advanced spot process for counterparty risk and …...march 2012 version 1.0 abstract geometric...

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Advanced Spot Process for Counterparty Risk and CVA via

Stochastic Volatility

Ignacio Ruiz∗, Ricardo Pachon†

March 2012

Version 1.0

Abstract

Geometric Brownian Motion is the standard model in the market for Equity andFX modelling in counterparty risk. Being a good starting point, this model showssome strong limitations. Ignacio Ruiz and Ricardo Pachon revise those shortcomingsand propose a new model based on stochastic volatility. The authors show thatcounterparty risk measures become more accurate, which has strong implications inrisk management, risk weighted assets calculations and CVA pricing. The authorsillustrate this with some practical examples where the more accurate stochasticmodel that is proposed decreases counterparty risk measures to half of what is oftengiven by GBM models.

Counterparty risk has now become a central focus of new developments in the financialindustry. The most widely used measures are PFEt, as a percentile-like measure of futureportfolio value, and EPEt, as the average of the possible exposures after zero-flooring.In risk management, these measures are mainly used in assessments of portfolio creditrisk, in capital calculations for risky assets and in CVA pricing. They are central toolsfor accounting for the counterparty risk embedded in a bank’s book of derivatives.

A key aspect of counterparty risk measurement is the modelling of the underlying riskfactors. The relevant models can be split into two families: models that simulate spotvalues (Equity and FX spots) and models that simulate term structures (yield curves,credit curves, commodity curves, etc). The most widely used model for spot simulation

∗Founding Director, iRuiz Consulting Ltd, London. Ignacio is a contractor and independent consul-tant in quantitative risk analytics and CVA. Prior to this, he was the head quant for Counterparty Risk,exposure measurement, at Credit Suisse, and Head of Market and Counterparty Risk Methodology forequity at BNP Paribas. Contact: [email protected]†Risk Model Validation, Credit Suisse, London. This work has been done in the context of a personal

collaboration with Ignacio Ruiz. Ricardo is in no way linked to iRuiz Consulting Ltd. The contentof this paper reflects the views of the author, it does not represent his employer’s views. Contact:[email protected]

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is GBM. This model, being a good starting point, shows a number of limitations, whichincludes:

1. Lack of netting with volatility

In a simple Geometric Brownian Motion (GBM) model, the volatility is a constantparameter. For that reason, volatility products, like variance swaps, cannot benetted off with spot driven products like forwards. So, counterparty risk set-ups using GBM to model the spot tend to aggregate those risks after netting,which produces artificially inflated PFEt and EPEt profiles. As explained later,stochastic volatility models overcome this issue.

2. Slow reactivity to market crises

Counterparty risk models tend to be calibrated to the real measure via historicaltime series for Risk Weighted Assets (RWA) and risk management. As a result,models tend to be insufficiently reactive to swinging market conditions and, hence,can misestimate risk during swift changes in market conditions. For example, inearly 2009, evaluating the risk of a 6-month forward using 5 years of historical datain the simple GBM-type calibration would have underestimated risk. If the volatil-ity is then “inflated” to account for this risk underestimation, this might cause thelong-term risk measure to become over-evaluated, hence causing misestimation oflong-term trades.

A potential solution to this is calibrating the models to the market-implied risk-neutral measure, as CVA desks tend to do, but then the models tend to fail thebacktesting as needed for RWA and risk management. We will show how themodelling framework we propose overcomes these limitations.

3. Lack of fat tails and skew

GBM models deliver no skew and no excess kurtosis. However, historical time seriesindeed show those characteristics. We will illustrate how the stochastic volatilitywe proposed, embedded within a GBM model, produces a skew and kurtosis whichmatch well the values measured in the market. In fact, quite remarkably, we willshow how this is achieved without any explicit model calibration to skew andkurtosis.

4. Path clustering

Under a GBM framework, the negative σ2

2 drift in the logarithm of the spot processmakes the diffusion quantiles drift downwards1 after a period of time. This isillustrated in Figure ??. It is quite counter-intuitive to have a risk model whichdelivers lower risk after a certain time horizon.

This effect is especially relevant in the current times of high volatility. We willshow how a spot-volatility diffusion model offers a framework for better control

1σ represents the diffusion volatility.

