, the reader canrefer to Chronopoulou and Viens (2012). In this context, our goal in this subsection is toverify if the volatility of the S&P 500 index is driven by two factors simultaneously: onefactor mean-reverting on a short scale and the other on a long scale.
At this point, based on the procedure described in Section 2.3, we use the variogram toestimate the volatility mean-reversion times: 1/ε and δ.4 To do that, we fit the variogramgiven by Equation (24) by weighted least squares solving the following problem:
minθ
∑j
ωj(V Nj − Vj(θ)
)2, (40)
where θ is the vector of parameters to be estimated. We choose ωj as the Cressie (1985)weights, given by an approximation for the variance of V N
j :
ωj = 1Var(V N
j ) ≈Nj
[Vj(θ)]2. (41)
Note that the Cressie’s weights put the highest emphasis on variogram estimates thatare based on a large number of points (where Nj is large) and on values near j = 0(where Vj(θ) is small). In Figure 1, we present the empirical and the fitted variogramsfor the S&P 500, using the discrete model described in Section 2. The dashed line is afitted exponential obtained by nonlinear weighted least squares regression. It is worthmentioning that, if the variograms in Figure 1 were flat, we would conclude there is nofast and/or slow time scale on the S&P 500 index.
To obtain the sample distribution of the estimated parameters, we bootstrap the pairsof residuals of the non linear regressions solved to obtain the variograms given by Equa-tions (27) and (28). We use the circular block bootstrap of Politis and Romano (1992),with 10,000 bootstrap samples. The block size was chosen according to the procedure de-veloped by Politis and White (2004). Table 1 presents the estimated values and standarddeviations for the parameters, as well as the 95% bootstrapped confidence interval. The
4We refer to Fouque, Papanicolaou, Sircar, and Solna (2011) for further details about the propertiesof this estimator.
13
days0 2 4 6 8 10 12 14 16
0
0.2
0.4
0.6
0.8
Empirical VariogramFitted Variogram
years0 5 10 15 20 25
0
0.2
0.4
0.6
0.8
1
1.2
Empirical VariogramFitted Variogram
Figure 1: The empirical variogram for the S&P 500. The top graph shows the estimate of thevariogram model for 5-min log normalized fluctuations after applying a 9 point median filter tocompensate for the singular noise log |εn|, while the bottom graph shows the empirical variogramfor 5-day log normalized fluctuations after applying a 3 point median filter. The dashed lines areexponential fits from which the rates of mean-reversion can be obtained. Since both variogramsshare a parameter, they were fitted simultaneously by using weighted least squares regression.The number of lags in the variogram model was chosen to be equal to j = 1, 275.
14
sample distribution of the estimated parameters is shown in Figure 2.
Table 1: Summary statistics of the estimates of the model variogram (36).
Parameter Estimate Mean Std Dev95% Confidence Interval
LB UB
γ 0.4643 0.4614 0.0233 0.4128 0.4998νY 0.4752 0.4783 0.0199 0.4417 0.51741/ε 0.2860 0.2933 0.0600 0.1908 0.4240νZ 0.3158 0.3091 0.0186 0.2690 0.3421δ 0.7114 0.7321 0.3104 0.2775 1.2590
Note: This table presents the summary statistics of the estimated parameters of the S&P500 variogram given by Equation (26). The parameters were estimated using weightedleast squares regression. We fit Equation (27) using 5-min data and Equation (28) using5-day data simultaneously by performing the procedure described in Section 2.3. The mean,standard deviation and 95% confidence interval of each parameter were obtained using acircular block boostrap with 10,000 bootstrap samples. LB indicates the lower bound of theconfidence interval, while UB indicates the upper bound.
From Table 1, observe that the estimated fast time scale, 1/ε, is 0.286, with stan-dard deviation equal to 0.023, meaning an average decoupling time of 3.497 days, witha 95% confidence interval [2.358, 5.241]. The estimated slow time scale, δ, is 0.711,which corresponds to a mean-reversion time of 16.868 months, with a confidence in-terval [9.531, 43.243]. Notice that these mean-reversion times indicate that the volatilityprocess of the S&P 500 is driven by a fast mean-reverting diffusion and a slowly varyingmean-reverting process. The volatility of the invariant distribution of the OU process Ywas estimated to be 0.475, while the volatility of the OU process Z was estimated to be0.316.
Note the oscillatory nature of the variograms in Figure 1. This characteristic is relatedto intra-day variation of volatility, and this was already noticed in Fouque et al. (2000).For a discussion of this cycles from an implicit-volatility viewpoint, see Fouque et al.(2004).
3.3 Calibration Results
In this section, we report the results of the calibration of the group market parametersto the market implied volatility data on a specific day, following the procedure outlined inSection 2.4. Our goal is to demonstrate the improvement in fit of the models’s predicted
15
.
