W. van Ruitenburg (boven) isInsurance Risk Officer bij INGReinsurance.D. Schrager PhD is directeurPricing & Product Hedgingbij Nationale-Nederlanden.

thema beleggingen

de actuaris september 201434

Explanation Liability Movement in Greeks:Guarantees underlying e.g. variable annuity contractsand separate account guarantees in pension contractsare often on the performance of a basket of assetspotentially including a variety of equity, exchange ratesand bond exposures.

[Eq 1]: dBSK (X1,..,XN,t ) = �i wi dXi,t

The market value of the embedded guarantee (MVL) ona guarantee level K can be expressed as:

[Eq 2]: MVL(BSK;K,T,Ω) = ΕQ � DFT ∙ Max { K - BSKT } �

The parameter Ω is the implied covariance matrix forthe basket elements. This matrix is typically calibratedusing historically observed correlations and impliedvolatilities for the individual asset classes, if available.

First generation hedge programs target the change inMVL due to a change in the assets in the basket or Deltarisks. Delta risks can be hedged using futures, forwardsor the asset itself. Once Delta risks are hedged specificoption, or non-linear, risks remain. Consider that whenhedging an option one has to adjust the Delta hedgeupon a move in the assets. This is due to convexity ofthe MVL as a function of BSK. In other words, MVL is nota linear function of BSK and hence a Delta hedge of aguarantee is not static but requires rebalancing. Thisarticle focusses on these non-linear risks, specificallyGamma.

Just as Gamma in practice is defined as the change innotional Delta of the underlying due to a change in theunderlying, Cross-Gamma is defined as the change innotional Delta of an underlying due to a change inanother underlying. We will now provide moremathematical background to this explanation of Cross-Gamma.

The market value movement of the MVL only as afunction of immediate return on the basket can beapproximated by a 2nd order Taylor approximation like:

H E D G I N G C O M P L E X C R O S S -G A M M A E X P O S U R E S :A N E L E G A N T V A N I L L A A L T E R N A T I V E

Abstract:Over the last decennia many insurancecompanies have sold insurance contracts,either in the pension or retail markets, withembedded guarantees on the performance ofmarkets and/or investment funds. More andmore insurance companies will report P&L onthese books using Market Consistentvaluation assumptions. Hedge programs areinitiated to reduce P&L volatility. As a firststep linear hedges, using e.g. futures andforward contracts, are often considered toeliminate the directional exposure towardsmarket movements. More advanced hedgeprograms will include vanilla options to(partly) protect against bigger market returnsand/or higher return volatility of single assetclasses, i.e. so-called gamma/convexitymatching, and to hedge for movements inmarket implied volatilities affecting the(Market Consistent) reserves of the embeddedoptions in the insurance contracts. Theinvestments underlying the guarantees oftencontain exposures in multiple equity indices,exchange rates and potentially bond funds. Insurance companies are therefore exposedto simultaneous movements across thedifferent markets. The option risk ismultivariate, rather than univariate innature. In this article we will discuss anelegant approach to hedge these so-calledcross-gamma exposures using vanilla optionhedge strategies.

by Willem van Ruitenburg and David Schrager

Valu

e

-20% -15% -10% -5% 0% 5% 10% 15% 20%

Gamma

Liability Net ResultLinear Hedge

[Eq 3]: dMVL(dBSK) = ∂MVL ∙ BSK ∙ dBSK + 1 ∙ ∂2MVL ∙ (BSK∙dBSK )2 + Ο-3

Equation [3] is more often displayed in terms of Greeks as:

[Eq 4]: dMVL(dBSK) = ΔBSK ∙ dBSK + 1 ∙ ΓBSK ∙ (dBSK) 2 + Ο-3

Substituting [Eq 1] into [Eq 3] results in:

[Eq 5]: dMVL(dX1,..,XN ) = �i � ∂MVL ∙ Xi ∙ dXi � + 1 ∙ ��i �j∂2MVL ∙ ( Xi Xj ∙ dXi dXj ) � + Ο-3

In terms of Greeks [Eq 5] translates into:

[Eq 6]: dMVL(dX1,..,XN ) = ⌊�i Δ Xi ∙ dXi ⌋+ 1 ∙⌊�i ΓXi ∙ (dXi )2⌋+ 1 ∙ ��i �j<>i ΓXiXj ∙ dXi dXj � + Ο-3

The movement of the liability option value caused by immediate returns in the underlying assetscontains a delta return, a gamma return and a cross-gamma return.

For continuously rebalanced baskets it can be derived using the chain-rule of calculus that:

ΔXi = ΔBSK ∙ wi ΓXi = ΓBSK ∙ wi

2

ΓXi,Xj = ΓBSK ∙ wiwj

Following above relations the more elements the basket contains the greater the sum of all thecross-gamma exposures is relative to the single asset gamma exposures. For example in anequally weighted basket of N underlying components the single asset gammas only represent 1/Npart of all the 2nd order exposures in the basket option.

Hedging programs that focus on hedging the linear exposures will have a short position ingamma. This means options have to be bought to manage the risks in these portfolios. As thesedynamically delta hedged positions will always end up in 'buying high and selling low' the deltahedge will typically lose money on a daily base. Typically the more futures/forwards need to betraded to remain delta neutral the higher the realized gamma loss.

From Figure 1 it can be seen how short gamma positions leave hedge programs exposed to thesize of the market return or better said to the level of realized market return volatility1.

