Stochastic Local Volatility

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Stochastic Local VolatilityGrigore Tataru, Travis Fisher Quantitative Development Group, Bloomberg

[email protected], [email protected]

Version 1 – Feb 5, 2010

1. Introduction

Barrier prices and path dependent options in general are not determined by the vanilla market

quotes, they also depend on the dynamics of the market. Matching this extra market dimension isa modeling requirement to obtain good barrier prices. Classical models such as local volatility

(LV) and stochastic volatility (SV) are in fact calibrated to and completely determined by the

vanilla market and, as such, they offer little or no extra flexibility in matching the marketdynamics. The Stochastic Local Volatility (SLV) model we present here combines features of an

LV model with features of an SV model to give rise to dynamics that are unachievable by thesimpler models.

During the past few years, practitioners have settled on a consensus of using some variation of a

mixed stochastic/local volatility model as the standard for pricing barrier options in FX markets.

Details vary widely from implementation to implementation. The most naive approach is simplyto calibrate a local volatility model and, completely independently, calibrate a stochastic

volatility model. Then the prices for exotic options are estimated as being some % of the SVmodel price and 100− % of the LV model price. There is an obvious flaw to this approachin that the only dynamics consistent with the model are completely unnatural — randomly

choose at time zero to follow the SV model with probability

or the LV model with probability

1 − . Another approach extends the LV model to include several volatility states, with somedynamics to jump between these states. For more sophisticated approaches which modelrealistic dynamics, the challenge is how to perform a good calibration of the model in a

reasonable amount of time. Some practitioners resort to carefully chosen functional forms for

the local volatility surface to allow semi-analytic pricing of vanilla options. There are problems

that arise from that approach: the functional form required (typically quadratic) may imply thatthe exchange rate can reach zero or infinity. Calibration speed is still an issue, as is the stability

of the best fit.

In contrast, we allow a non-parametric form for the local volatility component of our SLV

model: the ―leverage surface‖. We calibrate it quickly and stably using a fixed point approach

centered around the solution of the forward Kolmogorov PDE for the transition density. Thisgives us near complete freedom of choice of stochastic parameters while still matching the entire

volatility surface of vanilla options.

In this paper we will describe in detail the SLV model as implemented in the Bloomberg OVML

function. We will give motivation for the choices we have made, particularly regarding thecalibration freedom of the model. We will also describe the mathematical approach we follow.

We will start in the next section with the description of our stochastic local volatility model, and

mailto:[email protected]:[email protected]:[email protected]:[email protected]


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then continue with extensive details regarding the model calibration. Finally, we will discuss

pricing and make comparisons with real-world data.

The first reference that we are aware of describing a SLV model is a paper by Jex, Henderson

and Wang [3]. Many other authors have contributed to this literature in the intervening years.

Our approach is similar to Lipton [4] and Ren, Madan, Qian [5] who describe a calibrationmethod based on the forward Kolmogorov PDE. The fixed point approach for the KolmogorovPDE that we use here is based on a method implemented at Bear Stearns in 2007 by the first

author, along with Christian Gilles and Graham Wells and with strong market insight coming

from Andrew Dexter. We offer thanks to Apollo Hogan, Ashish Midha, Peter Carr, MarkRubery, Leif Andersen, Peter Lee, and Oleg Kovrizhkin for helpful advice.

2. Model Details

In the SLV model, the foreign exchange rate dynamics are governed by the stochastic differentialequations: = − + , ∙ 1, (1) = − + 2,

1 ⋅ 2 = . Here S t represents the spot exchange rate process (given as units of domestic currency for a unit

of foreign currency),

is the stochastic volatility process, and

1 and

2 are independent

standard Brownian motions. We use deterministic rate curves for (the domestic interest rate)and (the foreign interest rate)., represents the local volatility component of the model, the leverage surface. We take anon-parametric form for , , calibrated numerically to match the implied volatility surface.The other parameters have the same meaning as in purely stochastic models: = speed of meanreversion, = long-term mean level of , = volatility of the volatility process , =correlation of the Brownian increments. We allow a term structure of these parameters with

piecewise constant values.