0.38 0.4 0.42 0.44 0.46 0.48 0.5 0.52
Fre
quen
cy
0
100
200
300
8Y
0.4 0.45 0.5 0.55
Fre
quen
cy
0
100
200
300
1/00.1 0.2 0.3 0.4 0.5 0.6
Fre
quen
cy
0
100
200
300
400
8Z
0.2 0.25 0.3 0.35 0.4
Fre
quen
cy
0
100
200
300
400
/
0 0.2 0.4 0.6 0.8 1 1.2 1.4
Fre
quen
cy
0
200
400
600
Figure 2: Bootstrap distributions of the sample parameters obtained by fitting Equations(27) and (28) to two-different datasets. The red line correspond to the estimated parameter,while the red lines correspond to the lower and upper bounds of the 95% bootstrapped confidenceinterval. In order to obtain the sample distributions, we use the circular block bootstrap ofPolitis and Romano (1992) with 10,000 bootstrap samples. The block length was chosen usingthe procedure suggested by Politis and White (2004).
16
implied volatility, given by Equation (32), to market data.
In order to to that, we consider three different models:
• Model 1 : in this model, we assume that the volatility is driven by a single mean-reverting OU process, Zt, fluctuating on a slow time scale;
• Model 2 : at the opposite extreme, in this model we assume that the volatilityprocess is only driven by a fast mean-reverting OU process, Yt;
• Model 3 : in this model, we assume that the volatility is driven by two differentfactors, one fluctuating on a fast time scale, Yt, and a other fluctuating on a longertime scale, Zt.
We first look at the performance of Model 1, which considers only a slow volatilityfactor. In Figure 3a, we show the results of the calibration using only the slow-factorapproximation given by Equation (39), which corresponds to assuming ε = 0. Corrobo-rating with the findings of Fouque, Papanicolaou, Sircar, and Solna (2011), we observethat Model 1 fails to capture the range of maturities.
In Figure 3b, we look at the performance of Model 2, which uses only the fast-factorapproximation given by Equation (38). Again, note that Model 2 fails to capture therange of maturities.
Finally, in fitting the Model 3, which uses the two-factor volatility approximation (32),we follow the two-step procedure of Fouque, Papanicolaou, Sircar, and Solna (2011). First,we fit the skew to obtain ai and bi for different maturities τi. In the second stage, theseestimates are then fitted to an affine function of τ to give estimates of b∗, bδ, aε, and aδ. Aplot of this second term-structure fit on November 03, 2015, is shown in Figure 4a alongwith the calibrated multiscale approximation (32) to all the data on November 03, 2015in Figure 4b.
From Figure 4b, note that the ability of Model 3 to capture the range of maturities ismuch improved when compared to Models 1 and 2. Thus, the two-scale volatility modelwith its additional parameters performs better than either of the one-scale models.
Table 2 reports the estimated parameters of the correction to the Black-Scholes pricefor each one of the three models presented in this section. Observe that in the regimewhere our approximation is valid, the parameters aε, aδ, and bδ are expected to be small,while b∗ is the leading-order magnitude of volatility. The first parameter, σ∗, can beinterpreted as an indicator of the “average” S&P 500 volatility level. Here, r is the risk-free rate, which we assume to be known and constant. Throughout our analysis, we user = 0.34%, which is the value of the interest rate released by the Federal Reserve (FED)
17
LM-0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05
Impl
ied
Vol
atili
ty
0.09
0.1
0.11
0.12
0.13
0.14
0.15
0.16
0.17LM Fit to Residual: Slow Mean Reversion
(a) τ -adjusted implied volatility I − bδτ .
LMMR-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
Impl
ied
Vol
atili
ty
0.08
0.1
0.12
0.14
0.16
0.18
0.2Pure LMMR Fit: Fast Mean Reversion
(b) S&P 500 implied volatilities.
Figure 3: Data and calibrated fit on November 03, 2015. In Figure (a), we plot the maturityadjusted implied volatility as a function of the log-moneyness, LM. The plus signs are from S&P500 data and the line σ∗+aδ(LM) shows the fit using the estimated parameters from only a slowfactor fit using ordinary least squares regression. In figure (b), we present the S&P 500 impliedvolatilities as a function of the LMMR. The plus signs are from S&P 500 data, and the lineb∗+aε(LMMR) shows the result using the estimated parameters from only a fast factor fit usingordinary least squares regression. In both figures, the maturities increase going counterclockwisefrom the top-leftmost strand.
18
= (years)0 0.5 1 1.5 2 2.5
a
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1a = a0 + a/=
= (years)0 0.5 1 1.5 2 2.5
b
0.12
0.13
0.14
0.15
0.16
0.17
0.18
0.19b = b$ + b/=
(a) Term-structure fits.
LMMR-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4
Impl
ied
Vol
0.06
0.08
0.1
0.12
0.14
0.16
0.18
0.2Fitted Surface and Data
(b) Data and calibrated fit.
