stochastic skew in currency optionsfaculty.baruch.cuny.edu/lwu/890/carrwu2004_ssmov.pdfotc currency...
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Stochastic Skew in Currency Options
PETER CARR
Bloomberg LP and Courant Institute, NYU
L IUREN WU
Zicklin School of Business, Baruch College
Citigroup Wednesday, September 22, 2004
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Overview
• There is a huge market for foreign exchange (FX), much larger than theequitymarket ... an understanding of FX dynamics is economically important.
• FX option prices can be used to understand risk-neutral FX dynamics, i.e. howthe market prices various path bundles.
• Despite their greater economic relevance, FX options are not aswidely studiedas equity index options, probably due to the fact that the FX options market isnow primarily OTC.
• We obtain OTC options data on 2 currency pairs (JPYUSD, GBPUSD).
• We use the options data to study the dynamics of the 2 currencies:
– Document the behavior of the currency options.
– Propose a new class of models to capture the behavior.
– Estimate the new models and compare to older models such as Bates (1996).
– Study the implications of the new model class for currency dynamics.
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OTC FX Option Quoting Conventions
• Quotes are in terms of BS model implied volatilities rather thanon option pricesdirectly.
• Quotes are provided at a fixed BS delta rather than a fixed strike. In particular,the liquidity is mainly at 5 levels of delta:
10 δ Put, 25δ Put, 0δ straddle, 25δ call, 10δ call.
• Trades include both the option position and the underlying, where the positionin the latter is determined by the BS delta.
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A Review of the Black-Scholes Formulae
• BS call and put pricing formulae:
c(K,τ) = e−r f τStN(d1)−e−rdτKN(d2),
p(K,τ) = −e−r f τStN(−d1)+e−rdτKN(−d2),
with
d1,2 =ln(F/K)
σ√
τ± 1
2σ√
τ, F = Se(rd−r f )τ.
• BS Deltaδ(c) = e−r f τN(d1), δ(p) = −e−r f τN(−d1).
|δ| is roughlythe probability that the option will expire in-the-money.
• BS Implied Volatility (IV): the σ input in the BS formula that matches the BSprice to the market quote.
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Data
• We have 8 years of data: January 1996 to January 2004 (419 weeks)
• On two currency pairs: JPYUSD and GBPUSD.
• At each date, we have 8 maturities: 1w, 1m, 2m, 3m, 6m, 9m, 12m, 18m.
• At each maturity, we have five option quotes at the five deltas.
All together, 40 quotes per day, 16,760 quotes for each currency.
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OTC Currency Option Quotes
• The five option quotes at each maturity are in the following forms:
1. Delta-neutral straddle implied volatility (ATMV)
– A straddle is a sum of a call and a put at the same strike.– Delta-neutral meansδ(c)+δ(p) = 0→ N(d1) = 0.5→ d1 = 0.
2. ten-delta risk reversal (RR10)
– RR10= IV (10c)− IV (10p).– Captures the slope of the smile (skewnessof the return distribution).
3. ten-delta strangle margin (butterfly spread) (SM10)
– A strangle is a sum of a call and a put at two different strikes.– SM10= (IV (10c)+ IV (10p))/2−ATMV.– Captures the curvature of the smile (kurtosis of the distribution ).
4. 25-delta RR and 25-delta SM
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Convert Quotes to Option Prices
• Convert the quotes into implied volatilities at the five deltas:
IV (0δs) = ATMV;IV (25δc) = ATMV+RR25/2+SM25;IV (25δp) = ATMV−RR25/2+SM25;IV (10δc) = ATMV+RR10/2+SM10;IV (10δp) = ATMV−RR10/2+SM10.
• Download LIBOR and swap rates on USD, JPY, and GBP to generate therelevantyield curves (rd, r f ).
• Convert deltas into strike prices
K = F exp
[
∓IV (δ,τ)√
τN−1(±er f τδ)+12
IV (δ,τ)2τ]
.
