Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework

Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework Kathrin Glau, Nele Vandaele, Michèle Vanmaele Bachelier Finance Society World Congress 2010 June 22-26, 2010 Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 1/42

Aim A compact representation for the pricing formula by using the Jamshidian decomposition Hedging strategies with default-free zero coupon bonds (delta-hedging quadratic hedging) Numerical implementation and results Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 2/42

Outline 1 Introduction 2 Pricing of swaption 3 Hedging of swaption 4 Numerical results Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 3/42

Outline 1 Introduction Model Swaption Tools for option pricing and hedging 2 Pricing of swaption 3 Hedging of swaption 4 Numerical results Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 4/42

Zero-coupon bond B(t, T ) B(T, T ) = 1 No coupons, No default B(t, T ) < 1 for every t < T f (t, u) instantaneous forward rate: B(t, T ) = exp( T f (t, u)du) t Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 5/42

Original HJM model Dynamics of forward interest rate df (t, T ) = α(t, T )dt + σ (t, T )dw t with W standard d-dimensional Brownian motion under P α and σ adapted stochastic processes in R, resp R d denotes transpose Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 6/42

Original HJM model Dynamics of zero coupon bonds db(t, T ) = B(t, T )(a(t, T )dt σ (t, T )dw t ) with a(t, T ) =f (t, t) α (t, T ) + 1 2 σ (t, T ) 2 α (t, T ) = σ (t, T ) = T t T t α(t, u)du σ(t, u)du. Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 7/42

Lévy driven HJM model Dynamics of forward interest rate df (t, T ) = α(t, T )dt σ(t, T )dl t with L: one-dimensional time-inhomogeneous Lévy process The law of L t is characterized by the characteristic function E[e izlt ] = e t 0 θs(iz)ds, t [0,T ] with θ s cumulant associated with L by the Lévy-Khintchine triplet (b s, c s, F s ): θ s (z) := b s z + 1 2 c sz 2 + (e xz 1 xz)f s (dx) with b t, c t R, c t 0, F t Lévy measure Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 8/42 R

Lévy driven HJM model Integrability assumptions: T ( ) b s + c s + (x 2 1)F s (dx) ds < 0 R There are constants M, ɛ > 0 such that for every u [ (1 + ɛ)m, (1 + ɛ)m]: T 0 { x >1} exp(ux)f s (dx)ds < L is an exponential special semimartingale Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 9/42

Interest rate products Savings account and default-free zero coupon bond prices: B t = B(0, t) exp( t B(t, T ) = B(0, T )B t exp( 0 A(s, t)ds t t 0 A(s, T )ds + 0 0 with for s T = min(s, T ) and s [0, T ] A(s, T ) = T s T α(s, u)du and Σ(s, T ) = Σ(s, t)dl s ) t T s T Σ(s, T )dl s ) σ(s, u)du, Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 10/42

Heath-Jarrow-Morton Unique martingale measure=spot measure A(s, T ) = θ s (Σ(s, T )) with θ the cumulant associated with L by (b s, c s, F s ) θ s (z) = b s z + 1 2 c sz 2 + (e xz 1 xz)f s (dx) Discounted zero-coupon bonds are martingales R Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 11/42

Forward MM dp T dp = 1 T B T B(0, T ) = exp( θ s (Σ(s, T )ds + 0 0 dp T dp = B(t, T ) t Ft B t B(0, T ) = exp( θ s (Σ(s, T ))ds + 0 T t L: time-inhomogeneous Lévy process under P T and special with characteristics (b P T s, c P T s, F P T s ): b P T s = b s + c s Σ(s, T ) + x(e Σ(s,T )x 1)F s (dx) c P T s = c s F P T s (dx) = e Σ(s,T )x F s (dx) R Σ(s, T )dl s ) 0 Σ(s, T )dl s ) Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 12/42

Interest rate derivative Swaption: option granting its owner the right but not the obligation to enter into an underlying interest rate swap. Interest rate swap: contract to exchange different interest rate payments, typically a fixed rate payment is exchanged with a floating one. A: Payer swaption B: Receiver swaption Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 13/42