Figure 4: Data and calibrated fit on November 03, 2015. In Figure (a), the plus signs in thetop plot are the slope coefficients ai of LMMR fitted in the first step of the regression. The solidline is the straight line (aε + aδτ) fitted in the second step of the regression. The bottom plotshows the corresponding picture for the skew intercepts bi fitted to the straight line b∗+ bδτ . InFigure (b), the plus signs are from S&P 500 data, and the lines are the formula (32) estimatedfrom the full fast and slow factor fit.
19
LMMR-0.5 0 0.5
Impl
ied
Vol
0.05
0.1
0.15
0.245 days
LMMR-0.5 0 0.5
Impl
ied
Vol
0.08
0.1
0.12
0.14
0.16
0.1873 days
LMMR-0.2 0 0.2
Impl
ied
Vol
0.1
0.12
0.14
0.16
0.18108 days
LMMR-0.2 0 0.2
Impl
ied
Vol
0.1
0.12
0.14
0.16
0.18136 days
LMMR-0.1 0 0.1
Impl
ied
Vol
0.12
0.14
0.16
0.18226 days
LMMR-0.05 0 0.05 0.1
Impl
ied
Vol
0.13
0.14
0.15
0.16
0.17
0.18318 days
LMMR-0.05 0 0.05
Impl
ied
Vol
0.14
0.15
0.16
0.17
0.18
0.19409 days
LMMR-0.05 0 0.05
Impl
ied
Vol
0.15
0.16
0.17
0.18
0.19444 days
LMMR-0.05 0 0.05
Impl
ied
Vol
0.16
0.17
0.18
0.19591 days
LMMR-0.02 0 0.02 0.04
Impl
ied
Vol
0.16
0.17
0.18
0.19
0.2773 days
Figure 5: Implied volatility fit to S&P 500 index options on November 03, 2015. In thisFigure, we present the fit of Equation (38) to implied volatility data. Note that this is the resultof a single calibration to all maturities and not a maturity-by-maturity calibration. Each panelshows the fit for a particular time to maturity.
20
on November 03, 2015.
Table 2: Estimated values of the group market parameters
σ∗ aε aδ b∗ bδ V δ0 V δ
1 V ε3
Model 1 0.1257 -0.1223 0.0308 0.0313 -0.0019Model 2 0.1434 -0.1192 0.1426 -0.0004Model 3 0.1318 -0.1383 -0.2049 0.1311 0.0274 0.0285 -0.0035 -0.0003
Note: In this table, we report the estimated parameters from the fit of implied volatilitiesto log-moneyness to maturity ratio for every model on November 03, 2015. For Models 1and 2, we fit Equations (38) and (39) to data using ordinary least squares regression. Theestimated parameters for Model 3 result from the fit of Equation (32) to data following the2-step procedure suggested by Fouque, Papanicolaou, Sircar, and Solna (2011). The lastthree columns were obtained through the inversion of Equations (34) to (37).
In Table 3, we compare the adjustment of the three models presented in this section tothe market prices using the absolute percentage error |Pobserved − Pmodel|/Pobserved, wherePmodel is the price given by the model and Pobserved is the observed market price of theoption. Since the volatility used to obtain the Black-Scholes prices differs in every model,we are not able to compare them unless we use the same volatility in the Black-Scholesformula across all the models.
In order to compare the corrected prices for all the three models, we calculate thehistorical volatility of the returns of the S&P 500. First, we obtain a time-series ofhistorical volatilities on this index by using a rolling-window of 63 days (three-months)during one year. Then, we use the average of this time-series as our estimator of thehistorical volatility, σ. For the time period from October 23, 2014, to October 23, 2015,the estimated historical volatility in the S&P 500 was 14%. In Panel A, we report thepercentage of times in which the model in question presented the best adjustment whencompared to the observed market prices. Since our model provides only corrected callprices, we use the put-call parity in order to obtain the put prices with the same strikeand time to maturity. Note that, for call prices, Model 1 presents the best adjustmentto data, while for put prices, Model 3 provides the best fit, followed by the Black-Scholesmodel.
In Panel B, we compare the adjustment of Model 1 to data. To obtain the Black-Scholes prices, we use the effective volatility σ∗ obtained from Equation (34) when wefit Equation (38) to data. Note that Model 1 presents the best adjustment in most ofthe time for put prices. When we look at call prices, we find that the Black-Scholesmodel performs better most of the time. The same result is obtained in Panel D when
21
we compare the adjustment of Model 3 to data, although the magnitudes are different.Finally, in Panel C we compare the adjustment of Model 2 to data. In this case, theBlack-Scholes model performs better when we look at put prices. For call prices, Model2 presents the best adjustment.
Notice that, when we compare the prices given by every model to the Black-Scholesprices, we find opposite results for call and put prices. This may be an evidence that theput-call parity does not hold, even in perfectly liquid markets as the American market.
