Asymptotic methods in risk management. Advances in Financial Mathematics

Transcription

1 Asymptotic methods in risk management Peter Tankov Based on joint work with A. Gulisashvili Advances in Financial Mathematics Paris, January 7 10, 2014 Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

2 Outline 1 Introduction 2 Multidimensional log-normal distribution 3 Numerics and Monte Carlo 4 Application: stress testing log-normal portfolios 5 Beyond the log-normal distribution Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

3 Introduction Outline 1 Introduction 2 Multidimensional log-normal distribution 3 Numerics and Monte Carlo 4 Application: stress testing log-normal portfolios 5 Beyond the log-normal distribution Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

4 Introduction Introduction Consider a portfolio of n dependent risks X = n i=1 X i The assets are dependent and non identically distributed We are interested in extremal behavior of X (tails of the distribution) Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

5 Introduction The insurance problem Classical setting: The risks X i represent losses (insurance claims) They are positive, unbounded from above, and one is interested in the right tail of X. Meta-theorem: if the variables X i have sufficiently fat (subexponential) tails, the tail behavior of X is determined by the single variable with the fattest tail, even if X i are dependent. Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

6 Introduction The insurance problem Classical setting: The risks X i represent losses (insurance claims) They are positive, unbounded from above, and one is interested in the right tail of X. Meta-theorem: if the variables X i have sufficiently fat (subexponential) tails, the tail behavior of X is determined by the single variable with the fattest tail, even if X i are dependent. A positive random variable ξ is subexponential if lim x P[ξ ξ n > x] P[max(ξ 1,..., ξ n ) > x] = 1 for all n, where ξ 1,..., ξ n are independent copies of ξ. This implies that the distribution function of ξ decays slower than exponential. Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

7 Introduction The finance problem Our setting: The risks X i represent asset prices in a long-only portfolio. One is interested in the left tail of X. Very different problem: For X x it is enough that at least one of X i satisfies X i x For X x is is necessary that all X i satisfy X i x Dependence and diversification effects are expected Very few results in the literature: Wüthrich (2003) gives left tail asymptotics of the distribution function of d identically distributed risks with dependence given by an Archimedean copula. Gao et al. (2009) treat the case of 2 correlated log-normal variables. Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

8 Introduction Asymptotic problems in risk management Relevant questions for risk management Asymototic behavior of distribution function and density of X in the left tail Value at Risk, OTM basket put price / implied volatility Conditional law of (X 1,..., X n ) given X x Stress scenarios Estimation of tail event probabilities by Monte Carlo importance sampling Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

9 Introduction Asymptotic problems in risk management Relevant questions for risk management Asymototic behavior of distribution function and density of X in the left tail Value at Risk, OTM basket put price / implied volatility Conditional law of (X 1,..., X n ) given X x Stress scenarios Estimation of tail event probabilities by Monte Carlo importance sampling Our goals and results: Quantify dependence and diversification effects in the asymptotic setting Sharp asymptotics for the multidimensional log-normal distribution Logarithmic large deviations asymptotics for a large class of distributions Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

10 Multidimensional log-normal distribution Outline 1 Introduction 2 Multidimensional log-normal distribution 3 Numerics and Monte Carlo 4 Application: stress testing log-normal portfolios 5 Beyond the log-normal distribution Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

11 Multidimensional log-normal distribution Multidimensional log-normal distribution Consider the random variable X = where Y = (Y 1,, Y n ) is a n-dimensional Gaussian vector with mean µ = (µ 1,, µ n ) and covariance matrix B = (b ij ) with det B 0 n k=1 e Y k Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

12 Multidimensional log-normal distribution Multidimensional log-normal distribution Consider the random variable X = where Y = (Y 1,, Y n ) is a n-dimensional Gaussian vector with mean µ = (µ 1,, µ n ) and covariance matrix B = (b ij ) with det B 0 Long-only buy and hold portfolio in the multidimensional Black-Scholes model Toy model for understanding dependence / correlation / diversification Popular model in other domains (insurance, economics, signal processing... ) n k=1 e Y k Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

