A variance reduction method for computing VaR 1. Computing Value at - PowerPoint PPT Presentation

A variance reduction method for computing VaR 1. Computing Value at Risk by Monte Carlo simulations 2. Importance Sampling for variance reduction 3. Interacting Particle Systems for Importance Sampling (IPS-IS) 4. Simulation results Nadia Oudjane - EDF R&D- Journ´ ees MAS 2008 1

1. Computing Value at Risk by Monte Carlo simulations 2 Value at Risk and quantile � T ◮ P&L of a portfolio on [0 , T ] ∆ V ( X ) = V T − V 0 + 0 CF X ∈ R d the risk factors impacting the portfolio value on [0 , T ] with V aR α = | inf { s ∈ R | P (∆ V ≤ s ) ≥ 1 − α } | ◮ Value at Risk V aR α = | F − ( α ) | , F ( s ) = P (∆ V ≤ s ) , for all s ∈ R where N. Oudjane - EDF R&D- Journ´ ees MAS 2008 2

1. Computing Value at Risk by Monte Carlo simulations 3 Monte Carlo method for VaR estimation ◮ The distribution function F can be viewed as an expectation F ( s ) = E [ I ∆ V ( X ) ≤ s ] , s ∈ R for all ◮ Traditional Monte Carlo Method for computing VaR 1. Monte Carlo simulations give an approximation of F ( s ) : N F N ( s ) = 1 � ˆ I (∆ V ( X i ) ≤ s ) , for all s ∈ R N i =1 ⇒ Too many evaluations of ∆ V for a given accuracy 2. Inversion of ˆ F N and interpolation for approximating VaR N. Oudjane - EDF R&D- Journ´ ees MAS 2008 3

2. Importance Sampling for variance reduction 4 Importance Sampling for variance reduction p − → q where q dominates Hp ◮ Change of measure m = E p [ H ( X )] = E q [ H ( Y ) p q ( Y )] , X ∼ p Y ∼ q where and → q ∗ p − achieves zero variance if H ≥ 0 ◮ Optimal change of measure Hp Hp q ∗ = H ( x ) p ( x ) dx = E p [ H ( X )] = H · p � ◮ Monte Carlo approximation M M = 1 H ( Y i ) p � m q E p [ H ( X )] ≈ ˆ q ( Y i ) , ( Y 1 , · · · , Y M ) i.i.d. ∼ q where M i =1 ⇒ How to simulate and evaluate approximately q ∗ ? N. Oudjane - EDF R&D- Journ´ ees MAS 2008 4

2. Importance Sampling for variance reduction 5 Variance of the Importance Sampling estimate ◮ Let q be a (possibly random) importance probability density dominating q ∗ � � � � m q m q m q V ar ( ˆ M ) = E V ar [ ˆ M | F q ] + V ar E [ ˆ M | F q ] � �� � =0 F q denotes the sigma-algebra generated by the random variables involved in q ◮ The variance of the IS estimate depends on the ”distance” between q and q ∗ � � � M ) = m 2 [( q ∗ − q ) q ∗ m q V ar ( ˆ q ]( x ) dx M E ◮ Idea : use Interacting Particle Systems for Importance Sampling (IPS-IS) to approximate q ∗ by q N based on an N -particle system to achieve C m q N V ar ( ˆ M ) ≤ 0 < α < 1 / 2 with MN α N. Oudjane - EDF R&D- Journ´ ees MAS 2008 5

2. Importance Sampling for variance reduction 6 Some alternative approaches ◮ Large deviation approximation for rare events simulation ◮ Approximation of H to obtain a simple form fo q ∗ ex : [Glasserman&al00] for computing VaR, ∆ - Γ approximation of the portfolio ◮ Cross-entropy [Homem-de-Mello&Rubinstein02] q θ is chosen in a parametric family such as to minimize the entropy K ( q θ , q ∗ ) ◮ Interacting Particle Systems whithout Importance Sampling [DelMoral&Garnier05], [Cerou&al06] Interacting Particle Systems for Importance Sampling (IPS-IS) can be viewed as a non parametric version of cross entropy approach N. Oudjane - EDF R&D- Journ´ ees MAS 2008 6

2. Importance Sampling for variance reduction 7 Progressive correction [Musso&al01] ◮ We introduce a sequence of non negative functions ( G k ) 0 ≤ k ≤ n such that  G 0 ( x ) = 1    x ∈ R d , G 0 ( x ) · · · G n ( x ) = H ( x ) for all The product    G k ( x ) = 0 G k +1 ( x ) = 0 If then H ( x ) = I (∆ V ( x ) ≤ s ) ◮ In our case then we choose G k ( x ) = I ∆ V ( x ) ≤ s k , s = s n ≤ · · · ≤ s 0 = + ∞ with ( ν k ) 0 ≤ k ≤ n ◮ Dynamical system on the space of probability measures  ν 0 = p dx   G k ν k − 1 ν k = R d G k ( x ) ν k − 1 ( x ) dx = G k · ν k − 1 , 1 ≤ k ≤ n for all �   ⇒ ν n = q ∗ dx N. Oudjane - EDF R&D- Journ´ ees MAS 2008 7