One should note that by setting = 0 the SLV model degenerates to a local volatility model. Atthe other extreme, setting

,

≡1 gives a purely stochastic vol model. The mixing of the LV

and SV features in the general SLV is mostly controlled by the volatility of volatility , followedin importance by the correlation ρ. To make it more intuitive, we simplify this mixing to a one- parameter family: = ⋅ ρ = λ ⋅ ρmax 0% ≤ ≤ 100%, where and ρmax correspond to what we will define later as the set of maximally stochastic parameters. Again, we allow a term structure of these parameters, including the mixing fraction

λ , with piecewise constant values.


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The dependence of barrier prices on the mixing of stochastic and local components is shown in

Figure 1. Different mixing fractions of the SLV model give different prices, all consistent withthe same implied volatility surface. The graph shows spot scenario value curves for a double-no-

touch option. As discussed later, our calibration includes matching of the spot/smile dynamics to

choose between these different possibilities.

We have chosen a lognormal process for the volatility process, as opposed to the familiar square-root process found, for example, in the Heston model. We get more realistic behavior for the

paths of the volatility process and for the dynamics of the volatility surface. Also, the volatility

process does not reach 0, whereas it is possible in some cases for the Heston model. Oneadvantage of the Heston model, due to the square-root variance process, allows for analytic

formulas for vanillas. This is lost in the more general case of stochastic local volatility models,

though approximate formulas can be obtained for very particular choices of the leveragefunction. Since we are using a general leverage surface and numerical solutions of the PDEs, ourmethod is not resting on the existence of analytical formulas, and as such we have made a choice

for the volatility process that we believe is more in line with the market behavior.

Models with a lognormal volatility process can have infinite moments and thus give infinite

prices for some instruments. This is not an issue for the model described here, mostly due to the

choice of a non-parametric leverage surface and its construction. Any tendency of the stochasticvolatility to produce high volatility in the wings will be annihilated by a leverage surface that

decreases fast in the wings.

Figure 1

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

1.58 1.63 1.68

100% Stochastic

50% Stochastic

0% Stochastic(Local Vol)

Black-Scholes


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3. Calibration Overview

Given a choice of stochastic parameters, one can calibrate the leverage surface , to matchthe vanilla market. A different set of stochastic parameters, along with the recalibrated , ,will again match the vanilla market, but will correspond to a different dynamics.

The calibration process has two main steps: find stochastic parameters to match the dynamics(not to be recalibrated often) and then calibrate the leverage surface , to match the vanillas(recalibrated more often to the continuously changing market quotes for vanillas).

Both steps are essential for a successful model, but the second one is more important: not

matching the vanillas gives a bigger mispricing error of barrier options than a similar error in

stochastic parameters (the vanillas are the ones that every market participant agrees on, withoutthe same being true about exotics, or at least not to the same degree). The second step is also

more technically complex and is the one that corrects any misspecification of the stochastic

parameters.

The first calibration step — choosing stochastic parameters — is as much of an art as a science forour situation. We will achieve an equally good fit to the current vanilla market regardless of thischoice, so there is no information in the current volatility surface that forces a particular set of

stochastic parameters. In many cases market makers who use versions of the SLV model may

hand-tune one or more parameters for a given market, or even for different instruments tradingon the same market. Also market participants may calibrate their models to barrier option prices

available to them. Neither of these approaches is available for Bloomberg’s default calibration.

Instead we have focused on matching particular statistics relating the historical dynamics of

volatility skew and spot. Details of this step will be presented later in the document.

4. Leverage Surface Calibration

Given a set of stochastic parameters, interest rate curves, and an implied volatility surface, wecalibrate a non-parametric leverage surface which matches the entire volatility surface. This

calibration is carried out by solving the forward Kolmogorov partial differential equation for the

SLV model while simultaneously solving for the unknown leverage surface in order to match theconditional forward spot densities implied by the vanilla volatility surface.

The computation is organized into two main steps. First match the market with a local volatility

model. Second, for the given stochastic parameters, find a leverage function such that the SLVhas the same marginal distributions as the local volatility model.

Local Volatility

We start by calibrating a local volatility model: = − + , . Using the Dupire formula to strip the implied volatility surface we get the local volatility , .The implied volatility surface is interpolated from available data; in the usual case this is the

Bloomberg generic volatility data which is itself a filtered, smoothed, and averaged version ofthe data obtained directly from contributors.


