Modelling Dividends

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    Q U A N T I T A T I V E R E S E A R C H

    Stochastic Dividend ModelingFor Derivatives Pricing and Risk Management

    Global Derivatives Trading & Risk Management Conference 2011

    Paris, Thursday April 14th, 2011

    Hans Buehler, Head of Equities QR EMEA, JP Morgan.

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    Part IVanilla Dividend Market

    Part IIGeneral structure of stock price models with dividends

    Part IIIAffine Dividends

    Part IVModeling Stochastic Dividends

    Part VCalibration

    Presentation will be under http://www.math.tu-berlin.de/~buehler/

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    Part I

    Vanilla Dividend Market

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    Vanilla Dividend Market

    Dividend Futures

    Dividend future settles at the sum of dividends paid over aperiod T1 to T2 for all members of an index such as STOXX50E.

    Standard maturities settle in December, so we have Dec 13, Dec

    14 etc trading.

    2

    1

    T

    Ti

    i

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    Vanilla Dividend Market

    Vanilla Options

    Refers to dividends over a period T1

    to T2

    .Listed options cover Dec X to Dec Y.

    Payoff straight forward

    note that dividends are not accrued. Note in particular that a Dec 13 option does not overlap with a Dec 14

    option ... makes the pricing problem somewhat easier than forexample pricing options on variance.

    Market

    Active OTC market in EMEA

    EUREX is pushing to establish alisted market for STOXX50E

    At the moment much lessvolume than in the OTC market

    KT

    Ti

    i2

    1

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    Vanilla Dividend Market

    Quoting

    The first task at hand is now to provide a Quoting mechanismfor options on dividends this does not intend to model

    dividends; just to map market $ prices into a more general

    implied volatility measure.

    For our further discussion let t* be t* :=max{T1,t} and

    The simplest quoting method is as usual Black & Scholes:

    ]Fut[E:EFut,Past,Fut

    PastFut

    *

    * 1

    2

    2

    1

    t

    t

    Ti

    iT

    ti

    i

    T

    Ti

    i

    BS forward equal toexpected future dividends

    divT

    Ti

    i

    t tTKK s,;Past,EFutBS:E 22

    1

    Imply volatility

    from the market.

    Adjust strike by

    past dividends.

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    Quoting

    ... term structure looks a bit funny though.

    Vanilla Dividend Market

    Ugly kink

    Graph shows

    ATM prices for

    option son divfor the period

    T1=1 and T2=2

    at various

    valuation times.

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    Vanilla Dividend Market

    Quoting

    Basic issue is that dividends are an average so using straightBlack & Scholes doesnt get the decay right.

    Alternative is to use an average option pricer for simplicity,

    use the classic approximation

    and define the option price using BS formula as

    ]E[11 2

    0 61

    3

    0

    31

    00

    xWxx

    i

    iYxdsWxx dsWx

    i

    i x

    x

    s

    s eeeexx

    ssss

    s

    Basically the average pricing

    translates to a new scaling in time.

    div

    T

    Ti

    i

    t

    tTtTK

    K

    s,3/)()(;Past,EFutBS:

    E

    *2*1

    2

    1

    Imply a different

    volatility from themarket.

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    Vanilla Dividend Market

    Quoting

    ... gives much better theta:

    Average optionmethod yieldsdecent theta,

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    Vanilla Dividend Market

    Quoting

    ... market implied vols by strike:

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    Vanilla Dividend Market

    Quoting

    Using plain BS gives rise to questionably theta, in particulararound T1 using an average approximation leads to much

    better results.

    After that, market quotes can be interpolated with any implied

    volatility model.

    At that level no link to the actual stock price

    let us focus on that now.