Advanced Spot Process for Counterparty Risk and CVA via Stochastic Volatility 2

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Figure 1: Example of a Monte Carlo simulation using a standard GBM process, with a volatilityof 80%. The thick black line corresponds to the 95th percentile.

over this feature.

A Joint Spot-Volatility Stochastic Model for CounterpartyRisk and CVA

The stochastic differential equations which we propose in using to simulate the spotvalue X are

dXt = Xtσt dWXt

d(lnσt) = θ (lnµ− lnσt) dt+ η dW σt

dWXt dW σ

t = ρdt(1)

where dWXt and dW σ

t are Brownian motions, σt is the process volatility and θ, µ, η andρ are model parameters that will be discussed in the next sections.

In order to highlight the effect of the stochastic volatility, the drift of the spot was chosento be zero. However, the practitioner could set this drift appropriately.

There have been several models proposed and studied for stochastic volatility. However,to our knowledge, most if not all of them are applied in the context of derivative pricing[?, ?, ?, ?, ?, ?]. The requirements of models in derivative pricing tend to be differentthan in risk management, where good model backtesting is essential. The main driverof “quality” as well as the required regulatory approval both depend on backtestingresults.

Figure ?? shows a time series for the volatility index VIX. A mean-reverting pattern toa level of around 20% is clearly observed.


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0

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90VIX (Adjusted Close)

Jan90

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Figure 2: VIX time series

There are a number of obvious mean-reverting candidates to model the instantaneousvolatility. The Heston model is the most popular one, arguably because it has a tractableanalytical solution. However, if we let our choice of model be dictated by backtestingrequirements, there is evidence in the literature that a Black-Karasinski (BK) processcould be a better choice for modelling the volatility: in an empirical study assuming avolatility process

dσ ∼ σγdW, (2)

the DJ index time series is calibrated to historical time series (i.e., the best γ is found).The obtained value of γ = 1.05 indicates that the distribution is close to log-normal[?].

The purpose of our research is not to investigate the optimal stochastic volatility processfor a given asset class. However, these preliminary results indicate that a BK process isa good option for volatility modelling in the context of counterparty risk. By “good” itis meant that the model backtests successfully. We hence chose to use that process inthe research presented in this paper.

Gaining model intuition

First, we need to gain an understanding of the role of each parameter in the model. Wefirst examine how a stochastic volatility induces changes to the spot diffusion. We thenprovide intuition on the roles of the ρ, θ η, and µ parameters in the spot diffusion too,as these roles are clear for the volatility diffusion process, but not straightforward forthe spot diffusion process.


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The introduction of a stochastic volatility in a GBM process for a spot alters the distri-bution of the log-spot values during the simulation. The log-spot is generally not normalany more under a stochastic volatility framework.

To illustrate this, suppose for a moment, for the sake of simplicity, that

Xt ∼ X0 + σt ∆WXt (3)

and thatσt ∼ σ0 + η ∆W σ

t (4)

with

∆WXt ∆W σ

t ∼ ρ ∆t, (5)

where ∆WXt ∆W σ

t are discrete differences of Brownian motions.

Then,

Xt ∼ X0 + (σ0 + η ∆W σt ) ∆WX

t . (6)

If we say that

∆W σt = ρ ∆WX

t +√

1− ρ2 ∆W ∗t (7)

where

∆WXt ∆W ∗t ∼ 0, (8)

then, up to first order in t,

Xt ∼ X0 + ηρt+ σ0 ∆WXt + η

√1− ρ2 ∆WX

t ∆W ∗t , (9)

or

Xt ∼ Xt + ηρt+ η√

1− ρ2 ∆WXt ∆W ∗t , (10)

where Xt would be the value of Xt if the volatility were a constant equal to σ0.

Equation ?? illustrates how adding stochasticity to the volatility (i) adds a drift (ηρ ∆t)

to the spot price process and (ii) adds a non-normal random component (η√

1− ρ2 ∆WXt ∆W †t )


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to the process. It can be shown how an equivalent effect is delivered by the model wepropose for the stochastic volatility.

