some aspects of post-crisis equities modellingamijatov/events/lucic.pdf · some aspects of...

Some aspects ofpost-crisis

equitiesmodelling

Vladimir Lucic

Outline

Volatility-targetindices

Intro

Mechanics ofVolatilityControl

Modeldependence

Hedging

A RegulatoryModellingchallenge

References

Some aspects of post-crisis equities modelling

Vladimir Lucic

Quantitative Analytics, Barclays

17 October 2014

Copyright c© 2014 Barclays - Quantitative Analytics, London

Disclaimer: This paper represents the views of the authors alone, and not

the views of Barclays Bank Plc.


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References

Outline

Volatility-target indicesIntroMechanics of Volatility ControlModel dependenceHedging

A Regulatory Modelling challenge

References


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References

Intro

Vol target structures find favour post-UK budget(Derivatives Week, 6/6/14)

“Sellside firms are seeing increased appetite for structured productswith volatility target mechanisms as a way of providing a primarysource of income for UK individuals during retirement. The spike ininterest comes on the back of Chancellor George Osborne’s 2014budget that set out greater investment flexibility for individuals atretirement from April 2015.”

Related studies

Horfelt (2014), Nelte (2013), Morrison and Tadrowski (2013)


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References

Intro

What has changed post-crisis?

I myriad of tailored topical indices with built-in risk controls

I clients willing/not willing to take particular risks (e.g. volatility)

I a typical pre-crisis high-flier: straight-out exposure to volatility


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References

Intro

I Investment strategies are self-financing

It =m∑

i=1

h(i)t A

(i)t , α

(i)t := h

(i)t A

(i)t−/It−

dIt =m∑

i=1

h(i)t dA

(i)t dIt/It− =

m∑i=1

α(i)t dA

(i)t /A

(i)t−

I discrete case (∆Zn := Zn − Zn−1)

In =m∑

i=1

h(i)n A(i)

n α(i)n := h(i)

n A(i)n /In

∆In =m∑

i=1

h(i)n−1∆A(i)

n In/In−1 =m∑

i=1

α(i)n−1A(i)

n /A(i)n−1


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References

Intro

I Assume A(1) is the risky asset and A(2) is the cash account,

R(i)n = A

(i)n /A

(i)n−1 − 1

I Total Return indices evolve according to

In+1 = In(

1 + α(1)n R

(1)n+1 + (1− α(1)

n )R(2)n+1

)I this is just the definition of self-financing, rewrite

In+1 = In(

1 + α(1)n R

(1)n+1 − α

(1)n R

(2)n+1

)︸︷︷︸

Excess Return index

+ InR(2)n+1︸︷︷︸

financing

I both versions are offered, with Excess Return having a biggershare


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References

Mechanics of Volatility Control

single-asset volatility controlled investment strategies

I α(1)t — investment in a risky asset — function of asset volatility

I α(2)t ≡ 1− α(1)

t — investment in cash

volatility estimationI simple variance estimator

vwn =

252

w − 1

n∑j=n−w+1

(R

(1)j − Rn

)2

I Rn = 1w

∑nk=n−w+1 R

(1)k

I volatility estimated as σwn :=

√vw

n

I exponential forgetting is often used to enhance accuracy


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References

Crux of the matterI Assume lognormal asset with vol σ1

I Also assume that risky asset is total-return, and can be fundedat overnight rate r

I portfolio for target volatility σTV : α(1)t = σTV /σ1,

α(2)t = 1− α(1)

t

dItIt

=σTV

σ1

dA(1)t

A(1)t

+

(1− σTV

σ1

)dA

(2)t

A(2)t

=σTV

σ1(r dt + σ1 dWt) +

(1− σTV

σ1

)r dt

= σTV dWt + r dt

I It is lognormal with volatility σTV

I practitioners often use σTV + ε in Black-Scholes formula whenpricing options on I