Table 3: Comparison of prices generated by the models and observed market prices
Model Call Prices Put Prices
Panel A: Model Comparison using Historical Volatility
Model 1 36.3636 13.5802Model 2 16.2338 17.9012Model 3 29.8701 37.6543Black-Scholes 17.5325 30.8642
Panel B: Model 1 versus Black-Scholes
Model 1 4.5455 96.2963Black-Scholes 95.4545 3.7037
Panel C: Model 2 versus Black-Scholes
Model 2 56.4935 47.5309Black-Scholes 43.5065 52.4691
Panel D: Model 3 versus Black-Scholes
Model 3 31.1688 71.6049Black-Scholes 68.8312 28.3951
Note: In this table, we compare the adjustment of each model to the observed market pricesof the options. In Panel A, we compute the Black-Scholes prices using the historical volatilityσ = 0.14. From Panel B to Panel D, we compute the Black-Scholes prices using the effectivevolatility σ∗ estimated from every model. All the values are expressed in percentage terms.
3.4 Simulating Derivative Prices
The goal of this section is to present a numerical experiment using synthetic data (i.e.,synthetic option prices) in order to show the validity of the asymptotic theory of Fouque,Papanicolaou, and Sircar (2000) under different scenarios. We test the behavior of thealgorithm suggested by these authors when the estimated model is the one generating the
22
data and also when we try to estimate using the wrong model (wrong number of scales,for example). We will also test the performance of the algorithm under different samplesizes and parameter settings.
We will denote as N∗ the true number of volatility scales that generate the data, whileN will be the value with which we do the estimation. We will generate synthetic dataunder four possible parameter settings:
• Setting A: The first data set will be obtained by generating the option prices withthe Black–Scholes formula (29), assuming a constant volatility.
• Setting B: The second data set will be obtained by generating the option pricesassuming that Model 1 is the Data Generating Process (DGP) and N∗ = 1.
• Setting C : The third data set will be obtained by generating the option prices usingModel 2 as the GDP, which assumes that N∗ = 1.
• Setting D: Finally, the fourth data set will be obtained by generating the optionsprices with Model 3 as the DGP and N∗ = 2.
In the estimation, we will use N = 1 or N = 2, with two different sample sizes: T = 500or T = 1000. Note that any combination of the above forms an experiment.
4 Application: Pricing an Exotic Derivative
In this section, we present an application of the model for pricing a binary or digitaloption with no exercise option, using the calibration obtained in Section 3.3 for optionsin and close to the money.
Consider a cash-or-nothing call that pays a fixed amount Q on the date T if XT > K,and zero otherwise. Its payoff function is
h(x) = Q1x>K, (42)
where 1A is the indicator function of a set A.
According to the perturbation theory developed in Fouque, Papanicolaou, Sircar, andSolna (2011), the stochastic volatility-corrected price is given by
P (t, x, z) = P0(t, x, z) + P1(t, x, z) = Qe−r(T−t)N(d2) + P1(t, x, z), (43)
23
whereP1(t, x, z) = (T − t)
(V δ
0∂
∂σ+ V δ
1 D1
(∂
∂σ
)+ V ε
3D1D2
)P ∗BS, (44)
and P ∗BS = Qe−r(T−t)N(d2).
Using the simulated prices from Section 3.4, the goal of this section is to obtain theprices of this exotic derivative under the four different settings considered in the lastsection, and then compare the models across different scenarios.
5 Concluding Remarks
The main idea of the asymptotic models of Fouque, Papanicolaou, Sircar, and Solna(2011) is to use perturbation techniques to correct constant volatility models in order tocapture the effects of stochastic volatility. In this paper, we compare the performance ofthree different specifications of the Fouque, Papanicolaou, and Sircar (2000) stochasticvolatility model for pricing European call options with the Black-Scholes model. So far,we have found no evidence that these asymptotic models perform better than the Black-Scholes model. Moreover, we find opposite results when we apply this theory to pricingcall and put options, what may indicate that the put-call parity does not hold.
Following Fouque, Papanicolaou, and Sircar (2000), we derive a first-order asymptoticapproximation for European options under multiscale stochastic volatility models withfast and slow factors. The price approximation was translated to an implied volatilitysurface approximation which is linear in log-moneyness. We show that the extracted pa-rameters (group market parameters) are small, in accordance with the asymptotic analysisdeveloped by these authors.
We use S&P 500 data to demonstrate that the volatility of this index is driven by twodiffusions: one fast mean-reverting and one slow-varying. We find a fast time scale of 3.5days, and a slow time scale of 16.9 months. These time-scales are statistically significant,and therefore cannot be ignored in option pricing and hedging.
In what follows, we will perform the simulation study described in Section 3.4, andcompare the prices of exotic derivatives based on these simulated prices. Our goal isto demonstrate that the first-order approximation fits the data well across strikes andmaturities.
24
References
Alizadeh, S., M. W. Brandt, and F. X. Diebold. 2002. Range-based estimation of stochas-tic volatility models. Journal of Finance 06:1047–1092.