    Dec 12 Dec 13 Dec 14

    a0 25% 25% 31%

    r -0.85 -0.84 -9.59

    n 102% 47% 28%

    tttt

    tttt

    dWdWd

    dWd

    21 rrnaa

    a

    Using SABR tointerpolate

    impliedvolatilities

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    Part II

    The Structure of Dividend Paying Stocks

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    The Structure of Dividend Paying Stocks

    Assumptions on Dividends

    We assume that the ex-div dates 0

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    The Structure of Dividend Paying Stocks

    Stock Price Dynamics

    In the absence of friction cost, the stock price under risk-neutral dynamics has to fall by the dividend amount in thesense that

    For example, we may consider an additional uncertainty risk inthe stock price at the open:

    For the case where Shas almost surely no jumps at tkweobtain the more common

    i

    kkkSS ttt

    k

    kkSS tt

    221

    )(wwYi eSS

    kk

    tt

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    The Structure of Dividend Paying Stocks

    Stock Price Dynamics

    In between dividend dates, the risk-neutral drift under any risk-neutral measure is given by rates and repo, i.e. we can write the

    stock price between dividend dates for t:tk

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    The Structure of Dividend Paying Stocks

    Stock Price Dynamics - Warning

    This gives

    the martingaleZcan nothave arbitrary dynamics but needs to be

    floored to ensure that the stock price never falls below any future

    dividend amount.

    kk

    tt ZRSSk

    kk

    )1(1

    1

    t

    tt

    Funding ratebetweendividends

    (Local) martingale partbetween dividends

    )(k

    ttt ZRSSi

    k

    t

    t

    k

    kkSS tt

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    The Structure of Dividend Paying Stocks

    Towards Stock Price Dynamics with Discrete Dividends

    In other words, any generic specification of the form

    does not work either - a common fix in numerical approachesis to set

    Intuitively, the restriction is that the stock price at any time

    needs to be above the discounted value of all future dividends: otherwise, go long stock and forward-sell all dividends

    lock-in risk-free return.

    k

    k

    t

    tttttt dtZ

    dZSdtrSdS

    k)(tm

    )~

    min{: kk

    S t

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    The Structure of Dividend Paying Stocks

    Theorem (extension of Buehler 2007 [2])

    The stock price process remains positive if and only if it has the form

    where the positive local martingaleZis called the pure martingale of

    the stock price process.

    The extension over [2] is that this actually also holds in the presence of

    stochastic interest rates and for anydividend structure, not just affine

    dividends as in [2].

    tk

    k

    tttt

    k

    DZSRSt:

    0

    ~:

    Discounted value ofall future dividends

    Ex-dividendstock price

    k

    t

    k

    tk

    k k

    kRDDSS

    t

    t0

    0:000

    ::

    ~

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    The Structure of Dividend Paying Stocks

    Consequences

    In the case of deterministic rates and borrow, we get [1], [2]:

    with forward

    This structure is not an assumption it is a consequence of the

    assumption of positivity S>0 !

    All processes with discrete dividends look like this.

    tk

    k

    tttttttt

    k

    DRAAZAFSt:

    :

    tk

    k

    ttttt

    k

    DRZSRFt:

    0

    The stochasicity of the equitycomes from the excess valueof S over its future dividends.

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    The Structure of Dividend Paying Stocks

    Structure of Dividends

    A consequence of the aforementioned is that we can write alldividend models as follows:

    We decompose

    so that we can split effectively the stock price into a fixed

    cash dividend part and one where the dividends are

    stochastic:

    kkkkk

    kkk

    minmin

    min

    :~,min:

    ~:

    tk

    k

    tt

    tk

    k

    tttt

    kk

    DRDZSRStt :

    min,

    :

    0

    ~~:

    DeterministicdividendsRandom dividends,

    floored at zero

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    The Structure of Dividend Paying Stocks

    Exponential Representation Theorem

    Every positive stock price process S>0 which pays dividends kcan be written in exponential form as

    whereA is given as before and whereXis given in terms of aunit martingaleZas

    with stochastic proportional dividends

    tk

    tktt

    k

    XdXt:

    )()Zlog(

    min,0

    ~: t

    X

    tt AeSRSt

    kk X

    k

    keS

    Xdt

    t

    0

    ~

    ~

    1log:)(

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    Part III

    Affine Dividends

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    Affine Dividends

    Affine Dividends

    Black Scholes Merton: inherently supports proportionaldividends.

    Plenty of literature on general affine dividends, i.e.