Interest rate derivative Swaption: option granting its owner the right but not the obligation to enter into an underlying interest rate swap. Interest rate swap: contract to exchange different interest rate payments, typically a fixed rate payment is exchanged with a floating one. A: Payer swaption B: Receiver swaption A fixed rate B floating rate Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 13/42

Jamshidian Closed-form expression for European option price on coupon-bearing bond P(r, t, s): Price at time t of a pure discount bond maturing at time s, given that r(t) = r and R r,t,s is a normal random variable ( ) + aj P(R r,t,t, T, s j ) K = aj (P(R r,t,t, T, s j ) K j ) + with K j = P(r, T, s j ) and r is solution to equation a j P(r, T, s j ) = K Holds for any short rate model as long as zero coupon bond prices are all decreasing (comonotone) functions of interest rate Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 14/42

Fourier transformation Theorem Eberlein, Glau, Papapantoleon (2009) If the following conditions are satisfied: (C1) The dampened function g = e Rx f (x) is a bounded, continuous function in L 1 (R). (C2) The moment generating function M XT (R) of rv X T exists. (C3) The (extended) Fourier transform ĝ belongs to L 1 (R), E[f (X T s)] = e Rs e ius ϕ XT ( u ir)ˆf (u + ir)du, 2π R with ϕ XT characteristic function of the random variable X T. Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 15/42

Outline 1 Introduction 2 Pricing of swaption 3 Hedging of swaption 4 Numerical results Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 16/42

Pricing of swaption Assumptions on volatility structure Volatility structure σ: bounded and deterministic. For 0 s and T T : 0 Σ(s, T ) = T s T σ(s, u)du M < M, For all T [0, T ] we assume that σ(, T ) 0 and σ(s, T ) = σ 1 (s)σ 2 (T ) 0 s T, where σ 1 : [0, T ] R + and σ 2 : [0, T ] R + are continuously differentiable. inf s [0,T ] σ 1 (s) σ 1 > 0 Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 17/42

Pricing of swaption Payer swaption can be seen as a put option with strike price 1 on a coupon-bearing bond. Payer swaption s payoff at T 0 : (1 n c j B(T 0, T j )) +, j=1 T 1 < T 2 <... < T n : payment dates of the swap with T 1 > T 0 δ j := T j T j 1 : length of the accrual periods [T j 1, T j ] κ: fixed interest rate of the swap coupons c i = κδ i for i = 1,..., n 1 and c n = 1 + κδ n Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 18/42

Pricing of swaption Start from 1 PS t = B t E[ (1 B T0 n c j B(T 0, T j )) + F t ] t [0, T 0 ] j=1 with expectation under risk-neutral measure P Change to forward measure P T0 eliminating instantaneous interest rate B T0 under expectation PS t = B(t, T 0 )E P T 0 [(1 n c j B(T 0, T j )) + F t ] t [0, T 0 ] j=1 Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 19/42

Pricing of swaption Define: g(s, t, x) = D s t e Σ t s x 0 s t T ( D s t B(0, t) s ) = B(0, s) exp [θ u (Σ(u, s)) θ u (Σ(u, t))] du Σ t s = t s 0 σ 2 (u)du and X s = g(s, t, X s ) = B(s, t) and price payer swaption PS t = B(t, T 0 )E P T 0 [(1 s 0 σ 1 (u)dl u 0 s t T n c j g(t j, T 0, X T0 )) + F t ] by volatility structure assumptions functions x g(t 0, T j, x) are non-decreasing functions for j = 1,..., n j=1 Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 20/42

Pricing of swaption n PS t = B(t, T 0 )E P T 0 [(1 c j g(t j, T 0, X T0 )) + F t ] = B(t, T 0 ) j=1 n c j E P T 0 [(bj g(t 0, T j, X T0 )) + F t ] j=1 Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 21/42

