Numerical Methods with Lévy Processes

Transcription

1 Numerical Methods with Lévy Processes 1

2 Objective: i) Find models of asset returns, etc ii) Get numbers out of them. Why? VaR and risk management Valuing and hedging derivatives Why not? Usual assumption: Returns are normally distributed, Joint distributions are normal. Alas, not true. Marginal distributions: Joint distributions: Alas, not ignorable. Effects are too great. Not normal Not jointly normal. (not a normal copula).

3 Modelling joint returns distributions M assets, S i t, i = 1,,M, Returns, R i t = ln( S i t/s i 0 ). Marginal distributions: F i t(r) = Pr[ R i t < r ], Joint distribution: F t ( r 1,, r M ) = Pr[ R 1 t < r 1,, R M t < r M ], Separate out the marginals and the dependency. Note: F i t : R [0,1], u i t = F i t(r i t) is uniform. Use each F i t to map F t onto a joint distribution function C on I M = [0,1] M, F t ( r 1,, r M ) = C t ( F 1 t(r 1 ),, F M t(r M ) ). C : I M [0,1] is a probability distribution on I M with uniform marginals. C is called a copula. 3

4 Jointly normal distributions? F i t(r) are normal, C t is the normal copula. Generalise: Model marginals as Lévy processes Choose a non-normal copula. This talk: The problems and joys of Lévy processes. 4

5 Lévy Processes X = {X t } t 0, X 0 = 0, is a Lévy process if i) Increments are independent of the past : 0 s < t <, X t - X s is independent of I s ii) Increments are stationary : 0 s < t <, the distribution of X t - X s is the same as the distribution of X t-s. iii) X t is continuous in probability: (ie ε > 0, Pr[ X t - X s > ε ] 0 as s t) X t will have a modification which is càdlàg. This is the Lévy process. Additive process: has properties (i) and (iii) 5

6 The Lévy-Khintchine representation Characteristic function of µ, a prob measure on R d : µˆ (z) = d R eiz x µ(dx) = E µ [e iz x ], z R d. If X t is a Lévy process and X 1 X 0 ~ F X1 = µ, then µˆ (z) = exp( φ(z) ), z R, with φ(z) = -½z Az + iz γ + R d (e iz x iz x.1 D (x))ν(dx). where D = { x x 1} is the unit ball in R d, and A is a symmetric non-negative definite matrix, γ R d, ν is a measure on R d, such that ν{0} = 0, R d ( x 1)ν(dx) ) <, (A,ν,γ) is the generating triplet of µ ν isn t a probability measure. May not be integrable. i) (A,ν,γ) is unique ii) (A,ν,γ) Lévy processes 6

7 Notes: If µ X 1 has generating triplet (A,ν,γ) then µ t X t has generating triplet (ta,tν,tγ). ν is called the Lévy measure of µ. If ν(dx) = k(x)dx has a density, k is called the Lévy density of µ. The Lévy-Khintchine representation is not unique. Can have: φ(z) = -½z Az + iz γ c + R d (e iz x iz x.c(x))ν(dx). where, eg, c(x) = (1 + x ) -1, c(x) = 1 { x ε} (x), ε > 0, etc when γ c = γ + R d x( c(x) - 1 D (x) )ν(dx). Write (A, ν, γ c ) c. 7

8 Centre and Drift Suppose that x 1 x ν(dx) <, then can set c(x) = 0 and φ(z) = -½z Az + iz γ 0 + R d (e iz x - 1)ν(dx). This γ 0 is the drift of µ. Suppose that x > 1 x ν(dx) <, then can set c(x) = 1 and φ(z) = -½z Az + iz γ 1 + R d (e iz x iz x)ν(dx). This γ 1 is the centre of µ. If γ 1 exists then γ 1 = R d x µ(dx) is the mean of µ. µ Gaussian then ν = 0 and γ 0 = γ 1. Brownian motion with drift, γ 0 is the drift of the Brownian motion. 8

9 Observations µ compound Poisson, jumps arriving at a rate c, jump sizes distributed according to σ, then A = 0, ν = cσ, γ 0 = 0. Γ-distribution, parameters c, α > 0, then φ(z) = c [0, )(e ixz - 1) αx so A = 0, ν(dx) = c e x This ν has infinite mass. e αx x dx, 1 [0, ) (x)dx, γ 0 = 0. X t additive, continuous sample paths as iff X t has Gaussian distribution t, ie, X t is Brownian motion. A is Gaussian covariance of µ. ν = 0 iff µ is Gaussian. A = 0, then is purely non-gaussian A, γ = 0, then µ is pure jump. 9