• Convert implied volatilities into out-of-the-money option prices using the BSformulae.
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Time Series of Implied Volatilities
97 98 99 00 01 02 03 04
10
15
20
25
30
35
40
45
Imp
lied
Vo
latil
ity,
%
JPYUSD
97 98 99 00 01 02 03 04
4
6
8
10
12
14
Imp
lied
Vo
latil
ity,
%
GBPUSD
Stochastic volatility — Note the impact of the 1998 hedge fundcrisis on dollar-yen:During the crisis, hedge funds bought call options on yen to cover their yen debt.
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The Average Implied Volatility Smile
10 20 30 40 50 60 70 80 9011
11.5
12
12.5
13
13.5
14
Put Delta, %
Ave
rag
e Im
plie
d V
ola
tility
, %
JPYUSD
10 20 30 40 50 60 70 80 908.2
8.4
8.6
8.8
9
9.2
9.4
9.6
9.8
Put Delta, %
Ave
rage
Impl
ied
Vol
atili
ty, %
GBPUSD
1m (solid), 3m (dashed), 12m (dash-dotted)
• The mean implied volatility smile is relatively symmetric ...
• The smile (kurtosis) persists with increasing maturity.
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Time Series of Strangle Margins and Risk Reversals
97 98 99 00 01 02 03 04−30
−20
−10
0
10
20
30
40
50
60
RR
10 a
nd S
M10
, %A
TMV
JPYUSD
97 98 99 00 01 02 03 04
−20
−15
−10
−5
0
5
10
15
20
RR
10 a
nd S
M10
, %A
TMV
GBPUSD
RR10 (solid), SM10 (dashed)
• The strangle margins (kurtosis measure) are stable over time, ...
• But the risk reversals (return skewness) vary greatly over time.⇒ Stochastic Skew.
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Correlogram Between RR and Return
−20 −15 −10 −5 0 5 10 15 20−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
Number of Lags in Weeks
Sa
mp
le C
ross
Co
rre
latio
n
[JPYUSD Return (Lag), ∆ RR10]
−20 −15 −10 −5 0 5 10 15 20−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
Number of Lags in Weeks
Sa
mp
le C
ross
Co
rre
latio
n
[GBPUSD Return (Lag), ∆ RR10]
• Changes in risk reversals are positively correlated with contemporaneous cur-rency returns ...
• But there are no obvious lead-lag effects.
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How Does the Literature Capture Smiles?
Two ways to generate a smile or skew:
1. Add jumps: Merton (1976)’s jump-diffusion model
dSt/St− = (rd− r f )dt+σdWt +∫ ∞
−∞(ex−1)[µ(dx,dt)−λn(µj ,σ j)dxdt]
• The arrival of jumps is controlled by a Poisson process with arrivalrateλ.
• Conditional on one jump occurring, the percentage jump sizex is normallydistributed, with densityn(µj ,σ2
j ).
• Nonzeroµj generates asymmetry (skewness).
⇒ Stochastic skew would requireµj to be stochastic ...
Not tractable!
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How Does the Literature Capture Smiles?
Two ways to generate a smile or skew:
1. Add jumps
2. Stochastic volatility: Heston (1993)
dSt/St = (rd− r f )dt+√
vtdWt,
dvt = κ(θ−vt)dt+σv√
vtdZt, ρdt = E[dWtdZt]
• Vol of vol (σv) generates smiles,
• Correlation (ρ) generates skewness.
⇒ Stochastic skew would require correlationρ to be stochastic ...
Not tractable!
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How Do The Two Methods Differ?
• Jump diffusions (Merton) induce short term smiles and skews that dissipatequickly with increasing maturity due to the central limit theorem.
• Stochastic volatility (Heston) induces smiles and skews that increase as maturityincreases over the horizon of interest.
• Bates (1996)combines Merton and Heston to generate stochastic volatilityandsmiles/skews at both short and long horizons ... but NOTstochastic skew.