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Leverage Surface

The marginal distribution of the SLV model is (see Gyöngy [1]) the same as the distribution of a

local volatility model with , given by:2 (, )

2

= ( , )2

2

= = (, )2

2

= . We want (, ) to be the local volatility calibrated above which will guarantee that our SLV iscalibrated to the market. The difficulty is to find (, ) such that the relation (2) holds. Thedifficulty is increased by the fact that the conditional expectation above, 2 = , itselfdepends on (, ) in a nontrivial way.To see this, consider the forward Kolmogorov PDE for the transition probability density, , of (1) from the initial state 0, 0,0 to , ,:

3 – 1

2 2

2 2

, 2

2

−1

2 2

2

2 2

− 2

, 2

+ − + − = 0, 0, , = − 0 − 0, where is the Dirac distribution centered at = 0. We calculate the conditional expectation of the instantaneous variance given future spot and time

as an integral against the density:

2 = = 2, ,∞0

,

,

∞0

and can rewrite (2) as:4 (, )2 = (, )2 2, ,∞0 , ,∞0

.

Both (, ) and , , are unknown at this stage, while all the other terms in the equation (3)and (4) are known, including the stochastic parameters and the local volatility. Note that

knowing gives by (4), while knowing gives by solving the PDE (3). We will set up aniterative argument in the next section to find , ,, and hence (, ). This will give an SLVmodel calibrated to the vanilla market for the given set of stochastic parameters.

Fixed point reformulationFinding

,

,

to satisfy (3) and (4) amounts to solving the nonlinear PDE:

5 − 12 22 22, 2 , ,∞0 2, ,∞

0

− 122 22 2

− 2 2, , ,∞0 2, ,∞

0

1 2 + − + − = 0.


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This can be reformulated as a fixed point problem → = where the operator ∙ is thesolution of the linear PDE:

6 − 12 22 22 , 2 , ,∞0 2, ,∞

0

− 122 22 2

− 2

2, , ,∞0 2, ,∞0 12

+ − + − = 0. One can construct a recursive sequence () →() = (+1) by solving the above PDErepeatedly. Corresponding to each (), , we have a leverage function ()(, ) defined by(4) and the procedure above can be understood alternatively as a fixed point problem for , . We will solve the previous PDE numerically. Convergence of the sequences () and (), atleast numerically, will give us the calibrated leverage surface.

Note that we expect

,

=

,

,

∞0

to be the transition density of the local volatility

model, assumed to be known, for example by solving the forward Kolmogorov PDE for the LVmodel:

7 − 12 22 2, 2 + − = 0. We could have replaced ∞

0 and ∞

0 with in the (5) and (6) to end up with a slightly

different construction for and (). As a sanity check, one would like to know that, = , ,∞0 is indeed true for obtained by solving (5) (or the modified versionof (5)). This follows by simple integration of (5) in the direction and the uniqueness ofsolutions for equation (7).

Finite difference scheme

To solve (6) we use a finite difference method, a variation of the Douglas scheme as described in

in ’t Hout and Foulon [2]. This is an operator splitting scheme of ADI type with a predictor-

corrector treatment of the mixed derivative. To solve the one-dimensional problems in the and direction we use fully implicit time-stepping, while for the mixed partial derivative in and representing the correlation term we use fully explicit time-stepping.

5. Risk-Reversal Dynamics in the SLV model

In a typical local volatility model, the risk-reversal will increase when spot increases and

decrease when spot decreases. Schematically, this can be seen as a result of the curvature of the

local volatility surface as illustrated in Figure 2. Suppose the local volatility surface is a functionof spot as given by the curve shown. For a lower level S1 of the spot, the 25 delta call and put

strikes are C1 and P1, respectively. For a slightly higher level S2 of the spot, the 25 delta call and

put strikes are C2 and P2, respectively. The implied volatility for a vanilla option is related to a

weighted average of the local volatilities between the spot and strike levels. Thus the impliedvolatility for strike C2 will be higher than the implied volatility for strike C1, and the implied

volatility for strike P2 will be lower than the implied volatility for strike P1. Since the risk-


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reversal is given by the difference of the implied volatility for the call minus the implied

volatility for the put, the risk-reversal is increased by increasing spot.

In the stochastic volatility model, the correlation between spot and risk-reversal is neutralized.Our SLV model, with the leverage surface set equal to one, becomes the following stochasticvolatility model: = − + 1 = − + 2 Note that this model has a clear spot scaling property. Rescaling the spot process results in

rescaled but otherwise unchanged dynamics. The volatility smile expressed in terms of deltas

shares the same spot scaling property and is unchanged by rescaling spot. So for the pure

stochastic volatility model, the risk-reversal is unchanged by changing the initial spot level.

