Double Mean-Reversion in FX

Philippe Balland

Quan Minh Tran

Gabriela Hodinic

P. Balland 1 June 2010

Contents 1. Market Observations 2. Single Mean-Reversion 3. Double Mean-Reversion 4. ATM Minimum 5. Smile Calibration 6. Two-Factor SLV Model 7. Simplification 8. Calibration

P. Balland 2 June 2010

Market Observations Three key observations:

(1) Short-dated ATM expiring within a month are:

• highly volatile • weakly correlated to ATM expiring

beyond one year (when considering monthly changes).

P. Balland 3 June 2010

(2) The ATM curve has often a local minimum within the first three months.

(3) Strangle margins are persistent at the short-end of the curve and the implied volatility-of-volatility is therefore large at the short-end.

USDJPY - May 2010

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0.00%

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0 1 2 3 4 5 6

mat

EqvolvolMktEqrhoMkt

P. Balland 4 June 2010

Eqvolvol, Eqrho for expiry T are respectively the SABR volatility-of-volatility and the SABR correlation :

Teqγ

Teqρ

dtdWdW

dWtWXdX

XtFS

Teq

fxtt

fxt

Teqt

Teq

Ttt

tt

ρ

γγσσ

σ

>=<

−=

=

,

))(exp(/

)(2

21

0

where S is the spot process, F is the forward, and T

0σ is the initial volatility for expiry T. We calculate and using the local-time approximation as this is more accurate than the expansion formula in the vicinity of the forward:

Teqγ T

eqρ

221

2

)()(2

02

20

)]([

)]([)(])[(

kItkt

Att

Ttt

kT

tekXE

dtkXEkXkXE

−

++

≈−

−+−=− ∫κδσ

δσ

See E. Benhamou, O. Croissant (2007).

P. Balland 5 June 2010

Single Mean-Reversion We consider for each expiry T the following one-factor mean-reverting dynamic:

dtdWdW

dWeZ

ZZ

dWXdX

fxtt

tu

tut

ttt

fxtt

Ttt

ρ

γ

υ

υσ

σ

σλσ

σσ

>=<

=

−=

=

∫ −

,

)var22exp(

/

0)(

0

The volatility-of-volatility γ , correlation ρ and mean-reversion λ are calibrated to the entire smile-surface by moment matching:

Teq

duE

duEF

TeqFF

Tu

TuTK

TKTK

ρρ

λγ

γλγ

σ

σ

=

=

=

∫

∫

022

022

)][(

])[(1

11

)](,[

)](0,[)](,[

P. Balland 6 June 2010

As illustrated in the example below, the calibration loses accuracy at the short-end. This could be prevented by using a time-dependent volatility-of-volatility, but at the cost of losing time-homogeneity.

USDJPY - May 2010λ=1

-100.00%

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0.00%

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100.00%

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200.00%

250.00%

300.00%

350.00%

400.00%

0 1 2 3 4 5 6

mat

EqvolvolMkt

EqrhoMkt

EqvolvolAnalytic 1F

EqrhoAnalytic 1F

Eqvolvol MC

Eqrho MC

P. Balland 7 June 2010

Double Mean-Reversion We could improve the calibration’s accuracy at the short-end and maintain time-homogeneity by using a two-factor stochastic volatility model:

sdtsd

sdtsd

sdt

ldtldt

sdtldt

ttt

fxtt

Ttt

dWdtZZdZ

dWdtZZdZ

ZZ

dWXdX

γλ

γλ

υ

υσ

σσ

σσ

+−=

+−=

−=

=

∞ )(

)(

)var22exp(

/ 0

By direct integration, we obtain:

sdt

ldtt

t sdu

tusd

t ldu

tutuld

ttttt

ZZZE

dWe

dWee

eeZZeZeZZ

sd

ldsdld

ldsd

sd

sdld

sd

ldsdsdsd

++=

+

−+

−−+−+=

∫

∫−

−−−−

−−−∞

−∞

−

][

)1(

)()()1(

0)(

0))(()(

00

σ

λ

λλλλλ

λ

λλλ

λλσλλσσ

γ

γ

Typically, the dynamic is based on two non-overlapping time-scales:

yearweek lddlsdsd 1/1,1/1 ∝=∝= λτλτ .

P. Balland 8 June 2010

Hence, the dynamic is similar to the two-factor volatility model considered in Balland (2006) and Bergomi (2008, 2005):

∫ −= −−−−

t ldu

tutuld

ldt dWeeZ ldsdld

ldsd

sd0

))(()( )1( λλλλλ

λ γ

∫≈ −t ldu

tuld dWe ld

0)( λγ

∫= −t sdu

tusd

sdt dWeZ sd

0)(λγ

P. Balland 9 June 2010

ATM Minimum We can approximate short-dated ATM levels as follows:

∫=∫

∫≈T

uTT

uT

TuTT

duZEduE

duEATM

01

021

021

])[2exp(][

][σσ

σ

The ATM curves generated by the double mean-reversion model admit typically local minima.