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References

Back to realityI contractual vol target index looks more like this

In+1 = In(

1 + α(1)n R

(1)n+1 + (1− α(1)

n )R(2)n+1

)I participation in the risky asset is capped

α(1)n = min

(cap, σTV /

√vw

n

)I vol estimation

I the size of averaging window w varies from 20 to 60 daysI frequently estimator is defined as max of two estimators with

different wI there are products in which VIX is used instead of the estimator

I underlyings vary across asset classesI often rebalancing is triggered only when moves in the market are

above a threshold


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References

discrete vs. continuousI the contractual version of vol target does not arise from

discretization in the usual way

I for corlol g , semimartingale X and partition0 = tn

1 , tn2 , . . . , t

nn = T

Y nt := Πk

(1 + gtn

k(Xt∧tn

k+1− Xt∧tn

k))

converges in law on [0,T ] to Doleans-Dade exponentialE(∫ ·

0gu− dXu)t as n→∞ (Emery (1978))

I ties in with the notion of multiplicative stochastic integraldeveloped in ’60s ’70s (McKean, Emery,...)


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References

Master Formula

I for the general case of semimartinglae risky asset A(1)t , put

gt := σTV A(2)t√

vwt A

(1)t−

, the vol target strategy converges to

ertE

(∫ ·0

σTV A(2)u√

vw (u)A(1)u−

d(A(1)/A(2))u

)t

I case of diffusive asset with (stochastic) vol σt , the vol targetstrategy converges to

ertE(∫ ·

0

σTVσu√vwu

dWu

)t

I the quality of the estimator is essential for tracking σTV


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References

How does it work in practice?

Source: Bloomberg 2014

I volatility visibly reduced, with trend following and smallerdrawdowns

I what happens with realized volatility?


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References

How does it work in practice?

Source: Bloomberg 2014

I long-term window estimation yields vol around 10%

I short-term window shows vol-vol: the mechanism fails to dealwith large moves


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References

Model dependence – basic tests

I fix w = 20 throughout the tests

I to check the approximation, look at 1Y ATM option on 10%target vol

I for simplicity, take underlying with 20% flat volI replace the vol estimation with exact simulated vol – the implied

ATM vol is exactI introducing vol estimator yields errors in excess of 1/2 vol point

fly in the ointmentI the volatility estimator is biased (Jensen’s inequality)

I Holtzman’s correction

σ = CN

√v

CN = Γ

(N − 1

2

)√N − 1

2

/Γ

(N

2

)


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References

Model dependence – basic tests

I simplification, Brugger (1969): since limN→∞N−1.5N−1 C 2

N = 1,replace N − 1 by N − 1.5 in the estimation formula

I for simplicity, take underlying with 20% flat vol

I local vol results very similar, with extra 10bps errors


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References

Model dependence – Jumps

Jump-diffusion model

dSt/St− = · · · − λSt dt +

√vt dWt + dNS

t , NSt :=

Nt∑i=1

(eYi − 1)

I Merton jumpsI Poisson Nt with intensity λI jump sizes are Gaussian, Yi ∼ N(α, δ2)

I v can be any vol model, we focus on local vol

CalibrationI for a choice of jump parameters (λ, α, δ), local vol is calibrated

I calibration instruments: short-dated options, crash-cliquets


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References


I the “master formula” gives

I (t) = ertE(∫ ·

0

σTVσu√vwu

dWu +

∫ ·0

σTV√vwu

dNSu

)t

(1)

I assume for simplicity σimpl = 30%, take the Merton JD modelcalibrated to var swap prices

σ2 + λ(α2 + δ2) = VarSwap1Y = 30%

I assuming√

vwt ≈ E[

√vwt ] =

√VarSwap1Y, we find

It = ertE

∫ ·

0

σTVσ√σ2 + λ(α2 + δ2)︸︷︷︸

:=σI

dWu +

∫ ·0

σTV√σ2 + λ(α2 + δ2)︸︷︷︸

:=γI

dNSu

t


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References


I the 1Y var swap price of the index is precisely σ2TV :