For our SLV model, we never use a completely pure stochastic volatility model. We alwayshave a leverage surface calibrated to meet the residual prices of the vanilla options from the fullvolatility surface. Still the overall relationship is the same: on the local volatility side of the

spectrum the SLV model has a strong positive relationship between the risk-reversal and the spot

level. On the maximally stochastic end of the spectrum, the SLV model has risk-reversal levels

nearly unaffected by changes of the spot level. For mixing fractions in between, there areintermediate relationships between the risk-reversal and spot dynamics.

Figure 3 shows normalized risk-reversal vs. spot for the calibrated SLV model, together with

local volatility and maximally stochastic volatility extremes of the SLV model. This uses theUSDJPY volatility surface as of 1 Feb 2010. Note the expected trend for the local volatility

model compared to the near-constant trend for the maximally stochastic model. The calibrated

SLV model gives an intermediate result.

C2

C1P1

P2

Figure 2


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6. Risk-Reversal Dynamics and One-Touch OptionsThe dynamics of the risk-reversal in relation to the spot are also closely related to barrier option pricing. For example consider a down one-touch option paying in domestic currency. In the

Black-Scholes model an approximate hedge for it would be a European digital put option struck

at the barrier level paying double the touch option’s notional. If the barrier is hit, the digital

option is at-the-money and so has approximately (ignoring rates, etc.) 50% chances of paying. A50% chance at double the payoff for the digital option vs. 100% chance at the payoff for the

touch option at the barrier hit would give these the same value. In a model with a skew,

however, this hedge will fail because the digital put price depends on the implied volatility skewat its strike: the higher the skew, the higher the value of the digital option. Thus the necessary

notional amount of the hedge is no longer double the one-touch notional, but is instead related to

the at-the-money implied volatility skew that will be in effect when the spot has moved to the barrier level. If the skew is higher the digital price is higher and less notional is required,whereas if skew is lower the value of the digital is lower and more notional is required. In

conclusion, if the one-touched is hedged as above, its value on trade date depends on the notional

of the hedging digital, which depends on the implied volatility skew when the spot reaches the barrier.

-35.00%

-30.00%

-25.00%

-20.00%

-15.00%

-10.00%

-5.00%

0.00%

5.00%

80 82 84 86 88 90 92 94 96 98 100

R R V o l a s P e r c e n t o f A T M V

o l

USDJPY Spot

Calibrated SLV Local Vol Stochastic Vol

Figure 3


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7. Historical Trends of Risk-Reversal Dynamics

Particularly for skewed currency pairs, there is often a very clear relationship between the spot

exchange rate and the volatility skew. This relationship is immediately clear in a scatter-plot ofrisk-reversal level vs. spot. A market like USDJPY may stay in the same regime for many

months, revisiting the same spot and risk-reversal levels. Unfortunately for modeling, the market

may also shift to a different regime, re-normalizing around a different base range of spot and adifferent base level of risk-reversal. This re-normalizing is beyond the range of market dynamicsthe SLV model can capture. Still we are interested in historical calibration, and for that purpose

we want to answer the question: to the extent that the risk-reversal dynamics are predictable,what is the best prediction of the relationship of spot to risk-reversal?

We have developed an approach which helps to isolate the predictable trends from the

unpredictable regime changes. To do this we study deviations of the risk-reversal from its

exponentially weighted moving average, in comparison to deviations of the exchange rate fromits exponentially weighted moving average. Figure 4 illustrates the advantage of this approach.

On the left, we look at the normalized risk reversal (one month RR vol as a percent of ATM vol)

compared to exchange rate. We split the data into three consecutive time periods. In each time period, the normalized risk-reversal shows a strong trend with spot. But for the entire time period, the market shifts overwhelm the trends. On the right, we look at the deviation from

exponentially weighted moving average, using a half-life of 12 business days. Here the trend is

clear.