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Our Models Based on Time-Changed Levy Processes
lnSt/S0 = (rd− r f )t +(
LRTRt−ξRTR
t
)
+(
LLTLt−ξLTL
t
)
, (1)
• LRt is a Levy process that generates positive skewness
(diffusion + positive jumps)
• LLt is a Levy process that generates negative skewness
(diffusion + negative jumps)
• [TRt ,TL
t ] randomize the clock underlying the two Levy processes so that
– [TRt +TL
t ] determines total volatility: stochastic
– [TRt −TL
t ] determines skewness (risk reversal): ALSO stochastic
⇒ Stochastic Skew Model (SSM)
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SSM In the Language of Merton and Heston
dSt/St− = (rd− r f )dt ↼ risk-neutral drift
+σ√
vRt dWR
t +∫ ∞
0(ex−1)[µR(dx,dt)−kR(x)dxvR
t dt] ↼ right skew
+σ√
vLt dWL
t +∫ 0
−∞(ex−1)[µL(dx,dt)−kL(x)dxvL
t dt] ↼ left skew
dvjt = κ(1−vt)dt+σv
√vtdZj
t , ρ jdt = E[dW jt dZj
t ], j = R,L ↼ activity rates
• At short term, the Levy densitykR(x) has support onx ∈ (0,∞) 7→ Positiveskew . The Levy densitykL(x) has support onx∈ (−∞,0) 7→ Negative skew .
• At long term,ρR > 0 7→ Positive skew . ρL < 0 7→ Negative skew .
• Stochastic skewis generated via the randomness in[vRt ,vL
t ], which randomizesthe contribution from the two jumps and from the two correlations.
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Our Jump Specification
• The arrival rates of upside and downside jumps (Levy density) follow exponen-tial dampened power law (DPL):
kR(x) =
{
λe− |x|
vj |x|−α−1, x > 0,
0, x < 0., kL(x) =
{
0, x > 0,
λe− |x|
vj |x|−α−1, x < 0.(2)
• The specification originates in Carr, Geman, Madan, Yor (2002), and capturesmuch of the stylized evidence on both equities and currencies (Wu, 2004).
• A general and intuitive specification with many interesting special cases:
– α = −1: Kou’s double exponential model (KJ),finite activity .
– α = 0: Madan’s VG model,infinite activity, finite variation .
– α = 1: Cauchy dampened by exponential functions (CJ),infinite variation.
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Option Pricing Under SSM
• Our specifications are within the very general framework of time-changed Levyprocesses (Carr&Wu, JFE 2004).
• Following the theorem in Carr& Wu, we can derive the generalized Fourier trans-form function (FT) of the currency return in closed forms.
– Derive the characteristic exponent of the Levy process
– Convert the FT of the currency return into the Laplace transform of theran-dom time change.
– Derive the Laplace transform following the bond pricing literature.
• Given the FT, we price options using fast Fourier transform (FFT) (Carr&Madan,1999; Carr&Wu, 2004).
• Analytical tractability and pricing speed are similar to that for the Heston modeland the Bates model.
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The Characteristic Exponents of Levy Processes
• The Levy-Khintchine Theorem describesall Levy processes via their character-istic exponents:
ψx(u) ≡ lnEeiuX1 = −iub+u2σ2
2−
∫
R0
(
eiux−1− iux1|x|<1
)
k(x)dx.
• TheLevy density k(x) specifies the arrival rate as a function of the jump size:
k(x) ≥ 0,x 6= 0,∫
R0(x2∧1)k(x)dx< ∞.
• To obtain tractable models, choose the Levy density of the jump component sothat the above integral can be done in closed form.