USDJPY ATM curve

0.12

0.125

0.13

0.135

0.14

0.145

0.15

0.155

0.16

0 0.5 1 1.5 2 2.5

double-mrev lnsv

double-mrev heston

USDJPY Nov09

P. Balland 10 June 2010

Smile Calibration We calibrate the parameters ldsdldsd λλγγ ,,, to the smile surface by moment matching:

Teq

duE

duEldsdldsdF

TeqFldsdldsdF

Tldeq

Tsdeqldsd

Tldeq

Tsdeq

Tldeqfxld

Tsdeqfxsd

Tu

TuTK

TKTK

ρ

λλγγ

γλλγγ

γγργγ

γργρ

σ

σ

=

=

=

++

+

∫

∫

,,,2

,2

,

,,,,

022

022

2)()(

)][(

])[(2

12

)](,,,[

)](0,[)](,,,[

where and are obtained by solving the following equations:

Tsdeq,γ T

ldeq,γ

)](,[)](0,[

)](,[)](0,[

1,1

1,1

TKTK

TKTK

ldldFT

ldeqF

sdsdFT

sdeqF

λγγ

λγγ

=

=

P. Balland 11 June 2010

USDJPY - May 2010

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0 1 2 3 4 5 6

mat

EqvolvolMktEqrhoMktEqvolvolAnalytic 2FEqrhoAnalytic 2FEqvolvolMCEqrhoMC

The moment-matching technique used to compute effective volatility-of-volatilities and correlations is accurate when compared to the Monte-Carlo method. We observe that the fit to market data is substantially improved by using double mean-reversion.

P. Balland 12 June 2010

As illustrated in the example below, the model parameters sdλ , ldλ , sdγ , ldγ , ldsd ,ρ and fxld ,ρ appear relatively stable over time.

USDJPYλsd=40, λ ld=0.4

-1

0

1

2

3

4

5

Dec-09 Jan-10 Jan-10 Feb-10 Mar-10 Mar-10 Apr-10 May-10 May-10

volvol[sd]

volvol[ld]rho[sd,ld]

rho[sd,fx]

rho[ld,fx]

The correlation parameter fxsd ,ρ appears less stable. This is expected since the model does not include any local volatility.

P. Balland 13 June 2010

Two-Factor SLV Model We control the joint evolution between risk-reversal and spot by including a local volatility component )ln,( tXtσ in the dynamic:

)var22exp(

)ln,(/0

σσυ

υσ

ttt

fxttttt

ttt

ZZ

dWXtXdX

FXS

−=

=

=

∫

∫⊥−⊥

−

−+

=

+=

t fxu

tuldfxld

ldu

t fxu

tuldfxld

ldu

ldt

sdt

ldtt

dWem

dWemZ

ZZZ

ld

ld

0)(2/12

,

0)(

,

)1( λ

λ

σ

γρ

γρ

2/12,

,,,

)1(

0)(2/122

,

0)(

0)(

,

)1(

fxld

fxldfxsdldsd

sd

sd

sd

sd

t fxu

tusdsdfxsd

sdu

t fxu

tusdsd

sdu

t fxu

tusdfxsd

sdu

sdt

dWem

dWem

dWemZ

ρ

ρρρ

λ

λ

λ

α

γαρ

γα

γρ

−

−

⊥⊥−⊥

⊥−⊥

−

=

−−+

+

=

∫

∫

∫

where ⊥⊥ ldldsdsd mmmm ,,, are the mixing-weight parameters.

P. Balland 14 June 2010

As these parameters vary between zero and one while maintaining fixed the target smile, the dynamic varies from local to stochastic volatility dynamic. The parameters control the amount of volatility-of-volatility parallel to the spot motion. As they increase from zero to one while the target smile is fixed, the slope of the local volatility decreases to compensate for the increase in volatility-of-volatility parallel to spot. Despite these parameters affecting the backbone of the dynamic, they have in fact little effect on the valuation of exotics.

ldsd mm ,

An asymptotic calculation shows that we have for all mixing weights:

volatility-implied :)/ln,(ln

0/lnln )0,)(ln(2

=Σ

∆∆ Σ∂=

FKF

FKFATM

T

F

P. Balland 15 June 2010

The ATM-speed coefficient FATMln∆

∆ is to be understood in the sense of Malliavin derivative:

⊥∆∆ += fx

tfx

tFATM

ATMdATM dWdW

t

t )(ln L The parameters ⊥⊥ ldsd mm , control the amount of volatility-of-volatility orthogonal to the spot motion. As they increase from zero to one while the target smile is fixed, the convexity of the local volatility decreases to compensate for the increase in volatility-of-volatility. Hence, the mixing-weights ⊥⊥ sdld mm , control the convexity of the local volatility and thus control the joint evolution of risk-reversal (slope of smile) and spot. They are therefore critical to the valuation of Barrier and DNT products as these parameters affect directly the expected slope of the smile prevailing when spot hits the barrier level. An asymptotic calculation shows that the RR25 speed F

RRln

25∆∆ depend on the level of mixing

weights.