    All known approaches either:

    approximate by approach (i.e., the dividends are not affine). approximate by numerical methods

    but they fit well in out framework.

    )0(: iii dS

    it

    iSiii

    ta:

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    Affine Dividends

    Structure of the Stock Price

    Direct application in our framework can be done but it is easier

    to simply write the proportional dividend effectively as part of

    the repo-rate m(this is what happens in Merton 1973 [5]) i.e.

    write

    All previous results go through [1,2], i.e. we get

    with our new funding factorR.

    iSiii ta:

    ti

    i

    dsr

    t

    i

    tsseR

    t

    m

    :

    )1(: 0

    tk

    k

    tttttttt

    k

    DRAAZAFSt:

    :

    Again thisstructure is theonlycorrect

    representationof a stock price

    which paysaffine

    dividends.

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    Affine Dividends

    Impact on Pricing Vanillas

    The formula

    really means that we can model a stock price which pays affinedividends by modeling directlyZsince:

    which means that we can easily compute option prices on Sif weknow how to compute option prices onZ.

    Hence,Zcan be any classic equity model Black-Scholes

    Heston, SABR, l-SABR

    Levy/Affine

    Numerical Models (LVSV) ....

    TTTTTTT

    AF

    AKKKZAFKS

    :

    ~

    ~E)(E

    ttttt AZAFS

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    Affine Dividends

    Implied Volatility Affine Dividends

    The reverse interpretation allows us to convert observed marketprices back into market prices onZ:

    which in turn allows us to computeZs implied volatility fromobserved market data.

    TTTTTT

    Z AKAFTAF

    KT

    ~

    )(,MarketCall)(DF

    1:)

    ~,(Call

    H

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    Affine Dividends

    Implied Volatility and Dupire with Affine Dividends

    Case 1: Market is given as a flat40% BS world.

    We imply the pure equityvolatility forZif we assume thatdividends are cash for 3Y, then

    blended and purely proportionalafter 4Y

    Case 2: Market is given as anaffine dividend world with a 40%

    vol onZ(3Y cash, proportionalafter 4Y).

    We imply the equivalent BSimplied volatility for S.

    H

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    Affine Dividends

    Implied Volatility and Dupire with Affine Dividends

    Once we have the implied volatility from

    we can compute Dupires local volatility for stock prices with

    affine dividends as

    Similarly, numerical methods are very efficient, see [2]

    Simple credit risk

    Variance Swaps with Affine Dividends

    PDEs

    TTTTTT

    Z AKAFTAF

    KT

    ~

    )(,MarketCall)(DF

    1:)

    ~,(Call

    ),(Call

    ),(Call2:),(

    22

    2

    xtx

    xtxt

    Zxx

    XtX

    s

    H

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    Affine Dividends

    Main practical issues

    Since the stock price depends on future dividends, anymaturity-Toption price has a sensitivity to any cash

    dividends past T.

    The assumption that a stock price keeps paying cash

    dividends even if it halves in value is not really realistic

    Black & Scholes assumes at least that the dividend falls

    alongside the drop in spot price

    Hence, assuming we are structurally long dividends it is more

    conservative on the downside to assume proportional

    dividends rather than cash dividends.

    All in all, it would be desirable to have a dividend model

    which allows for spot dependency on the dividend level.

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    Part IV

    Modeling Stochastic Dividends

    H

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    Modeling Stochastic Dividends

    Basics

    From the market of dividend swaps, we can imply a future levelof dividends.

    The generally assumed behavior is roughly

    The short end is cash (since rather certain)

    The long end is yield (i.e. proportional dividends)

    H

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    Modeling Stochastic Dividends

    Dividends as an Asset Class

    H

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    Modeling Stochastic Dividends

    Modeling

    Before we looked at cash dividends.However, following our remarks before we can focus on the

    exponential formulation

    This proportional dividend approach makes life much

    easier basically, to have a decent model, we only have to

    ensure that dremains positive.

    We will present a general framework for handling dividend

    models on 2F models.

    We start with a BS-type reference model

    )1( idi eSSS

    ttt

    H

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    Modeling Stochastic Dividends

    Modeling

    What do we want to achieve: Very efficient model for test-pricing options on dividends

    Black-Scholes-type reference model.