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Characteristic Exponents For Dampened Power Law
Model α Right (Up) Component Left (Down) Component
KJ -1 ψD− iuλ[
11−iuv j
− 11−v j
]
ψD + iuλ[
11+iuv j
− 11+v j
]
VG 0 ψD +λ ln(1− iuv j)− iuλ ln(1−v j) ψD +λ ln(1+ iuv j)− iuλ ln(1+v j)
CJ 1 ψD−λ(1/v j − iu) ln(1− iuv j) ψD−λ(1/v j + iu) ln(1+ iuv j)
+iuλ(1/v j −1) ln(1−v j) +iuλ(1/v j +1) ln(1+v j)
CG α ψD +λΓ(−α)[(
1v j
)α−
(
1v j− iu
)α]
ψD +λΓ(−α)[(
1v j
)α−
(
1v j
+ iu)α]
−iuλΓ(−α)[(
1v j
)α−
(
1v j−1
)α]
−iuλΓ(−α)[(
1v j
)α−
(
1v j
+1)α]
ψD = 12σ2
(
iu+u2)
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CF of Return as LT of Clock
φs(u) ≡ EQeiu ln(ST/S0)
= eiu(rd−r f )tEQ
[
eiu
(
LRTRt−ξRTR
t
)
+iu
(
LLTLt−ξLTL
t
)]
= eiu(rd−r f )tEM[
e−ψ⊤Tt
]
≡ eiu(rd−r f )tLMT (ψ) ,
• The new measureM is absolutely continuous with respect to the risk-neutralmeasureQ and is defined by a complex-valued exponential martingale,
dM
dQ t≡ exp
[
iu(
LRTRt−ξRTR
t
)
+ iu(
LLTLt−ξLTL
t
)
+ψRTRt +ψLTL
t
]
.
• Girsanov’s Theorem yields the (complex) dynamics of the relevant processesunder the complex-valued measureM.
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The Laplace Transform of the Stochastic Clocks
• We chose our two new clocks to be continuous over time:
T jt =
∫ t
0v j
sds, j = R,L,
where for tractability, the activity ratesv j follow square root processes.
• As a result, the Laplace transforms are exponential affine:
LMT (ψ) = exp
(
−bR(t)vR0 −cR(t)−bL(t)vL
0−cL(t))
,
where:
b j(t) =2ψ j
(
1−e−η j t)
2η j−(η j−κ j)(
1−e−η j t),
c j(t) = κσ2
v
[
2ln(
1− η j−κ j
2η j
(
1−e−η j t))
+(η j −κ j)t]
,
and:
η j =
√
(κ j)2+2σ2vψ j , κ j = κ− iuρ jσσv, j = R,L.
• Hence, the CFs of the currency return are known in closed form for our models.
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Estimation
• We estimate six models:HSTSV, MJDSV, KJSSM, VGSSM, CJSSM, CGSSM.
• Using quasi-maximum likelihood method with unscented Kalman filtering (UKF).
– State propagation equation: The time series dynamics for the 2 activity rates:
dvt = κP(θP−vt)dt+σv√
vtdZ, (2×1)
– Measurement equations:yt = O(vt;Θ)+et, (40×1)
∗ y — Out-of-money option prices scaled by the BS vega of the option.∗ Turn option price into the volatility space.
– UKF generates efficient forecasts and updates on conditional mean and co-variance of states and measurements sequentially (fromt = 1 to t = N).
– Maximize the following likelihood function to obtain parameter estimates:
L (Θ)=N
∑t=1
lt+1(Θ)=−12
N
∑t=1
[
log∣
∣At
∣
∣+(
(yt+1−yt+1)⊤ (
At+1)−1
(yt+1−yt+1))]
.
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Full Sample Model Performance Comparison
HSTSV MJDSV KJSSM VGSSM CJSSM CGSSM
JPY: rmse 1.014 0.984 0.822 0.822 0.822 0.820L ,×103 -9.430 -9.021 -6.416 -6.402 -6.384 -6.336
GBP: rmse 0.445 0.424 0.376 0.376 0.376 0.378L ,×103 4.356 4.960 6.501 6.502 6.497 6.521
• SSM models with different jumps perform similarly.
• All SSM models perform much better than MJDSV, which is better than HSTSV.
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Likelihood Ratio Tests
Under the nullH0 : E[l i − l j ] = 0, the statistic (M ) is asymptoticallyN(0,1).