13 Multidimensional log-normal distribution Related work Right tail: Asmussen and Rojas-Nandayapa (2008): the asymptotics is correlation-independent and satisfies P[X > x] mf µ,σ 2, σ = max k=1,...,n σ k, µ = max k:σ k =σ µ k. where F is the one-dimensional log-normal survival function and m = #{k : σ k = σ, µ k = µ}. Similar result holds for dependent subexponential variables (Geluk and Tang, 2009). Left tail: Asymptotics depend on correlation; only partial results are available in the literature. Gao et al. (2009) treat the case n = 2 and the case of arbitrary n for a subclass of covariance matrices Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

14 Multidimensional log-normal distribution Notation and the quadratic programming problem Let and n n := {w R n : w i 0, i = 1,..., n, and w i = 1} n E(w) = w i log w i, for w n with 0 log 0 = 0. i=1 i=1 Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

15 Multidimensional log-normal distribution Notation and the quadratic programming problem Let and n n := {w R n : w i 0, i = 1,..., n, and w i = 1} n E(w) = w i log w i, for w n with 0 log 0 = 0. i=1 i=1 Let w n be the unique point such that w T B w = min w n w T Bw. (QP) Markowitz minimum variance portfolio Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

16 Multidimensional log-normal distribution Notation and the quadratic programming problem Let and n n := {w R n : w i 0, i = 1,..., n, and w i = 1} n E(w) = w i log w i, for w n with 0 log 0 = 0. i=1 i=1 Let w n be the unique point such that w T B w = min w n w T Bw. (QP) Markowitz minimum variance portfolio With Ī := {i {1,..., n} : w i > 0} and n := Card Ī, assume WLOG that Ī = {1,..., n}. We let µ R n with µ i = µ i and B M n(r) with b ij = b ij ; the elements of B 1 are denoted by ā ij and Āk := n j=1 ākj. Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

17 Multidimensional log-normal distribution Assumption (A) Our main result requires the following non-degeneracy assumption: (A) for inequality constraints in problem (QP) which are saturated, Lagrange multipliers are strictly positive For i = n + 1,..., n, (e i w) T B w 0, where e i R n satisfies ej i = 1 if i = j and ej i = 0 otherwise. Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

18 Multidimensional log-normal distribution Main result Let Assumption (A) hold true. Then, as x 0, ( P[X x] = C log 1 ) 1+ n { 2 x w T µ+e( w) w x T B w exp where 1 w T C = B w 2π B Ā1 Ā n exp 1 2 n i,j=1 log2 x 2 w T B w } ( ( )) O, log x ā ij ( µ i log w i ) ( µ j log w j ) + ( w T µ + E( w)) 2 2 w T B w. Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

19 Multidimensional log-normal distribution Main result Let Assumption (A) hold true. Then, as x 0, ( P[X x] = C log 1 ) 1+ n { 2 x w T µ+e( w) w x T B w exp where 1 w T C = B w 2π B Ā1 Ā n exp 1 2 n i,j=1 log2 x 2 w T B w } ( ( )) O, log x ā ij ( µ i log w i ) ( µ j log w j ) + ( w T µ + E( w)) 2 2 w T B w. Leading term depends on correlation through min w n w T Bw Asymptotic diversification present when w T B w < min i b ii Only the stocks with w i > 0 contribute to diversification Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