    Modeling assumptions

    Deterministic rates (for ease of exposure) We know the expected discounted implieddividendsD and

    therefore the forward

    Our model should match the forward and

    Drops at dividend dates

    tkk

    ttt

    kDSRF t:

    0:

    H

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    Modeling Stochastic Dividends

    Proportional Dividends

    Black & Scholes with proportional discrete dividends:

    such that

    to match the market forward we choose

    i

    itttttt dtddtdWdtrSd )2

    1log 2

    tssm

    tkk

    t

    ssst

    t

    s

    t

    sstt

    k

    ddWrF

    dsdWFS

    t

    m

    ss

    :0

    0

    2

    21

    0

    )(exp

    exp

    kF

    Dd

    k

    k

    t

    01log

    H

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    Modeling Stochastic Dividends

    Proportional Dividends

    Since we always want to match the forward, consider theprocess

    which has to have unit expectation in order to match themarket.

    This approach has the advantage that we can take the

    explicit form of the forward out of the equation.

    In Black & Scholes, the result is

    t

    tt

    F

    SX log:

    dtdWdX tttt2

    2

    1ss

    H

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    Stochastic Proportional Dividends (Buehler, Dhouibi, Sluys 2010 [3])

    Let u solve

    and define

    The volatility-like factor ekexpresses our (static) view on thedividend volatility:

    ek= 1 is the normal

    ek= dk is the log-normal case

    The constant c is used to calibrate the model to the forward,i.e. E[ exp(Xt) ] = 1.

    The deterministic volatility s is used to match a term structureof option prices on S.

    Modeling Stochastic Dividends

    kkktttt

    dtcuedtdWdX )()(2

    1 2 tt

    ss

    ttt dBdtkudu n

    H

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    Modeling Stochastic Dividends

    21:

    E1

    log:TTk

    kt

    t

    t

    kS

    yt

    Regimewith mean-reverting

    yield

    Trending

    yield

    CH

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    Modeling Stochastic Dividends

    Stochastic Proportional Dividends

    We have

    Note log S/Fis normalmean and variance ofSare analytic.

    Step 1: Find cksuch that E[St] = Ft.

    Step 2: Given the stochastic dividend parameters for u, find ssuch that Sreprices a term-structure of market observable option

    prices on Smodel is perfectly fitted to a given strike range.

    t

    tk

    kks

    t

    sstt

    k

    kcuedsdWFS

    0:

    2

    21

    0exp

    t

    tss

    CH

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    Modeling Stochastic Dividends

    Stochastic Proportional Dividends

    DynamicsThe short-term dividend yield

    is approximately an affine function ofu, i.e.

    A strongly negative correlation therefore produces very

    realistic short-term behavior (nearly fixed cash) while

    maintaining randomness for the longer maturities.

    kt

    t

    tS

    y t E1

    log:

    tt buay

    CH

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    Modeling Stochastic Dividends

    Stochastic Proportional Dividends

    CH

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    Modeling Stochastic Dividends

    Stochastic Proportional Dividends

    Good Very fast European option pricing calibrates to vanillas

    We can easily compute future forwards Et[ST] and therefore

    also future implied dividends.

    Very efficient Monte-Carlo scheme with large steps since(X,u)are jointly normal.

    CH

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    Modeling Stochastic Dividends

    Stochastic Proportional Dividends

    Not so great Dividends do become negative

    No skew for equity or options on dividends.

    Dependency on stock relatively weak

    try a more advanced

    version

    Very littleskew in the

    optionprices

    CH

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    From Stochastic Proportional Dividends to a General Model

    where this time we specify a convenient proportional factor

    function. A simple 1F choice

    q= 1/S* controls the dividend factor as a function spot: For S >> S* we get 1/Sand therefore cash dividends.

