Pricing Inflation Vanillas and Exotics
Pricing Inflation Vanillas and Exotics
Yann Ticot
Bank of America Merrill Lynch, London
June 2011
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Pricing Inflation Vanillas and Exotics
Outline
1 Inflation market overview
2 Popular inflation models
3 Pricing challenges
4 Inflation vanilla model
5 Inflation exotic model
6 Conclusion
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Pricing Inflation Vanillas and Exotics
Inflation market overview
Outline
1 Inflation market overview
2 Popular inflation models
3 Pricing challenges
4 Inflation vanilla model
5 Inflation exotic model
6 Conclusion
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Pricing Inflation Vanillas and Exotics
Inflation market overview
Inflation index
measures cost of living, price of a representative basket ofgoods
RPI (UK), HICP (EU), CPI (US)...
short-term drivers : energy, agriculturals, commodities
long-term drivers : economic outlook, central bank policy
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Pricing Inflation Vanillas and Exotics
Inflation market overview
Inflation linear products
outstanding inflation-linked debt: $1.7 trillion1, i.e. about0.5% of total outstanding debt on markets
basic instrument: zero-coupon, pays at maturity IT /It , withIt the inflation index and t the pricing date
PV = Pt ,TIt ,TIt
with It ,T the inflation forward, and Pt ,T the discount factor
1source: Barclays Capital as at 21/05/20105 / 32
Pricing Inflation Vanillas and Exotics
Inflation market overview
Year-on-year rate
zero-coupon is a single payment of the compounding ofinflation over a whole period
It ,T = It exp
(∫ T
tit ,u du
)with it ,u the instantaneous inflation forward rate
some investors prefer a payoff in a “swap” format
year-on-year rate (yoy) is defined as
yoyT =IT
IT−1y− 1
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Pricing Inflation Vanillas and Exotics
Inflation market overview
Inflation derivatives: vanilla options
derivatives represent 10% to 15% of the inflation market
yoy options: call/put on yoy rate
zero-coupon options: call/put on zero-coupon rate
investors have appetite for zero-strikes
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Pricing Inflation Vanillas and Exotics
Inflation market overview
Inflation derivatives: exotics
Limited Price Indexation (LPI): liquid due to UK pensionsregulation, maturities up to 50y
n∏i=1
(1 +
[yoyTi
]capfloor
)
LPIs have sensitivity to all yoy smiles & correlations
callables, range accruals
apart from LPIs, exotics are not liquidly traded
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Pricing Inflation Vanillas and Exotics
Popular inflation models
Outline
1 Inflation market overview
2 Popular inflation models
3 Pricing challenges
4 Inflation vanilla model
5 Inflation exotic model
6 Conclusion
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Pricing Inflation Vanillas and Exotics
Popular inflation models
Popular inflation models: nominal-realapproach (Jarrow-Yildirim)
Choose as numeraire the value of inflation index paid at T
Pt ,T It ,T = It exp
(−∫ T
t(nt ,u − it ,u) du
)with nt ,u the instantaneous nominal forward rate.FX analogy:
exp(−∫ T
t (nt ,u − it ,u) du)
is a “real” discount factor
inflation index is the nominal-real FX
for an index-linked payoff IT HT
PV = Pt ,T It ,T Er ,Tt [HT ]
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Pricing Inflation Vanillas and Exotics
Popular inflation models
Popular inflation models: modeling inflationforwards (Belgrade-Benhamou-Koehler)
all index-linked payoffs can be expressed as a function ofinflation forwards It ,T
inflation forwards are modeled directly (martingale underforward-neutral measure)
for an index-linked payoff IT HT
PV = Pt ,T ETt[IT ,T HT
]
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Pricing Inflation Vanillas and Exotics
Pricing challenges
Outline
1 Inflation market overview
2 Popular inflation models
3 Pricing challenges
4 Inflation vanilla model
5 Inflation exotic model
6 Conclusion
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Pricing Inflation Vanillas and Exotics
Pricing challenges
Market yoy smile
market normal yoy volatilities exhibit a steep skewstandard models & dynamics struggle to reproduce yoymarket smile
Table: Market smile for RPI yoy, 2y expiry as of 10th Sep 2010
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Pricing Inflation Vanillas and Exotics
Pricing challenges
LPI market prices
difficult to reproduce LPI market prices
however, better match of LPI market if yoy options marketprices are matched
need for a model consistent with the yoy marginaldistributions
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Pricing Inflation Vanillas and Exotics
Inflation vanilla model
Outline
1 Inflation market overview
2 Popular inflation models
3 Pricing challenges
4 Inflation vanilla model
5 Inflation exotic model
6 Conclusion
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Pricing Inflation Vanillas and Exotics
Inflation vanilla model
Inflation vanilla model
idea: mimic what is done in the rates world by usingseparate models for vanillas and exotics
use a term distribution approach for pricing yoy options
PV = Pt ,T BlackNormal(yoyt ,T , K , σ (K , T ) , T − t
)
Mercurio suggests a similar approach, using directlymarket volatilities instead of a term distribution
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Pricing Inflation Vanillas and Exotics
Inflation vanilla model
Inflation vanilla model: consistency
challenge # 1: yoy forward rate is model-dependent, with aconvexity adjustment
ETt
[IT
IT−1y
]−
It ,TIt ,T−1y
⇒ yoy convexity adjustment must be consistent with yoydistribution
challenge # 2: find a term-distribution which fits the yoysmile
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Pricing Inflation Vanillas and Exotics
Inflation vanilla model
Inflation vanilla model: consistency
For lognormal inflation forwards, the convexity adjustment ofyoyTn is proportional to
∫ Tn−1
0
n−1∑i, Ti≥t
(ρY ,Y
i,n (t)σYi (t)σY
n (t)− ρY ,Li,n−1(t)σ
Yi (t)σL
n−1(t))
dt
withσL
j the normal volatility of the 1y libor starting at Tj
σYi the lognormal volatility of
It,TiIt,Ti−1y
(homogeneous to a yoynormal volatility)ρY ,Y
i,j , ρY ,Li,j the yoy-yoy and yoy-rates correlations
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Pricing Inflation Vanillas and Exotics
Inflation vanilla model
Inflation vanilla model: consistency
large number of degrees of freedom makes convexityadjustment & optionality “orthogonal”
consider a set of “reasonable” rates & yoy marginals
also consider a set of yoy convexity adjustments
it should be possible to find a set of correlations whichmake the yoy convexity adjustment consistent with themarginals
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Pricing Inflation Vanillas and Exotics
Inflation vanilla model
Inflation vanilla model: consistency
liquid market quotes available for expiries2y , 3y , 5y , 7y , 10y , 12y , 15y , 20y , 30y
use a simple term-structure model as interpolation tool tocompute yoy forwards at any expiry
handle the optionality with a term-distribution
assume that the convexity adjustment and optionality areconsistent
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Pricing Inflation Vanillas and Exotics
Inflation vanilla model
Inflation vanilla model: year-on-year distribution
look for a term-distribution which could fit the yoy smile,related to a Levy process (generic class)
diffusion & jump-diffusion processes (finite activity) :shifted lognormal, SABR ?
