Post on 03-Apr-2018
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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
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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)
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Modeling Stochastic Dividends
Dividends as an Asset Class
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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
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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:
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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
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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
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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
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Modeling Stochastic Dividends
21:
E1
log:TTk
kt
t
t
kS
yt
Regimewith mean-reverting
yield
Trending
yield
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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
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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
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Modeling Stochastic Dividends
Stochastic Proportional Dividends
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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.
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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
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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),
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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 ),
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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
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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
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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/
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C lib ti
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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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C lib ti
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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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Jordinson Scaling
Calibration
Imprecise forshort maturities
Constantprecision overthe entire time
line
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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
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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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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
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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
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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
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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
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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
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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
ARCH
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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
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Calibration
Withoutsmoothing, the
solution actually
blows up in 10Y
Smoothingbrings the fitback into line
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Calibration
ARCH
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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
hans.buehler@jpmorgan.com
*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