    For S

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    From Stochastic Proportional Dividends to a General Model

    Modeling Stochastic Dividends

    Proportionaldividends on

    the very shortend

    Cashdividends onthe high end.

    xexu

    q

    1

    1),

    CH

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    From Stochastic Proportional Dividends to a General Model

    2F version which avoids negative dividends

    various choices are available, but there are limits ... for example

    yields a cash dividend model with absorption in zero.

    future research into the allowed structure for .

    Modeling Stochastic Dividends

    xe

    uxu

    q

    a

    1

    )1)(tanh(

    2

    1),

    xexu ),

    CH

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    Definition: Generalized Stochastic Dividend Model

    The general formulation of our new Stochastic Dividend Model is

    Note that following our Exponential Representation Theorem

    this model is actually very general:

    it covers allstrictly positive two-factor dividend models where

    future dividends are Markov with respect to stock and another

    diffusive state factor ... in particular those of the form:

    as long asS>0

    .

    Modeling Stochastic Dividends

    kkk

    ttt

    tttttt

    dtcXue

    dtcXu

    dtXdWXdX

    kk )(),

    )),(

    )(2

    1)(

    discrete

    yield

    2

    ttt

    ss

    k

    t

    k

    ttttt dtuSdWSSdS )(),()( tt s

    CH

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    Generalized Stochastic Dividend Model

    This model allows a wide range of model specification including cash-

    like behavior if(u,s) 1/s.

    Compared to the Stochastic Proportional Dividend model, this model has

    the potential drawbacks that

    Calibration of the fitting factors c is numerical.

    Calculation of a dividend swap (expected sum of future dividends)

    conditional on the current state (S,u) is usually not analytic.

    Spot-dependent dividends introduce Vega into the forward !!

    Modeling Stochastic Dividends

    k

    kk

    ttt

    tttttt

    dtcXue

    dtcXu

    dtXdWXdX

    kk)(),

    )),(

    )(2

    1)(

    discrete

    yield

    2

    ttt

    ss

    CH

    d l h d d

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    The rest of the talk will concentrate on a general calibration

    strategy for such models using Forward PDEs.

    Modeling Stochastic Dividends

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    Part V

    Calibration

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    Calibrating the Generalized Stochastic Dividend model

    We aim to fit the model to both the market forward and a market ofimplied volatilities.

    The main idea is to use forward-PDEs to solve for the density and

    thereby to determine

    i. The drift adjustments c and

    ii. The local volatility s. We assume that the volatility market is described by a Market Local

    Volatility which is implemented using the classic proportional dividend

    assumptions of Black & Scholes (or affine dividends).

    Discussion topics:

    Forward PDE and Jump Conditions

    Various Issues

    Calibrating c and susing a Generalized Dupire Approach

    A few results.

    Calibration

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    Forward PDE

    Recall our model specification

    On t tkthis yields the forward PDE

    with the following jump condition on each dividend date:

    Calibration

    dBdtudu tt n

    k

    kk

    ttt

    tttttt

    dtcXue

    dtcXu

    dtXdWXdX

    kk)(),

    )),(

    )(2

    1)(

    discrete

    yield

    2

    ttt

    ss

    ),()(),(2

    1),()(

    2

    1

    ),(),(),;)(2

    1),(

    22222

    yield

    2

    uxpxvuxpvuxpx

    uxpuuxpcuxtxuxp

    txutuuttxx

    tutttxtt

    srs

    s

    )),,((),( discrete uuxxpuxp

    tt

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    Issue #1: Jump Conditions

    The jump conditions require us to interpolate the densityp on thegrid which is an expensive exercise hence, we will approximate

    the dividends by a local yield.

    This simply translates the problem into a convection-dominance

    issue which we can address by shortening the time step locally (in

    other words, we are using the PDE to do the interpolation for us).

    Definitely the better approach for Index dividends.

    Calibration

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    Issue #2: Strong Cross-Terms

    The most common approach to solving 2F PDEs is the use of ADI schemes

    where we do a q-step in first thex and then the u direction and alternate

    forth.

    The respective other direction is handled with an explicit step and

    that step also includes the cross derivative terms.

    If |r|

    1, this becomes very unstable and ADI starts to oscillate ... inour cases, a strongly negative correlation is a sensible choice.