2. Importance Sampling for variance reduction 8 Space exploration ( Q k ) 0 ≤ k ≤ n ◮ We introduce a sequence of Markov kernels such that � x ∈ R d ν k ≈ ν k Q k ν k ( dx ) ≈ R d ν k ( du ) Q k ( u, dx ) , i.e. for all G k ( x ) = I ∆ V ( x ) ≤ s k , if p is Gaussian then Q k is ◮ In our case where easily obtained from a Gaussian kernel Q reversible for p , � � Q k ( x, dx ′ ) = Q ( x, dx ′ ) I ∆ V ( x ) ≤ s k + 1 − Q ( x, ∆ V − (( −∞ , s k ])) δ x ( dx ′ ) ( ν k ) 0 ≤ k ≤ n ◮ Dynamical system on the space of probability measures   ν 0 = p dx ν k = G k · ( ν k − 1 Q k − 1 ) , 1 ≤ k ≤ n  for all ⇒ ν n = q ∗ dx N. Oudjane - EDF R&D- Journ´ ees MAS 2008 8

3. Interacting Particle Systems for Importance Sampling (IPS-IS) 9 Approximation of the dynamical system ◮ The idea is to replace at each iteration k , ν k − 1 Q k − 1 by its N -empirical S N ( ν k − 1 Q k − 1 ) measure such that N S N ( ν k − 1 Q k − 1 ) = 1 � ( X 1 k , · · · , X N δ X i k ) are i.i.d. ∼ ν k − 1 Q k − 1 where N k i =1 ◮ Dynamical system on the space of dicrete probability measures ( ν N k ) 0 ≤ k ≤ n  ν N 0 = S N ( ν 0 )  ν N k = G k · S N ( ν N k − 1 Q k − 1 ) , 1 ≤ k ≤ n  for all n ≈ q ∗ dx ν N ⇒ One can show that [DelMoral] N. Oudjane - EDF R&D- Journes MAS 2008 9

3. Interacting Particle Systems for Importance Sampling (IPS-IS) 10 Algorithm ◮ Initialization : Generate independently N 0 = 1 � ( X 1 0 , · · · , X N ν N 0 ) ∼ p δ X i i.i.d. then set N 0 i =1 ◮ Selection : Generate independently N � ( ˜ k , · · · , ˜ X 1 X N ν N ω i k ) ∼ k = k δ X i i.i.d. k i =1 ◮ Mutation : Generate independently for each i ∈ { 1 , · · · , N } , Q k ( ˜ X i X i ∼ k , · ) k +1 ◮ Weighting : For each particle i ∈ { 1 , · · · , N } , compute N G k +1 ( X i k +1 ) � ω i ν N ω i k +1 = k +1 = k +1 δ X i then set � N j =1 G k +1 ( X j k +1 ) k +1 i =1 N. Oudjane - EDF R&D- Journes MAS 2008 10

3. Interacting Particle Systems for Importance Sampling (IPS-IS) 11 Adaptive choice of the sequence ( G k ) 0 ≤ k ≤ n [Musso&al01], [Hommem-de-Mello&Rubinstein02], [C´ erou&al06] ◮ The performance of Interacting particle systems is known to deteriorate max G k when the quantities are big S N ( ν N k − 1 Q k − 1 )( G k ) N The idea is then to chose G k such that 1 � G k ( X i k ) is not to small N i =1 G k ( x ) = I ∆ V ( x ) ≤ s k , the threshold s k is chosen as a ◮ In our case where r.v. depending on the current particle system and on a parameter ρ ∈ (0 , 1) : � � N � s k = inf s I ∆ V ( X i ) ≤ s ≥ ρN such that i =1 ◮ This choice of s k is not prooved to guarantee that the algorithms ends in a finite number of iterations but this point does not seem to be a problem in our simulations N. Oudjane - EDF R&D- Journes MAS 2008 11

3. Interacting Particle Systems for Importance Sampling (IPS-IS) 12 Density estimation n ≈ q ∗ dx ◮ At the end of the algorithm, we get ν N But Importance Sampling requires a smooth approximation with density q N K ◮ Kernel of order 2 � � � K ≥ 0 K = 1 x i K = 0 | x i x j | K < ∞ K h ( x ) = 1 h d K ( x K h h ) ◮ Rescaled kernel � � ν N = ω i δ X i q N,h = ω i K h ( · − X i ) Density estimation − − − − − − − − − → ◮ K h ∗ · E � q N − q ∗ � 1 ≤ C ◮ Optimal choice of h = > 4 2( d +4) N W3 W4 W5 <−−−−−−−− WEIGHTS W2 <−−−−−−−− DENSITY ESTIMATE W1 <−−−−−−−− KERNELS <−−−−−−−− SAMPLE <−−−−−−−− SAMPLE N. Oudjane - EDF R&D- Journes MAS 2008 12

3. Interacting Particle Systems for Importance Sampling (IPS-IS) 13 Some simulation results : Variance ratio ◮ Several test cases depending on the form of function x �→ ∆ V ( x ) have been studied : results are all comparable ◮ X is a d dimensional Gaussian variable and m = E p [ I ∆ V ( X ) ≤ s ] ◮ Particles N = 500 Iterations n ≈ 10 to 60 Simulations M = 10 000 d = 1 d = 2 d = 3 d = 4 d = 5 150 m = 10 − 2 50 50 30 25 10 − 1 1000 m = 10 − 3 300 300 200 140 2 2 . 10 5 10 5 10 5 5 . 10 4 2 . 10 4 m = 10 − 6 200 400 300 460 480 d = 6 d = 7 d = 8 d = 9 · · · d = 30 m = 10 − 2 5 . 10 − 3 22 14 11 8 · · · m = 10 − 3 10 − 3 100 70 55 40 · · · 10 4 2 . 10 3 2 . 10 3 4 . 10 3 1 m = 10 − 6 · · · 250 480 300 300 360 N. Oudjane - EDF R&D- Journes MAS 2008 13