Pricing Inflation Vanillas and Exotics
Transcript of Pricing Inflation Vanillas and Exotics
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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