Model Risk Embedded in Yield-Curve Construction Methods

Model Risk Embedded in Yield-Curve Construction Methods Areski Cousin ISFA, Université Lyon 1 Joint work with Ibrahima Niang Bachelier Congress 2014 Brussels, June 5, 2014 Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 1/35

Introduction What is understood as a yield-curve in this presentation? Term-structure construction consists in finding a function T P(t 0, T ) given a small number of market quotes S 1,..., S n Market information only reliable for a small set of liquid products with standard characteristics/maturities We have to rely on interpolation/calibration schemes to construct the curve for missing maturities Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 2/35

Introduction Andersen (2007), curves based on tension splines Le Floc h (2012), examples of one-day forward curves Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 3/35

Introduction What is a good yield curve construction method? (Hagan and West (2006)) Ability to fit market quotes Arbitrage freeness Smoothness Locality of the interpolation method Stability of forward rate Consistency of hedging strategies : Locality of deltas? Sum of sequential deltas close enough to the corresponding parallel delta? (Le Floc h (2012)) Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 4/35

Market-fit condition At time t 0, the term-structure T P(t 0, T ) is built from market quotes of standard products n : number of products S = (S 1,..., S n) : set of market quotes at t 0 T = (T 1,..., T n) : corresponding set of increasing maturities t = (t 1,, t m) : payment time grid The two time grids T and t coincide at indices p i such that t pi = T i Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 5/35

Market-fit condition Let P = (P(t 0, t 1),..., P(t 0, t m)) be the vector formed by the values of the curve at payment dates t 1,..., t m Assumption : Linear representation of present values Presents values of products used in the curve construction have a linear representation with respect to P For i = 1,..., n where p i k=1 A ik P(t 0, t k ) = B i A = (A ij ) is a n m matrix with positive coefficients B = (B i ) is a n 1 matrix with positive coefficients A and B only depend on current market quotes S, on standard maturities T, on payment dates t and on products characteristics. Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 6/35

Market-fit condition Market-fit condition The market-fit condition can be restated as a rectangular system of linear equations A P = B where P = (P(t 0, t 1),..., P(t 0, t m)) A is a n m matrix with positive coefficients B is a n 1 matrix with positive coefficients A and B only depend on current market quotes S, on standard maturities T, on payment dates t and on products characteristics. Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 7/35

Market-fit condition Example 1 : Corporate or sovereign debt yield curve S i : market price (in percentage of nominal) at time t 0 of a bond with maturity T i c i : fixed coupon rate t 1 <... < t pi = T i : coupon payment dates, δ k : year fraction of period (t k 1, t k ) p i c i k=1 δ k P B (t 0, t k ) + P B (t 0, T i ) = S i where P B (t 0, t k ) represents the price of a (fictitious default-free) ZC bond with maturity t k Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 8/35

Market-fit condition Example 2 : Discounting curve based on OIS S i : par rate at time t 0 of an overnight indexed swap with maturity T i t 1 < < t pi = T i : fixed-leg payment dates (annual time grid) δ k : year fraction of period (t k 1, t k ) S i p i 1 k=1 δ k P D (t 0, t k ) + (S i δ pi + 1) P D (t 0, T i ) = 1, i = 1,..., n where P D (t 0, t k ) is the discount factor associated with maturity date t k Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 9/35

Market-fit condition Example 3 : credit curve based on CDS S i : fair spread at time t 0 of a credit default swap with maturity T i t 1 < < t p = T i : premium payment dates, δ k : year fraction of period (t k 1, t k ) R : expected recovery rate of the reference entity p i S i k=1 δ k P D (t 0, t k )Q(t 0, t k ) = (1 R) Ti t 0 P D (t 0, u)dq(t 0, u) where u Q(t 0, u) is the F t0 -conditional (risk-neutral) survival distribution of the reference entity. We implicitly assume here that recovery, default and interest rates are stochastically independent. Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 10/35

Market-fit condition Example 3 : credit curve based on CDS (cont.) Using an integration by parts, the survival function u Q(t 0, u) satisfies a linear relation : p i S i k=1 δ k P D (t 0, t k )Q(t 0, t k ) + (1 R)P D (t 0, T i )Q(t 0, T i ) + (1 R) Ti t 0 f D (t 0, u)p D (t 0, u)q(t 0, u)du = 1 R where f D (t 0, u) is the instantaneous forward (discount) rate associated with maturity date u. Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 11/35

