Table Of ContentModeling stochastic skew of FX options using SLV models
with stochastic spot/vol correlation and correlated jumps
A. Itkina
aTendon School of Engineering, New York University,
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12 Metro Tech Center, RH 517E, Brooklyn NY 11201, USA
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Abstract
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It is known that the implied volatility skew of FX options demonstrates a
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stochastic behavior which is called stochastic skew. In this paper we create
] stochastic skew by assuming the spot/instantaneous variance correlation to
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be stochastic. Accordingly, we consider a class of SLV models with stochastic
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correlation where all drivers - the spot, instantaneous variance and their cor-
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relation are modeled by L´evy processes. We assume all diffusion components
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f to be fully correlated as well as all jump components. A new fully implicit
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q splitting finite-difference scheme is proposed for solving forward PIDE which
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isusedwhencalibratingthemodeltomarketpricesoftheFXoptionswithdif-
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ferent strikes and maturities. The scheme is unconditionally stable, of second
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order of approximation in time and space, and achieves a linear complexity in
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2 each spatial direction. The results of simulation obtained by using this model
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demonstrate capacity of the presented approach in modeling stochastic skew.
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Keywords: stochastic correlation, FX options, stochastic skew, SLV models,
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1 correlated jumps, 3D PIDE, finite-difference, forward equations, fully
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implicit splitting scheme, unconditional stability
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1 JEL: C6, C61, G17
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1. Introduction
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It is known that in the FX world skew of the implied volatility of the
FX options demonstrates some kind of stochastic behavior. For instance,
Carr and Wu (2007) report that for two currency pairs (which are the U.S.
dollar prices of the Japanese yen and the British pound) by analyzing the
∗Corresponding author
Email address: [email protected] (A. Itkin )
Preprint submitted to January 20, 2017
market implied volatilities of option series they find the slope of the smile to
greatly vary over time. Namely, the sign of the slope switches several times
in the considered sample. Therefore, although the risk-neutral distribution
of the currency return exhibits persistent fat-tail behavior, the risk-neutral
skewness of the distribution experiences strong time-variation which can be
very positive or very negative on any given date.
To capture this effect when doing option pricing, new models are required
because, as mentioned in Carr and Wu (2007), the jump-diffusion stochastic
volatility models like that in Bates (1996), are able to capture the average
shape of implied volatility smiles and the time-variation in their levels but
cannot generate strong time variation in the risk-neutral skewness of currency
returns. We can mention at least two approaches that are used to attack this
problem.
The firstone isproposedin Carrand Wu (2004) whereskew is modeled by
separate the up jumps and the down jumps in spot. These jumps are modeled
by two L´evy processes. The stochastic volatility and skewness are intro-
duced by a separate time change in each L´evy component. The stochastic
variation in the relative proportion of up and down jumps generates stochas-
tic variation in the risk-neutral skewness of currency returns. The model is
analytically tractable for pricing European vanilla options. However, as per
Higgins (2014) it often requires jumps in spot that are much larger than those
actually experienced. Therefore, it can misprice barrier options if the proba-
bility of spot jumping through the barrier is significant Also pricing of exotic
options requires numerical methods, see e.g., Itkin and Carr (2011).
Anotherapproachistorandomizethosemodelparametersthatgovernthe
risk-neutral skewness. In jump-diffusion models they are the mean jump size
and the correlation coefficient between the currency return and the stochastic
variance process. The first parameter governs the risk-neutral skewness at
short maturities, and the second one - at long maturities. Therefore, in a
set of papers the correlation between the spot and instantaneous variance is
consideredtobestochasticanddrivenbyaseparatestochasticprocess,seevan
Emmerich (2006); Da Fonseca et al. (2007); Ma (2009); Ahdida and Alfonsi
(2013); Higgins (2014); Zetocha (2015a) and references therein. Moreover,
in Zetocha (2015b) where a pricing problem is considered for basket equity
options, the standard correlation model, which is based on the Jacobi process,
is extended by adding jumps in the correlation process. This is done based on
theanalysisofthemarketdataforbasketoptionswhichdemonstratesthatthe
jump to high level of correlation occurs after several down moves that drag
the basket below some comfort zone of normality. The jumps are modeled
in a simplified way by adding to the Jacobi process a term proportional to
2
dN with N being a Poisson process whose intensity depends on the basket
performance and time.