Curr M HSTSV MJDSV KJSSM VGSSM CJSSM CGSSMJPY HSTSV 0.00 -2.55 -4.92 -4.88 -4.75 -4.67JPY MJDSV 2.55 0.00 -5.39 -5.33 -5.22 -5.07JPY KJSSM 4.92 5.39 0.00 -1.11 -0.86 -1.20JPY VGSSM 4.88 5.33 1.11 0.00 -0.72 -1.21JPY CJSSM 4.75 5.22 0.86 0.72 0.00 -1.59JPY CGSSM 4.67 5.07 1.20 1.21 1.59 0.00GBP HSTSV 0.00 -2.64 -4.70 -4.68 -4.63 -4.71GBP MJDSV 2.64 0.00 -3.85 -3.86 -3.89 -4.19GBP KJSSM 4.70 3.85 0.00 -0.04 0.34 -0.37GBP VGSSM 4.68 3.86 0.04 0.00 0.56 -0.39GBP CJSSM 4.63 3.89 -0.34 -0.56 0.00 -0.51GBP CGSSM 4.71 4.19 0.37 0.39 0.51 0.00
The outperformance of SSM models over MJDSV/HSTSV is statistically significant.
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Out of Sample Performance Comparison
HSTSVMJDSVKJSSMVGSSMCJSSMCGSSM HSTSVMJDSVKJSSMVGSSMCJSSMCGSSM
JPYUSD GBPUSD
In-Sample Performance: 1996-2001
rmse 1.04 1.02 0.85 0.85 0.85 0.85 0.47 0.44 0.41 0.41 0.41 0.41L /N -23.69 -23.03 -16.61 -16.60 -16.57 -16.47 8.36 10.06 12.27 12.27 12.26 12.28M
HSTSV 0.00 -2.14 -4.44 -4.41 -4.33 -4.17 0.00 -3.34 -4.42 -4.39 -4.24 -4.33MJDSV 2.14 0.00 -4.74 -4.70 -4.61 -4.42 3.34 0.00 -3.40 -3.39 -3.33 -3.33KJSSM 4.44 4.74 0.00 -0.49 -0.51 -0.84 4.42 3.40 0.00 0.08 0.36 -0.42VGSSM 4.41 4.70 0.49 0.00 -0.51 -0.89 4.39 3.39 -0.08 0.00 0.51 -0.42CJSSM 4.33 4.61 0.51 0.51 0.00 -1.14 4.24 3.33 -0.36 -0.51 0.00 -0.55CGSSM 4.17 4.42 0.84 0.89 1.14 0.00 4.33 3.33 0.42 0.42 0.55 0.00
Similar to the whole-sample performance
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Out of Sample Performance Comparison
HSTSVMJDSVKJSSMVGSSMCJSSMCGSSM HSTSVMJDSVKJSSMVGSSMCJSSMCGSSM
JPYUSD GBPUSD
Out-of-Sample Performance: 2001-2002
rmse 1.06 1.00 0.90 0.90 0.89 0.89 0.39 0.37 0.27 0.27 0.27 0.27L /N -24.01 -21.75 -18.47 -18.35 -18.23 -18.11 14.36 15.85 23.30 23.29 23.26 23.25M
HSTSV 0.00 -6.01 -5.90 -6.01 -6.08 -6.12 0.00 -4.88 -7.06 -7.06 -7.05 -7.05MJDSV 6.01 0.00 -3.11 -3.23 -3.32 -3.48 4.88 0.00 -5.98 -5.99 -5.99 -5.97KJSSM 5.90 3.11 0.00 -7.76 -6.81 -5.27 7.06 5.98 0.00 0.64 1.47 4.51VGSSM 6.01 3.23 7.76 0.00 -4.39 -3.67 7.06 5.99 -0.64 0.00 1.63 4.19CJSSM 6.08 3.32 6.81 4.39 0.00 -3.11 7.05 5.99 -1.47 -1.63 0.00 0.23CGSSM 6.12 3.48 5.27 3.67 3.11 0.00 7.05 5.97 -4.51 -4.19 -0.23 0.00
Similar to the in-sample performance: SSMs>> MJDSV>> HSTSV
Infinite variation jumps are more suitable to capture large smiles/skews.