20 Multidimensional log-normal distribution Example Let n = 2, and b 11 = σ 2 1, b 22 = σ 2 2 and b 12 = ρσ 1 σ 2 ; assume σ 1 σ 2. Then, w = ( v, 1 v) T with v = σ 2(σ 2 ρσ 1 ) σ σ2 2 2ρσ 1σ 2 0. If ρ < σ 2 σ 1, both weights are strictly positive, assumption (A) holds, and C P[X z] e 1 log z 3 2 (µ 1+x log z,µ 2 +y log z)b 1 (µ 1 +x log z,µ 2 +y log z) T. 2 If ρ > σ 2 σ 1, w = (0, 1) T, assumption (A) holds, and P[X z] 1 e (log z µ 2 ) 2 2σ 2 2. log z σ 2 2π asymptotics determined by the second component. If ρ = σ 2 σ 1, w = (0, 1) T but assumption (A) does not hold. Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

21 Numerics and Monte Carlo Outline 1 Introduction 2 Multidimensional log-normal distribution 3 Numerics and Monte Carlo 4 Application: stress testing log-normal portfolios 5 Beyond the log-normal distribution Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

22 Numerics and Monte Carlo Quality of the asymptotic approximation A 4 4 covariance matrix with elements of the form b ij = σ i σ j ρ (constant correlation) with σ = {2; 2.3; 3; 3}. The asymptotic approximation F a (x) of the distribution function P[X x] = P[e Y 1 + e Y 2 + e Y 3 + e Y 4 x] is compared with its Monte Carlo estimate F mc (x). We plot the ratio Fmc(x) F a(x) the correlation ρ. for a wide range of values of x and two values of Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

23 Numerics and Monte Carlo Quality of the asymptotic approximation As expected, the ratios converge very slowly to one. The asymptotic formula gives the right order of magnitude for a wide range of probabilities (the values shown correspond to probabilities from 10 1 to ). Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

24 Numerics and Monte Carlo Variance reduction of Monte Carlo estimates The standard estimate of the distribution function F(x) = P[X x] is F N (x) = 1 N N k=1 1, n (k) i=1 ey i x where Y (1),..., Y (N) are i.i.d. vectors with law N(µ, B). Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

25 Numerics and Monte Carlo Variance reduction of Monte Carlo estimates The standard estimate of the distribution function F(x) = P[X x] is F N (x) = 1 N N k=1 1, n (k) i=1 ey i x where Y (1),..., Y (N) are i.i.d. vectors with law N(µ, B). The variance of F N (x) is given by and the relative error Var F N (x) = F (x) F 2 (x) N Var F N (x) F(x) F(x) N, x 0, 1 NF(x) explodes very quickly as x 0 (as e c log2 x for some constant c). Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

26 Numerics and Monte Carlo Variance reduction of Monte Carlo estimates Rewrite the formula for F as follows: F(x) = E[e ΛT B 1 (Y µ) 1 2 ΛT B 1Λ 1 n i=1 ey i +Λ i x ], where Λ R n is a vector to be chosen such that the corresponding estimate F Λ N(x) = 1 N N e ΛT B 1 (Y (k) µ) 1 2 ΛT B 1Λ 1 n i=1 k=1 has variance smaller than the standard estimate. (k) +Λ ey i i x Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

27 Numerics and Monte Carlo Variance reduction of Monte Carlo estimates Rewrite the formula for F as follows: F(x) = E[e ΛT B 1 (Y µ) 1 2 ΛT B 1Λ 1 n i=1 ey i +Λ i x ], where Λ R n is a vector to be chosen such that the corresponding estimate F Λ N(x) = 1 N N e ΛT B 1 (Y (k) µ) 1 2 ΛT B 1Λ 1 n i=1 k=1 has variance smaller than the standard estimate. for optimal variance reduction, we need to minimize [ n ] V (Λ, x) = e ΛT B 1Λ P e Y i Λ i x. i=1 (k) +Λ ey i i x Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

28 Numerics and Monte Carlo Variance reduction of Monte Carlo estimates To obtain an explicit estimate, replace the probability by its asymptotic equivalent. This amounts to minimizing Ṽ (Λ, x) = Λ T B 1 Λ 1 2 n i,j=1 The optimal value of Λ is given by Λ k = ā ij (log(x w i ) µ i + Λ i ) (log(x w j ) µ j + Λ j ). n i,j=1 b ki ā ij (log(x w j ) µ j ), Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

