Negative Swap Spreads and Limited Arbitrage

Transcription

1 Negaive Swap Spreads and Limied Arbirage Urban Jermann Wharon School, Universiy of Pennsylvania

2 Ineres rae swap Pay LIBOR, ge fixed rae periodically Swap spread Fixed swap rae minus Treasury of same mauriy Why a spread? LIBOR vs TBill rae Defaul/Liquidiy Limis o arbirage

3 Swap spreads year 5 year 10 year 20 year 30 year Jan98 Jan99 Jan00 Jan01 Jan02 Jan03 Jan04 Jan05 Jan06 Jan07 Jan08 Jan09 Jan10 Jan11 Jan12 Jan13 Jan14 Jan15 Jan16 Jan17 Jan18

4 This paper Develops model wih limied arbirage for swaps Analyically, wih srong fricions negaive swap spread Quaniaive model wih negaive swap spreads Addiional empirical suppor for model

5 Lieraure Swap spreads: Liu, Longsaff & Mandell (2006), Feldhueer & Lando (2008), Hanson (2014), Gupa & Subrahmanyam (2000), Collin-Dufresne and Solnik (2001), Eom, Subrahmanyam & Uno (2002), Johannes & Sundaresan (2007), Smih (2015), Klinger & Sundaresan (2016) Limied arbirage: Shleifer & Vishny (1997), Dow & Goron (1994), Garleanu, Pedersen & Poeshman (2009), Gabaix, Krishnamurhy & Vigneron (2007), Vayanos & Vila (2009), Liu & Longsaff (2004), Tuckman & Vila (1992), Gromb & Vayanos (2010) Effecs of recen financial regulaion: Duffi e (2016), Du, Tepper & Verdelhan (2016), Boyarchenko e al. (2018) Firs limied arbirage model for swaps

6 Model ouline Dealer selecs bonds and swaps Bond prices exogenous, swap price endogenous Holding coss for bonds Laer exensions: Capial requiremens, demand effecs, swap holding coss

7 Model Shor-erm riskless deb q ST (z) = exp ( y ST (z)) LIBOR deb q LIB (z) = exp ( (y ST (z) + θ (z))) wih θ he TED spread Long-erm deb pays c LT + λ + (1 λ) q LT ( z ) wih price and yield q LT (z) = wih erm spread τ (z) c LT + λ exp (y LT (z)) 1 + λ y LT (z) = y ST (z) + τ (z)

8 Swap Swap pays (o fixed receiver) 1 c Sw ( 1) + (1 λ) m q LIB (z) wih m value nex period For new swap, find c Sw so ha and is he swap rae m (z, ω) = 0, y Sw (z, ω) = c Sw If zero ne demand for swaps, can creae a-marke swap and y Sw for each (z, ω). Oherwise assume here is only one swap.

9 Fricions The cos for holding long-erm deb is given by j ( α LT ) κ = LT 2 wih α LT amoun of long-erm deb Similarly for shor-erm deb h ( α ST ) κ = ST 2 ( α LT ) 2 ( α ST ) 2

10 Dealer s problem V (ω, z) = max u (c) + β (ω, z) E ( V ( ω, z )) c,α ST,α LT,s subjec o c = ω α ST q ST (z) α LT q LT (z) s m h ( α ST ) ( ) j α LT and ω = α ST e µ(z ) + α [ ( LT clt + e µ(z λ + (1 λ) q ) LT z )] [ ( ) ] + s 1 e µ(z c ) Sw q LIB (z) 1 + (1 λ) m +π ( z )

11 Equilibrium Swap marke clears s = d (z) Firs-order condiions h 1 (α ST ) = βe u 1(c ) u 1 (c)e µ(z ) q ST ( ) j 1 (α LT ) = βe u 1 (c ) u 1 (c)e µ(z ) [c LT + λ + (1 λ) q LT (z )] ( m = βe u 1 (c ) u 1 (c)e µ(z ) [ c Sw ( 1 q LIB (z) 1 ) + (1 λ) m ]) q LT