Figure 4


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8. Historical Stochastic Parameter Calibration—Implementation Details

As discussed above, the key degree of freedom of the SLV model is the ability to interpolate

between a local volatility model and a stochastic volatility model. In terms of the dynamics ofthe volatility smile, this degree of freedom is reflected in different relationships of how the risk-

reversals change when spot moves. Stylistically, for a pure local volatility model with a smile in

the volatility surface, the risk-reversal increases as spot increases and decreases as spotdecreases. For a purely stochastic volatility model, the risk-reversal is largely unchanged bymoves of the spot. This stylistic view is made a little cleaner even if we normalize the market

25% delta risk-reversal to a ―normalized risk -reversal‖ by dividing by the at-the-moneyvolatility. The separation of the risk-reversal strikes is approximately proportional to the overalllevel of volatility, so the normalization gives a value that more cleanly reflects the degree of

skewness of the volatility smile.

We calibrate stochastic parameters in a two-phase process. The first phase is to find a set of―maximally stochastic‖ parameters, where the residual shape of the leverage surface is nearly flat

in the main region of interest. The second phase is to find a term structure of mixing fractions to

match the desired relationship of normalized risk-reversal move to spot move.

The calibration of maximally stochastic parameters proceeds in a bootstrap fashion. For each

maturity in the term-structure of parameters, a Levenberg-Marquardt nonlinear least squares

optimization is used. The mean reversion rate is fixed to 1.0. The initial 0 and long-termmean level of are both set to 1.0 as well, reflecting a scaling of the leverage surfacecomparable to the local volatility surface. The parameters to be calibrated are and . For eachtest set of parameters a leverage surface calibration is performed, and the resulting leverage

surface is measured for flatness. The objective function consists of three components. For

calibrating the and ρ in effect from time to +1, we take a vega-weighted average of 20leverage surface points distributed between the 10 delta put and the 10 delta call strikes. The

objective function penalizes deviation of leverage surface points from the average level. Thesecond component is forward-looking regularization that penalizes deviation of the leverage

surface points from the average deviation at the succeeding maturity +2. The third componentis a backward-looking regularization that penalizes deviation of from −1 and deviation of from −1.After the maximal stochastic parameter calibration, a further regularization is done to reduce

intraday variation. Instead of the calibrated parameters being used directly, an exponentially

weighted moving average is used with the new calibration weighted at 20%.

The mixing fraction calibration computes a trend of how normalized risk-reversal moves are

related to moves of the forward exchange rate. It performs a Levenberg-Marquardt nonlinearleast squares optimization to find a term structure of mixing fractions to match the model to the

historical relationship in both the short and long term. Daily historical data of the risk-reversal,

at-the-money, and forward exchange rate are used. The computed trend measures deviation ofnormalized risk-reversal from its exponentially weighted moving average. This is compared to

deviation of forward exchange rate from its exponentially weighed moving average. A linear

trend of these quantities is estimated, again using exponentially decreasing weights. The weights

used correspond to a halflife in days of 50*t , where t is the tenor as a year fraction, truncated to a


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minimum of 12 business days for short tenors and a maximum of 35 business days for long

tenors.

Currently the term structure of mixing fraction is calibrated to be controlled by two points — the

historical trends computed at the one month and one year tenors. The term structure of

parameters varies linearly in time for intermediate parameter points.

For each parameter choice, the leverage surface is calibrated corresponding to those parameters.

A series of option scenario pricings are performed to find the model normalized risk-reversal

values at a future time and across a range of spot levels. For each tenor t , the scenario horizontime is set to (1/2)t and option maturity is set to (3/2)t . A set of strikes are chosen in the range

between the currently 1% delta put and 1% delta call range. For each strike, a scenario pricing is

carried out giving prices at a series of spot levels chosen in the range between the currently 25%

delta put and 25% delta call. The implied volatility is computed for each strike and spot pair,giving an implied volatility smile for the strikes involved. For each spot, the 25% delta risk-

reversal and at-the-money volatilities are interpolated from this scenario volatility smile. A

linear trend is estimated between the scenario normalized risk-reversals and the scenario spotlevels.

The optimization penalizes differences between the scenario normalized risk-reversal trend and

the computed historical trend. The objective function also includes regularization penalties fordeviation of the mixing fraction parameter from 50%.

After the mixing fraction calibration, a further regularization is done to reduce intraday variation.Instead of the calibrated mixing fraction being used directly, an exponentially weighted moving

average is used with the new calibration weighted at 20%.

