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
ZCIS and Seasonality 7 If the estimation date of the forward CPI required for the trade is not on pillars curve, you have to take into account inflations 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.
Inflation-Linked Cashflows and Real Bonds 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 8
Inflation-Linked Cashflows and Real Bonds 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% 9
Inflation-Linked Cashflows and Real Bonds 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 Fixed Rate of ZCIIS is published by EUROSTAT Note: I(0) is CPI at (t-3 Months) Most attractive CPI instrument for market traders is HICP Ex-Tob Start delay: +2 Business days 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 11
Inflation Curve Details The inflation curve is completely independent from the rate curve (its 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. 12
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)) 13
YoY Inflation Swap Details Start Delay: +2 Business days 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. 16
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. Where: -ω is 1 for Caplet, -1 otherwise -K Inflation Rate Strike -Φ Year fraction between [T(i-1),T(i)] 17
Inflation-Indexed Cap & Floor According to the previous slide we have Cap/Floor Analytic Payoff: 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: 18
Inflation-Indexed Cap & Floor Black Normal Model assumes Normality of Inflation Rate Variations: thats 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. 19
Brief Review Topics About Black Normal 20 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.
Inflation-Indexed Cap & Floor 21 Market Sources (Brokers and Counteparties) quote Cap/Floor Premium, so you have to compute implied volatility according to your input curve data. As a rule you have to bootstrap caplet/floorlet volatilities under non arbitrage conditions. Black Normal Implied Volatilities are smaller than log Normal Volatilities.
Inflation-Indexed Cap & Floor 22 Its 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.
Inflation-Indexed Cap & Floor 23 Black Normal HICP Ex Tob Caplet/Floorlet Volatility Surface
Inflation-Indexed Cap & Floor 24 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.
Inflation-Indexed Cap & Floor 25 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 1Y Floorlets HICP Ex Tob YoY
26 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(FWDK) In general remember Volatilities have informations about underlying implied distribution, with smile convexity gives us info about Kurthosis, Smile Slope about Skewness. 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 alert about a potential regimes shift with significant underlying jump. Considering C&F Market Quotes for each remarkable Strike we expected a sticky strike volatility regime for YoT Swap Rate underlying.
27 1Y HICP Ex Tob C&F Volatility Smile shows a strong smile.
28 Conclusions: Thats 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 its a first step to see derivatives like a repository of news about key data of our economy.