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
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.
2
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.
3
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.
4
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.
5
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
6
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.
7
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.
8
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.
9
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.
10
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.
11
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!
12
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!
13
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.
14
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)
15
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.
16
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.
17
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.
18
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.
19
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)
20
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.
21
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.
22
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))]
.
23
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.
24
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.
25
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
26
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.
27
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.
28
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).
29
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
30
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.
31