2. Modeling stochastic correlation
We mention two basic approaches to model the stochastic correlation ρ .
t
Considerfirstonlythediffusionprocesses. Sincebydefinitionρ isboundedin
t
[−1,1], thefirstapproachwouldbetomodelitasaboundeddiffusion. Oneof
the popular choices is the Jacobi process used, e.g., in van Emmerich (2006);
Zetocha (2015a). However, this requires the process to be mean-reverting,
while it is not obvious why the correlation should be mean-reverted.1 Other
bounded diffusions can also be used.
This approach is tractable for European vanilla options, but not for the
exoticoptions. Becauseofthat,inHiggins(2014)ananalyticalapproximation
for barrier and one-touch options is developed which is based on semi-static
vega replication. It is shown that this approximation works well in markets
where the risk neutral drift is modest.
The second approach, van Emmerich (2006), uses a transformation ρ =
t
g(X ) where X is an arbitrary diffusion process (perhaps driftless), and func-
t t
tion g(X ) transforms the support of the X outcomes to [−1,1]. Popular
t t
transformationsarehyperbolictangent, Tengetal.(2016), normalCDF,Carr
(2012), normalized inverse tangent, van Emmerich (2006), etc. In principle,
any continuous map R → [−1,1] can be used for this purpose, so the real
choice becomes a matter of having some additional properties, e.g., tractabil-
ity, tobetakenintoaccount. Also, despitethisapproachisabitlessintuitive,
itopensthedoortovarioussophisticatedmodelsofthestochasticcorrelations.
One of such possible extensions would be a natural idea to consider the
underlyingprocessX tobeajump-diffusionora L´evyprocess. Tothebestof
t
our knowledge, there is the only attempt to model correlation by introducing
jumps which is presented in Zetocha (2015a). However, this model is very
simplisticandempirical. Itisintroducedbasedontheanalysisofsomemarket
data on baskets of stocks, with an observation that the jump to high level of
correlation might happen after several down moves that put the basket below
some level of normality. Therefore, when modeling these baskets using the
stochasticcorrelationmodel,Zetocha(2015a)definesρ tofollowtheextended
t
1Perhaps,theonlyreasonwouldbeifonewantsanexpectationofthestochasticcorrela-
tion to be stationary at long time horizons. However, we don’t know any paper that would
justified this based on the available market data.
3
Jacobi process
(cid:112)
dρ = α(ρ−ρ )+β (1−ρ )(ρ −ρ )dW +(1−ρ )dN ·1 . (1)
t t t t min t t t It<I0
Hearα,β,ρ aresomeconstantmodelparameters,W isthestandardBrow-
min t
nian motion, N is the Poisson jump process with the intensity λ given by the
t
following expression
(cid:18) 1−I (cid:19)κ
t
λ = λ , (2)
0
1−I (t)
c
where λ ,κ are constants, I (t) is the critical level defined as n standard
0 c
deviations of the basket performance I
t
√
Ic(t) = F(t)e−12σ2t−nσ t, (3)
Here σ is the at-the-money volatility of the basket at time t, and F(t) is the
basket forward at time t. So the jumps can occur if the basket performance
I hits the level I = I at t > 0. Otherwise, if I −I is positive the jump
t t 0 t 0
doesn’t occur. The jump size (1−ρ ) in Eq.(1) is chosen in such a way, that
t
if the jump occurs, ρ always jumps from ρ to 1 (full correlation).
t t−
Obviously, this model is empirical with a lot of parameters introduced,
which financial meaning is not entirely transparent. Therefore, calibration
of such a model could be a hard issue. Also the assumption that the jump
always results in the full correlation to be reached immediately after the jump
occurs, seems to not be justified at all.
Therefore, in this paper we utilize a different approach. We use a general
L´evymodelsframeworktointroducejumps, seee.g., ContandTankov(2004).
Since a general L´evy process is not bounded, we use a certain transformation
as this is described in above, to bound it to [−1,1]. Once this is done, we
considerthreestochasticprocesses: thespot, instantaneousvarianceandtheir
correlation each driven by some L´evy process. We assume that the Brownian
components of each process are correlated, and the jump component are cor-
related similar to how this is done in Itkin and Lipton (2015); Itkin (2016b).
However, the Brownian motions and the jump components remain uncorre-
lated. This model has a clear financial meaning, and can be calibrated in
few steps. For instance, idiosyncratic components of jumps can be calibrated
separately to the corresponding market; the local volatility functions can be
calibrated to a set of FX European vanilla options; then stochastic volatility
and correlation parameters (the diffusion part) of the model can be calibrated
to FX exotic options which demonstrate a smile. Finally, the common jumps
parameters of the model can be calibrated to the implied volatility skew of
the exotic options. This procedure runs in a loop until it converges.