    We therefore employ an Alternating Direction Explicit ADE scheme as

    proposed by Dufffie in [9].

    Calibration

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    ADE Scheme

    Assume we have a PDE in operator form

    we split the operatorA=L+Uinto a lower triangular matrixL and an upper

    triangular matrix U, where each carries half the main diagonal.

    Then we alternate implicit and explicit application of each of those operators:

    However, since both U and L are tridiagonal, solving the above is actually

    explicit hence the name.

    This scheme is unconditionally stable and therefore good choice for

    problems like the one discussed here.

    In our experience, the scheme is more robust towards strongly correlated

    variables ... and much faster for large mesh sizes.

    However, ADI is better if the correlation term is not too severe.

    Calibration

    Apdt

    dp

    dttdttdttdtt

    tdtttdtt

    dtUpdtLppp

    dtUpdtLppp

    2/2/

    2/2/

    RCH

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    ADE vs. ADI

    Stochastic Local Volatility where we

    additionally cap and floor the total volatility term.

    In the experiments below, the OU process parameters where =1,

    n=200% and correlationr=-0.9 (*).

    Calibration

    tt

    u

    tt

    dWSeStdS t21

    );(s

    Instabilityon the

    short end Blows up afteroscillations

    from thecross-term.

    __

    (*) we usedX=logS, not scaled. Grid was 401x201 on 4 stddev with a 2-day step size and q=1 for ADI

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    Issue #3: Grid scaling

    We wish to calibrate our joint density for both short and long maturities

    from, say, 1M up to 10Y.

    A classic PDE approach would mean that we have to stretch our available

    mesh points sufficiently to cover the 10Y distribution of our process ...

    but then the density in the short end will cover only very few mesh

    points.

    The basic problem is that the processXin particular expands with sqrt(t)

    in time (u is mean-reverting and therefore naturally bound).

    We follow Jordinson in [1] and scale both the processXand the OU

    process u by their variance over time this gives (in the no-skew case) a

    constant efficiency for the grid.

    Calibration

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    Calibration

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    Jordinson Scaling

    Calibration

    Imprecise forshort maturities

    Constantprecision overthe entire time

    line

    RCH

    Calibration

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    Jordinson Scaling

    It is instructive to assess the effect of scalingX.SinceXfollows

    we get

    the rather ad-hoc solution is tostart the PDE in a state dt where

    this effect is mitigated.

    Calibration

    dtcXudtdWdX tttttttt ),2

    1 2 ss

    dtXt

    dtctXudtt

    dWt

    Xd ttttttt

    t

    ~1)

    ~,

    2

    11~ 2

    s

    s

    Very strongconvection

    dominance fort0

    RCH

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    Calibration

    Let us assume that our forward PDE scheme converges robustly.

    The next step is to use it to calibrate the model to the forward and

    volatility market.

    Calibration

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    Calibration

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    Generalized Dupire Calibration

    Assume we are given

    A state process u with known parameters.

    A jump measureJwith finite activity (e.g. Merton-type jumps; dividends;

    credit risk ...) and jumps wt(St-,ut) which are distributed conditionally

    independent onFt- with distribution qt(St-,ut;.)

    We aim at the class of models of the type

    where we wish to calibrate

    c to fit the forward to the market i.e. E[St] =Ft.

    sto fit the model to the vanilla option market we assume that this is

    represented by an existing Market Local Volatility S.

    Calibration

    v

    tt

    ttt

    uS

    t

    ttttttttt

    dWdtdu

    uSdtJeS

    dWSutStdtScuStdS

    ttt

    )()(

    ),;()1(

    );();()),;((),(

    a

    smw

    The drift c will be used to fitthe forward to the market

    Note theseparablevolatility.

    ARCH

    Calibration

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    Generalized Dupire Calibration

    Let us also introduce m such that

    where m may have Dirac-jumps at dividend dates.

