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