P. Balland 16 June 2010

As illustrated in the example below, the spot and risk-reversal are strongly correlated.

dRR25/dlnF

∆RR25 = 0.0767* ∆lnF - 2E-05

-0.006

-0.004

-0.002

0

0.002

0.004

0.006

-0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04

∆F/F

∆R

R25

The mixing weight parameters ⊥⊥ sdld mm , can be set to match historical RR25 speeds or DNT prices.

P. Balland 17 June 2010

Simplification Typical short-dated products do not depend strongly on the mixing-weights and as these parameters control whether the skew implied by the dynamic originates from local or stochastic volatility.

sdm ldm

Consequently, we can simplify the dynamic by assuming and to be zero: sdm ldm

)var22exp(

)ln,(/0

σσυ

υσ

ttt

fxttttt

ttt

ZZ

dWXtXdX

FXS

−=

=

=

2/12,

,,,

)1(

0)(2/122

,

0)())((2/12

,

)1(

])1([

fxld

fxldfxsdldsd

sd

ldldsd

sd

t fxu

tusdsdfxsd

sdu

t fxu

tutusdsd

sduldfxld

ldut

dWem

dWeemmZ

ρ

ρρρ

λ

λλλσ

α

γαρ

γαγρ

−

−

⊥⊥−⊥

⊥−−−⊥⊥

=

−−+

+−=

∫

∫

Using our moment matching technique, we can approximate the dynamic of the volatility driver using either a one-factor or a two-factor dynamic:

P. Balland 18 June 2010

(i) ∫∫ ⊥⊥−⊥⊥⊥−⊥ += t fxu

tuu

t fxu

tuut dWedWeZ sdld

0)(

0)( λλσ γγ

(ii) ∫ ⊥−⊥= t fx

utu

ut dWeZ ld0

)(λσ γ The orthogonalisation allows fast backward and forward inductions. In particular, we can approximate the volatility drivers using Markov chains:

(i) ⊥⊥⊥⊥⊥⊥ Σ+Σ= tttttZ ξξσ

(ii) ⊥⊥Σ= tttZ ξσ where ⊥⊥⊥

tt ξξ , are independent N(0,1)-processes characterized by their auto-correlation functions. The version (ii) is sufficient for first generation exotic products. We can calibrate the local-volatility to the smile assuming (ii) in particular.

P. Balland 19 June 2010

Calibration Parametric Local Volatility We parameterize the local volatility:

)(ln )(ln

)(ln )()(ln 0

tkilling

tconvex

tskew

t

XX

XtX

σσσ

σσ

×××

=

We choose the local volatility skew using a ratio of CEV. This ensures that the skew component has a CEV-like shape near the forward while being bounded:

)ln)1(tanh(

)(ln

11

21

21

)()(

1

1

tqqqq

bXaX

tskew

X

Xt

t

−+=

=

−+−+

++

−

−

β

σ β

β

1)(ln << tskew Xq σ

P. Balland 20 June 2010

The local volatility skew component has a functional form similar to that suggested by Brown and Randall (2003):

)1(

)/lntanh()(ln

)/lncosh(1

smiletsmile XXasmile

skewtskewskewatmt XXaX

−×+

×+=

σ

σσσ

Note however that the BR functional form is additive while our parameterisation is in fact multiplicative. The convex local volatility shares the same short-dated asymptotic as SABR in order to minimize changes in the smile when the mixing-weight parameters vary:

22 )(lnln21)(ln tttconvex XbXaX ++=σ The killing component ensures finite moments by exponentially decreasing the spot volatility outside the boundaries and : )(tX low )(tXup

))(ln)(lnexp()(ln )(

)(++ −−=

tlow

upt

XtX

tXX

tkilling kkXσ

P. Balland 21 June 2010

Finally, we calibrate the function )(0 tσ to the prevailing ATM curve by forward induction. Parametric Smile We can parameterize the smile surface using asymptotic expansions of the previous diffusion ( 0,0 == ldsd λγ ) which is a direct extension of SABR for FX:

))(exp(

)(ln)(ln /2

21

0 tγWγ

dWXXXdXT

tTT

t

ttconvextskewttt

−=

××=σσσ

σσσ

In this case, the local volatility is obtained by forward induction using the equation:

2/1]ln|[/)ln,()ln,( var22t

ZZtDt XeEXtXt tt

σσσσ −=

where )ln,( tD Xtσ is the Dupire local volatility obtained from the parameterized smile.

P. Balland 22 June 2010