Pricing of swaption PS t = B(t, T 0 )E P T 0 [(1 = B(t, T 0 ) n c j g(t j, T 0, X T0 )) + F t ] j=1 n c j E P T 0 [(bj B(T 0, T j )) + F t ] j=1 weighted sum of put options with different strikes on bonds with different maturities Tj with b j such that j Σ T z DT T 0 e 0 = g(t 0, T j, z ) = b j and z is the solution to the equation n j=1 c jg(t 0, T j, z ) = 1 Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 21/42

Pricing of swaption PS t B(t, T 0 ) n j=1 c j e RXt 2π R e iuxt ϕ P T 0 X T0 X t (u + ir)ˆv j ( u ir)du with ϕ P T 0 X T0 X t (z) = exp and where T0 t [θ s (Σ(s, T 0 ) + izσ 1 (s)) θ s (Σ(s, T 0 ))]ds ˆv j ( u ir) = b j e ( iu+r)z j ΣT T 0 ( iu + R)( iu + Σ T j T 0 + R) Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 22/42

Outline 1 Introduction 2 Pricing of swaption 3 Hedging of swaption Delta-hedging Mean variance hedging strategy 4 Numerical results Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 23/42

Assumptions Integrability assumptions Volatility structure assumptions σ 1 < σ 1 for a certain σ 1 R u ϕ P T 0 X T0 X t (u + ir) is integrable Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 24/42

Delta-hedging of PS Theorem The optimal amount, denoted by j t, to invest in the zero coupon bond with maturity T j to delta-hedge a short position in the forward payer swaption is given by: j t = B(t, T 0) B(t, T j ) Σ T j t with for l = 0, 1 l H k (t,xt ) X l t n k=1 c k ( Σ T 0 t H k (t, X t ) + X t H k (t, X t )), = 1 2π R e( R+iu)X t ϕ P T 0 X T0 Xt (u+ir)ˆv k ( u ir)( R+iu) l du. Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 25/42

Self-financing delta B(t, T 0 ): bond used as cash account, depends also on X B(t, T j ): bond in which to invest, with T j T 0 solving system of equations for j t and 0 t to obtain discrete hedging strategy: V t X t = PS t X t + j B(t, T j ) t + 0 B(t, T 0 ) t = 0 X t X t ( j t j t 1 )B(t, T j) + ( 0 t 0 t 1)B(t, T 0 ) = 0 Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 26/42

MVH strategy Quadratic hedge in terms of discounted assets S MVH strategy is self-financing = optimal amount of discounted assets is sensible amount to invest in non-discounted assets Minimizing the mean squared hedging error defined as E[(H (v + (ξ S) T )) 2 ] Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 27/42

MVH strategy Unrealistic to hedge with risk-free interest rate product = choose bond B(, T 0 ) as numéraire MVH strategy for payer swaption under forward measure P T0 using numéraire B(, T 0 ) Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 28/42

MVH strategy Self-financing strategy minimizing E P T 0 [(PST0 ṼT 0 ) 2 ] = E P T 0 [(PST0 (Ṽ0 + T0 0 ξ j ud B(u, T j ))) 2 ] with PS T0 = PS T 0 B(T 0, T 0 ) : (discounted) price of PS at time T 0 V Ṽ = : (discounted) portfolio value process B(, T 0 ) Value of self-financing portfolio V : V t =ξ 0 t B(t, T 0 ) + ξ j tb(t, T j ) =V 0 + (ξ 0 B(, T 0 )) t + (ξ j B(, T j )) t Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 29/42

Determination Ideas of Hubalek, Kallsen and Krawczyk (2006). Variance-optimal hedging for processes with stationary independent increments. Annals of Applied Probability, 16:853-885 adapted to present setting GKW decomposition of special type of claims: H(z) = B(T 0, T j ) z for a z C Express PS T0 as f ( B(T 0, T j )) with f : (0, ) R and f (s) = s z Π(dz) for some finite complex measure Π on a strip {z C : R Re(z) R} Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 30/42