Andersen, T. G., and T. Bollerslev. 1997. Intraday Periodicity and Volatility Persistencein Financial Markets. Journal of Empirical Finance 4:115–158.
Black, F., and M. Scholes. 1973. The Pricing of Options and Corporate Liabilities. Journalof Political Economy 81:7637–659.
Chernov, M., R. Gallant, E. Ghysels, and G. Tauchen. 2003. Alternative models for stockprice dynamics. Journal of Econometrics 116:225–257.
Chronopoulou, A., and F. G. Viens. 2012. Estimation and pricing under long-memorystochastic volatility. Annals of Finance 8:379–403.
Cressie, N. 1985. Fitting variogram models by weighted least squares. J. Int. Ass. Math.Geol. 17:563–586.
Engle, R. F. 1982. Autoregressive conditional heteroskedasticity with estimates of thevariance of U.K. inflation. Econometrica 50:987–1008.
Engle, R. F., and A. Patton. 2001. What good is a volatility model? Quantitative Finance1:237–245.
Figlewski, S. 2009. Estimating the Implied Risk Neutral Density for the U.S. MarketPortfolio. In Volatility and Time Series Econometrics: Essays in Honor of Robert F.Engle. Oxford University Press.
Fouque, J. P., G. Papanicolaou, and K. R. Sircar. 2000. Derivatives in Financial Marketswith Stochastic Volatility. NY: Cambridge University Press.
Fouque, J. P., G. Papanicolaou, K. R. Sircar, and K. Solna. 2003a. Short time-scale inS&P 500 volatility. Journal of Computational Finance 6:1–23.
Fouque, J. P., G. Papanicolaou, K. R. Sircar, and K. Solna. 2003b. Singular Perturbationsin Option Pricing. SIAM Journal on Applied Mathematics 63:1648–1681.
Fouque, J. P., G. Papanicolaou, K. R. Sircar, and K. Solna. 2004. Maturity cycles inimplied volatility. Finance and Stochastics 8:451–477.
Fouque, J. P., G. Papanicolaou, R. Sircar, and K. Solna. 2003c. Short Time-Scale in S&P500 Volatility. Journal of Computational Finance 6:1–23.
25
Fouque, J. P., G. Papanicolaou, R. Sircar, and K. Solna. 2011. Multiscale StochasticVolatility for Equity, Interest-Rate and Credit Derivatives. USA: Cambridge UniversityPress.
Gatheral, J. 2006. The Volatility Surface: a Practitioner’s Guide. Inc: John Wiley andSons.
Heston, S. L. 1993. A Closed-Form Solution for Options with Stochastic Volatility withApplications to Bond and Currency Options. The Review of Financial Studies 6:327–343.
Hillebrand, E. 2005. Neglecting parameter changes in GARCH models. Journal of Econo-metrics 1-2:121–138.
LeBaron, B. 2001. Stochastic volatility as a simple generator of apparent financial powerlaws and long memory. Quantitative Finance 1:621–631.
Machuca, G. E. G. 2010. Volatilidade Estocástica Multiescala: Modelagem, Estimação eAplicação com Preços da Petrobras. Master Thesis, IMPA.
Merton, R. C. 1969. Lifetime portfolio selection under uncertainty: The continuous-timecase. Review of Economics and Statistics 51:247–257.
Politis, D. N., and J. P. Romano. 1992. A circular block-resampling procedure for sta-tionary data. In R. LePage and L. Billard (eds.), Exploring the Limits of Bootstrap,pp. 263–270. New York: John Wiley.
Politis, D. N., and H. White. 2004. Automatic block-length selection for the dependentbootstrap. Econometric Rev. 23:53–70.
26
Appendix
A Properties of the Variogram Estimator
In this section, we will study some properties of the variogram estimator given byEquation (26). In order to do that, consider the following lemmas:
Lemma 1. Let s < t, then
E[
(Yt − E[Yt]) (Ys − E[Ys])]
= ν2Y e−α(t−s)
(1− e−2αs
), (A.1)
E[
(Zt − E[Zt]) (Zs − E[Zs])]
= ν2Ze−δ(t−s)
(1− e−2δs
). (A.2)
Proof.
E[
(Zt − E[Zt]) (Zs − E[Zs])]
= E(
νY√
2α∫ t
0e−α(t−u)dW (1)
u
)(νZ√
2α∫ s
0e−α(s−u)dW (1)
u
)= 2αν2
Y e−α(t+s)E
(∫ t
0eαudW (1)
u
)(∫ s
0eαu)dW (1)
u
)= 2αν2
Y e−α(t+s)E
(∫ s
0eαudW (1)
u
)(∫ s
0eαudW (1)
u
)+
+ 2αν2Y e−α(t+s)E
(∫ t
seαudW (1)
u
)(∫ s
0eαudW (1)
u
)= 2αν2
Y e−α(t+s)E
(∫ t
0eαudW (1)
u
)(∫ s
0eαudW (1)
u
)= 2αν2
Y e−α(t+s)E
(∫ t
0eαudW (1)
u
)2 = 2αν2
Y e−α(t+s)E
(∫ t
0e2αudt
)2 = ν2
Y e−α(t+s)(e2αs−1)
= ν2Ze−δ(t−s)
(1− e−2δs
).