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The main contribution of the paper consists in several results. First, a
SLV model for pricing exotic FX options which demonstrate the existence
of stochastic skew is proposed which assumes the spot/InV correlation to be
stochastic. Moreover, the model also includes correlated L´evy jumps in each
ofthestochasticdrivers. Second, anewfullyimplicitfinite-differencesplitting
schemeisproposedtosolveaforwardKolmogorovequationwhichisofsecond
order in both temporal and each spatial dimension, is unconditionally stable
and preserves both positivity and norm of the solution while complexity-wise
is linear in the number of grid nodes in each spatial dimension. We also
formulate and prove a theorem that the proposed discretization preserves the
norm of the solution.
The rest of the paper is organized as follows. In Section 3 we formu-
late the model which considers stochastic correlation of Brownian drivers and
deterministic correlation of L´evy jump processes. Section 4 discusses the
corresponding backward PIDE and boundary and initial conditions. In Sec-
tion 5 same consideration is given to the Forward Kolmogorov equation. In
Section 6 we introduce a new fully-implicit finite-difference scheme for this
forward PIDE and analyze its properties. Results of some numerical experi-
ments obtained by using this scheme are provided in Section 7. An Appendix
provides a detailed derivation of the forward FD scheme.
3. Model
WeconsideranLSVmodelwithstochasticspot/InVcorrelationandjumps
by introducing stochastic dynamics for variables S ,v ,R . Here S is the spot
t t t t
price, v is the instantaneous variance, and
t
R = arctanh(ρ ). (4)
t t
In other words, the stochastic correlation ρ is represented as ρ = tanh(R ).
t t t
Therefore, since R ∈ R, this map guarantees that ρ ∈ [−1,1].
t t
Similar to Itkin (2016b), we consider a model where the stochastic SDE
for each variable S ,v ,R includes both diffusion and jumps components, as
t t t
follows:
√
dS = (r −r )S dt+Σ (S ,t)Sc v W +S dL , (5)
t d f t s t t t s,t t S,t
dv = κ (t)[θ (t)−v ]dt+ξ vaW +v dL ,
t v v t v t v,t t v,t
dR = κ (t)(θ (t)−R )dt+ξ W +dL ,
t r r t r r,t r,t
subject to the following initial conditions
(cid:12) (cid:12) (cid:12)
S (cid:12) = S , v (cid:12) = v , R (cid:12) = R = arctanh(ρ ). (6)
t(cid:12) 0 t(cid:12) 0 t(cid:12) 0 0
t=0 t=0 t=0
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Here r ,r are the domestic and foreign interest rates, t is the time, Σ is the
d f s
local volatility function, W ,W ,W are the correlated Brownian motions,
s,t v,t r,t
κ ,θ ,ξ are the mean-reversion rate, mean-reversion level and volatility of
v v v
volatility (vol-of-vol) for the instantaneous variance v ,κ ,θ ,ξ are the corre-
t r r r
sponding parameters for R , 0 ≤ a < 2, 0 ≤ c < 2 are some constants (power
t
constants) which are introduced to add an additional flexibility to the model
as compared with the popular Heston (α = 0.5), lognormal (α = 1) and 3/2
(α = 1.5) models. As mentioned in Itkin (2016b), one has to be careful if she
wants to determine these parameters by calibration. This is because having
both the vol-of-vol and the power constant in the same diffusion term brings
an ambiguity into the calibration procedure. However, it can be resolved
if some additional financial instruments are used in calibration, e.g., exotic
option prices are combined with the variance swaps prices, see Itkin (2013).
In the last equation of Eq.(5) the drift term is introduced to be mean-
reverting. However, this might not be necessary2. Therefore, when doing so
we rely on calibration of the model to the market data. In other words, if the
data exhibits mean-reversion, we expect the mean-reversion level θ (t), found
r
by the calibration, to differ from zero. On contrary, if θ (t) = 0, there is no
r
mean-reversion, and the sign of the drift term depends on the sign of κ (t).
r
Processes L ,L ,L are pure discontinuous jump processes with gen-
s,t v,t r,t
erator A
(cid:90)
(cid:0) (cid:1)
Af(x) = f(x+y)−f(x)−y1 µ(dy),
|y|<1
R
with µ(dy) be a L´evy measure, and
(cid:90)
eyµ(dy) < ∞.
|y|>1
Atthis stage, thejumpmeasureµ(dx)isleft unspecified, so alltypesofjumps
includingthosewithfiniteandinfinitevariation,andfiniteandinfiniteactivity
could be considered here.
The last line of Eq.(5) is a combination of the Hull-White model, Hull and
White(1990),forthediffusioncomponentwitha(arithmetic)Levyprocessfor
the jump component. As such, this model allows both positive and negative
values of R .
t
3.1. Correlations
Below for a better transparency we consider three cases.