29 Numerics and Monte Carlo Variance reduction of Monte Carlo estimates To obtain an explicit estimate, replace the probability by its asymptotic equivalent. This amounts to minimizing Ṽ (Λ, x) = Λ T B 1 Λ 1 2 n i,j=1 The optimal value of Λ is given by and it can be shown that Λ k = ā ij (log(x w i ) µ i + Λ i ) (log(x w j ) µ j + Λ j ). n i,j=1 b ki ā ij (log(x w j ) µ j ), V (Λ, x) CF 2 (x) ( log 1 ) n x relative error grows only logarithmically in x as x 0. Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

30 Numerics and Monte Carlo Variance reduction of Monte Carlo estimates ρ = 0.2 ρ = 0.8 x P[X x] red. factor x P[X x] red. factor Standard deviation reduction factors obtained with the variance reduction estimate (10 6 trajectories). Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

31 Application: stress testing log-normal portfolios Outline 1 Introduction 2 Multidimensional log-normal distribution 3 Numerics and Monte Carlo 4 Application: stress testing log-normal portfolios 5 Beyond the log-normal distribution Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

32 Application: stress testing log-normal portfolios Multidimensional Black-Scholes model Assume that the assets S 1,..., S n follow a n-dimensional Black-Scholes model: log S t = bt diag(b)t 2 + B 1 2 Wt, where W is a standard n-dimensional Brownian motion, B is a covariance matrix, b R denotes the drift vector and diag(b) is the main diagonal of B. Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

33 Application: stress testing log-normal portfolios Multidimensional Black-Scholes model Assume that the assets S 1,..., S n follow a n-dimensional Black-Scholes model: log S t = bt diag(b)t 2 + B 1 2 Wt, where W is a standard n-dimensional Brownian motion, B is a covariance matrix, b R denotes the drift vector and diag(b) is the main diagonal of B. Let X t = n ξ i St. i where ξ 1,..., ξ n are positive weights, represent a market index. i=1 Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

34 Application: stress testing log-normal portfolios Stress testing The fund manager holds a portfolio of assets S 1,..., S n with weights v 1,..., v n, whose value will be denoted by n V t = v i St. i Goal: understand the behavior of this portfolio under adverse market scenarios i=1 Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

35 Application: stress testing log-normal portfolios Stress testing The fund manager holds a portfolio of assets S 1,..., S n with weights v 1,..., v n, whose value will be denoted by n V t = v i St. i i=1 Goal: understand the behavior of this portfolio under adverse market scenarios Assume that the scenario corresponds to the fall of 1 α per cent in the index Systematic method to build stress scenarios: conditional expected value given the scenario: E[V t X t = αx 0 ] = n n v i e i (t, αx 0 ), e i (t, X) = E[St i ξ k St k i=1 k=1 = X]. Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

36 Application: stress testing log-normal portfolios Stress testing: main result Let Assumption (A) hold true. Then, as X 0, if i Ī, the i-th asset decays proportionnally to the index: e i (t, X) = w ( ( ( ix 1 + O log 1 ) )) 1, ξ i X Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

37 Application: stress testing log-normal portfolios Stress testing: main result Let Assumption (A) hold true. Then, as X 0, if i Ī, the i-th asset decays proportionnally to the index: e i (t, X) = w ( ( ( ix 1 + O log 1 ) )) 1, ξ i X if i / Ī, the i-th asset decays faster than the index: n ( ) e i (t, X) = X 1+ λ i exp b i t b pi ā pq log Ā1 + + Ā n + µ q where we write exp t 2 n p,q=1 p,q=1 ( ā pq b pi b qi 1 + O Ā q ( ( log 1 ) )) 1 X µ q = log ξ q + b q t t 2 b qq and λ i = [B w] i w T 1 > 0, i / Ī. B w Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