Arbitrage-free bounds We studied two types of curves : Interest-rate curves : P = P B (price of zero-coupon bond), P = P D (discount factors) Credit curves : P = Q (risk-neutral survival probability) Arbitrage-free condition A curve T P(t 0, T ) is said to be arbitrage-free if the two following conditions hold P(t 0, t 0) = 1 T P(t 0, T ) is a non-increasing function Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 12/35

Arbitrage-free bounds Market fit condition : p 1 k=1 Arbitrage-free inequalities : A ik P(t 0, t k ) + + p i k=p i 1 +1 A ik P(t 0, t k ) = B i P(t 0, T 1) P(t 0, t k ) 1 for 1 k p 1. P(t 0, T i ) P(t 0, t k ) P(t 0, T i 1 ) for p i 1 + 1 k p i 1 Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 13/35

Arbitrage-free bounds Proposition (arbitrage-free bounds) For i = 1,..., n, where P min (t 0, T i ) = 1 P max(t 0, T i ) = P min (t 0, T i ) P(t 0, T i ) P max(t 0, T i ) A ipi i 1 B i H ij P(t 0, T j 1 ) (H ii A ipi )P(t 0, T i 1 ) j=1 1 i 1 B i H ij P(t 0, T j ) H ii j=1 and where H ij := p j k=p j 1 +1 A ik Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 14/35

Arbitrage-free bounds Iterative computation of model-free bounds Step 1 : where Step 2 : For i = 2,..., n, where P min (t 0, T i ) = 1 A ipi P max(t 0, T i ) = P min (t 0, T 1) P(t 0, T 1) P max(t 0, T 1) P min (t 0, T 1) = 1 A 1p1 (B 1 (H 11 A 1p1 )) P max(t 0, T 1) = B1 H 11 P min (t 0, T i ) P(t 0, T i ) P max(t 0, T i ) i 1 B i H ij Pmax(t 0, T j 1 ) (H ii A ipi ) P max(t 0, T i 1 ) j=1 1 i 1 B i H ij Pmin (t 0, T j ) H ii j=1 Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 15/35

Arbitrage-free bounds : OIS discount curves We consider OIS par rates as of t 0 = May 31st 2013 Market quotes S available for n = 14 maturities T = (1y, 2y,..., 10y, 15y, 20y, 30y, 40y) t = (1y, 2y,..., 10y, 11y,..., 40y) : payment time grid A is a 14 40 rectangle matrix, B is a 14 1 column vector We are looking for bounds on OIS discount factors P D (t 0, T i ), i = 1,..., n Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 16/35

Arbitrage-free bounds : OIS discount curve Bounds for OIS discount factors P D (t 0, T i ) are sharp 1 Bounds on OIS discount factors 0.9 OIS discount factor 0.8 0.7 0.6 0.5 0.4 0 5 10 15 20 25 30 35 40 time to maturity Input data : OIS swap rates as of May, 31st 2013 Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 17/35

Arbitrage-free bounds : OIS discount curves Bounds for the associated discount rates 3.5 Bounds on OIS discount rates 3 OIS spot rate (percentage) 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 30 35 40 time to maturity Input data : OIS swap rates as of May, 31st 2013, 1 T log(pd (t 0, T )) Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 18/35

Arbitrage-free bounds : OIS discount curves Range of arbitrage-free market-consistent OIS discount curves 1 Bounds for OIS discounting curves 0.9 OIS discount factors 0.8 0.7 0.6 0.5 0.4 0 5 10 15 20 25 30 35 40 time to maturity Input data : OIS swap rates as of May, 31st 2013 Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 19/35

Arbitrage-free bounds : CDS-implied default curves We consider AIG CDS spreads as quoted at t 0 = Dec 17, 2007 Market quotes S available for n = 4 maturity times T = (1y, 3y, 5y, 10y) t = the whole time interval (0, 10y) A is a 4 rectangle matrix (the present value of CDS protection legs involves an integral instead of a sum) B is a 4 1 column vector We are looking for bounds on risk-neutral survival probabilities Q(t 0, T i ), i = 1,..., n Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 20/35

Arbitrage-free bounds : CDS-implied default curves Range of market-consistent survival curves 1 Bounds for CDS implied survival probabilities 0.99 CDS implied survival probability 0.98 0.97 0.96 0.95 0.94 0.93 0.92 0.91 0 1 2 3 4 5 6 7 8 9 10 time to maturity Input data : CDS spreads of AIG as of December 17, 2007, R = 40%, P D (t 0, t) = exp( 3%(t t 0)) Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 21/35