Similarly, we can get (A.1).
By using the previous Lemma, we can conclude that
E[
log f1(Yj) log f1(Y0)]≈ ν2
Y e−αj∆t,
E[
log f1(Y )2]≈ ν2
Y .(A.3)
27
Similarly, we haveE[
log f1(Zj) log f1(Z0)]≈ ν2
Ze−αj∆t,
E[
log f1(Z)2]≈ ν2
Z ,(A.4)
where Y and Z denote the asymptotic distributions of the OU processes given by (1).
Lemma 2. For the model presented in this paper, if s < t, then
E[
(Yt − E[Yt]) (Zs − E[Zs])]
= 2√α√δ
α + δρ1,2νY νZ
(e−α(t−s) − e−αte−δs
). (A.5)
Otherwise, if s > t,
E[
(Yt − E[Yt]) (Zs − E[Zs])]
= 2√α√δ
α + δρ1,2νY νZ
(e−δ(s−t) − e−αse−δt
). (A.6)
Proof. Suppose that s < t, then
E[
(Yt − E[Yt]) (Zs − E[Zs])]
= E(
νY√
2α∫ t
0e−α(t−u)dW (1)
u
)(νZ√
2δρ1,2
∫ s
0e−δ(s−u)dW (1)
u
)+ E
(νY√
2α∫ t
0e−α(t−u)dW (1)
u
)(νZ√
2δ√
1− ρ1,2
∫ s
0e−δ(s−u)dW (2)
u
)= E
(νY√
2α∫ s
0e−α(t−u)dW (1)
u
)(νZ√
2δρ1,2
∫ s
0e−δ(s−u)dW (1)
u
)+ E
(νY√
2α∫ s
0e−α(t−u)dW (1)
u
)(νZ√
2δ√
1− ρ1,2
∫ s
0e−δ(s−u)dW (2)
u
)=√
2ανY√
2δνZ ρ1,2e−αte−δs
∫ s
0e(α+δ)udu
= 2√α√δ
α + δνY νZ ρ1,2e
−αte−δs(e(α+δ)s − 1
)= 2√α√δ
α + δρ1,2νY νZ
(e−δ(s−t) − e−αse−δt
).
By following Fouque, Papanicolaou, Sircar, and Solna (2011) and using the previousLemma, we get
E[
log f1(Y ) log f2(Z)]≈ 2√α√δ
α + δρ1,2νY νZ , (A.7)
E[
log f1(Yj) log f2(Z0)]≈ 2√α√δ
α + δρ1,2νY νZe
−αj∆t, (A.8)
E[
log f1(Y0) log f2(Zj)]≈ 2√α√δ
α + δρ1,2νY νZe
−δj∆t. (A.9)
28
Proposition 1.
E
(Fn+j − Fn)2≈ 2c2
Y + 2ν2Y (1− e−αj∆t) + 2ν2
Z(1− e−δj∆t)+
+ 4√α√δ
α + δρ1,2νY νZ
(2− e−αj∆t − e−δj∆t
).
(A.10)
Proof.
E
(Fn+j − Fn)2
= E
(Fj − F0)2
= E[ (
log f1(Yj)− log f1(Y0))
+(log f2(Zj)− log f2(Z0)
) ]+
+ E[
(log | εj | − log | ε0 |)]
= E (
log f1(Yj)− log f1(Y0))
+ E (
log f2(Zj)− log f2(Z0))
+
+ E
(log | εj | − log | ε0 |)
+
+ 2E (
log f1(Yj)− log f1(Y0)) (
log f2(Zj)− log f2(Z0))
= 2E
log f1(Y )2− 2E
log f1(Yj) log f1(Y0)
+ 2E
log f2(Z)2
− 2E
log f2(Zj) log f2(Z0)
+ 2Var (log (ε)) + 4E
log f1(Y0) log f2(Z)
− 2E
log f1(Yj) log f2(Z0)
− 2E
log f1(Y0) log f2(Z)
≈ 2c2
Y + 2ν2Y
(1− e−αj∆t
)+ 2ν2
Z
(1− e−δj∆t
)+ 8√α√δ
α + δρ1,2νY νZ
− 4√α√δ
α + δρ1,2νY νZe
−αj∆t − 4√α√δ
α + δρ1,2νY νZe
−δj∆t
≈ 2c2Y + 2ν2
Y
(1− e−αj∆t
)+ 2ν2
Z
(1− e−δj∆t
)+
+ 4√α√δ
α + δρ1,2νY νZ
(2− e−αj∆t − e−δj∆t
)
In the previous proof, we used Equations (A.3), (A.4), and Equations (A.7) to (A.9).Since we assume that δ 1/ε, by Equation (A.10) we have that
√α√δ
α + δ≈√α√δ
α≈√δ√α≈ 0. (A.11)
Therefore, the previous equation allows us to estimate the variogram model (24) byusing two reduced models, one for each time scale.
