Toward a coherent Monte Carlo simulation of CVA

Transcription

1 Toward a coherent Monte Carlo simulation of CVA Lokman Abbas-Turki (Joint work with A. I. Bouselmi & M. A. Mikou) TU Berlin January 9, 2013 Lokman (TU Berlin) Advances in Mathematical Finance 1 / 16

2 Plan Introduction Formulation Model families Conditional expectation using Malliavin Calculus Complexity and accuracy Lokman (TU Berlin) Advances in Mathematical Finance 2 / 16

3 Introduction Plan Introduction Formulation Model families Conditional expectation using Malliavin Calculus Complexity and accuracy Lokman (TU Berlin) Advances in Mathematical Finance 3 / 16

4 Introduction Formulation Unilateral default In a financial transaction between a party A that has to pay another party B some amount V, the CVA value is the price of the insurance contract that covers the default of party A to pay the whole sum V. CVA t,t = (1 R)E t ( V + τ 1 t<τ T ) (1) R is the recovery to make if the counterparty defaults (Assume R = 0), τ is the random default time of the counterparty, T is the protection time horizon. Numerical approximation N 1 ( ) CVA 0,T E V t + 1 τ (tk k+1,t k+1 ], (2) k=0 N the number of time steps used for SDEs discretization. References D. Brigo & al, Counterparty Credit Risk, Collateral and Funding: With Pricing Cases For All Asset Classes, John Wiley and Sons, G. Cesari & al, Modelling, Pricing and Hedging Counterparty Credit Exposure, Springer Finance, Lokman (TU Berlin) Advances in Mathematical Finance 4 / 16

5 Various kind of contracts Introduction Formulation Simulating assets S t = (St 1,..., Sd t ) trajectories then contracts trajectories to get V t as a sum: V t = ie φ exp ie (St) + ii φ eui ii (S t) + id φ eud id,t (St) + ia φ am ia,t (St), (3) where ie, ii, id and ia are the exposure indices and: φ exp explicit function, for example: φ exp (S tk ) = St 1 S 2 k t. k φ eui is a path-independent European contract, φ eui (S tk ) = E(f eui (S T ) S tk ). (4) φ eud t is a path-dependent European contract, φ eud t (S tk ) = E(f eud k t (S tk+1 ) S tk ), (5) k φ am t for example: ft eud (S tk+1 ) = ( max k i=0,..,k S1 t i St 1 S 2 k+1 t ) k+1 +. is an American contract, involving an optimal stopping problem φ am t (S tk ) = f (S tk ) E(φ am k t (S tk+1 ) S tk ) (6) k+1 with f an explicit payoff that generally does not depend on the asset path. Common problem ϕ(x) = E(f (Stk+1 ) S tk = x) and xi ϕ(x) with x i {x 1,..., x d }. (7) Lokman (TU Berlin) Advances in Mathematical Finance 5 / 16

6 Introduction Model families Assets model Starting model ds i t S i t = r i dt + i j=1 σ ij (t)dw j t, Si 0 = z i, i = 1,.., d, (8) r i are constants and σ(t) = {σ ij (t)} 1 i,j d is a deterministic triangular matrix, invertible, bounded and uniformly elliptic. Jump extension ds i t S i t = r i dt + i j=1 σ ij (t)dw j t + dji t, Si 0 = z i, i = 1,.., d, where J = (J 1,..., J d ) is a jump process independent from W. Then ( ϕ(x)=e E[f (S tk+1 ) σ ( ) ) (J u) 0 u t, Stk = x] S tk = x. Stochastic volatility extension ds i t S i t = r i dt + i j=1 σ ij (t, W t)dw j t, Si 0 = z i, i = 1,.., d, where W is a multidimensional Brownian motion correlated to W. Then ( ) ) ϕ(x)=e E[f (S tk+1 ) σ (( W u) 0 u t, S tk = x] S tk = x. Lokman (TU Berlin) Advances in Mathematical Finance 6 / 16

7 Introduction Model families Intensity models Hazard function t Λ(t) = λ sds, (9) 0 λ represents the intensity or hazard rate. ( P(τ > t) = E e ) t 0 λ s ds. (10) Hull & White example J. Hull and A. White, CVA and Wrong Way Risk, Financial Analysts Journal, 68(5): 58 69, F t-adapted hazard rate λ t. For f some "known" positive function: λ(t) = f (t, V t). (11) Our choice: λ t = a 1 + a 2 (V t) +, with a 2 0 and a 1 0. (12) Lokman (TU Berlin) Advances in Mathematical Finance 7 / 16