9. Pricing

Assume that we have a calibrated model (1). Pricing is done by solving backwards the following

PDE for the price , , of a claim: + 12 2, 22 22 + , 2 2 + 12 22 22 + − + − = ,

with the appropriate boundary conditions and final value for the payoff. We solve this PDE

numerically using a variation of the Douglas finite difference scheme, just as we use for the

leverage surface calibration PDE.

10. Calibration Speed and Accuracy

As a representative example, we calibrated the SLV model to market data from February 1,

2010. The calibrated term structure of stochastic parameters is as follows. Calibration of the set

of maximal stochastic parameters followed by the historical estimation of mixing fraction takes

about one minute. The parameters not shown are mean reversion (equal to 100% across theentire term structure) and the initial value 0 and the long-term mean level of (both equal to1.0).


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Vol-of-Vol Rho Mixing Fraction

0-2W 128.71% -0.1455 62.39%

2W-1M 79.60% -0.2063 62.39%

1M-2M 57.97% -0.2843 62.37%

2M-3M 52.35% -0.2787 61.90%

3M-6M 61.39% -0.254 61.42%

6M-1Y 64.99% -0.2626 59.96%1Y-2Y 64.05% -0.2587 57.05%

Using these parameters, we calibrate to the implied volatility surface. Our non-parametric

calibration matches the entire volatility surface to the very distant wings. Here we show the

results of testing the calibration on the key tenors and market strikes. Calibration time is about 1

second, using a single iteration of the fixed point solution. Option pricing times vary bymaturity, with an average pricing time in the table below taking about one half second. The mean

absolute volatility error in this example is 5.6 basis points while the root mean square error is 6.7

basis points. The maximum error of 17 basis points occurs in the high-volatility corner at the 2Ymaturity. In all cases the calibrated prices are within the bid/ask spread of the market data.

Table entries in the following table show the market level of volatility, followed in square

brackets by the error in implied volatility achieved when re-pricing each option using the SLV

pricing PDE.

10 Put 25 Put ATM 10 Call 25 Call

1W 14.14 [0.07] 13.36 [-0.08] 12.88 [-0.13] 12.89 [-0.05] 13.31 [0.14]

2W 13.95 [0.04] 13.08 [-0.05] 12.43 [-0.08] 12.31 [-0.02] 12.58 [0.10]

3W 14.35 [0.02] 13.37 [-0.04] 12.60 [-0.06] 12.43 [-0.01] 12.63 [0.08]

1M 14.42 [0.02] 13.39 [-0.03] 12.56 [-0.04] 12.31 [0.00] 12.45 [0.07]

2M 15.26 [-0.02] 13.94 [-0.07] 12.92 [-0.07] 12.54 [-0.02] 12.67 [0.06]

3M 16.06 [-0.04] 14.42 [-0.07] 13.20 [-0.07] 12.66 [-0.02] 12.77 [0.05]

4M 16.55 [-0.04] 14.74 [-0.06] 13.43 [-0.06] 12.81 [-0.02] 12.89 [0.04]

6M 17.17 [-0.04] 15.14 [-0.05] 13.70 [-0.05] 12.98 [-0.01] 13.05 [0.04]

9M 17.91 [-0.05] 15.57 [-0.06] 13.97 [-0.06] 13.13 [-0.02] 13.18 [0.03]

1Y 18.46 [-0.08] 15.85 [-0.08] 14.11 [-0.06] 13.17 [-0.02] 13.18 [0.02]

18M 19.06 [-0.12] 16.23 [-0.11] 14.35 [-0.09] 13.28 [-0.04] 13.29 [0.01]

2Y 19.36 [-0.17] 16.41 [-0.15] 14.49 [-0.11] 13.26 [-0.05] 13.24[0.00]

References

1. Gyöngy, I., Mimicking the One-Dimensional Marginal Distributions of Processes Havingan Ito Differential , Probability Theory and Related Fields 71 (1986)

2. in ’t Hout, K.J., Foulon, S., ADI finite difference schmes for option pricing in the Hestonmodel with correlation, Working paper (2007)

3. Jex, M., Henderson, R., Wang, D., Pricing Exotics Under the Smile, J.P.Morgan (1999)4. Lipton, A., The vol smile problem, Risk (2002)5. Ren, Y., Madan, D., Qian, M.Q., Calibrating and pricing with embedded local volatility

models, Risk (2007)

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