Inflation-indexed Swaps and Swaptions

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Inflation-indexed Swaps and Swaptions Mia Hinnerich Aarhus University, Denmark Vienna University of Technology, April 2009 M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 1 / 36

Outline Introduction: Markets, Instruments & Literature Foreign-Exchange Analogy Pricing & Hedging of Inflation Swaps Pricing Inflation Swaptions with a Market Model M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 2 / 36

Introduction to the Inflation Market & Instruments Overview 1 Introduction to the Inflation Market & Instruments 2 Foreign-Exchange Analogy 3 Pricing & Hedging of Inflation Swaps 4 Pricing Inflation Swaptions with a Market Model M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 3 / 36

Introduction to the Inflation Market & Instruments Inflation An increase in the economy s price level is known as inflation. Inflation reduces the purchasing power, i.e. the value of money decrease. A consumer price index (CPI) is the price of a particular basket consisting of consumer goods and services. The price index is a measure of the general price level in the economy. Inflation is typically measured as the percentage rate at which the consumer price index changes over a certain period of time. Negative inflation is known as deflation. M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 4 / 36

Introduction to the Inflation Market & Instruments Inflation Protected Bonds Many names: Inflation-indexed bonds, Inflation linked bonds, Real bonds, TIPS (US), Index-linked gilts (UK). The payoff is linked to a price index. (CPI, RPI) Typically coupon bonds. Can be floored. The issuer of an inflation protected bond has an incentive to keep inflation low. Useful for governments. These bonds are typically issued by Treasuries. Typical investors are pension funds, mutual funds. World wide outstanding nominal amount 2007: 1000 billion dollar. M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 5 / 36

Introduction to the Inflation Market & Instruments Markets UK (1981) Australia (1983) Canada (1991) Sweden (1994) United States (1997) Greece (1997) France (1998) Italy (2003) Japan (1904) Germany (2006) Earlier: Chile, Brazil, Columbia, Argentina. First inflation protected bond issue: Massachusetts Bay Company 1780. M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 6 / 36

Introduction to the Inflation Market & Instruments Inflation Derivatives Swaps Caps & Floors Swaptions Bond options... M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 7 / 36

Introduction to the Inflation Market & Instruments Inflation Indexed Swaps & Swaptions Inflation Indexed Swap Agreement between two parties A and B to exchange cash flows in the future Prespecified dates for when the cash flows are to be exchanged At least one of the cash flows is linked to inflation (CPI) A B Inflation Indexed Swaption It is an option to enter into an inflation indexed swap at pre specified date at a pre determined swap rate. M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 8 / 36

Introduction to the Inflation Market & Instruments Main References Hughston (1998) General theory Foreign-currency analogy Jarrow & Yildirim (2003) 3-factor HJM model TIPS (coupon bonds) Option on Inflation index Mercurio (2005) YYIIS, Caplets, Floorlets (ZCIIS) JY version of HJM with Hull-White vol 2 Market Models M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 9 / 36

Introduction to the Inflation Market & Instruments Contribution HJM model with jumps YYIIS Inflation Swap Market Models ZCIISwaptions YYIISwaptions HJM model ZCIISwaptions TIPStions Verify the foreign-currency analogy for an arbitrary process YYIIS= Year-on-Year Inflation Indexed Swaps ZCIIS= Zero Coupon Inflation Indexed Swaps M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 10 / 36

Foreign-Exchange Analogy Overview 1 Introduction to the Inflation Market & Instruments 2 Foreign-Exchange Analogy 3 Pricing & Hedging of Inflation Swaps 4 Pricing Inflation Swaptions with a Market Model M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 11 / 36

Foreign-Exchange Analogy Price and Payoff I(t) : p n (t, T) : p IP (t, T) : An arbitrary stochastic process Price in dollar at t of a contract that pays out 1 dollar at T. Price in dollar at t of a contract that pays out I(T ) dollar at T. Assume : There exist a market for p n (t, T ) and p IP (t, T ) for all T Define : p r (t, T ) = pip (t,t ) I(t) M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 12 / 36