The volatility-spot correlation ρ delivers an interesting and important effect. Figure ??shows the skew of the one-year moves for a simulation under the proposed spot-volatilitydiffusion framework, for different values of the correlation. We observe how negativecorrelation introduces a negative skew and vice-versa. The correlation parameter isprecisely what controls the asymmetry between the spot and volatility processes.

rho = −1

skew = −0.8

rho = −0.5

skew = −0.4

rho = −0.1

skew = −0.1

rho = 0.1

skew = 0.1

rho = 0.5

skew = 0.4

rho = 1

skew = 0.8

Figure 3: Probability distribution of a standard normal (black line) and of the BK spot-volatilitydiffusion model (blue bars) for different values of ρ.

The value of the mean reversion speed θ and the vol-of-vol η also have a notable effecton the spot diffusion. This is depicted in Figure ?? for the case of η; a similar graphcan be obtained for a ranging of θ.

The distribution of the log-spot process gets leptokurtic as θ decreases and as η increases,and tends to the normal distribution when they increase and decrease respectively. Thisis because low values of θ and high values of η result in higher stochastic variation ofthe volatility process. As a result, high kurtosis is induced2.

2In fact, we can loosely say that low values of θ and high values of η introduce a quasi-student-T effect.


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eta = 0.1

kurt = 3.0

eta = 1

kurt = 3.3

eta = 2

kurt = 4.7

eta = 3

kurt = 8.9

eta = 4

kurt = 27.9

eta = 4.95

kurt = 961.2

Figure 4: Probability distribution of a standard normal (black line) and of the BK spot-volatilitydiffusion model (blue bars) for different values of η.

The effect of the mean reversion parameter µ in the spot diffusion is perhaps the easiestto depict directly from Equation ?? since it is strongly liked to the long run limit of thevolatility of the spot. Hence, from a risk perspective, µ mainly determines the long-termrisk for the spot3.

Overcoming GBM Limitations

What are the implications of this new modelling framework with respect to the fourissues of GBM models we highlighted above? Let’s analyse them one by one.

The first obvious positive outcome is the netting effect between volatility and spot trades.With a spot-volatility diffusion model, the strong correlation between the volatility and

This is because a student-T distribution is created when there is uncertainty regarding the volatility [?],which is precisely what we are introducing when reducing θ or increasing η.

3The mean reversion level of a BK process is exp(

lnµ− η2

4θ

).


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the spot is accounted for within the same model, and so the institution can benefit fromnetting effects in a positive way.

Figure ?? shows both the collateralised4 and uncollateralised EPE profiles for a simplespot forward and variance swap. PFE profiles show the same behaviour.

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Figure 5: EPE profiles for a forward (black), a variance swap (red), trades with both pro-files added (blue) and trades properly netted (green) per scenario. This is shown both for thecollateralised (top panels) and the uncollateralised (bottom panels) cases.

The risk profile of each individual trade is shown on the left column. The netting effectis displayed in the right column by comparing added profiles versus profiles of tradeswhich were netted first. The model was calibrated to S&P5005.

The resulting netting benefit is substantial. In both the PFE and EPE cases, the peakof the profile more than halves when using the spot-volatility diffusion model.

Another important limitation of a simple GBM model is how little it reacts to swinging

4Building a collateral algorithm is well beyond the purpose of this paper, so we are using 10-day riskas an indication of collateralised risk. This is equivalent to assuming an ideal CSA with daily margining,zero threshold, zero minimum transfer amount, no independent amount, no rounding, etc.

5It was calibrated to S&P500 corrected for dividends. This was done to avoid the semi-deterministicjumps which occur when an expected dividend is paid. Hence, the calibration of this model leaves outdividend risk for the case of equities. If a dividend model does not exist, this calibration can be done toinclude dividends without any major impact to the methodology.


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market conditions. The proposed stochastic volatility-spot diffusion model solves thisproblem as well. If historical calibration is needed, the parameters θ, µ, and η arecalibrated to long time series which capture full business cycles (e.g., 10 years), whilethe spot volatility is input into the model on a daily basis as part of the daily marketdata update.

Figure ?? shows the difference in collateralised and uncollateralised PFE profiles fora standard GBM process with a volatility of 40%, compared to a volatility-spot diffu-sion process with different short-term (instantaneous) volatilities and with a long-termvolatility of 30%. This is a situation often encountered in practice, where simple GBMvolatility is calibrated to medium-term time horizons even if the short-term volatilitymay strongly vary. A similar result could be shown for EPE.