10 Connections X t a Lévy process on R d, generating triple (A,ν,γ). Is type A: if A = 0, ν(r d ) <, type B: if A = 0, ν(r d ) =, x 1 x ν(dx) <, type C: if A 0, or x 1 x ν(dx) =. Sample paths of X t are: cts iff ν = 0, Piecewise constant iff i) X t is type A with γ 0 = 0, or ii) X t is compound Poisson ν(r d ) =, then jump times are countable, dense in [0, ). 0 < ν(r d ) <, then jump times are countable, but not dense as. Time to first jump is exponential, mean ν(r d ) -1. X t is type A or B then has finite variation on (0,t], as. X t is type C then has infinite variation on (0,t], t. 10

11 Modelling with Lévy Processes Stock Model, eg Under risk-neutrality suppose that S t = S 0 exp( rt + σx t - ωt ), where r is the short rate and ω compensates for the drift in X t, so that S t /exp( rt ) is a martingale, E[e σx t ] = exp( tω ) (ie, X t specified under the pricing measure). Interest Rate Model, eg dr t = α(µ(t) - r t )dt + σdx t, Note: In each case can assume that X 1 has zero mean and unit variance in unit time. Need to price options, etc, on S t, r t, etc 11

12 How to Solve? Gaussian case: Explicit/analytic solutions? PDE Monte Carlo integration Lattices Lévy case: Analytic solutions: May involve tricky numerical integration PDE: Method of lines? Is it general? Monte Carlo: Tricky if Lévy density unbounded near zero. Lattices: Need very high order branching? Problem: Too many small jumps Too many big jumps 1

13 The Generalised Hyperbolic Distribution (Barndorff-Nielsen (01), Eberlein (01), Rydberg (99)) The density is: f GH (x λ,α,β,δ,µ) α β ( ) = λ 1 λ ( πα δ K δ α β ) λ λ.(δ + (x-µ) ) (λ-½)/ K λ-½ (α(δ +(x-µ) ) -½ )exp(β(x-µ)), ( + )dy ν 1 where K ν (z) = ½ y exp 1 z( 1 y y ) 0 is the modified Bessel function of the third kind. Parameters: α > 0, shape, 0 β < α, skewness, λ R, class of the distribution, µ R, location, δ > 0, scale. Reparameterise: replace α, β by ξ = (1 + δ α β ) -½, χ = ξβ/α, so 0 χ < ξ < 1. ξ and χ are invariant under X ax + b. 13

14 l = 1: Hyperbolic distribution Then K ½ (z) = (π/z) ½ e -z, and f H (x) = f H (x α,β,δ,µ) = f GH (x 1,α,β,δ,µ) ( α β ) = αδk 1 1 ( ) δ α β exp(-α(δ +(x-µ) ) ½ + β(x-µ)) Centred if µ = 0, symmetric if β = 0. Get special cases (ξ - χ parameterisation): ξ 0, Normal ξ 1, Laplace χ ±ξ, Generalised inverse Gaussian χ 1, Exponential l = -½: Normal Inverse Gaussian distribution f NIG (x α,β,δ,µ) = f GH (x -½,α,β,δ,µ) αδ ( ) K1 α δ + ( x µ ) ( ) = exp δ α β + β( x µ ) π δ + ( x µ ) Distribution of first hitting times of a -dim BM, starting at (µ,0) to R {δ}, drift (β, α β ), vol (1,1). 14

15 Variance-gamma Special case of the GH distribution f VG (x σ, ν, µ) = f GH (x ν Explicitly this becomes: f VG (x t,µ,ν) = Γ(µ t/ν, µ/ν) σ, θ 4 σ ν +, θ, 0, 0) ( ) µ t ν µ t ν 1 µ ( ) = x µ exp x ( ) ν Γ µ t ν ν σ, x > 0 Convolutions: NIG and VG processes: closed under convolutions f NIG (α,β,δ 1,µ 1 )*f NIG (α,β,δ,µ ) = f NIG (α,β,δ 1 +δ,µ 1 +µ ) 15

16 Lévy Processes and Time Changes X t a 1-dimensional semi-martingale: Representable as a time-changed Brownian motion. X t = w h(t), w t a Brownian motion h(t) a (stochastic) time change. Lévy Processes are semimartingales Variance gamma process: Brownian motion subordinated to Γ. x t ~ Γ(t,νt) a gamma variate, density f(x t ) = ( t ν) 1 x e t ν ν Γ x ( t ν) Then if X t is VG have: X t X VG (t σ,ν,θ) = θx t + σw x(t). ν, Normal inverse Gaussian process: Brownian motion subordinated to IG. x t ~ IG(δ, γ) an inverse Gaussian variate, density f IG (x δ, γ) = γ ( ) x-3/ exp δ x x γ δ π Then X t X IG (t δ, γ) = w x(t)., 16