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Theory (Bates) and Evidence on Stochastic Volatility
Top panels: Implied vol (data). Bottom panels: Activity rates (fromBates model)
97 98 99 00 01 02 03 04
10
15
20
25
30
35
40
45
Impli
ed Vo
latility
, %
JPYUSD
97 98 99 00 01 02 03 04
4
6
8
10
12
14
Impli
ed Vo
latility
, %
GBPUSD
Jan97 Jan98 Jan99 Jan00 Jan01 Jan02 Jan03 Jan040
1
2
3
4
5
6
7
8
Activ
ity R
ates
Currency = JPYUSD; Model = MJDSV
Jan97 Jan98 Jan99 Jan00 Jan01 Jan02 Jan03 Jan040
0.5
1
1.5
2
2.5Ac
tivity
Rate
sCurrency = GBPUSD; Model = MJDSV
Demand for calls on yen drives up the activity rate during the 98 hedge fund crisis.
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Theory (SSM) and Evidence on Stochastic Volatility
Top panels: Implied vol (data). Bottom panels: Activity rates (fromKJSSM)
97 98 99 00 01 02 03 04
10
15
20
25
30
35
40
45
Impli
ed Vo
latility
, %
JPYUSD
97 98 99 00 01 02 03 04
4
6
8
10
12
14
Impli
ed Vo
latility
, %
GBPUSD
Jan97 Jan98 Jan99 Jan00 Jan01 Jan02 Jan03 Jan040
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Activ
ity R
ates
Currency = JPYUSD; Model = KJSSM
Jan97 Jan98 Jan99 Jan00 Jan01 Jan02 Jan03 Jan040
0.5
1
1.5
2
2.5
Activ
ity R
ates
Currency = GBPUSD; Model = KJSSM
The demand foryen callsonly drives up the activity rate ofupward yen moves(solid line), but not the volatility of downward yen moves (dashed line).
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Theory and Evidence on Stochastic Skew
Three-month ten-delta risk reversals: data (dashed lines), model (solid lines).
Bates:Jan97 Jan98 Jan99 Jan00 Jan01 Jan02 Jan03 Jan04
−20
−10
0
10
20
30
40
50
10−D
elta
Risk
Rev
ersa
l, %AT
M
Currency = JPYUSD; Model = MJDSV
Jan97 Jan98 Jan99 Jan00 Jan01 Jan02 Jan03 Jan04
−15
−10
−5
0
5
10
10−D
elta
Risk
Rev
ersa
l, %AT
M
Currency = GBPUSD; Model = MJDSV
SSM: Jan97 Jan98 Jan99 Jan00 Jan01 Jan02 Jan03 Jan04
−20
−10
0
10
20
30
40
50
10−D
elta
Risk
Rev
ersa
l, %AT
M
Currency = JPYUSD; Model = KJSSM
Jan97 Jan98 Jan99 Jan00 Jan01 Jan02 Jan03 Jan04
−15
−10
−5
0
5
10
10−D
elta
Risk
Rev
ersa
l, %AT
MCurrency = GBPUSD; Model = KJSSM
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Conclusions
• Using currency option quotes, we find that under a risk-neutral measure, cur-rency returns display not only stochastic volatility, but alsostochastic skew.
• State-of-the-art option pricing models (e.g. Bates 1996) capture stochastic volatil-ity andstatic skew, but not stochastic skew.
• Using the general framework of time-changed Levy processes, we propose aclass of models (SSM) that capture both stochastic volatilityand stochasticskewness.
• The models we propose are also highly tractable for pricing and estimation. Thepricing speed is comparable to the speed for the Bates model.
• Future research: The economic underpinnings of the stochastic skew.
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