38 Beyond the log-normal distribution Outline 1 Introduction 2 Multidimensional log-normal distribution 3 Numerics and Monte Carlo 4 Application: stress testing log-normal portfolios 5 Beyond the log-normal distribution Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

39 Beyond the log-normal distribution Preliminary remarks and notation In the first part we have computed the exact left-tail asymptotics for X = n f (X i ) i=1 when X 1,..., X n is a Gaussian vector and f is the exponential function. The leading term can be obtained under much weaker assumptions on f. Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

40 Beyond the log-normal distribution Preliminary remarks and notation In the first part we have computed the exact left-tail asymptotics for X = n f (X i ) i=1 when X 1,..., X n is a Gaussian vector and f is the exponential function. The leading term can be obtained under much weaker assumptions on f. We write f g as x a whenever lim x a f (x) g(x) = 1. Recall that f is slowly varying as x 0 whenever lim x 0 f (λx) f (x) = 1 λ > 0. f is regularly varying as x 0 whenever lim x 0 f (λx) f (x) = λ α for some index α, and we write f R α. Similar definitions can be given for x. Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

41 Beyond the log-normal distribution Non-linear functions of Gaussian risk factors Theorem Let f 1,..., f n be increasing continuous functions from R to R + such that f 1 1,..., f 1 n are slowly varying at 0 and for some function f, f 1 k (x) f 1 (x) as x 0 for k = 1,..., n Let (X 1,..., X n ) be centered Gaussian with covariance matrix B > 0. Then, H(z) = P[f 1 (X 1 ) + + f n (X n ) f (z)] satisfies z 2 log H(z) 2 inf w n w T Bw, z. Model-free result quantifying the diversification property of Gaussian copula Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

42 Beyond the log-normal distribution Example: portfolio of calls in the Black-Scholes model Asset prices at date T are given by S i = e X i for i = 1,..., n, where (X 1,..., X n ) is Gaussian with covariance BT The portfolio contains exactly one option on each asset with log-strikes (k 1,..., k n ) and maturity dates (T 1,..., T n ), where T i > T for i = 1,..., n. Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

43 Beyond the log-normal distribution Example: portfolio of calls in the Black-Scholes model Asset prices at date T are given by S i = e X i for i = 1,..., n, where (X 1,..., X n ) is Gaussian with covariance BT The portfolio contains exactly one option on each asset with log-strikes (k 1,..., k n ) and maturity dates (T 1,..., T n ), where T i > T for i = 1,..., n. The price of i-th option at date T is given by the Black-Scholes formula: P i = e X i N(d 1 ) e k i N(d 2 ), d 1,2 = X i k i σ i Ti T ± σ i Ti T, σ i = B ii. 2 Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

44 Beyond the log-normal distribution Example: portfolio of calls in the Black-Scholes model Asset prices at date T are given by S i = e X i for i = 1,..., n, where (X 1,..., X n ) is Gaussian with covariance BT The portfolio contains exactly one option on each asset with log-strikes (k 1,..., k n ) and maturity dates (T 1,..., T n ), where T i > T for i = 1,..., n. The price of i-th option at date T is given by the Black-Scholes formula: P i = e X i N(d 1 ) e k i N(d 2 ), d 1,2 = X i k i σ i Ti T ± σ i Ti T, σ i = B ii. 2 Then, as z 0, log 1 z log P[P P n z] inf w n w T Rw, where R is a n n matrix with elements R ij = B ij T σ i σ j (Ti T )(T j T ). Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

45 Beyond the log-normal distribution Random variables with Gaussian copula Theorem Let X 1,..., X n be positive random variables with distribution functions G 1,..., G n and Gaussian copula with correlation matrix R, det R > 0. Assume that for each k, g k (x) := log G k (x) is slowly varying at 0 g k (x) σ 2 k g(x) as x 0, for σ k > 0 and some function g. Then, log P[X X n z] g(z) inf w n w T Bw as z 0, where the matrix B is defined by B ij = σ i σ j R ij, 1 i, j n. Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