2Or, as it was already mentioned, there is no evidence that the stochastic correlation
should be mean-reverting.
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No jumps. Consider first the case when there is no jumps in Eq.(5). Then
(cid:104)dW dW (cid:105) = ρ dt, (7)
s,t r,t (cid:101)sr
(cid:104)dW dW (cid:105) = ρ dt,
v,t r,t (cid:101)vr
(cid:104)dWs,tdWv,t(cid:105) = ρ(cid:101)tdt = tanh(R(cid:101)t),
whereweusethesymbol tomarkparametersandvariablesforapurediffusion
(cid:101)
case. The last line in Eq.(7) is explained in detail in van Emmerich (2006). It
means that
(cid:20)(cid:90) t (cid:21)
E[W W ] = E ρ ds (cid:54)= ρ t
s,t r,t (cid:101)s (cid:101)t
0
as it would be if ρ is constant.
(cid:101)t
Weassumethatρ isconstant,butnotρ . Indeed,thecorrelationmatrix
(cid:101)vr (cid:101)sr
1 ρ ρ
(cid:101)t (cid:101)sr
R = ρ(cid:101)t 1 ρ(cid:101)vr (8)
ρ ρ 1
(cid:101)sr (cid:101)vr
must be positive-semidefinite. This provides restrictions on, at least, one of
the correlation coefficients ρ ,ρ . For instance, let us assume that ρ ∈
(cid:101)sr (cid:101)vr (cid:101)sr
[−1,1] and then, since also ρ ∈ [−1,1], the semi-definiteness is defined by the
(cid:101)t
condition
det(R) = 1−ρ2−ρ2 −ρ2 +2ρ ρ ρ ≥ 0, (9)
(cid:101)t (cid:101)sr (cid:101)vr (cid:101)t(cid:101)vr(cid:101)sr
which is solved by
(cid:113)
−B−ρ ρ ≤ ρ ≤ B−ρ ρ , B = (1−ρ2)(1−ρ2 ).
(cid:101)t(cid:101)sr (cid:101)vr (cid:101)t(cid:101)sr (cid:101)t (cid:101)sr
These allowed values of ρ are depicted in Fig. 1. So the domain of definition
(cid:101)vr
of ρ depends on ρ , and since ρ is stochastic, it also changes stochastically.
(cid:101)vr (cid:101)t (cid:101)t
Therefore, ρ cannot be a constant.
(cid:101)vr
To resolve this issue, following the idea of Higgins (2014), further we as-
sume that ρ = const, ρ = ρ ρ . Thus, the spot/correlation correlation is
(cid:101)vr (cid:101)sr (cid:101)t(cid:101)vr
also stochastic, but driven by the ρ process multiplied by a constant. This
(cid:101)t
provides
det(R) = (1−ρ2)(1−ρ2 ) ≥ 0, ∀ (ρ ∈ [−1,1])∪(ρ ∈ [−1,1]).
(cid:101)t (cid:101)vr (cid:101)t (cid:101)vr
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Figure 1: A volume with the allowed values of ρ (inside the shaded area) as a function of
(cid:101)vr
ρ ,ρ .
(cid:101)t (cid:101)sr
Pure jump processes. Itisknownthatmodelswithjumpsinthespotprice
better predict the observed market data, especially at short maturities. It is
also known, that jumps in v might also be needed. For instance, in Sepp
t
(2011b,a) exponential and discrete jumps are investigated in both S and v .
t t
The author concludes that infrequent negative jumps in v are necessary to
t
fit the market data on equity options. Definitely, jumps in both spot and
the instantaneous variance could be correlated. Then it would be natural
to extend this point and introduce the correlation between the jumps in ρ
t
and that in S , as well as between jumps in ρ and that in v . For instance,
t t t
Clements and Liao (2013) considered the links between co-jumps within a
group of large stocks, the volatility of, and correlation between their returns.
They found that, despite the occurrence of common, or co-jumps between the
stocks is unrelated to the level of volatility or correlation, both volatility and
correlation are lower subsequent to a co-jump. Therefore, it does make sense
to consider jumps in the spot, instantaneous variance and their correlation to
be fully correlated.