How to construct admissible yield curves? Mean-reverting term-structure models as generators of admissible yield curves The risk-neutral dynamics of short-term interest rates (or default intensities) is assumed to follow either a OU process driven by a Lévy process dx t = a(b(t; p, T, S) X t)dt + σdy ct, where Y is a Lévy process with cumulant function κ and parameter set p L or an extended CIR process dx t = a(b(t; p, T, S) X t)dt + σ X tdw t, where W is a standard Browian motion Depending on the context, p = (X 0, a, σ, c, p L ) will denote the parameter set of the Lévy-OU process and p = (X 0, a, σ) the parameter set of the CIR process Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 22/35

How to construct admissible yield curves? In both cases, b is represented by a step function : b(t; p, T, S) = b i (p, T, S) for T i 1 < t T i, i = 1,..., n The vector b = (b 1,..., b n) solves a triangular system of non-linear equations. Market-fit linear conditions The rectangular market-fit system translates into a triangular system of non-linear equations A P(b) = B where P(b) = (P(t 0, t k ; b)) k=1,...,m is the m 1 vector of discount factors, ZC bond price or survival probabilities (depending on the context). A is a n m matrix, B is a n 1 matrix A and B only depend on current market quotes S, on standard maturities T and on products characteristics. Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 23/35

How to construct admissible yield curves? Proposition (Discount factors in the Lévy-OU approach) Let T i 1 < t T i. In the Lévy-OU model, the current value of the discount factor or of an assimilated quantity with maturity time t is given by [ ( t )] P(t 0, t; b) := E exp X udu = exp ( I (t 0, t, b)) t 0 where i 1 I (t 0, t, b) := X 0φ(t t 0) + b k (ξ(t T k 1 ) ξ(t T k )) k=1 and functions φ, ξ and ψ are defined by + b i ξ(t T i 1 ) + cψ(t t 0) φ(s) := 1 ( 1 e as ) a (1) ξ(s) := s φ(s) ψ(s) := s 0 κ ( σφ(s θ)) dθ Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 24/35

How to construct admissible yield curves? Proposition (Discount factors in the CIR approach) Let T i 1 < t T i. In the CIR model, the current value of the discount factor or of an assimilated quantity with maturity time t is given by [ ( t )] P(t 0, t; b) := E exp X udu = exp ( I (t 0, t, b)) t 0 where i 1 I (t 0, t, b) := X 0ϕ(t t 0) + b k (η(t T k 1 ) η(t T k )) + b i η(t T i 1 ) k=1 and functions ϕ and η are defined by where h := a 2 + 2σ 2 2(1 e hs ) ϕ(s) := (2) h + a + (h a)e hs [ s η(s) := 2a h + a + 1 ] h + a + (h a)e hs log σ2 2h Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 25/35

How to construct admissible yield curves? Construction of (b 1,..., b n) by a bootstrap procedure For any i = 1,..., n, the present value of the instrument with maturity T i only depends on b 1,..., b i is a monotonic function with respect to b i The vector b = (b 1,..., b n) satisfies a triangular system of non-linear equations that can be solved recursively : Find b 1 as the solution of p 1 j=1 A 1j P(t 0, t j ; b 1) = B 1 Assume b 1,..., b k 1 are known, find b k as the solution of p k j=1 A kj P(t 0, t j ; b 1,..., b k ) = B k Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 26/35

How to construct admissible yield curves? Proposition (smoothness condition) A curve t P(t 0, t) constructed from the previous approach is of class C 1 and the corresponding forward curve (or default density function) is continuous. Proof : Let b( ) be a deterministic function of time, instantaneous forward rates are such that Lévy-driven OU f (t 0, t) = X 0e a(t t 0) + a where φ is defined by (1) extended CIR t f CIR (t 0, t) = X 0ϕ (t t 0) + a where ϕ is the derivative of ϕ given by (2) t 0 e a(t u) b(u)du cκ( σφ(t t 0)) t t 0 ϕ (t u)b(u)du Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 27/35

How to construct admissible yield curves? Assume that a curve has been constructed from a Lévy-OU term-structure model with positive parameters (X 0, a, σ, c, p L ) : i 1 f (t 0, t) = X 0e a(t t0) + a b k (φ(t T k 1 ) φ(t T k )) for any T i 1 t T i, i = 1,..., n. k=1 + ab i φ(t T i 1 ) cκ( σφ(t t 0)) Proposition (arbitrage-free condition in the Lévy-OU approach) Assume that the derivative of the Lévy cumulant κ exists and is strictly monotonic on (, 0). The curve is arbitrage-free on the time interval (t 0, T n) if and only if, for any i = 1,..., n, f (t 0, T i ) > 0 and one of the following condition holds : f t (t0, T i 1) f t (t0, T i ) 0 f t (t0, T i 1) f t (t0, T i ) < 0 and f (t 0, t i ) > 0 where t i is such that f t (t0, t i ) = 0, Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 28/35