Determination (H t (z)) t [0,T0 ] := E P T 0 [ B(T 0, T j ) z F t ] Optimal number of risky assets related to claim H T0 (z) for every t [0, T 0 ]: ξ j t(z) = d H(z), B(, T j ) P T0 t d B(, T j ), B(, T j ) P T 0 t = ξ j t = ξ j t(z)π(dz) Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 31/42

Determination Lemma PS T0 = n k=1 can be expressed as PS T0 = with Π(du) = n k=1 1 c k e (iu R)X T 0 ˆv k ( u ir)du 2π R R B(T 0, T j ) iu R Σ T j T 0 Π(du), iu R c k 2π (f j Σ T 0 ) T j T 0 ˆv k ( u ir)du, f j T 0 = B(0, T 0) B(0, T j ) exp( T0 0 [θ s (Σ(s, T j )) θ s (Σ(s, T 0 ))]ds). Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 32/42

Determination Theorem If additionally 3M M and if R is chosen in ]0, M 2σ 1 ] GKW-decomposition of the PS exists. Optimal number ξ j t to invest in B(, T j ) is according to the MVH strategy given by R e T0 t κ X j s ( iu R iu R Σ T )ds j Σ T 0 B(t, Tj ) T j T 0 1 κ X j t j +1) κ X t T 0 ( iu R Σ T j κ X j t (2) ( iu R Σ T j T 0 ) Π(du), with Π(du) as in previous lemma and with for w c = 1 w κ X j s (w)=θ s(wσ(s,t j )+w c Σ(s,T 0 )) wθ s(σ(s,t j )) w c θ s(σ(s,t 0 )), Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 33/42

Outline 1 Introduction 2 Pricing of swaption 3 Hedging of swaption 4 Numerical results Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 34/42

Numerical results Receiver swaption Normal Inverse Gaussian Vasiček volatility structure a(t s) σ(s, T ) = ˆσe Maturity in 10 years Tenor=10 years Two payments/year hedging swap 0 10 20 Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 35/42

Hedging strategies B(, T 1 ) B(, T 10 ) B(, T 20 ) Delta 9.51 (0.77) 3.02 (0.24) 2.30 (0.22) Delta-gamma 87.93 (5.78) 35.19 (2.63) 30.01 (2.64) MVH 4.36 (0.40) 3.88 (0.39) 3.28 (0.38) Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 36/42

Delta- and gamma hedging a = 0.02 a = 0.06 δ = 0.1 30.01 (2.64) 20.92 (1.80) δ = 0.06 17.68 (1.53) 12.32 (1.07) Characteristic function of the NIG model φ(z) = exp( δ( α 2 (β + iz) 2 α 2 β 2 )), Vasiček volatility model a(t s) σ(s, T ) = e Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 37/42

Hedging strategies B(, T 1 ) B(, T 10 ) B(, T 20 ) Delta 9.51 (0.77) 3.02 (0.24) 2.30 (0.22) Delta-gamma 87.93 (5.78) 35.19 (2.63) 30.01 (2.64) MVH 4.36 (0.40) 3.88 (0.39) 3.28 (0.38) Full risk: 3.29 (0.41) Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 38/42

Intentions 6 3.6 5 4 3 2 1 0 0 2 4 6 8 10 Delta-hedge 3.4 3.2 3 2.8 2.6 2.4 2.2 2 1.8 0 2 4 6 8 10 Mean-variance hedge Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 39/42

B(, T 20 ) 12 0.2957 11 10 9 0.2956 8 7 6 0.2956 5 4 3 0 2 4 6 8 10 Delta-hedge on average 0.2955 0 2 4 6 8 10 MVH on average Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 40/42

B(, T 0 ) 4 3.13 2 0 2 4 6 8 0 2 4 6 8 10 Delta-hedge on average 3.128 3.126 3.124 3.122 3.12 3.118 3.116 3.114 0 2 4 6 8 10 MVH on average Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 41/42

Thank you for your attention This study was supported by a grant of Research Foundation-Flanders Nele Vandaele Hedging of swaptions in a Lévy driven Heath-Jarrow-Morton framework 42/42