Evaluation of Credit Value Adjustment with a Random Recovery Rate via a Structural Default Model

Transcription

1 Evaluation of Credit Value Adjustment with a Random Recovery Rate via a Structural Default Model Xuemiao Hao and Xinyi Zhu University of Manitoba August 6, 2015 The 50th Actuarial Research Conference University of Toronto Xuemiao Hao (University of Manitoba) CVA with a random recovery rate August 6, / 19

2 Credit value adjustment Consider a financial derivative instrument held between two parties, the institution and its counterparty. Risky value = risk-free value CVA CVA (1 R) K k=1 (1 R): the loss given default DF(t k ) EE(t k ) PD(t k 1, t k ) DF(t k ): the risk-free discount factor for time t k EE(t k ): the expected risk exposure for the institution at t k PD(t i 1, t i ): the default probability between dates t i 1 and t i See, for example, Grogery (2012, Chapter 12) and Feng and Volkmer (2012). Xuemiao Hao (University of Manitoba) CVA with a random recovery rate August 6, / 19

3 Denoting by τ the default time of the counterparty, we propose CVA Assumptions: K k=1 [ DF(t k ) EE(t k ) E The institution itself cannot default. Risk-free valuation can be performed. ] (1 R τ )1 {tk 1 <τ t k }. (1) }{{} main focus The credit exposure is independent of default probability and loss given default. Xuemiao Hao (University of Manitoba) CVA with a random recovery rate August 6, / 19

4 The Lévy first-passage model Under the risk-neutral setting: The counterparty s asset process V t, t 0, follows V t = V 0 e Z t, where Z t is a spectrally negative Lévy process. E(V t ) = V 0 e rt with r the constant interest rate. For a threshold level L < V 0, default time is defined as τ = inf {t : V t L} = inf {t : ln(v 0 /L) + Z t 0}. Xuemiao Hao (University of Manitoba) CVA with a random recovery rate August 6, / 19

5 Denote X t = ln(v 0 /L) + Z t. τ is the ruin time of process X t starting from ln(v 0 /L). X τ specifies the default severity. Assume R τ = R( X τ ), where R( ) [0, 1] is loss settlement function which is non-increasing defined on [0, ). For instance, R τ = V τ /V 0 = (L/V 0 ) e X τ (0, L/V 0 ). The idea of introducing a loss settlement function can also be found in Tang and Yuan (2013) and van Damme (2011). Xuemiao Hao (University of Manitoba) CVA with a random recovery rate August 6, / 19

6 Some remarks of the model: According to the empirical study by Carr et al. (2002), risk-neutral processes for equity prices should be processes of infinite activity and finite variation. A firm s asset value is exposed to shocks (represented by negative jumps), which is the main concern in risk management practice. The process Z t should be such that the expectation in CVA (1) can be efficiently calculated with enough accuracy. Xuemiao Hao (University of Manitoba) CVA with a random recovery rate August 6, / 19

7 Meromorphic Lévy processes We assume Z t = µt S t with µ > 0 and S t a pure-jump meromorphic process with only upward jumps. Lévy measure of Z t : Π(dx) = π(x)dx, where π(x) = 1 {x<0} m=1 b m e ρ mx, b m > 0, ρ m > 0. Laplace exponent of Z t : 0 ψ(s) := ln E(e sz 1 ) = µs + = µs + m=1 (e sx 1) Π(dx) b m ( 1 ρ m + s 1 ρ m ). Xuemiao Hao (University of Manitoba) CVA with a random recovery rate August 6, / 19

8 Examples: Z t is an θ-process if Now b m = 2 π cβm2, ρ m = β(α + m 2 ), c, α, β > 0. ψ(s) = µs c α + s/β coth ( π ) α + s/β + c α coth(π α). Z t is a β-process with λ (1, 2) if ( ) m + λ 2 b m = cβ, ρ m = β(α + m), c, α, β > 0. m 1 Now ψ(s) = µs + cb(1 + α + s/β, 1 λ) cb(1 + α, 1 λ). See Kuznetsov (2010a, b) for properties of these processes. Xuemiao Hao (University of Manitoba) CVA with a random recovery rate August 6, / 19

9 Generalized expected discounted penalty function Consider the process X t = x + Z t, with x 0. Definition 1 The generalized expected discounted penalty function (EDPF) of X t is φ(x; r) := E [ e rτ w( X τ, X τ, X τ )1 {τ< } X 0 = x ], and the generalized finite-time EDPF of X is φ t (x; r) := E [ e rτ w( X τ, X τ, X τ )1 {τ<t} X 0 = x ], with r 0 and w a bounded measurable function on R 3 + = [0, ) 3. Biffis and Morales (2010) and Kuznetsov and Morales (2014) have introduced the generalized EDPF into actuarial literature. Xuemiao Hao (University of Manitoba) CVA with a random recovery rate August 6, / 19

10 Let us go back to the main focus in CVA (1). [ ] E (1 R τ )1 {tk 1 <τ t k } ( ) ( ) = E 1 {tk 1 <τ t k } E R τ 1 {tk 1 <τ t k } = Q (τ t k ) Q (τ t k 1 ) [ ) (L/V 0 ) E E ( e X τ 1 {τ tk } ( )] e X τ 1 {τ tk 1 }. In terms of generalized EDPF, Q(τ t) = φ t (x; 0) with w( X τ, X τ, X τ ) = 1 and ) E (e X τ 1 {τ t} = φ t (x; 0) with w( X τ, X τ, X τ ) = e X τ. Xuemiao Hao (University of Manitoba) CVA with a random recovery rate August 6, / 19