Figure 1: Example of typical pay-off profile of linear hedge program

Equivalently the expected result of open cross-gamma exposures is a function of the covarianceamong the pairs of underlying asset classes. On a daily base such positions will lose moneywhenever the different assets show big market movements in the same direction.

de actuaris september 2014

thema beleggingen

35

∂BSK

∂xi ∂xixj

∂BSK22

2

2

2 2

1 – It should be noted that thesegamma losses are offset byreleases of the market valuereserve over time as time value("Theta") runs out.

�

thema beleggingen

de actuaris september 201436

Increases in covariance can either be caused by 1) increasedvolatilities of individual assets or 2) increased correlation between theasset classes.

One of the better known hedge instruments to replicate thesecomplex liability exposures are basket options. Basket options arehowever rather illiquid instruments for which the pricing is nottransparent. This illiquidity typically increases when less standard equity indicesand/or bonds are part of the basket.Traders who sell these baskets will typically hedge their shortpositions by going long options on the individual asset classes andwill charge the buyer of the basket options generously for the openposition/risk that the trader faces.

As traders will hedge their short cross-gamma positions using vanillahedge strategies insurance companies could take a similar tactic toreplicate cross-gamma exposure. In this article the 'Realized Gamma'approach will be further explained.

Realized Gamma approach:From [Eq 3] and [Eq 4] it is seen that for an option with price P onsingle underlying S the delta and gamma in money terms are definedby:

ΔS = ∂P ∙ S

ΓS = ∂2P ∙ S2

Via some calculus using the chain formula it's straightforward toderive that the changes in the notional- or money-deltas can beapproximated by:

[Eq 7]: dΔS ≈ ( ΓS+ΔS ) ∙ dS or dΔS - ΔS ∙ dS ≈ ΓS ∙ dS

The above relationship is very sound: the change in the delta of theoption is driven by the gamma of the option and the return of theunderlying2.

Replacing S with the basket BSK and substituting Eq[1] shows theexplanation of option liability delta movement dΔXi

as:

[Eq 8]: dΔXi= wi ∙ dΔBSK ≈ wi ∙ (ΓBSK + ΔBSK ) ∙ �iwidXi

The change in delta for asset component Xi is driven by the return ofthe basket as a whole and not by the return of the individualexposures.

The Realized Gamma approach is a regression based approach wherethe observed changes of the individual asset deltas are regressed onthe individual asset returns.

∂S

∂S2

Daily

Chan

ge D

elt

a I

ndex X

-3% -2% -1% 0% 1% 2% 3% 4% 5%

Effective Gamma Example (2)

Valu

e G

am

ma

Time

Effective Gamma Example (1)

Fitted Realized Gamma

Single assed-factor Gamma

The following regression will be performed:

[Eq 9]: dΔXi,t- ΔXi,t

∙ dXi,t = ci + ai ∙ dXi,t + εi,t

The fitted coefficient âi can be seen as the gamma that has beenrealized.

Gamma exposures change on a daily base as markets move and themoneyness of the guarantee changes. This implies that the regressionshould be based on a limited interval of recent historical observationsand that this window is being rolled forward.

In Figure 2 the outcomes of conducted analyses are shown. The greendots are the actually observed corrected delta movements, i.e. dΔXi,t

- ΔXi,t ∙ dXi,t . The red dots represent the delta movements

explained by the individual gamma exposure, i.e. ΓXi,t ∙ dXi,t and the

blue points show the fitted delta movements, ci + ai ∙ Xi,t

Figure 2: Example of observed versus fitted delta movements

In the analyses that have been conducted it has been observed thatthe correlation between the actual delta movements and the fitteddelta movements is typically 70%-80% for baskets containing equityand exchange rates exposures.

In Figure 3 it is shown that for embedded insurance guarantees on abasket of assets the force that drives the delta changes (and thereforetrading in/out of linear hedges) may be a multiple of the theoreticallycalculated gamma for that single component as calculated by theliability model in case of average positive correlations.

Figure 3: Fitted short realized versus calculated single asset gamma exposures

over time

2 – It should be noted that inpractice the absolute gamma inmoney terms is a much bigger thanthe absolute money-deltaexposure.

�

thema beleggingen

Insurance companies that are currently hedging the single assetgamma exposures can improve their hedge program by focusinghedging the realized gamma rather than the effective gamma.

Advantages of this approach are:– Cross-gamma exposures can be hedged using simple vanilla

options, no need to buy expensive and illiquid basket options.– Easy to implement, no need to explicitly calculate cross-gammas.

Calculation time efficient as it re-uses Delta hedge program output.– Easy to understand concept, can be explained to management. Can

be introduced as simple risk measure alongside use as explicit hedging tool.

– Method relies on recent observed correlations among the risk-drivers and hence is an up-to-date measure of non-linear risks (see also disadvantages).

Some disadvantages are:– Method relies on historically observed correlations among the risk-

drivers, no guarantee that this correlation will remain in-tact.– Movements in (aggregated) liability delta can also be caused by

e.g.: assumption changes, changes in in-force etc. This may eitherdistort the validity of the observed fitted gamma or may require thetime series to be cleaned-up.

ConclusionInsurance companies write or have written guarantees on multipleunderlying assets or asset classes. In this article we discuss a newtechnique for risk management of these products.The proposed 'Realized Gamma' approach may be the best solutionfor companies seeking (computational) efficient risk measurementand a hedge based on simple instruments. �

Reacties op dit artikel graag naar redactie.actuaris@ag-ai.nl@