Orthonormal polynomial expansions and lognormal sum densities

Transcription

1 1/28 Orthonormal polynomial expansions and lognormal sum densities Pierre-Olivier Goffard Université Libre de Bruxelles February 22, 2016

2 2/28 Introduction Motivations Definition Goal: Recover the PDF of the sum of n lognormally distributed random variables. Dependence The lognormals in the sum may be correlated. Extreme Value The lognormal distribution is heavy tailed. Actuarial Science Modelization of the total claim amounts of non life insurance portfolios.

3 3/28 Introduction Motivations Definition Further aplications... Finance Black & Scholes model security prices are lognormally distributed, The value of the portfolio is distributed as a sum of lognormally distributed random variable, Pricing of Asian option. The pay off is determined by the average of the underlying price. Telecommunication The inverse of the signal to noise ratio can be modeled as a sum of i.i.d. lognormals.

4 4/28 Introduction Motivations Definition Definition The random variable defined as S = e X e Xn where (X 1,..., X n ) MN (µ, Σ) admits a sum of lognormals distribution SLN (µ, Σ). The PDF is given by

5 4/28 Introduction Motivations Definition Definition The random variable defined as S = e X e Xn where (X 1,..., X n ) MN (µ, Σ) admits a sum of lognormals distribution SLN (µ, Σ). The PDF is given by f S (x) =???

6 4/28 Introduction Motivations Definition Definition The random variable defined as S = e X e Xn where (X 1,..., X n ) MN (µ, Σ) admits a sum of lognormals distribution SLN (µ, Σ). The PDF is given by f S (x) =??? The PDF is unknown even in the case of the sum of two i.i.d. lognormals.

7 5/28 Let X be a random variable governed by P X, with unknown PDF f X. I. Pick a reference probability measure ν P X absolutely continuous w.r.t. ν. f ν as first approximation of f X. II. {Q k } k N sequence of orthonormal polynomials w.r.t. ν Gram-Schmidt orthogonalization procedure. III. Orthogonal projection of f X /f ν onto {Q k } Adjustment of the first approximation.

8 6/28 {Q k } k N must be complete in a set of functions in which f X /f ν belongs. The L 2 (ν) space The set of square integrable functions with respect to ν. Inner product, f, g = f (x)g(x)dν(x), where f, g L 2 (ν). {Q k } k N is a sequence of orthogonal polynomials w.r.t. ν in the sense that Q k, Q l = Q k (x)q l (x)dν(x) = δ kl.

9 7/28 Sufficient Condition If e α x dν(x) < +, then {Q k } k N is complete in L 2 (ν). Orthonormal Polynomial Expansion If f X /f ν L 2 (ν) then f X (x) = + k=0 a k Q k (x)f ν (x), where a k = Q k, f X = f ν Q k (x) f X (x) f ν (x) dν(x) = E [Q k(x )].

10 8/28 Orthonormal polynomial approximation The approximation follows from simple truncation K fx K (x) = a k Q k (x)f ν (x). k=0 The accuracy relies on the decay of the a k s.

11 9/28 Lognormal distribution as reference f ν (x) = Associated Orthogonal Polynomials ( ) 1 xσ ln(x) µ 2 2π e σ 2, x > 0. Q k (x) = e k2 σ 2 2 k i=0 ( 1) k+i e iµ i2 σ 2 2 [e σ 2, e σ2] k e k i (1,..., e (k 1)σ2) x i, for k N where e i (X 1,..., X k ) are the elementary symmetric polynomials and [x, q] n = n 1 ( i=0 1 xq i ) is the Q-Pochhammer symbol.

12 10/28 Let S be SLN (µ, σ)-distributed. Integrability Condition If ( ) f S (x) = O e b log2 x for x 0 and, where b > ( 4σ 2) 1, then fs /f ν L 2 (ν)

13 10/28 Let S be SLN (µ, σ)-distributed. Integrability Condition If ( ) f S (x) = O e b log2 x for x 0 and, where b > ( 4σ 2) 1, then fs /f ν L 2 (ν) Asymptotics for f S f S (x) = O(exp{ c 1 ln(x) 2 }) as x 0, where c 1 = [ 2 min w w T Σ 1 w ] 1, see Tankov et al [GT15]. and f S (x) = O(exp{ c 2 ln(x) 2 }) as x where c 2 = [ 2 max i=1,...,n Σ ii ] 1, see Asmussen et al. [ARN08]

14 11/28 The integrability condition is satisfied as long as σ 2 > max Σ ii. 2 PB1 Is {Q k } k N complete in L 2 (ν)?