8 Introduction Model families Structural models Merton model Black & Cox extension R. Merton, On the pricing of corporate debt: The risk structure of interest rates, Journal of Finance, 3: , Merton s example assumes that default occurs if, at the final time T, the firm value X T is below a given threshold L which generally represents a promised terminal payoff. F. Black and J. C. Cox, Valuing corporate securities: Some effects of bond indenture provisions, Journal of Finance, 31: , τ = inf{t 0 X t L t}. (13) Our choice: L t constant. Lokman (TU Berlin) Advances in Mathematical Finance 8 / 16

9 Conditional expectation using Malliavin Calculus Plan Introduction Formulation Model families Conditional expectation using Malliavin Calculus Complexity and accuracy Lokman (TU Berlin) Advances in Mathematical Finance 9 / 16

10 Conditional expectation using Malliavin Calculus Define E b (R d ) = { f M(R d ) : C > 0 & m N; f (y) C(1 + y d ) m ) } Theorem 1 For every s (0, t), g E b (R d ) and x > 0 ( ) E g(s t) S s = x = Ts,t[g](x) T, (14) s,t[1](x) ( ) xi E g(s t) S s = x = Ri s,t [g](x)ts,t[1](x) Ts,t[g](x)Ri s,t [1](x) T s,t[1](x) 2, (15) ( ) with T s,t[f ](x) = E f (S t)γ s,t Ĥ x (S s), Ĥ x (S s) = d Hk x (Sk s ) S k, k=1 s ( f (St) ( and Rs,t i [f ](x) = E Ss i Ĥ x (S s) Γ s,t(1+π i,d s,t ) d k=1 Γ k s,t C k,i ) ). Lokman (TU Berlin) Advances in Mathematical Finance 10 / 16

11 Conditional expectation using Malliavin Calculus Γ s,t Expression Denote S 1,d the set of the second order permutations Defining for k {1,..., d} Then S 1,d = {p S 1,d, p p = Id}, (16) where S 1,d is the set of permutations on {1,..., d}, Id is the identity. d t π k,d s,t = 1 + ϕ jk (u)dwu j, ρ(u) = σ 1 (u), j=k 0 ϕ jk (u) = 1 s ρ jk(u)1 u (0,s) 1 t s ρ jk(u)1 u (s,t). Γ s,t = d ɛ(p) A i,p(i), ɛ(p) is the signature of p (17) p S i=1 1,d A = π 1,d s,t C 1,2 C 1,3 C 1,d 1 π 2,d s,t C 2,3 C 2,d π d 1,d s,t C d 1,d π d,d s,t ( ), C k,l = Cov π k,d s,t, πl,d s,t. Lokman (TU Berlin) Advances in Mathematical Finance 11 / 16

12 Complexity and accuracy Plan Introduction Formulation Model families Conditional expectation using Malliavin Calculus Complexity and accuracy Lokman (TU Berlin) Advances in Mathematical Finance 12 / 16

13 Complexity and accuracy Monte Carlo square Monte Carlo with regression Comparison of the different algorithms Monte Carlo Monte Carlo Monte Carlo with square with regression Malliavin calculus Complexity O(T dm 2 N 2 ) O((T d + KM 3 )MN) O(T C d M 2 N) American Op. No Yes Yes Dimensions Unlimited! Very limited Limited Delta Expensive Not possible Limited Parallelization Easy Very limited Possible References P. Glasserman and B. Yu, Number of Paths Versus Number of Basis Functions in American Option Pricing, The Annals of Applied Probability, 14(4): , L. A. Abbas-Turki, S. Vialle, B. Lapeyre and P. Mercier, Pricing derivatives on graphics processing units using Monte Carlo simulation, Concurrency and Computation: Practice and Experience, Lokman (TU Berlin) Advances in Mathematical Finance 13 / 16

14 Conclusion Summary Future work References Developing a benchmark method for European contracts Malliavin Calculus provides an attractive theoretical framework for CVA Very accurate CVA values for d 5 when independence is assumed Sufficiently accurate CVA values for d 5 for both intensity and structural model Sufficiently accurate sensitivity values for d 2 and for 3 d 5 if the asset is heavily involved Extend this work to American contracts Approximating/Simulating the backward conditional density L. A. Abbas-Turki and B. Lapeyre, American Options by Malliavin Calculus and Nonparametric Variance and Bias Reduction Methods, SIAM Journal on Financial Mathematics, 3(1): , L. A. Abbas-Turki, A. I. Bouselmi and M. A. Mikou, Toward a coherent Monte Carlo simulation of CVA. Lokman (TU Berlin) Advances in Mathematical Finance 14 / 16

15 Conclusion Lokman (TU Berlin) Advances in Mathematical Finance 15 / 16

16 The End Thank you Questions? Lokman (TU Berlin) Advances in Mathematical Finance 16 / 16