46 Beyond the log-normal distribution Application: pricing a multi-asset option Popular approach for pricing European multi-asset options: calibrate complex models for the marginal distributions, use a copula to model the dependence structure. Gaussian copula is often used since reliable information about dependence is often limited to correlations. Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

47 Beyond the log-normal distribution Application: pricing a multi-asset option Popular approach for pricing European multi-asset options: calibrate complex models for the marginal distributions, use a copula to model the dependence structure. Gaussian copula is often used since reliable information about dependence is often limited to correlations. Our contribution: logarithmic asymptotics for option prices sharp asymptotics of implied volatilities in the spirit of tail wing formulas of Benaim and Friz (2009) and Lee (2004) T=1 month T=3 months T=1 year Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

48 Beyond the log-normal distribution Multidimensional tail wing formula Let S i = e X i be asset prices at T, denote by G i the distribution function of X i, and let the copula of X be Gaussian with correlation matrix R. Assume that for each i = 1,..., n, log G i ( k) R α and log G i ( k) σ 2 i log G( k) as k + for some G and some σ 1,..., σ n Then the implied volatility I as function of log-strike of a basket put option on S 1,..., S n satisfies I 2 ( k)t k [ ψ log G( k) k inf w n w T Bw ( x ) where B ij = σ i σ j R ij and ψ(x) = x x. ], as k +, Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

49 Beyond the log-normal distribution Example: Normal Inverse Gaussian marginals Assume that for all i, X i NIG(α i, β i, µ i, δ i ) with moment generating function { } ) M i (z) = exp (δ i αi 2 βi 2 α 2i (β i + z) 2 + µ i z. From Theorem 2 in Benaim and Friz (2009): our assumptions are satisfied with σ i = 1 αi β i and G(k) = e k. Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

50 Beyond the log-normal distribution Example: Normal Inverse Gaussian marginals Assume that for all i, X i NIG(α i, β i, µ i, δ i ) with moment generating function { } ) M i (z) = exp (δ i αi 2 βi 2 α 2i (β i + z) 2 + µ i z. From Theorem 2 in Benaim and Friz (2009): our assumptions are satisfied with σ i = 1 αi β i and G(k) = e k. It follows that I 2 ( k)t k [ ψ 1 inf w d w T Bw ], k +, where the matrix B is defined by B ij = R ij (αi β i )(α j β j ). Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

51 Beyond the log-normal distribution Concluding remarks Fully explicit sharp asymptotics for buy and hold portfolios in the multidimensional Black-Scholes model Quantitative understanding of the diversification effect of the Gaussian dependence in a model-free setting New applications of asymptotic methods which were up to now mostly used in single-asset option pricing (building stress scenarios) Next step: more general dynamic multidimensional models (Wishart?) Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35

52 Beyond the log-normal distribution References S. Asmussen and L. Rojas-Nandayapa, Asymptotics of sums of lognormal random variables with Gaussian copula, Statistics & Probability Letters, vol. 78, no. 16, pp , P. Friz and S. Benaim, Regular variation and smile asymptotics, Mathematical Finance, 19 (2009), pp X. Gao, H. Xu and D. Ye, Asymptotic Behavior of Tail Density for Sum of Correlated Lognormal Variables, Int. J. Math. and Math. Sciences, vol J. Geluk and Q. Tang, Asymptotic tail probabilities of sums of dependent subexponential random variables, Journal of Theoretical Probability, vol. 22, no. 4, pp , E. Hashorva and J. Hüsler, On multivariate Gaussian tails, Annals of the Institute of Statistical Mathematics, 55 (2003), pp M. Wüthrich, Asymptotic value-at-risk estimates for sums of dependent random variables. Astin Bulletin 33 (2003), no. 1, Peter Tankov (Université Paris Diderot) Asymptotic methods in risk management Paris, January 7 10, / 35