15 11/28 The integrability condition is satisfied as long as σ 2 > max Σ ii. 2 PB1 Is {Q k } k N complete in L 2 (ν)? Proposition No, it s not!

16 11/28 The integrability condition is satisfied as long as σ 2 > max Σ ii. 2 PB1 Is {Q k } k N complete in L 2 (ν)? Proposition No, it s not! PB2 The lognormal distribution is not characterized by its moments and neither is the sum of lognormals distribution.

17 12/ K = 0 K = 1 K = 40 f Fig. 1 orthogonal polynomial approximations of LN (0, ) using a LN (0, ) reference.

18 13/28 Normal distribution N (µ, σ) as reference f ν (x) = 1 ( ) 2 σ x µ 2π e σ 2, x R. Associated Orthonormal Polynomials Q k (x) = 1 k!2 k/2 H k ( x µ σ 2 ), where {H k } k N are the Hermite polynomials, see Szego [Sze39].

19 f S /f ν / L 2 (ν). We consider a transformation of S Integrability Condition Z = ln(s). f Z (x) = O ( e ax2 ) as x ±, where a > ( 4σ 2) 1. Asymptotics for f Z f Z (z) = O(exp{ c 1 z 2 }) as z and f Z (z) = O(exp{ c 2 z 2 }) as z + Extension of the work of Gao et al. [GXY09]. 14/28

20 15/28 The integrability condition is satisfied as long as σ 2 > max Σ ii. 2 f Z is approximated by f Z (z) = K a k Q k (z)f ν (z), k=0 where the coefficient a k = E {Q k [ln(s)]} are evaluated using Crude Monte Carlo for k = 1,..., K. f S is approximated by f S, N (x) = 1 x f Z (ln(x)).

21 16/28 Gamma Distribution Γ(m, r) As Reference f ν (x) = e x/m x r 1 Γ(r)m r, x > 0. Associated Orthonormal Polynomials [ ] Γ(n + r) 1/2 Q n (x) = ( 1) n L r 1 n (x/m), Γ(n + 1)Γ(r) where the {L r 1 n } k N are the generalized Laguerre polynomials defined in Szego [Sze39].

22 17/28 f S /f ν / L 2 (ν). We consider the PDF of the exponentially tilted distribution of S f θ (x) = e θx f S (x), L(θ) where L(θ) = E ( e θs). Integrability Condition If ( f θ (x) = O e δx) where δ > 1 2m, and for x +, ( f θ (x) = O x β) for x 0, where β > r/2 1, then f θ /f ν L 2 (ν).

23 18/28 The integrability condition is satisfied as long as m > 1 2θ. f θ is approximated by f θ (x) = K a k Q k (x)f ν (x), k=0 where the coefficient a k = E [Q k (S θ )] with S θ f θ. f S is approximated by f S, Γ (x) = L(θ)e θx fθ (x).

24 19/28 Regarding the evaluation of the a k s, we have a k = E [Q k (S θ )] = q k0 + q k1 E(S θ ) q kk E(S k θ ) where the q ki s are the coefficients of Q k, and E(S i θ ) = E ( S i e θs) L(θ) = L i(θ) L(θ) where L i (θ) denotes the i th derivative of L(θ). We adapt techniques developed in Asmussen et al. [AJRN14, LAJRN16] to access L i (θ).

25 20/28 Polynomial Aproximation Settings Normal distribution as reference µ = E(Z) and σ 2 = V(Z) 10 5 replications for the CMC procedure Gamma distribution as reference θ = 1 Moment matching of order 2 between fθ and f ν to get m and r.

26 21/28 Some Challengers The Fenton-Wilkinson approximation f FW, see [Fen60] The log skew normal approximation f SK, see [HB15] The conditionnal Monte Carlo approximation f Cond, cf. Example 4.3 on p. 146 of [AG07]. A benchmark is obtained through numerical integration. f S (x) and f S (x) together with ( fs (x) f S (x)) are plotted over x [0, 2E(S)], The L 2 error are computed over [0, E(S)].

27 22/ f FW f Sk f Cond f N f Γ f f FW fsk fcond fn f Γ Test 1 mu = (0, 0), diag(σ) = (0.5, 1), ρ = 0.2. Reference distributions used are N (0.88, ) and Gamma(2.43, 0.51) with K = 32.