processes with infinite activity
Generalised Hyperbolic processes tend to fit well empiricaldistributions with fat tails and leptokurtic behaviour
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Pricing Inflation Vanillas and Exotics
Inflation vanilla model
Inflation vanilla model: Normal InverseGaussian distribution (Barndorff-Nielsen)
NIG is a sub class of Generalised HyperbolicX is NIG if
X |Z ∼ N (µ + βZ , Z )
Z ∼ IG(δ,√
α2 − β2) where 0 ≤ |β| ≤ α
with N (), IG() Gaussian & Inverse Gaussian distributionsthe density is
p(x ;α, β, δ, µ) =αδK1(α
√δ2 + (x − µ)2)
π√
δ2 + (x − µ)2exp (δγ + β(x − µ))
where γ =√
α2 − β2 and K1 the modified Bessel functionof second kind and index 1.
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Pricing Inflation Vanillas and Exotics
Inflation vanilla model
Inflation vanilla model: Normal InverseGaussian distribution
NIG has finite moments at all orders
NIG is closed under convolution
its characteristic function is given by
φNIG(u) = exp(
iuµ + δ
(γ −
√α2 − (β + iu)2
))
option price is computed efficiently by Fourier transform
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Pricing Inflation Vanillas and Exotics
Inflation vanilla model
NIG smile on RPI yoy, expiry 2Y as of 10th Sep2010
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Pricing Inflation Vanillas and Exotics
Inflation vanilla model
NIG smile on HICP yoy, expiry 20Y as of 10thSep 2010
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Pricing Inflation Vanillas and Exotics
Inflation exotic model
Outline
1 Inflation market overview
2 Popular inflation models
3 Pricing challenges
4 Inflation vanilla model
5 Inflation exotic model
6 Conclusion
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Pricing Inflation Vanillas and Exotics
Inflation exotic model
Inflation exotic model: LPI and non-callables
combine Gaussian copula and Monte Carlo pricing.
CDF of the joint distribution of the n yoy rates is
F (y1, ..., yn) = Cgaussn (F1 (y1) , ..., Fn (yn) ,Ω)
with Cgaussn the n-dimensional gaussian copula, Fi the
marginals, Ω the correlations.
the present value of LPI is
PV = Pt ,T ETt
[n∏
i=1
(1 +
[F−1
i (N (Xi))]cap
floor
)]with Xi correlated Gaussian variables, N() the gaussianCDF
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Pricing Inflation Vanillas and Exotics
Inflation exotic model
Inflation exotic model: callables
These trades require a term-structure model
must be able to calibrate to relevant strikes
must at least produce the right shape for the yoy smile
BGM, Cheyette, Quadratic Gaussian
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Pricing Inflation Vanillas and Exotics
Conclusion
Outline
1 Inflation market overview
2 Popular inflation models
3 Pricing challenges
4 Inflation vanilla model
5 Inflation exotic model
6 Conclusion
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Pricing Inflation Vanillas and Exotics
Conclusion
Conclusion
Possible extensions:a new generation of models is required to price inflationvanillas and exotics
a term-distribution approach for inflation vanillas may be astep in the right direction
in line with traders’ view of the market
satisfactorily reproduces the market smile
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Pricing Inflation Vanillas and Exotics
Conclusion
Bibliography
L. Andersen, A. Lipton, Levy processes and their vol smile.Short-term asymptotics, Bank of America Merrill Lynch &Imperial College, 2011.
N. Belgrade, E. Benhamou, E. Koehler, A Market Model forInflation, CDC Ixis Capital Markets, 2004.
X.Charvet, Y. Ticot Inflation vanillas: market overview andoption pricing using NIG distributions, Bank of AmericaMerrill Lynch, Internal document, 2010.
R. Jarrow, Y. Yildirim, Pricing treasury inflation protectedsecurities and related derivatives using a HJM model,Journal of Finance and Quantitative Analysis, 2003.
J. X. Zhang, F. Mercurio, Limited Price Indexation (LPI)Swap Valuation Ideas, Bloomberg L P, 2011.
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