11 Note that the Laplace transform of φ t becomes φ: 0 e qt φ t (x; r)dt = φ(x; r + q). q and 0 e qt Q (τ t) dt = q 1 c n e ζ nx n 1 0 where ) e qt E (e X τ 1 {τ<t} dt = Φ(q) q 2 m,n 1 c n ζ n b m e ζ nx (Φ(q) + ρ m )(ρ m ζ n )(ρ m + 1), { ζ n : n = 1, 2,...} are negative simple roots of ψ(z) = q, Φ(q) is the unique positive root of ψ(z) = q, c n = (1/ζ n + 1/Φ(q)) q/ψ ( ζ n ). Xuemiao Hao (University of Manitoba) CVA with a random recovery rate August 6, / 19

12 Computing the finite-time EDPF By an inverse Laplace transform based on the Gaver-Stehfest algorithm, the function φ t (x; q) is approximated by with M N large and φt GS 2M a n (x; q; M) = n φ(x; q + nt 1 ln 2) n=1 a n = ( 1) M+n n M j= (n+1)/2 j M+1 M! ( M j )( 2j j )( j n j ). Xuemiao Hao (University of Manitoba) CVA with a random recovery rate August 6, / 19

13 Numerical experiments Using the method of valuation in terms of bond prices, which has been discussed in Hull (2015), we study the following interest rate swap. Table: Hypothetical interest rate swap Effective date Termination date Notional principal Payment dates Fixed-rate payer Fixed rate Floating-rate payer Floating rate 18-Mar Mar-2018 USD 100 million From 18-Sep-2015 to and including 18-Mar-2018 Institution per annum Counterparty USD 6-month LIBOR a The fixed rate can be determined by setting the present value of the fixed rate payments equal to the present value of the floating rate payments. b All rates are quoted nominal continuously. Xuemiao Hao (University of Manitoba) CVA with a random recovery rate August 6, / 19

14 Table: Forward LIBOR rates Date Futures price Futures rate Convexity Forward rate adjustment 18-Mar Jun Sep Dec Mar Jun Sep Dec Mar Jun Sep Dec Mar a The futures prices and the volatility of the short rate (0.928%) are retrieved from the Bloomberg. Xuemiao Hao (University of Manitoba) CVA with a random recovery rate August 6, / 19

15 Table: LIBOR zero rates Date 6 m 12 m 18 m 24 m 30 m 36 m 18-Mar Sep Mar Sep Mar Sep Xuemiao Hao (University of Manitoba) CVA with a random recovery rate August 6, / 19

16 Table: Value of the interest rate swap Date B fixed B float EE 18-Mar Sep Mar Sep Mar Sep Xuemiao Hao (University of Manitoba) CVA with a random recovery rate August 6, / 19

17 If the counterparty is American Express Co: 1-y 2-y 3-y 5-y 7-y 10-y Market CDS Calibrated CDS a The calibrated process is a θ-process with c = 3.521, α = 1.24, β = 4.3, λ = 1.5. CVA fixed R = (1 R) = K k=1 DF(t k ) EE(t k ) PD(t k 1, t k ) K [ ] CVA random R = DF(t k ) EE(t k ) E (1 R τ )1 {tk 1 <τ t k } k=1 = % higher! Xuemiao Hao (University of Manitoba) CVA with a random recovery rate August 6, / 19

18 References Biffis, E.; Morales, M. (2010). On a generalization of the Gerber-Shiu function to path-dependent penalties. Insurance: Mathematics & Economics 46, no. 1, Carr, P.; Geman, H.; Madan, D. B.; Yor M. (2002). The fine structure of asset returns: an empirical investigation. Journal of Business 75, no. 2, Feng, R.; Volkmer, H.W. (2012). Modeling credit value adjustment with downgrade-triggered termination clause using a ruin theoretic approach. Insurance: Mathematics & Economics 51, no. 2, Gregory, J. (2012). Counterparty credit risk and credit value adjustment: a continuing challenge for global financial markets. John Wiley & Sons, UK. Hull, J.C. (2015). Options, futures, and other derivatives (9th edition). Pearson, US. Kuznetsov, A. (2010a). Wiener-Hopf factorization and distribution of extrema for a family of Lévy processes. Annals of Applied Probability 20, no. 5, Kuznetsov, A. (2010b). Wiener-Hopf factorization for a family of Lévy processes related to theta functions. Journal of Applied Probability 47, no. 4, Kuznetsov, A.; Morales, M. (2014). Computing the finite-time expected discounted penalty function for a family of Lévy risk processes. Scandinavian Actuarial Journal, no.1, Tang, Q.; Yuan, Z. (2013). Asymptotic analysis of the loss given default in the presence of multivariate regular variation. North American Actuarial Journal 17, no. 3, van Damme, G. (2011). A generic framework for stochastic Loss-Given-Default. Journal of Computational and Applied Mathematics 235, no. 8, Xuemiao Hao (University of Manitoba) CVA with a random recovery rate August 6, / 19

19 Thank you! Xuemiao Hao (University of Manitoba) CVA with a random recovery rate August 6, / 19