VarSwap1YI = σ2I + λγ2

I (α2 + δ2) = σ2TV

I so we’d expect little impact on vanillas, is this what we observe?I even with modest parameters λ = 0.5, α = −10.5%, δ = 4%

have significant differences


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References


I variance of the estimator plays an important role

I replacing√

vwt in the payoff with E[

√vwt ] =

√VarSwap1Y gives

much improved match


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References

Model dependence – Stochastic Volatility

LSV model

dSt/St = · · ·+√

vtσ(St , t) dWt

I variance follows Heston or lognormal dynamics

CalibrationI for a choice of stoch vol parameters σ(S , t) is calibrated to

vanillas

I forward-inductive PDE procedure

PricingI Again look at implied vol of I , but put nontrivial cap in risky

participationα(1)

n = min(cap, σTV /

√vwn

)


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References



equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References


I cap of 150% is ineffective with σTV = 10%, σimpl = 30%

I moving the cap further down, models produce different impact


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References

ConclusionsI volatility estimator subtleties have impact on tracking target vol

I any departure from pure lognormal case entails increase intracking error

I large impact of jumps

I presence of cap has different impact depending on spot dynamics

I on an intuitive levelI cap introduces concavity in the participation function

x 7→ min(cap, σTV /x

)I this “concavity” brings down the expected value of participation

in equityI the impact depends of the variance of the estimator, e.g.

increase of vol-vol increases the impact


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References

Hedging

I vol control indices provide directional exposure with controlledrisk

I buying options on those has become mainstream

I take example of vol control on SPX


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References

Hedging

I look at 1Y ATM put on SPX5UT, struck at 21/08/2010

I during stable market conditions Delta hedging works fine

I as we (the hedger) are short Γ, large moves will hurt

I more than third of initial premium (2%) is lost during the crash


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References

Hedging

I including vanilla options on the underlying can removesignificant amount of jump risk, Xia and Sahakyan (2014)

I the hedging portfolio has additional component – Delta-hedgedoption on SPX

I the exact amount of SPX options can be determined usingvarious criteria, say minimize the average portfolio move forjumps less than a threshold


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References

A Regulatory Modelling challenge

I post-crisis regulatory-related modelling has significantly increased

I CVA, FVA, COLVA..

I typically, need to provide various statistics of the derivativeportfolio in time under risk-neutral and physical measure

I eg. quantiles-based Potential Future Exposure (PFE) taken as amaximum amount of portfolio/trade exposure at a givenconfidence level

I assuming payoff VT at T , need Φ(X , t) s.t.

Φ(Xt , t) := E[VT |Ft ](= E[VT |FXt ]), 0 ≤ t ≤ T

I common approach American Monte Carlo (e.g. Sokol (2014)),entails

I choosing state variables for each trade XI proper choice of basis functions for each state variableI estimation (regression) of Φ


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References

I for simple trades entire state-space can be used for regressionI Asian option: spot and sum so far

I for path-dependent, multiasset trades, situation less rosy

I e.g. barrier option on 10-underlying basket/WO/BO withmemory coupons

I full state-space prohibitively largeI rich structure of Φ(·, t) due to presence of barriers, coupon

payments etc.

I ...on the other hand, the value of those trades is (intuitively)driven by a few factors (basket/WO/BO + some tradespecifics..)

I we are in need of a consistent framework forI choosing effective set of explanatory variables XI choice of the effective basis functionsI assessing the quality of approximation E[VT |F X

t ] ≈ E[VT |FXt ]


equitiesmodelling

Vladimir Lucic

Outline


Intro


Modeldependence

Hedging


References

References

R. M. Brugger (1969): A note on unbiased estimation of thestandard deviation, Amer. Statist.

M. Emery (1978): Stabilite des solutions des equationsdifferentielles stochastiques application aux integralesmultiplicatives stochastiques, Z. Wahrsch. Verw. Gabiete

P. Horfelt (2014): Options on vol target indices, ICBI GlobalDerivatives

S. Morrison and L. Tadrowski (2013): The guarantees and targetvolatility funds, Moody’s Analytics

M. Nelte (2013): Controlling volatility to reduce uncertainty,Risk Magazine, September

S. Xia and D. Sahakyan (2014): Gap risk hedging for options onrisk control indices, Barclays internal report

A. Sokol (2014): Long-Term Portfolio Simulation, Risk Books

some aspects of post-crisis equities modellingamijatov/events/lucic.pdf · some aspects of...

Documents