### Inflation derivatives

```Market Expectations Trough
Derivative Instruments
Stefano Caprioli
1° Lesson
Inflation Derivatives
1
Summary Lesson 1:
- Overview Inflation Derivatives
- Black Normal
- Market Sources
- Market Expectations through Vol Surface
2
Zero-Coupon Inflation Indexed Swap
Zero-Coupon Inflation Indexed Swap (ZCIIS or Breakeven Swap)
enables to swap a fixed lag vs a floating leg based on Consumer
Price Index (CPI) performance:
N[(1+K)^M-1] vs N[(I(M)-I(0))/I(0)]
Where:
-N is the Notional
-I(M) is CPI in t=M
-I(0) is CPI in t=0
At inception we have a fair swap if:
K=sqrt(I(M)/I(0),M)-1
3
Zero-Coupon Inflation Indexed Swap
Market Source: Bloomberg
4
ZC Sensitivity to CPI rates
dNPV
 disc. factor* No min al / CPI (0)
dCPI(T )
dCPI(t )
 d (CPI _ last * exp[CPI _ rate * T ])  T * CPI (t )
dCPI _ rate
dNPV
dNPV

* T * CPI (t )
dCPI _ rate dCPI(t )
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Sensitivity to Inflation Market Rate
dNPV
dNPV dCPI(T )

*
dK
dCPI(T )
dK
where :
CPI (T )  CPI (0) * (1  K )^ T
6
ZCIS and Seasonality
If the estimation date of the forward CPI required for the trade is not
on pillars curve, you have to take into account inflation’s seasonality
component.
For instance, considering the flow of a 3Y ZCIS where you receive
inflation return: CPI(T2)/CPI(T1)-1
Under hypothesis of CPI(T1) already fixed and CPI(T2) is on January
whereas the base month of inflation curve is September.
CPI(T2) becomes:
CPI(T2)=α(T2)*[CPI(T1)/α(T1)]*(1+s)^p
Where:
α(Month) is the seasonality factor of month m
S is the growth rate between T1 and T1+1Y
p is the number of months between T1 and T2
You can move from additive to multiplicative calculation mode as you
prefer. Market pratictioners usually compute seasonality according to
econometric models.
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Note you can obtain Real Rate according to nominal and Breakeven term
structure, remember that:
Real Yield=Nominal Yield-Inflation Rate
So we have:
Inflation Linked ZC Bond Yield = Real Rate
Rem.: ZCIS NPV=NOTIONAL*DF(r)*(1+K)^T=NOTIONAL*DF (r)/DF(K)
r: Nominal Rate
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Example:
We assume I(0)=100
Market IL-ZC Bond=98.04
Market Nominal Bond=96.15
Maturity T=1 Year
Nominal yield on nominal zero-coupon bond:
yn(0,T)= sqrt[(Dn(0,T)^-1),T]-1=(1/96.15%)-1=4%
Real yield on IL-ZC Bond:
yr(0,T)= sqrt[(Dr(0,T)^-1),T]-1=(1/98.04%)-1=2%
Given the growth of the inflation index, we can calculate the Real Yield on a nominal
bond and the nominal yield on an inflation-linked bond.
Assuming the inflation index grows to 102, I(T)=102, these can be computed in the
following manner:
Real yield on nominal zero-coupon bond (Real Rate for T=1):
[1xI(0)/I(T)]/96.15% -1=[(100/102)/96.15%] -1=1.96%
Nominal yield on IL-ZC Bond:
sqrt[I(T)/(I(0) Dr(0,T)),T]-1=(102/100)/98.04%=4.04%
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Nominal Bond has a certain nominal return but uncertain real return,
whereas IL Bond has a certain real return and an uncertain nominal return.
10
Zero-Coupon Inflation Indexed Swap
Note: I(0) is CPI at (t-3 Months)
Most attractive CPI instrument for market traders is HICP Ex-Tob
Duration: n exact years from start date
Payments: on the next business day after calculation end date
Fixed Leg: pays a fixed rate yearly compounded with ACT/ACT
yield rate convention
Floating Leg: receives the inflation over period with 3-Months Lag,
eventually CPI is interpolated
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Inflation Curve Details
The inflation curve is completely independent from the rate curve (it’s
different if you deal with NPVs).
The common interpolation method in order to build Inflation Curve is
Linear. The value to interpolate is the CPI (*) rate and the Zero rate
convention is EXP ACT/365.
(*) You can also choice to interpolate on (CPI*time) or on CPI Rate.
Note that a market practice is to roll the base month a couple of days
before month end. As a result, there is a short period during which the
fixing lag of the swaps in the curve is no longer 3 months but rather 2
months.