29
B First-Order Perturbation Theory
In this section, we derive the asymptotic approximation for options prices of Fouque,Papanicolaou, and Sircar (2000). In order to derive the corrected prices, as the termsassociated with δ are small when δ → 0 and give rise to a regular perturbation problem,we first expand P ε,δ in powers of
√δ:
P ε,δ = P ε0 +√δP ε
1 + δP ε2 + . . . . (B.1)
We insert the above expansion in equation (9) (both the PDE and the terminal condi-tion), grouping generators in term of powers of δ in the form
(1εL0 + 1√
εL1 + L2
)P ε
0 +√δ
(
1εL0 + 1√
εL1 + L2
)P ε
1 +(M1 + 1√
εM3
)P ε
0
+ . . . = 0.
(B.2)
Write Lε,δ and P ε,δ as
Lε,δ = Lε +√δMε + δM2, (B.3)
P ε,δ =∑j≥0
(√δ)jP ε
j , (B.4)
where
Lε = 1εL0 + 1√
εL1 + L2, (B.5)
Mε = 1√εM3 +M1, (B.6)
P εj =
∑i≥0
(√ε)iPi,j. (B.7)
Inserting (B.1) in (9) and collecting term of like-powers of√δ, we find that the lowest
order equations of the regular perturbation expansion are
O(1) : 0 = LεP ε0 , (B.8)
O(√δ) : 0 = LεP ε
1 +MεP ε0 , (B.9)
O(δ) : 0 = LεP ε2 +MεP ε
1 +M2Pε0 . (B.10)
Therefore, we can choose P ε0 and P ε
1 to satisfy the following PDEs with terminal con-
30
ditions (
1εL0 + 1√
εL1 + L2
)P ε
0 = 0,
P ε0(T, x, y, z) = h(x).
(B.11)
(
1εL0 + 1√
εL1 + L2
)P ε
1 = −(M1 + 1√
εM3
)P ε
0 ,
P ε1(T, x, y, z) = 0,
(B.12)
where the terminal payoff is assigned to the independent term P ε0 . Next, we expand P ε
0
and P ε1 in powers of
√ε.
B.0.1 Zeroth-Order Approximation P0
From a fast factor expansion of equation (9), we will find the zeroth order term P0,0,in our approximation (18).
Consider the expansion of P ε0 in powers of
√ε:
P ε0 = P0 +
√εP1,0 + εP2,0 + ε3/2P3,0 + . . . . (B.13)
By inserting (B.13) in (9) and rearranging terms, we get
1εL0P0 + 1√
ε(L0P1,0 + L1P0) + (L0P2,0 + L1P1,0 + L2P0) +
+√ε (L0P3,0 + L1P2,0 + L2P1,0) + . . . = 0.
(B.14)
By collecting term of like-powers of√ε, we get
O(1ε
): L0P0 = 0, (B.15)
O(
1√ε
): L0P1,0 + L1P0 = 0, (B.16)
O (1) : L0P2,0 + L1P1,0 + L2P0 = 0, (B.17)
O(√
ε)
: L0P3,0 + L1P2,0 + L2P1,0 = 0. (B.18)
We see from (11) and (12) that all terms in L0 and L1 take derivatives with respectto y. Thus, if we choose P0,0 and P1,0 to be independent of y, the above equations will
31
automatically be satisfied. Hence, we seek solutions of the form:
P0,0 = P0,0(t, x, z), (B.19)
P1,0 = P1,0(t, x, z). (B.20)
Note that Equation (B.17) is a Poisson equation for P2,0 with respect to y. For therebe a solution, L2P0 must satisfy the solvability condition:5
〈L2P0〉 = ∂P0
∂t+ 1
2 σ2x2∂
2P0
∂x2 + r
(x∂P0
∂x− P0
)= 0, (B.21)
which is the Black-Scholes pricing operator L(σ(z)) with effective averaged volatility σ(z):
σ(z) = 〈f 2(·, z)〉 =∫f 2(y, z)Φ(dy), (B.22)
where the level z of the slow factor appears as a parameter. Therefore, the leading-order term P0(t, x, z) = PBS(t, x; σ(z)) is the Black-Scholes price at volatility σ(z), whichsatisfies the following PDE and terminal condition:
LBS(σ(z))PBS = 0,
PBS(T, x; σ(z)) = h(x).(B.23)
B.0.2 Fast Time Scale Correction P ε1,0
The order√ε in (B.18) give the following Poisson equation for P3,0:
L0P3,0 + L1P2,0 + L2P1,0 = 0. (B.24)
The solvability condition for this equation is given by
〈L2P1,0 + L1P2,0〉 = 0. (B.25)
5Note that Equations (B.17) and (B.18) are Poisson equations of the form
0 = L0 + χ,
By the Fredholm alternative, the equation above, which is a linear ODE in y, admits a solution P inL2(Φ) only if the following solvability, or centering, condition holds:
〈χ〉 ≡∫χ(y)Φ(dy) = 0,
where Φ is the invariant distribution of Y .