    We will also look at the un-discounted option prices

    and for the model

    Calibration

    t

    st dsmSF0

    0 exp

    KtCalltDF

    KtC ,)(

    1:),(market

    KSKtC tE:),(model

    ARCH

    Calibration

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    Generalized Dupire Calibration - Examples

    Calibration

    ));(

    )()1(1]dt-E[);(

    );(

    );());((

    21

    dtdNSdtSdWSStdtSmdS

    dtNeSedWSStdtSmdS

    dWSeStdtSmdS

    dWSStdtScutrdS

    p

    tt

    tt

    t

    St

    p

    tttttttt

    k

    tttttttt

    tt

    u

    tttt

    tttttt

    l

    l

    ls

    ls

    s

    s

    Stochastic interestrates, see

    Jordinson in [1]

    Stochastic localvolatility c.f.Ren et al [7]

    Merton-type jumps

    Default riskmodeling with

    state-dependentintensity ala

    Andersen et al [8]

    We usedNl toindicate a

    Poisson-processwith intensity l.

    ARCH

    Calibration

    uS

    ttttttttt

    uSdtJeS

    dWSutStdtScuStdS

    ttt );()1(

    );();()),;((),(

    smw

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    Generalized Dupire Calibration

    We apply Ito to a call price and take expectations to get

    If the density of(S,u) is known at time t-, then all terms on the right handside are known except s and c.

    Ifc is fixed, then we have independent equations for eachK.

    The left hand side is the change in call prices in the model.

    The unknown there is

    Calibration

    ),;(E);(E);(

    2

    1

    1E)(),;(1EE1

    ),(

    222

    tttt

    uS

    t

    tKS

    KStttKStt

    uSdtJKSKeS

    utKtK

    StcuStSKSddt

    ttt

    t

    tt

    w

    s

    m

    KS dttE

    v

    tt

    tttt

    dWdtdu

    uSdtJeS ttt

    )()(

    ),;()1(),(

    a

    ARCH

    Calibration

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    Generalized Dupire Calibration Drift

    In order to fix c, we start with the caseK=0: we know that the zero strike

    call in the market satisfies

    On the other hand, our equation shows that

    hence we have two options to determine the left hand side:

    a. Incremental Fit:

    b. Total Fit :

    Calibration

    dtFmtCdt

    ttmarket )0,(1

    );()1(E1E)();(1EE1)( dtJeSStctSSddt ttKStKStt

    t

    tt

    wm

    dtFmSd ttt!

    E

    dttdtt FS !

    E

    ARCH

    Calibration

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    Generalized Dupire Calibration Drift

    Using the Total Fit approach is much more natural since it uses c to make

    sure that

    which is a primary objective of the calibration. The Incremental Match suffers from numerical instability: if the fitting

    process encounters a problem and ends up in a situation where E[St]Ft,

    then fitting the differential dE[St] will not help to correct the error.

    The Total Match, on the other hand, will start self-correcting any

    mistake by pulling back the solution towards the correct E[St+dt]Ft+dt.However, depending on the severity of the previous error, this may lead

    to a very strong drift which may interfere with the numerical scheme at

    hand.

    The optimal choice is therefore a weighting between the two schemes.

    Calibration

    dtStcdttSSdF KStKSttdtt tt 1E)();(1EE m

    ARCH

    Calibration

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    Generalized Dupire Calibration Volatility

    Recall

    where we now have determined the drift correction c - this leaves us with

    determining the local volatility st,K for each strikeK.

    We have again the two basic choices regarding dC(t,K):

    a. Incremental match (essentially Ren/Madan/Quing [7] 2007 for stochastic local

    volatility):

    b. Total Match (Jordinson mentions for his rates model in [1] 2006 ):

    Calibration

    ),;(E);(E);(

    2

    1

    1E)(),;(1E),(1

    ),(

    222

    model

    tttt

    uS

    t

    tKS

    KStttKSt

    uSdtJKSKeS

    utKtK

    StcuStSKtCdt

    ttt

    t

    tt

    w

    s

    m

    ),(1

    ),(1

    market

    !

    model KtCdtKtCdt

    ),(),( market

    !

    model KdttCKdttC

    ARCH

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    Generalized Dupire Calibration Volatility

    The incremental match

    It has the a nice interpretation in the case where we calibrate a stochastic local

    volatility model.

