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  • 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

    -100.00%

    -50.00%

    0.00%

    50.00%

    100.00%

    150.00%

    200.00%

    250.00%

    300.00%

    350.00%

    400.00%

    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 T0σ 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

    Teqρ

    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%

    -50.00%

    0.00%

    50.00%

    100.00%

    150.00%

    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 ld

    ututu

    ldldt dWeeZ ldsdldldsd

    sd0

    ))(()( )1( λλλλλλ γ

    ∫≈−t ld

    utu

    ld dWe ld0)( λγ

    ∫=−t sd

    utu

    sdsdt dWeZ sd0

    )(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

    Tldeq,γ

    )](,[)](0,[

    )](,[)](0,[

    1,1

    1,1

    TKTK

    TKTK

    ldldFT

    ldeqF

    sdsdFT

    sdeqF

    λγγ

    λγγ

    =

    =

    P. Balland 11 June 2010

  • USDJPY - May 2010

    -100.00%

    -50.00%

    0.00%

    50.00%

    100.00%

    150.00%

    200.00%

    250.00%

    300.00%

    350.00%

    400.00%

    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∂=

    FKF

    FKFATM

    T

    F

    P. Balland 15 June 2010

  • The ATM-speed coefficient FATMln∆

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

    ⊥∆∆ += fxt

    fxtF

    ATMATM

    dATM dWdWt

    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 fxutuut fx

    utu

    ut dWedWeZ sdld 0)(

    0)( λλσ γγ

    (ii) ∫ ⊥−⊥= t fxutuut 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 ββ

    1)(ln

  • 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,( var22 tZZ

    tDt XeEXtXt ttσσ

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

    P. Balland 22 June 2010

    Double Mean-Reversion in FXPhilippe BallandQuan Minh TranGabriela Hodinic

    ContentsMarket ObservationsTwo-Factor SLV Model

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