12 No-arbirage case m = E ( Λ+1 Λ [ ( 1 c Sw q ST Assume Λ prices all bonds ) ]) 1 θ + (1 λ) m +1 Then m = ( ) c Sw + λ Ω ({1}) 1 Ω ({θ }) wih Ω ({x }) = (1 λ) j Λ +1+j E x +j. Λ j=0

13 (A-marke) swap rae y Sw 0 =, ( ) y Sw + λ y Sw = 1 + Ω ({θ }) λ Ω (1) Ω (1) 1 Ω ({θ }) Price of long-erm bond ( ) q LT = c LT + λ Ω (1) Yield of long-erm bond y LT 1 = y LT =, ( ) y LT + λ Ω (1) 1 Ω (1) λ

14 No-arbirage case If θ = θ hen If θ 0, y Sw y LT = Ω ({θ }) Ω (1) y Sw y Sw y LT = θ y LT 0

15 Swap pricing wih very srong fricions Assume holding cos parameer, κ LT and κ ST, very large Consan endowmen of oher profis, π (z) = π Zero-ne demand for swaps, d (z) = 0 No-inflaion uncerainy, µ (z) = µ Then, swap price equals m = βe ([ c Sw y LIB + (1 λ) m +1 ]) Seing m = 0 defines he swap rae as y Sw ( ) = w j E y+j 1 LIB j=1

16 Taking uncondiional expecaion ( ) ( ) ( E = E = E y Sw y LIB y ST ) + E (θ ) Bond yield ( E y LT ) ( = E y ST ) + E (τ ) Swap spread ( E y Sw ) ( E y Sw y LT ) = E (θ ) E (τ ) Hisorically, E (θ ) = 0.6% and E (τ ) = 1.7%, so ha ( ) ( ) E E = 1.1% y LT

17 Swap pricing wih fricions from model Dealer s firs-order condiions q ST = E ( Λ+1 Λ ) κ ST α ST +1 q LT ( Λ+1 [ ] ) = E c LT + λ + (1 λ) q+1 LT κ LT α LT +1 Λ Implied swap pricing y Sw y LT = Ω ({θ }) Ω + Ω (1) Term spread, q LT ( κ ST {α ST +1} κ LT q LT Ω (1), { α LT +1}, Swap spread { } ) Λ Λ α LT +1 +1

18 Calibraion VAR(1) for four dimensional sae vecor Transiion: Var-Cov: 10 6 [y ST (z), τ (z), µ (z), θ (z)]

19 Calibraion Symbol Parameer Value ȳ ST Shor rae level τ Term spread level µ Inflaion level θ TED spread level γ Risk aversion 2 ν Discoun elasiciy 1 1/λ Mauriy of long-erm deb and swap 120

20 Quaniaive resuls 30Y SS κ LT TED E() Sd() E(α LT ) E() Daa 7/1997 9/ / / Model κ LT = κ LT = κ LT = Pos 10/2008 TED and κ LT = High ri. av., γ = Low disc.elas., ν = ST deb cos, κ ST = Consan TED Consan Inflaion

21 Holding cos esimae Enhanced Supplemenary Leverage Raio (US G-SIB) of 5% or 6% Wih κ ST = 0, ( E y Sw 5% 10% = 50, or 60 basis poins. +E wih χ 1 ) y LT = Eθ κ LT Eα LT +1 χ Ω ({θ E θ } κ LT q LT { Λ Λ (α LT E αlt +1) }) Ω (1)

22 Demand effecs and Swap holding coss Swap Demand LT bond Swap cos rae sensiiviy posiion impac Benchmark (κ LT = ) d = d = d =.2, κ Sw = Low fricion (κ LT = ) d = d = d =.2, κ Sw = High fricion (κ LT = 0.01) d = d = d =.2, κ Sw =

23 Leverage consrain max (α ST q ST (z), 0) + max (α LT q LT (z), 0) ξ ω E(30Y SS) freq(ss<0) freq(η > 0) ξ = ξ = ξ = ξ =

24

25 10/ : Swap Mauriy TERM TED MBSD MTB VIX R2adj

26 : Swap Mauriy TERM TED MBSD MTB VIX R2adj

27 Conclusion Model for swaps wih limied arbirage Wih reasonable fricions on bond holdings ge negaive swap spreads Negaive swap spreads even wihou explici demand effecs