    The market itself satisfies

    hence we can set

    Calibration

    22

    market

    2

    );(E

    E);(:);(

    tKS

    KS

    utKtKt

    t

    t

    ss

    ),(1

    ),(1

    market

    !

    model KtCdt

    KtCdt

    KSmarketmarket tKtKdtKtdC

    s E,2

    1),( 22

    ARCH

    Calibration

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    Generalized Dupire Calibration Volatility

    Good & bad for the Incremental Fit:

    This formulation suffers from the same numerical drawback of calibrating to a difference as we

    have seen for c: it does not have the power to pull itself back once it missed the objective.

    It suffers from the presence of dividends (if the original market is given by a classic Dupire LV

    model) or numerical noise.

    The upside of this approach is that it produces usually smooth local volatility estimates for

    stochastic local volatility and yield dividend models.

    Calibration

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    Calibration

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    Generalized Dupire Calibration Volatility

    The total match following a comment of Jordinson in [7] 2007 for stochastic rates means to

    essentially use

    Good & Bad

    As in the c-calibration case, it has the desirable self-correction feature which makes

    it very suitable for models with dividends which suffer usually from the problem thatthe target volatility surface is not produces consistently with the respective dividend

    assumptions.

    It also helps to iron out imprecision arising from the use of an imprecise PDE scheme.

    The downside is that the self-correcting feature is a local operation.

    It can therefore lead to highly non-smooth volatilities which in turn cause issues for

    the PDE engine.

    We therefore chose to smooth the local volatilities after the total fitting with a smoothing

    spline.

    Calibration

    ),(),( market

    !

    model KdttCKdttC

    ARCH

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    Calibration

    Withoutsmoothing, the

    solution actually

    blows up in 10Y

    Smoothingbrings the fitback into line

    ARCH

    Calibration

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    Calibration

    ARCH

    Calibration

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    Generalized Dupire Calibration - Summary

    Any model of the type

    can very efficiently be calibrated using forward-PDEs.

    First fit c to match the forward with incremental fitting

    Match swith a mixture of incremental and total fitting.

    Apply smoothing to the local volatility surface to aid the numerical

    solution of the forward PDE.

    The calibration time on a 2F PDE with ADE/ADI is negligible compared

    to the evolution of the density we can do daily calibration steps.

    Index dividends are transformed into yield dividends.

    Calibration

    v

    tt

    ttt

    uS

    t

    tttttttt

    dWdtdu

    uSdtJeS

    dWSutStdtStcuStdS

    ttt

    )()(

    ),;()1(

    );();())(),;((),(

    a

    smw

    ARCH

    Last Slide

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    Generalized Stochastic Dividend Model (Index version)

    Last Slide

    dtcXudtXdWXdX ttttttttt )),()(2

    1)( yield

    2 ss

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    Thank you very much for your attention

    [email protected]

    *1+ Bermudez et al, Equity Hybrid Derivatives, Wiley 2006

    *2+ Buehler, Volatility and Dividends, WP 2007, http://ssrn.com/abstract=1141877

    [3] Buehler, Dhouibi, Sluys, Stochastic Proportional Dividends, WP 2010, http://ssrn.com/abstract=1706758

    *4+ Gasper, Finite Dimensional Markovian Realizations for Forward Price Term Structure Models", Stochastic Finance, 2006, Part II,265-320

    [5] Merton "Theory of Rational Option Pricing," Bell Journal of Economics and Management

    Science, 4 (1973), pp. 141-183.

    [6] Brokhaus et al: Modelling and Hedging Equity Derivatives, Risk 1999[7] Ren et al, Calibrating and pricing with embedded local volatility models, Risk 2007[8] Andersen, LeifB. G. and Buffum, Dan, Calibration and Implementation of Convertible Bond Models (October 27, 2002).Available at SSRN: http://ssrn.com/abstract=355308

    [9] Duffie D., Unconditionally stable and second-order accurate explicit Finite Difference Schemes using Domain Transformation,2007

    http://ssrn.com/abstract=355308http://ssrn.com/abstract=355308http://ssrn.com/abstract=355308http://ssrn.com/abstract=355308