Foreign-Exchange Analogy If I(t) is the price of a hamburger A nominal bond: Pays out 1 dollar at maturity. p n (t, T ): the price of a nominal bond is in dollar A hamburger-indexed bond: At maturity it pays out a dollar amount that is enough to buy 1 hamburger. p IP (t, T ): the price of a hamburger-inflation protected bond is in dollar A hamburger-real bond: Pays out 1 hamburger at maturity p r (t, T ): the price of a real bond is in hamburgers Note: CPI M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 13 / 36

Foreign-Exchange Analogy Define Forward rates: f i (t, T ) = ln pi (t,t ) T for i = r, n. Short rates: r i (t) = f i (t, t) for i = r, n. Money Market Accounts: B i (t) = e t 0 ri (s)ds for i = r, n. M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 14 / 36

Foreign-Exchange Analogy HJM model with Jumps Assume: Under the objective probability measure P : df t r (T ) = αt r (T )dt + σt r (T )dw P + ξ r (t, v, T )µ P (dt, dv) V df t n (T ) = αt n (T )dt + σt n (T )dw P + ξ n (t, v, T )µ P (dt, dv) di t = I t µ I t dt + I t σ I t dw P + I t V V γ I t (v)µ P (dt, dv) M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 15 / 36

Foreign-Exchange Analogy Now calculate 1 Forward rates Bondprices (BKR) 2 Change measure from P to Q n (Girsanov) Now we have found the Q n -drift of p n (t, T ) and p IP (t, T ) which we call µ n Q (t, T ) and µip Q (t, T ) 3 By requiring p n (t, T ) B n (t) p IP (t, T ) B n (t) are Q n -martingales i.e. µ n Q (t, T ) = µip Q (t, T ) = rn (t) for all maturities T. 3 drift conditions One of the 3 conditions tells us that the Q n -drift of the index I is equal to r n r r M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 16 / 36

Foreign-Exchange Analogy Three drift conditions ( T ) α n (t, T ) = σ n (t, T ) σ r (t, s)ds h(t) t {δ n (t, v, T ) + 1} ξ n (t, v, T )λ t(dv) V ( T ) α r (t, T ) = σ r (t, T ) σ r (t, s)ds σ I (t) h(t) t ( ) 1 + γ I (t, v) (1 + δ r (t, v, T )) ξ r (t, v, T )λ t(dv) V µ I (t) = r n (t) r r (t) h(t)σ I (t) γ I (t, v)λ t(dv) V M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 17 / 36

Foreign-Exchange Analogy Result Under the nominal risk neutral measure Q n : dp n t (T ) p n t (T ) = rt n dt + βt n (T )dw + δt n (v, T ) µ(dt, dv) V dp IP p IP t (T ) t (T ) = rt n dt + βt IP (T )dw + V δt IP (v, T ) µ(dt, dv) where di t = (rt n rt r )dt + σt I dw + γt I (v) µ(dt, dv) I t V dp r t (T ) p r t (T ) = a(t, T )dt + βt r (T )dw + δt r (v, T ) µ(dt, dv) V µ(dt, dv) = µ(dt, dv) λ t (dv)dt Note: I has the same dynamics as an FX-rate! M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 18 / 36

Foreign-Exchange Analogy Foreign Currency Analogy Nominal vs Real p n (t, T ) : p r (t, T ) : I(t) : p IP (t, T ) : Domestic vs Foreign p n (t, T ) : p r (t, T ) : I(t) : Price of nominal T -bond in dollar Price of real T -bond in CPI units* Price level (dollar per CPI-unit) Price of a real T -bond in dollar denoted by p IP (t, T ) Price of domestic T -bond Price of foreign T -bond FX-rate (domestic per foreign unit) I(t)p r (t, T ) : Domestic price of foreign T -bond * M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 19 / 36

Pricing & Hedging of Inflation Swaps Overview 1 Introduction to the Inflation Market & Instruments 2 Foreign-Exchange Analogy 3 Pricing & Hedging of Inflation Swaps 4 Pricing Inflation Swaptions with a Market Model M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 20 / 36