In this paper we introduce these correlations as this is done in Itkin
(2016b); Itkin and Lipton (2015) by following the approach of Ballotta and
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Bonfiglioli (2014). They construct the jump process as a linear combination
of two independent L´evy processes representing the systematic factor and the
idiosyncratic shock, respectively (see also Cont and Tankov (2004)). It has an
intuitive economic interpretation and retains nice tractability, as the multi-
variate characteristic function in this model is available in closed form based
on the following proposition of Ballotta and Bonfiglioli (2014):
Proposition 3.1. Let Z , Y , j = 1,...,n be independent L´evy processes
t j,t
on a probability space (Q,F,P), with characteristic functions φ (u;t) and
Z
φ (u;t), for j = 1,...,n respectively. Then, for b ∈ R, j = 1,...,n
Yj j
X = (X ,...,X )(cid:62) = (Y +b Z ,...,Y +b Z )(cid:62)
t 1,t n,t 1,t 1 t n,t n t
is a L´evy process on Rn. The resulting characteristic function is
(cid:32) n (cid:33) n
(cid:88) (cid:89)
φ (u;t) = φ b u ;t φ (u ;t), u ∈ Rn.
X Z i i Yj j
i=1 i=1
By construction every factor X ,i = 1,...,n includes a common factor Z .
i,t t
Therefore, all components X ,i = 1,...,n could jump together, and loading
i,t
factors b determine the magnitude (intensity) of the jump in X due to the
i i,t
jump in Z . Thus, all components of the multivariate L´evy process X are
t t
dependent, and their pairwise correlation is given by (again see Ballotta and
Bonfiglioli (2014) and references therein)
b b Var(Z )
j i 1
ρ = . (10)
ij (cid:113) (cid:113)
Var(X ) Var(X )
j,1 i,1
Such a construction has multiple advantages, namely:
1. As sign(ρ ) = sign(b b ), both positive and negative correlations can
i,j i j
be accommodated
2. In the limiting case b → 0 or b → 0 or Var(Z ) = 0 the margins
i j 1
become independent, and ρ = 0. The other limit b → ∞ or b → ∞
i,j i j
represents a full positive correlation case, so ρ = 1. Accordingly,
i,j
b → ∞, b → ∞, i = 1,2 represents a full negative correlation case
i 3−i
as in this limit ρ = −1.
i,j
If we want the jump correlations to be constant, then this setting is sufficient.
However,sincewewantthecorrelationρ ≡ ρ tobestochastic,thisapproach
sv t
has to be modified. In this paper we don’t elaborate on it, and use the former
one, so assuming that the correlation between jumps is constant.
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General case. When the underlying L´evy processes have both the diffusion
and jumps components, the correlation matrix can be build as a combination
oftwoconstructionspresentedinabove. WeassumethatallBrownianmotions
W , i ∈ [S,v,r]areindependentofL . Inthissettingthetotalinstantaneous
i,t i,t
correlations read
ρ σ σ +b b Var(Z )
ρ = (cid:101)t s v s v 1 , (11)
sv (cid:113) (cid:113)
σ2+S2Var(L ) σ2+v2Var(L )
s s,1 v v,1
ρ σ σ +b b Var(Z )
ρ = (cid:101)vr v r v r 1 ,
vr (cid:113) (cid:113)
σ2+v2Var(L ) σ2+Var(L )
v v,1 r r,1
ρ ρ σ σ +b b Var(Z )
ρ = (cid:101)t(cid:101)vr s r s r 1 ,
sr (cid:113) (cid:113)
σ2+S2Var(L ) σ2+Var(L )
s s,1 r r,1
Var(L ) = Var(Y )+b Var(Z ), i ∈ [s,v,r],
i,1 i,1 i 1
where σ ,σ ,σ are volatilities of the diffusion processes of S ,v ,R , and Y
s v r t t t i,t
are the idiosyncratic jump processes. With such a definition all correlations
ρ ,ρ ,ρ couldtakeanyvaluewithintheinterval[−1,1]withnorestrictions.
vr sr t
As follows from Eq.(11) ρ is constant while ρ ,ρ are stochastic. If the
vr t sr
second term in the numerator of ρ in Eq.(11) is much bigger that the first
sv
one, this correlation also is constant and determined by the correlation of
jumps. If, on the contrary, the second term is small as compared with the
first one, ρ is stochastic and defined by the correlation of the Brownian
sv
motions. A similar argument is valid for ρ as well.
sr
4. Pricing PIDE
To price contingent claims, e.g., vanilla or exotic options written on the
underlying spot price, a multidimensional (backward) PIDE could be derived
by using a standard technique as in Cont and Tankov (2004). It describes the
evolution of the option price V under risk-neutral measure and reads
V = [D+J −r]V, (12)
τ
whereτ = T−tisthebackwardtime, T isthetimetothecontractexpiration,
r is the risk free interest rate, D is the three-dimensional linear convection-
10