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start vol = 30%

start vol = 40%

start vol = 60%

start vol = 30%

start vol = 40%

start vol = 60%

Figure 6: 95% PFE of spot, using constant volatility in a simple GBM model (left panel)and stochastic volatility in a BK spot-volatility diffusion model (right panel), displaying boththe collateralised (top panel) and the uncollateralised (bottom panel) cases. The stochasticvolatility runs each use a different volatility at the start of the simulation, all other parametersbeing equal.

The difference in results between the two models is remarkable. The left panels showthe well-known profiles using a standard GBM process for collateralised and uncollater-alised portfolios. The right panels show the equivalent profiles using the proposed spot-volatility diffusion model, for three different short-term market conditions. The readercan see how both the collateralised and the uncollateralised profiles are affected by the


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different short-term volatility values in a way which reflects the associated economicrisk: when short-term volatilities are high (low), collateralised profiles are substantiallyhigher (lower) for the first few months, but then they relax towards a common long-term10-day risk level. On the other hand, uncollateralised profiles accumulate the high (low)short-term volatility so that the overall profile gets pushed up (down).

As previously mentioned, market data analysis indicates that spot time series tend toshow skew and positive excess kurtosis. This is not captured at all by a simple GBM pro-cess. However, a spot-volatility diffusion model is able to display such properties.

In order to investigate this, we track how the first four moments provided by the stochas-tic volatility model evolve over time. First we do this with the spot distribution, whichdrives uncollateralised risk, and then we do the same with the distribution of 10-daymoves, which drives collateralised risk.

Figure ?? shows the evolution of those four moments, comparing the proposed spot-diffusion model (blue) to a simple GBM model (red), both calibrated to 10 years ofS&P500 historical data6.

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Figure 7: First four moments of the spot distribution function using a standard GBM process(red) and the proposed spot-volatility diffusion process (blue).

The following features are observed:

6Simple GBM model σ = 20%. Proposed spot-diffusion model θ = 3.8, µ = 20%, η = 1.0, σt0 = 20%.


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The stochastic volatility induces a negative drift in the spot (top left panel), consistentwith the explanation given in Equation ??. Importantly, it appears that that drift isconstant, so it can be easily managed if needed.

As for the standard deviation (top right panel), stochastic volatility seems to induce anextra “push” to the simulated volatility of the spot: the 95th percentile of the spot-diffusion model is above the

√t obtained with a GBM process.

A standard GBM process produces zero skew, but the spot-volatility diffusion processproduces a negative skew which, interestingly, tends to decrease (in absolute value) asthe simulation evolves (bottom left panel).

Finally, the spot-volatility diffusion process produces excess kurtosis which, similarly tothe skew, also tends to decrease over time (bottom right panel).

The above described the case of uncollateralised risk. For the collateralised case, we areinterested in the evolution of the distribution of the 10-day spot returns. This is shownin Figure ??.

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Figure 8: First four moments of the distribution of 10-day spot moves, using a standard GBMprocess (red) and the proposed spot-volatility diffusion process (blue).

The figure shows how the spot-volatility diffusion model adds a negative drift which isconstant and consistent with that seen in the uncollateralised case. We also observe anincrease in standard deviation, negative skew, and excess kurtosis. Interestingly, these


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changes are generated in a remarkable similar manner. In all three cases, the differencein the moment from the standard GBM value show a pattern of increasing throughoutthe simulation up to a stable state, which is achieved at approximately 0.5 years intothe simulation.

In fact, the reader may already anticipate how the approximate 0.5 year “relaxation”time is highly related to the mean-reversion speed of the stochastic volatility process. Itcan be shown that this relaxation time follows the equation T ∼ 3.197

2θ7 .

Before leaving the discussion on fat tails and skew, we would like to compare the skewand kurtosis delivered by the proposed model with those observed in the market. A fullin-depth backtest of this model for several market variables is outside the scope of thispaper, but it is important to compare the basic results with empirical facts, to makesure that the model can actually be relevant in practice.