28 23/ f FW f Sk f Cond f N f Γ f f FW fsk fcond fn f Γ Test 2 n = 4, µ i = 0, Σ ii = 1, ρ = 0.1. Reference distributions used are N (1.32, ) and Gamma(3.37, 0.51) with K = 32

29 24/ Test 6 Sum of 3 LN (0, 1) r.v.s with C Cl 10 ( ) copula (i.e., τ = 5 6 ). Reference distributions used are N (1.46, ) and Gamma(8.78, 0.25) with K = 40. The L 2 errors of f N and f Γ are and respectively. f N f Γ f

30 25/28 The orthonormal polynomial approximation is an efficicient numerical method: Accurate and easy to code. None of the method tested here seem to universally superior to the other. Good behavior in presence of a non-gaussian dependence structure: cf. test 3 with Clayton copula. For more details, the paper is available on ArXiv [AGL16].

31 26/28 Søren Asmussen and Peter W Glynn, Stochastic Simulation: Algorithms and Analysis, Stochastic Modelling and Applied Probability series, vol. 57, Springer, S. Asmussen, P.-O. Goffard, and P. J. Laub, Orthonormal polynomial expansion and lognormal sum densities, arxiv preprint arxiv: (2016). S. Asmussen, J. L. Jensen, and L. Rojas-Nandapaya, On the Laplace transform of the lognormal distribution, Methodology and Computing in Applied Probability (2014), Søren Asmussen and Leonardo Rojas-Nandayapa, Asymptotics of sums of lognormal random variables with Gaussian copula, Statistics & Probability Letters 78 (2008), no. 16,

32 27/28 Lawrence Fenton, The sum of log-normal probability distributions in scatter transmission systems, IRE Transactions on Communications Systems 8 (1960), no. 1, Archil Gulisashvili and Peter Tankov, Tail behavior of sums and differences of log-normal random variables, Bernoulli (2015), To appear, accessed online on 26th August 2015 at user/submissionfile/17119?confirm=ef Xin Gao, Hong Xu, and Dong Ye, Asymptotic behavior of tail density for sum of correlated lognormal variables, International Journal of Mathematics and Mathematical Sciences (2009), Volume 2009, Article ID , 28 pages. Marwane Ben Hcine and Ridha Bouallegue, Highly accurate log skew normal approximation to the sum of correlated lognormals, In the Proc. of NeTCoM 2014 (2015).

33 28/28 Patrick J. Laub, Søren Asmussen, Jens L. Jensen, and Leonardo Rojas-Nandayapa, Approximating the Laplace transform of the sum of dependent lognormals, Volume 48A of Advances in Applied Probability. (2016), In Festschrift for Nick Bingham, C. M. Goldie and A. Mijatovic (Eds.), Probability, Analysis and Number Theory. To appear. G. Szegö, Orthogonal polynomials, vol. XXIII, American Mathematical Society Colloquium Publications, 1939.

34 28/28 The symmetric polynomials are defined as { 1 j e i (X 1,..., X k ) = 1 <...<j i k X j 1... X ji, for i k, 0, for i > k, and [x, q] n = n 1 ( i=0 1 xq i ) is the Q-Pochhammer symbol.

35 28/28 Log Skew Normal Approximation X is governed by a skew normal distribution SN (λ, ω, ɛ) if its PDF is given by f X (x) = 2 ( ) ( x ɛ ω φ Φ λ x ɛ ), ω ω where φ and Φ are respectively the PDF and CDF of the standard normal distribution. Y = e X is governed by a log skew normal distribution LSN (λ, ω, ɛ) Central moment matching (the two first one) with the sum of lognormals distribution. Left tail slope matching using the asymptotics derived in [GT15].

36 28/28 Conditional Monte Carlo Approximation By definition, the conditional Monte Carlo is the Crude Monte Carlo estimator of the representation f S (x) = E {P(S dx Y )} Recall that S = e X e Xn, and set Y = e X 1, then the conditional Monte Carlo estimator is written as follows [ ] } f Cond = E {f e X1 x (S e X 1 X1 ), where f e X 1 is the PDF of a lognormal distribution LN [E(X 1 ), V(X 1 )]

37 28/28 Approximation of L(θ) Precisions regarding the evaluation of the a k s in the gamma case The Laplace transform of S is given by ( L(θ) = E e θs) exp ( h θ (x)) dx where h θ (x) = θ(e µ ) T e x xt Dx, where D = Σ 1.

38 Replace h θ by its second order Taylor expansion arround its unique minimizer x ( 1 1 ) T 2 x Dx (x x ) T (Λ + D) (x x ), where Λ = θdiag ( e µ+x ), to get the approximation L(θ). The reinjection of the Taylor expansion in the initial Laplace transform expansion permits to get a correction term L(θ) = L(θ)I (θ), where I (θ) = [ ( )] det(i + ΣΛ)E v Σ 1/2 Z, with v(u) = exp { (x ) T D(e u 1 u } and Z N (0, 1). More detail can be found in [LAJRN16] 28/28