thema beleggingen
by Willem van Ruitenburg and David Schrager
HEDGING COMPLEX CROSS-GAMMA EXPOSURES:
AN ELEGANT VANILLA ALTERNATIVE
W. van Ruitenburg (boven) is
Insurance Risk Officer bij ING
Reinsurance.
D. Schrager PhD is directeur
Pricing & Product Hedging
bij Nationale-Nederlanden.
Abstract:
Over the last decennia many insurance
companies have sold insurance contracts,
either in the pension or retail markets, with
embedded guarantees on the performance of
markets and/or investment funds. More and
more insurance companies will report P&L on
these books using Market Consistent
valuation assumptions. Hedge programs are
initiated to reduce P&L volatility. As a first
step linear hedges, using e.g. futures and
forward contracts, are often considered to
eliminate the directional exposure towards
market movements. More advanced hedge
programs will include vanilla options to
(partly) protect against bigger market returns
and/or higher return volatility of single asset
classes, i.e. so-called gamma/convexity
matching, and to hedge for movements in
market implied volatilities affecting the
(Market Consistent) reserves of the embedded
options in the insurance contracts. The
investments underlying the guarantees often
contain exposures in multiple equity indices,
exchange rates and potentially bond funds.
Insurance companies are therefore exposed
to simultaneous movements across the
different markets. The option risk is
multivariate, rather than univariate in
nature. In this article we will discuss an
elegant approach to hedge these so-called
cross-gamma exposures using vanilla option
hedge strategies.
Explanation Liability Movement in Greeks:
Guarantees underlying e.g. variable annuity contracts
and separate account guarantees in pension contracts
are often on the performance of a basket of assets
potentially including a variety of equity, exchange rates
and bond exposures.
[Eq 1]: dBSK (X1,..,XN,t ) = 兺i wi dXi,t
The market value of the embedded guarantee (MVL) on
a guarantee level K can be expressed as:
[Eq 2]: MVL(BSK;K,T,Ω) = ΕQ 关 DFT ∙ Max { K - BSKT } 兴
The parameter Ω is the implied covariance matrix for
the basket elements. This matrix is typically calibrated
using historically observed correlations and implied
volatilities for the individual asset classes, if available.
First generation hedge programs target the change in
MVL due to a change in the assets in the basket or Delta
risks. Delta risks can be hedged using futures, forwards
or the asset itself. Once Delta risks are hedged specific
option, or non-linear, risks remain. Consider that when
hedging an option one has to adjust the Delta hedge
upon a move in the assets. This is due to convexity of
the MVL as a function of BSK. In other words, MVL is not
a linear function of BSK and hence a Delta hedge of a
guarantee is not static but requires rebalancing. This
article focusses on these non-linear risks, specifically
Gamma.
Just as Gamma in practice is defined as the change in
notional Delta of the underlying due to a change in the
underlying, Cross-Gamma is defined as the change in
notional Delta of an underlying due to a change in
another underlying. We will now provide more
mathematical background to this explanation of CrossGamma.
The market value movement of the MVL only as a
function of immediate return on the basket can be
approximated by a 2nd order Taylor approximation like:
34 de actuaris september 2014
thema beleggingen
[Eq 3]: dMVL(dBSK) =
∂MVL
∂BSK
1
∙ BSK ∙ dBSK + 2 ∙
∂ 2 MVL
∙
∂BSK 2
(BSK∙dBSK )2 + Ο-3
Equation [3] is more often displayed in terms of Greeks as:
[Eq 4]: dMVL(dBSK) = ΔBSK ∙ dBSK + 21 ∙ ΓBSK ∙ (dBSK) 2 + Ο-3
Substituting [Eq 1] into [Eq 3] results in:
[Eq 5]: dMVL(dX 1 ,..,X N ) = 兺i
共 ∂MVL
∂xi ∙ Xi ∙ dXi 兲
+ 21 ∙
关 兺i 兺j ∂∂xMVL
∙ ( Xi Xj ∙ dXi dXj ) 兴 + Ο-3
ixj
2
In terms of Greeks [Eq 5] translates into:
[Eq 6]: dMVL(dX1,..,XN ) = ⌊兺i Δ Xi ∙ dXi ⌋+ 21 ∙⌊兺i ΓXi ∙ (dXi ) 2 ⌋+ 12 ∙
关 兺i 兺j<>i ΓXiXj ∙ dXi dXj 兴 + Ο-3
The movement of the liability option value caused by immediate returns in the underlying assets
contains 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:
Δ Xi = 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 the
cross-gamma exposures is relative to the single asset gamma exposures. For example in an
equally weighted basket of N underlying components the single asset gammas only represent 1/N
part of all the 2nd order exposures in the basket option.