Pricing & Hedging of Inflation Swaps Assumptions Exist a market for nominal T-bonds and nominal indexed-bonds for all maturity dates. The bond prices are differentiable wrt T. Forward rate dynamics according to HJM with jumps. Existence of martingale measure. All volatilities and the intensity are deterministic under the nominal risk neutral measure. M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 21 / 36

Pricing & Hedging of Inflation Swaps A Payer Swap starts at time T m At each payment date T j where j = m + 1, m + 2,, T M you pay α j K you receive αj [ ] I(Tj ) I(T j 1 ) 1 M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 22 / 36

Pricing & Hedging of Inflation Swaps The price The price at time t is: M j=m+1 [ ] I(T j ) Π t, α j I(T j 1 ) (K + 1) M j=m+1 α j p(t, T j ) Find: [ ] I(T j ) Π t, α j I(T j 1 ) i.e. the price of payoff α j I(T j ) I(T j 1 ) M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 23 / 36

Pricing & Hedging of Inflation Swaps Tool Let M(t) and N(t) be two martingales so that E t [M T ] = M t E t [N T ] = N t The key to price this swap is to find: Solution: E t [M T N T ] If M(t) and N(t) are independent, then E t [M T N T ] = M t N t What if they are NOT independent? E t [M T N T ] = M t N t G T t where G T t is the convexity correction. M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 24 / 36

Pricing & Hedging of Inflation Swaps Basic Result Let M(t) and N(t) be two martingales with dynamics: dm t = σt M dw t + δt M (v) µ(dt, dv) M t dn t N t = σ N t dw t + V V δ N t (v) µ(dt, dv) Assume that σ M, σ N, δ M, δ N, λ are deterministic. Then E t [M T N T ] = M t N t G T t where G T t = e T t (σu M σu N + V δm u (v)δu N (v)λ u(dv))du M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 25 / 36

Pricing & Hedging of Inflation Swaps The Inflation-linked Swap Leg The payoff function X 2 = I(T 2) I(T 1 ) The value at time t Π [t, X 2 ] = p n (t, T 1 )E T 1,n t [p r (T 1, T 2 )] where E T 1,n t [p r (T 1, T 2 )] = E T 1,r t hence [ p r (T 1, T 2 ) p r (T 1, T 1 ) ] L(T 1 ) = pr (t, T 2 ) L(t) p r (t, T 1 ) C(t, T 1, T 2 ) Π [t, X 2 ] = pn (t, T 1 )p r (t, T 2 ) p r C(t, T 1, T 2 ) (t, T 1 ) where C(t, T 1, T 2 ) is a convexity correction term. M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 26 / 36

Pricing & Hedging of Inflation Swaps The Inflation-linked Swap Leg The payoff function X 2 = I(T 2) I(T 1 ) The value at time t Π [t, X 2 ] = p n (t, T 1 )E T 1,n t [p r (T 1, T 2 )] where E T 1,n t [p r (T 1, T 2 )] = E T 1,r t hence [ p r (T 1, T 2 ) p r (T 1, T 1 ) ] L(T 1 ) = pr (t, T 2 ) L(t) p r (t, T 1 ) C(t, T 1, T 2 ) Π [t, X 2 ] = pn (t, T 1 )p r (t, T 2 ) p r C(t, T 1, T 2 ) (t, T 1 ) where C(t, T 1, T 2 ) is a convexity correction term. M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 26 / 36

Pricing & Hedging of Inflation Swaps The Inflation-linked Swap Leg The payoff function X 2 = I(T 2) I(T 1 ) The value at time t Π [t, X 2 ] = p n (t, T 1 )E T 1,n t [p r (T 1, T 2 )] where E T 1,n t [p r (T 1, T 2 )] = E T 1,r t hence [ p r (T 1, T 2 ) p r (T 1, T 1 ) ] L(T 1 ) = pr (t, T 2 ) L(t) p r (t, T 1 ) C(t, T 1, T 2 ) Π [t, X 2 ] = pn (t, T 1 )p r (t, T 2 ) p r C(t, T 1, T 2 ) (t, T 1 ) where C(t, T 1, T 2 ) is a convexity correction term. M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 26 / 36