32
Let Dk be defined asDk = xk
∂k
∂xk, (B.26)
then, introducing a solution φ(y, z) to the Poisson equation L0φ = f 2 − 〈f 2〉, we deducethe following expression for P2,0:
P2,0(t, x, y, z) = −12φ(y, z)D2P0,0(t, x, y, z) + c(t, x, z), (B.27)
up to an additive function c(t, x, z) which does not depend on y. Then, by using Equations(11) and (12) and the fact that P1,0 does not depend on y, we find that P ε
1 satisfied thefollowing PDE: LBS(σ(z))P ε
1 = AεPBS;
P ε1(T, x; z) = 0;
(B.28)
where the z-dependent operator Aε =√ε〈L1L−1
0 (L2−〈L2〉)〉. Using Equation (B.17), wecan rewrite Aε as
Aε = −V ε3 (z)D1D2 − V ε
2 (z)D2, (B.29)
where the two group parameters V ε2 (z) and V ε
3 (z) are given by
V ε2 (z) = −ρ1
√ε
2 〈βf(·, z)∂φ∂y
(·, z)〉, (B.30)
V ε3 (z) = −
√ε
2 〈βΛ1(·, z)∂φ∂y
(·, z)〉. (B.31)
With the previous notation, we have
LBS(σ(z)) = ∂
∂t+ 1
2 σ(z)2D2 + r(D1 − ·). (B.32)
Therefore, the first-order fast scale correction term P1,0(t, x, z) is given in terms of PBSby
P ε1,0(t, x, z) = −(T − t)AεPBS(t, x; σ(z)). (B.33)
B.0.3 Slow Time Scale Correction P δ0,1
Proceeding as in the last section, we insert expansions (B.7) into (B.9) and collect termsof like-powers of
√ε. We find that the lowest order equations of the singular perturbation
equation are given by
33
O(√
δ
ε
): L0P0,1 = 0, (B.34)
O(√
δ√ε
): L0P1,1 + L1P0,1 +M3P0,0 = 0, (B.35)
O(√
δ)
: L0P2,1 + L1P1,1 + L2P0,1 +M3P1,0 +M1P0,0 = 0, (B.36)
O(√
δ√ε)
: L0P3,1 + L1P2,1 + L2P1,1 +M3P2,0 +M1P1,0 = 0. (B.37)
Note thatM3P0,0 = 0 sinceM3 take derivatives in y and P0,0 is independent of y. Asall terms in L0 and L1 take derivatives in y, we seek solutions P0,1 and P1,1 independentof y.
Equations (B.36) and (B.37) are Poisson equations in y with respect to P2,1 and P3,1,respectively. Using the centering condition yields
O(√
δ)
: 〈L2〉P0,1 + 〈M1〉P0,0 = 0, (B.38)
O(√
δ√ε)
: 〈L1P2,1〉+ 〈L2〉P1,1 + 〈M3P2,0〉+ 〈M1〉P1,0 = 0. (B.39)
Again, since P0,0 and P0,1 do not depend on y, condition (B.38), after multiplying by√δ is given by
〈L2〉P δ0,1 = −
√δ〈M1〉P0,0.
This is an inhomogeneous Black-Scholes partial differential equation for P δ0,1. We can
rewrite the expression above as
√δ〈M1〉P0,0 ≡ 2AδPBS, (B.40)
where we define the operator
Aδ = V δ0 (z) ∂
∂σ+ V δ
1 (z)D1∂
∂σ, (B.41)
and the group parameters (V δ0 (z), V δ
1 (z)) by
V δ0 (z) = −g(z)
√δ
2 〈Λ2(·, z)〉σ′(z), (B.42)
V δ1 (z) = −ρ2g(z)
√δ
2 〈f(·, z)〉σ′(z). (B.43)
34
Therefore, the first order scale correction P δ0,1 is the unique classical solution to the
problemLBS(σ(z))P δ
0,1 = −2AδPBS,
P δ0,1(T, x, z) = 0.
(B.44)
The first-order slow scale correction P δ0,1(t, x, z) is given in terms of PBS(t, x; σ(z)) by
P δ0,1 = (T − t)APBS. (B.45)
By replacing Equations (B.45) and (B.33) into (18), we get the following first-orderapproximation for a European-style option price:
P ε,δ = PBS + (T − t)(V δ
0 (z) ∂∂σ
+ V δ1 D1
(∂
∂σ
))PBS+
+ (T − t) (V ε2 (z)D2 + V ε
3 (z)D1D2))PBS.(B.46)
35