17 Subordinator Approach to Lévy Processes Derivative, payoff H T at time T, value c t at time t. Martingale valuation: c t = E t [ c T ], where c T = H T p t /p T and p t is a numeraire, E t is expectations operator wrt p t. Suppose H T and p t depend on state variable S t, S t depends upon a Lévy process X t. Basis of research approach: Use subordinator representation of X t, X t = w(h t ). Iterated expectation: c t = E t [ c T ] = E t [ E t [ c T h t ] ]. Use numerical methods? Can value inner and outer expectations separately. 17

18 Various approaches Each expectation: MC, PDE, lattice? Appropriate, or not, for P or NP: Path dependent options A or NA: American or Bermudan options C or NC: Ease of calibration Overview of Outer methods MC Lattice PDE MC P, NA, NA NP, A, NC? Inner Lattice NP, ~A, C NP, A, C? PDE??? MC + MC: Ribeiro and Webber MC + lattice: Kuan and Webber (in progress) 18

19 MC + MC Generate a path H = {h j }for h t by Monte Carlo. Conditional on H, generate a path for X t. Straightforward. Trick is to apply speed-ups. Ribeiro and Webber: Stratified sampling + bridge VG process: working paper available NIG process: in progress. Get good speed-ups (up to a factor of ~800) MC + Lattice Kuan and Webber (in progress) Generate a path H = {h j }for h t by Monte Carlo. Conditional on H, generate a lattice for X t. Stock model: Easy to value European options, Barrier options Interest rate model: Can calibrate to a market term structure. 19

20 Simulating a Lévy Process Discretise time: 0 = t 0 < < t N = T. Set t j = t j+1 - t j. Stratified sampling: Suppose X ~ F X. u ~ U[0,1] uniform then (F X ) -1 (u) ~ F X. u i, i = 1,,M a stratified sample from U[0,1] then (F X ) -1 (u i ) is a stratified sample from F X. Stratified sample of U[0,1]? Let v i ~ U[0,1], i = 1,,M, then u i = (i +v i - 1)/M is a stratified sample of U[0,1]. Lévy process X t. Write X j ~ F tj. Suppose have a stratified sample X i,n, i = 1,,M. How to construct a set of paths 0 = X i,0,,x i,n with correct conditional properties? 0

21 The bridge Suppose X ~ F X, Y ~ F Y, Z ~ F Z, densities f X, f Y, f Z, and Z = Y + X Suppose have a draw z of Z, what is distribution X Z? Write f (X,Z) (x,z) for the joint density of X and Z. Have f X Z (x) = f (X,Z) (x,z)/f Z (z) = f X (x).f Y (z-x)/f Z (z) if X and Y = Z - X are independent. Our situation: X = F ti, Y = F tj, Z = F ti + t j, increments in a Lévy process X t. 1

22 Application to a Gamma Process Want a regular sample of a gamma process 0 = h 0,,h N. Are given h 0 = 0 and h N ~ Γ(t N,νt N ), want h i at time t i. Have z h N, x h i, y h N - h i. Have so f X (x) = ν f Y (z-x) = f Z (z) = ν t t i / x t i / ν 1 exp ( x / ν) ν Γ( ti / ν) ( ) ( tn ti z x ) / ν 1 ( tn ti )/ ν Γ ( t t ) / z x exp ν ν ν N z t / ν N Γ / ν 1 ( t / ν) N ( ) N i ( z ν) exp / tn ti ti f X Z (x) = 1 Γ + ν ν x 1 z ti tn ti z ν Γ Γ ν ν t t t i N i x z ν 1

23 The gamma bridge distribution Change variable to p = x/z, then p ~ B(t i /ν, (t N -t i )/ν) is a beta variate with parameters t i /ν and (t N -t i )/ν. ie, given h N and h 0, p h i h 0 ~ B(t i /ν, (t N -t i )/ν) hn h0 Procedure: i) Generate b i ~ B(t i /ν, (t N -t i )/ν) ii) Set h i = h 0 +b i.(h N - h 0 ) iii) Fill in the rest of the h j by binary chop. Stratify: i) Get h N by stratified inverse transform. ii) Get b i by stratified inverse transform Note: Can t do a fully stratified sample. Instead use low discrepancy sampling. 3

24 Results (for details see paper) Note: True se MC internal se. Estimate true se by replicating 100 times and finding actual sd of result. Comparison of two MC methods. Suppose: MC method 1 gives se σ 1 in time t 1, MC method gives se σ in time t. If: t is proportional to M σ is O(-1/) in M, then E 1, = (σ.t )/(σ 1.t 1 ) is the efficiency gain of method 1 over method. 4

25 Conclusions To use Lévy processes need good numerics. This area is very new with few results. Have outlined a general approach. Have results for part of this, working on other aspects. Much more to do. 5