Pricing & Hedging of Inflation Swaps The Payer Swap The price of the swap is at time t M j=m+1 (K + 1) α j p n (t, T j 1 )p IP (t, T j )C(t, T j 1, T j ) p IP (t, T j 1 ) M j=m+1 α j p n (t, T j ) Note The price does only depend on bonds in the nominal market. M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 27 / 36

Pricing & Hedging of Inflation Swaps Hedging the Inflation-linked Swap leg I When no jumps where g(s, T 1, T 2 ) = T1 t Π [t, X 2 ] = pn (t, T 1 )p IP (t, T 2 ) p IP e g(s,t 1,T 2 ) (t, T 1 ) ( β n (s, T 1 ) β IP (s, T 1 ) ) (β IP (s, T 2 ) β IP (s, T 1 ) ) ds Idea: Try to replicate the swap leg using p n (t, T 1 ), p IP (t, T 2 ) and p IP (t, T 1 ) from now on referred to as S 1, S 2 and S 3 respectively. M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 28 / 36

Pricing & Hedging of Inflation Swaps Hedging the Inflation-linked Swap leg II Define portfolio strategy (h 1 (t), h 2 (t), h 3 (t)) for t T 1 as h i (t) = Π [t, X 2] S i (t) for i = 1, 2, h 3 (t) = Π [t, X 2] S 3 (t) Then for t T 1 V h (t) = dv h (t) = 3 h i S i = Π(t) 1 3 h i ds i 1 At time T 1 we have V h (T 1 ) = pip (T 1,T 2 ) 1 I(T 1 ) which is just enough to buy I(T 1 ) T 2 IP -bonds which we keep until maturity and thus results in I(T 2) I(T 1 ) as it should! M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 29 / 36

Pricing Inflation Swaptions with a Market Model Overview 1 Introduction to the Inflation Market & Instruments 2 Foreign-Exchange Analogy 3 Pricing & Hedging of Inflation Swaps 4 Pricing Inflation Swaptions with a Market Model M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 30 / 36

Pricing Inflation Swaptions with a Market Model The Swap rate Recall that the swap price is: Y Y IIS M m (t, K) = M j=m+1 [ ] I(T j ) Π t, α j (K + 1)Sm M (t)) I(T j 1 ) where k Sm(t) k = α j p n (t, T j ) j=m+1 The par swap rate is: R M m (t) = ] M j=m+1 [t, Π I(T α j ) j I(T j 1 ) Sm M (t) Sm M (t) M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 31 / 36

Pricing Inflation Swaptions with a Market Model The HJM Swap rate The par swap rate at time t R M m (t) = M j=m+1 α jp n (t,t j 1)p IP (t,t j)c(t,t j 1,T j) p IP (t,t j 1) S M m (t) S M m (t) Where Note Nasty distribution! k Sm(t) k = α j p n (t, T j ) j=m+1 M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 32 / 36

Pricing Inflation Swaptions with a Market Model A payer YYIISwaption Payoff function Rewritten Payoff function Υ M m = max[y Y IIS M m (T m, K), 0] Υ = S T max[r T K, 0] where R T par swap rate (self-financing portfolio) S T sum of nominal bonds (self-financing portfolio) Note Easy if R T is lognormal! Black s pricing formula M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 33 / 36

Pricing Inflation Swaptions with a Market Model Swap Market Model Definition: For any given pair (m, k) of integers s.t. 0 m < k < M we assume that, under the measure for which S k m is numeraire, the forward swap rate R k m has dynamics given by where σ k m(t) is deterministic dr k m(t) = R k m(t)σ k m(t)dw k m(t) M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 34 / 36

Pricing Inflation Swaptions with a Market Model Summarizing Introduction: Markets, Instruments & Literature Foreign-Exchange Analogy Pricing & Hedging of Inflation Swaps Pricing Inflation Swaptions with a Market Model M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 35 / 36

Pricing Inflation Swaptions with a Market Model Remarks TIPS can have embedded option features Reverse: Swaps as given price TIPS The CPI index used is typically lagged The CPI index is typically only observed monthly (linearly interpolated) The suggested market model is not proved to exist M. Hinnerich (Aarhus University) Inflation-indexed Swaps and Swaptions April 2009 36 / 36

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