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Year on Year Inflation Indexed Swap
YYIIS pays a fixed leg, Nφ(i)K vs a floating leg, Nφ(i)[(I(i)-I(i-1))/I(i-1)]
Where φ(i) is year fraction between [T(i-1),T(i)]
I(i) is CPI in T=T(i)
I(i-1) is CPI in T=T(i-1)
N is the Notional
NOTE: YoY Inflation Rates can be obtained from ZCIIS according the further
analytics:
(1+K(i))^i=I(T(i))/I(T(0))
(1+K(i-1))^(i-1)=I(T(i-1))/I(T(0))
1  Ki i * I T0   1  I Ti   1
i 1
I Ti1 
1  Ki1  * I T0 
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YoY Inflation Swap
E (CPIt )  CPI0 (1  SwapRatet )t
FWDt 
E (CPIt )  E (CPIt 1 )
E (CPIt 1 )
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Break-Even Inflation
HICP ZC SWAP
2.700%
2.600%
2.500%
2.400%
2.300%
2.200%
2.100%
2.000%
1.900%
1.800%
1Y
2Y
3Y
4Y
5Y
6Y
7Y
8Y
9Y 10Y 12Y 15Y 20Y 25Y 30Y
Market Source: ICAP
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YoY Inflation Swap Details
Payment: on the next business day after the calculation end date of each
period
Fixed Leg: pays a fixed rate flow each year with linear ACT/ACT rate
convention
Floating Leg: receives the inflation over each period. Time Lag is 3 Months
and, if necessary, CPI is interpolated.
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Inflation-Indexed Cap & Floor
Inflation-Indexed Cap/Floor is a stream of Caplet/Floorlet on CPI.
Each Caplet/Floorlet (IICF) is a Call/Put option on Inflation Rate.
  Ii

Ni  
 1   

  I i1

Where:
-ω is 1 for Caplet, -1 otherwise
-K Inflation Rate Strike
-Φ Year fraction between [T(i-1),T(i)]
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Inflation-Indexed Cap & Floor
According to the previous slide we have Cap/Floor Analytic Payoff:

  I



N i P0,Ti ETi   i 1    
i1
 
  Ii1
M
NOTE: Cap/Floor Underlying could reach zero or negative values, so
Black Framework is inconsistent to evaluate this kind of pay-off
Market Pratictioners moved toward Black Normal Model:
dY( T )   (T )dW (T )
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Inflation-Indexed Cap & Floor
Black Normal Model assumes Normality of Inflation Rate Variations: that’s
a common market assumption independent from reality.
In this way Black Normal Volatility becomes “the wrong number into the
wrong formula for the right price” as every market implied volatility you
can observe on the market.
As usually Inflation Rate will be a martingale under its forward Measure
and you can evaluate your Cap/Floor through a closed formula.
Note you can move from Geometric Brownian motion of CPI(t) to Black
Normal Model for YoY Inflation Swap under assumptions of σ>>0 and
CPI(t)/CPI(t-1) >>1.
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Brief Review Topics About Black Normal
Suppose I(t) moves according to a Geometric Brownian Motion:
dI=μIdt+σIdW
If you use the appropriate risk measure and assume σ>>0 its easy to
arrive to a Gaussian Distribution for [I(t)/I(0)-1].
Those assumptions are realistic, so you can use Black Normal Model
in order to evaluate Caps and Floors about Breakeven Inflation Rates.
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Inflation-Indexed Cap & Floor
Market Sources (Brokers and Counteparties) quote Cap/Floor
Premium, so you have to compute implied volatility according to
As a rule you have to bootstrap caplet/floorlet volatilities under non
arbitrage conditions. Black Normal Implied Volatilities are smaller
than “log Normal Volatilities”.
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Inflation-Indexed Cap & Floor
It’s interesting to compare EURIBOR 6M Log Normal ATM Fwd vs
HICP-EX TOB NORMAL ATM Basis Point Vol.
On the long period (after 15Y) volatility term structures convey to
similar levels according to theory (Rebonato) that inflation is the first
driver to move Euribor long run volatilities.
22
Inflation-Indexed Cap & Floor
Black Normal HICP Ex Tob Caplet/Floorlet Volatility
Surface
23
Inflation-Indexed Cap & Floor
Remember this volatility surface is referred to YOY Swap, so on each smile
you can infer YoY market expectations under the risk neutral probability
measure.
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Inflation-Indexed Cap & Floor
1Y Floorlets HICP Ex Tob YoY
Remember this volatility surface is referred to YOY Fwds, you can infer YoY
market expectations under the risk neutral probability measure (Digitals
example on Floorlets) from each smile
25
Given a volatility smile is always possible to compute implied cumulative
density distribution pricing a put spread on “closest” strikes:
CDF(K)=(φ^-1) (dP/dK)
(**)
In this case this “exercise” is meaningful since Market quotes deep in/out
strikes with sufficient liquidity (0 Floor or 4% Cap for Instance).
(**): CDF(K)=Prob(FWD≤K)
In general remember Volatilities have informations about underlying
implied distribution, with smile convexity gives us info about Kurthosis,
Smile slope gives information on volatility/underlying correlation too. A
negative slope, for instance, means negative correlation between
underlying and expected volatility, otherwise, a pronounced smile gives
Considering C&F Market Quotes for each remarkable Strike we expected a
sticky strike volatility regime for YoT Swap Rate underlying.
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1Y HICP Ex Tob C&F Volatility Smile shows a strong
smile.
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Conclusions:
That’s an easy overview on Inflation Derivatives Market in order to underpin
useful concepts to analyze market perceptions about inflation future variations.
This review is not exhaustive but it’s a first step to see derivatives like a repository
of news about key data of our economy.
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