Hedging programs that focus on hedging the linear exposures will have a short position in
gamma. This means options have to be bought to manage the risks in these portfolios. As these
dynamically delta hedged positions will always end up in 'buying high and selling low' the delta
hedge will typically lose money on a daily base. Typically the more futures/forwards need to be
traded 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 the
size of the market return or better said to the level of realized market return volatility1.
Gamma
Liability
Net Result
Value
Linear Hedge
-20%
1 – It should be noted that these
gamma losses are offset by
releases of the market value
reserve over time as time value
("Theta") runs out.
-15%
-10%
-5%
0%
5%
10%
15%
20%
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 covariance
among the pairs of underlying asset classes. On a daily base such positions will lose money
whenever the different assets show big market movements in the same direction.
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Increases in covariance can either be caused by 1) increased
volatilities of individual assets or 2) increased correlation between the
asset classes.
The following regression will be performed:
One of the better known hedge instruments to replicate these
complex liability exposures are basket options. Basket options are
however rather illiquid instruments for which the pricing is not
transparent.
This illiquidity typically increases when less standard equity indices
and/or bonds are part of the basket.
Traders who sell these baskets will typically hedge their short
positions by going long options on the individual asset classes and
will charge the buyer of the basket options generously for the open
position/risk that the trader faces.
The fitted coefficient âi can be seen as the gamma that has been
realized.
Realized Gamma approach:
From [Eq 3] and [Eq 4] it is seen that for an option with price P on
single underlying S the delta and gamma in money terms are defined
by:
ΔS = ∂P ∙ S
∂S
2
ΓS = ∂ P2 ∙ S2
∂S
Gamma exposures change on a daily base as markets move and the
moneyness of the guarantee changes. This implies that the regression
should be based on a limited interval of recent historical observations
and that this window is being rolled forward.
In Figure 2 the outcomes of conducted analyses are shown. The green
dots are the actually observed corrected delta movements, i.e.
dΔX t - ΔX ∙ dXi,t . The red dots represent the delta movements
i,
i,t
explained by the individual gamma exposure, i.e. ΓX ∙ dXi,t and the
i,t
blue points show the fitted delta movements, ci + ai ∙ Xi,t
Effective Gamma Example (2)
Daily Change Delta Index X
As traders will hedge their short cross-gamma positions using vanilla
hedge strategies insurance companies could take a similar tactic to
replicate cross-gamma exposure. In this article the 'Realized Gamma'
approach will be further explained.
[Eq 9]: dΔX t - ΔX ∙ dXi,t = ci + ai ∙ dXi,t + εi,t
i,
i,t
-3%
Via some calculus using the chain formula it's straightforward to
derive that the changes in the notional- or money-deltas can be
approximated 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 the
option is driven by the gamma of the option and the return of the
underlying2.
Replacing S with the basket BSK and substituting Eq[1] shows the
explanation of option liability delta movement dΔX as:
i
0%
1%
2%
3%
4%
5%
In the analyses that have been conducted it has been observed that
the correlation between the actual delta movements and the fitted
delta movements is typically 70%-80% for baskets containing equity
and exchange rates exposures.
In Figure 3 it is shown that for embedded insurance guarantees on a
basket of assets the force that drives the delta changes (and therefore
trading in/out of linear hedges) may be a multiple of the theoretically
calculated gamma for that single component as calculated by the
liability model in case of average positive correlations.
Effective Gamma Example (1)
i
The Realized Gamma approach is a regression based approach where
the observed changes of the individual asset deltas are regressed on
the individual asset returns.
-1%
Figure 2: Example of observed versus fitted delta movements
[Eq 8]: dΔX = wi ∙ dΔBSK ≈ wi ∙ (ΓBSK + ΔBSK ) ∙ 兺 i wi dXi
The change in delta for asset component Xi is driven by the return of
the basket as a whole and not by the return of the individual
exposures.
-2%
Value Gamma
Fitted Realized Gamma
Single assed-factor Gamma
Time
Figure 3: Fitted short realized versus calculated single asset gamma exposures
over time
2 – It should be noted that in
practice the absolute gamma in
money terms is a much bigger than
the absolute money-delta
exposure.
36 de actuaris september 2014
thema beleggingen
Insurance companies that are currently hedging the single asset
gamma exposures can improve their hedge program by focusing
hedging 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 riskdrivers 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 riskdrivers, 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 either
distort the validity of the observed fitted gamma or may require the
time series to be cleaned-up.
Conclusion
Insurance companies write or have written guarantees on multiple
underlying assets or asset classes. In this article we discuss a new
technique for risk management of these products.
The proposed 'Realized Gamma' approach may be the best solution
for companies seeking (computational) efficient risk measurement
and a hedge based on simple instruments. @ Reacties op dit artikel graag naar redactie.actuaris@ag-ai.nl