As said, θ, µ, and η of the proposed spot-volatility diffusion model were calibrated fromthe VIX time series using data from 2001 to 2011 8. Importantly, the spot-diffusionmodel had no direct input regarding the skew and kurtosis of the S&P500 time series.The question is: how do the moments of the spot probability distribution of the pro-posed model compare to those historically observed from the S&P500 time series? Thefollowing table shows that comparison:

10-day moves S&P500 Simulation

Standard Deviation 0.039 0.042Skew -0.89 -0.60Kurtosis 9.1 5.5

We observe that both the standard deviation and the skew produced by the proposedmodel and are close to the values observed in the real S&P500 time series. This is quiteremarkable, particularly regarding the skew as, let’s reiterate, there is no direct inputin the model as to what is its real value. For this reason, arguably, this model providesinsight into the real dynamics of the S&P500. However, regarding kurtosis, the modelfalls short compared to the real kurtosis observed. It seems an extra model feature (e.g.,jumps) may be needed to match the fourth moment of the distribution.

Last but not least, we previously mentioned that a simple GBM process has “pathclustering” problems in the context of long-term uncollateralised profiles when volatilitiesare high. It appears that this problem can be somehow managed with the proposed spot-volatility diffusion model. After some exploration, we found that the vol-of-vol parameterη seems to have a big influence in the path clustering effect. However, the values of ηused to completely overcome the problem were somewhat off from those calibrated fromthe S&P500 index. Further research with other time series and asset classes would be

7In a BK model, the time dependence of the amplitude of the probability distribution is

η√

1−exp(−2θt)2θ

. So, the amplitude will reach 99% of its long term value at T = − ln(1−0.992)2θ

.8There is evidence that calibrating short-term volatility to VIX is as good as to past realised volatility

[?]


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needed to clarify this in detail.

In any case, it still appears that the proposed spot-volatility diffusion model providestools which can help manage the path clustering problem if needed.

Summary, Conclusions and Critique

For the reasons outlined in this paper, migration from a simple GBM model to a moreadvanced volatility-spot diffusion model seems to deliver important benefits for counter-party risk management, capital calculation, and CVA pricing.

Our research investigated the benefits and limitations of upgrading counterparty riskmodels for spots from being driven by a standard GBM process to following a moreadvanced spot-volatility diffusion process. A Black-Karasinski process was used for theinstantaneous volatility coupled with a log-normal process for the spot. Other volatility-spot diffusion models could be considered. We chose a BK model for the volatilitybased on some preliminary evidence that it performs well in the context of counterpartyrisk.

First of all, we showed that spot-volatility diffusion models can provide strong nettingcapabilities between spot and volatility driven trades. The benefits of this are obvious:without this feature, risk measures (e.g., PFE, EPE, etc) are usually added withoutnetting, hence overestimating risk profiles, capital, and CVA prices. We have shown thatthis overestimation can be substantial.

Secondly, simple GBM models for the spot limit our ability to differentiate betweenshort-term and long-term trades. As a result, the GBM volatility tends to be calibratedto an intermediate point, i.e. to some sort of “average” volatility between the short-and long-term. However, a volatility-spot diffusion model provides the capability toinclude both short-term and long-term trades within the same framework in an opti-mal and consistent manner. This way, the counterparty risk model can automaticallyadapt to short-term market swings with limited impact to long-term risk, resulting inmore accurate counterparty risk profiles. We have shown that this effect can be alsosubstantial.

Thirdly, we illustrated how the introduction of a stochastic volatility in a GBM modelintroduces a skew in the spot diffusion which approximately matches the empirical skewmeasured from the spot time series in the S&P500. The skew is mostly controlled bythe correlation factor between the volatility and spot moves. This is a positive resulton two fronts: it provides practitioners with an extra tool to fine-tune models andimprove backtesting results, and it can be interpreted as providing information aboutthe “fundamental” laws of asset price dynamics in the financial markets.

Finally, we discussed how the path clustering problem present in the case of standardGBM processes can potentially be managed by a spot-volatility diffusion model.


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Together with the main strengths outlined, the model has a number of potential weak-nesses which must be kept in mind. As seen, the kurtosis it delivers seems to be stilltoo low, so this model may not be appropriate yet to measure extreme tail risk. Also,for a given time horizon, the probability distribution of the spot under a spot-volatilitydiffusion process depends on the number of time steps taken by the simulation. However,we found that that dependence is generally small. Finally, more sophisticated correla-tion structures between the volatility and the spot could be investigated, as we onlyimplemented a locally gaussian copula dependency structure.


advanced spot process for counterparty risk and …...march 2012 version 1.0 abstract geometric...

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