How to construct admissible yield curves? Assume that a curve has been constructed from an extended CIR term-structure model with positive parameters (X 0, a, σ) : i 1 f CIR (t 0, t) = X 0ϕ (t t 0)+a b k (ϕ(t T k 1 ) ϕ(t T k ))+ab i ϕ(t T i 1 ) k=1 for any T i 1 t T i, i = 1,..., n. Proposition (arbitrage-free condition in the CIR approach) The constructed curve is arbitrage-free if, for any i = 1,, n, the implied b i is positive Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 29/35

How to construct admissible yield curves? Set of admissible OIS discount and forward curves : Lévy-OU short rates Parameters : a = 0.01, σ = 1, X 0 = 0.063% (fair rate of IRS vs OIS 1M). The Lévy driver is a Gamma subordinator with parameter λ = 1/50bps (mean jump size of 50 bps). c = {1, 10, 20,..., 100} Set of admissible discounting curves and associated forward curves 3 OIS discount rate (percentage) 2.5 2 1.5 1 0.5 0 0 5 10 15 20 25 30 35 40 time to maturity (year) Input data : OIS swap rates as of May, 31st 2013 Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 30/35

How to construct admissible yield curves? Set of admissible survival curves : CIR intensities Parameters : a = σ = 1, 100 X 0 = {0.01, 0.25, 0.49, 0.73, 0.97, 1.21, 1.45, 1.69, 1.94, 2.18, 2.42} 1 Set of admissible survival curves 0.99 0.98 survival probability 0.97 0.96 0.95 0.94 0.93 0.92 0.91 0 1 2 3 4 5 6 7 8 9 10 time to maturity Input data : CDS spreads of AIG as of December 17, 2007, R = 40%, P D (t 0, t) = exp( 3%(t t 0)) Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 31/35

Perspectives The proposed framework could be extended or used in several directions : Yield-curve diversity impact on present values (PV) and hedging stategies? max PV (C i ) PV (C j ) i,j p where the max is taken over all couples of admissible curves (C i, C j ) Risk management in the presence of uncertain parameters? dx t = ã(b(t; ã, σ, T, S) X t)dt + σ X tdw t, where Range(ã, σ) {(a, σ) b(t; a, σ, T, S) 0 t} Extension to multicurve environments? Impact on the assessment of counterparty credit risk (CVA, EE, EPE,...)? Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 32/35

References (curve construction methods) Andersen, 2007, Discount curve construction with tension splines Ametrano and Bianchetti, 2009, Bootstrapping the illiquidity Chibane, Selvaraj and Sheldon, 2009, Building curves on a good basis Fries, 2013, Curves and term structure models. Definition, calibration and application of rate curves and term structure market models Hagan and West, 2006, Interpolation methods for curve construction Iwashita, 2013, Piecewise polynomial interpolations Jerassy-Etzion, 2010, Stripping the yield curve with maximally smooth forward curves Kenyon and Stamm, 2012, Discounting, LIBOR, CVA and funding : Interest rate and credit pricing Le Floc h, 2012, Stable interpolation for the yield curve Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 33/35

References (model risk) Branger and Schlag, 2004, Model risk : A conceptual framework for risk measurement and hedging Cont, 2006, Model uncertainty and its impact on the pricing of derivative instruments Davis and Hobson, 2004, The range of traded option prices Derman, 1996, Model risk Eberlein and Jacod, 1997, On the range of option prices El Karaoui, Jeanblanc and Shreve, 1998, Robustness of the Black and Scholes formula Green and Figlewski, 1999, Market risk and model risk for a financial institution writing options Hénaff, 2010, A normalized measure of model risk Morini, 2010, Understanding and managing model risk Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 34/35

Cumulant function of some Lévy processes Cumulant Brownian motion Gamma process Inverse Gaussian process κ(θ) = θ2 2 κ(θ) = log ( 1 θ λ ) κ(θ) = λ λ 2 2θ Areski Cousin, ISFA, Université Lyon 1 Model Risk Embedded in Yield-Curve Const. Meth. 35/35