Modelling Interest Rates with Lévy Processes

Transcription

1 Modelling Interest Rates with Lévy Processes Grace Kuan and Warwick Business School University of Warwick Coventry CV4 7AL, UK 1

2 Modelling Interest Rates with Lévy Processes Introduction: Modelling stock returns and interest rates Lévy Processes Lévy Processes and Interest Rate Models The Random Lattice Numerical Results

3 Modelling Interest Rates with Lévy Processes Introduction: Modelling stock returns and interest rates 3

4 Empirical Regularities Want to model stock returns and interest rates. What should a model provide? 1) Fat tailed distributions ) Infinitely divisible distributions 3) Finite moments 4) Multivariate extensions 5) Tractability Almost all literature is on asset return distributions. Relatively little in interest rates: Lots of research waiting to happen 4

5 Interest Rate Models Interest rate models, typically driven by Wiener processes. Vasicek: dr t = α(µ - r t )dt + σdz t. Problems: Can t match volatility smiles. Can t account for empirical stochastic volatility. Empirically, r t jumps. Generalise: incorporate jumps? dr t = α(µ - r t )dt + σdz t + J t dn t, where N t is a Poisson process, intensity λ t J t is a random jump size. But: How to specify J t? Seems arbitrary, how to provide a context? 5

6 Solution? Replace z t, Wiener process, by L t, Lévy process. Lévy processes Generalise Wiener processes. Theoretically tractable Very flexible. (Contain compound Poisson) Potential to account for smiles, etc 6

7 What does a stochastic process look like? Probability measure on state space Ω, Ω = { f : R + R} Ω is huge. Almost every f is totally discontinuous, eg, different on Q and R\Q Space of continuous sample paths Ω c = { f : R + R f cts} Ω. Sample space for Brownian motions Tiny subset of Ω. 7

8 Lévy processes Lévy Processes: Stationary with independent increments, Continuous in probability, (starts at zero, càdlàg). Convergence in probability: ε > 0, Pr[ X t - X s > ε ] 0 as s t If not true then numerical methods might be hard Example: if t Q, then Pr[ X t = 0] = ½, Pr[ X t = ] = ½, if t Q, then Pr[ X t = 0] = ½, Pr[ X t = 1] = ½, Not convergent in probability. As t 0 then p l-k 0 for l - k large ( x constant). 8

9 Modelling Price and Rate Movements Different approaches: 1) Write down the SDEs. Stochastic volatility + jumps? ) Specify the conditional distributions: i) Give the distribution itself, F(S t S 0 ) ii) Give the density, f(s t S 0 ), if it exists iii) Give the inverse distribution, F -1 (S t S 0 ) 3) Specify the process as a time change 4) Specify as a Lévy process, with its Lévy measure, ν(dx). May have a Lévy density k(x), ν(dx) = k(x)dx. Approximate as compound Poisson? 5) Specify the process by its time copula. Specify by backing out from prices? Specify a functional form, then fit to prices? 9

10 Lévy Processes and No-arbitrage Pricing If there is no arbitrage: Price processes can be transformed into Martingales under the pricing measure. A semimartingale can be transformed to a Martingale under some equivalent measure. No-arbitrage? Asset price processes, rates, etc, must be semimartingales. A Lévy process is a semimartingale: X t a Lévy process then X t = Y t + Z t where Y t and Z t are Lévy, Y t is a martingale with bounded jumps, Z t has FV paths on compact intervals. 10

11 Modelling Stock Returns with Lévy Process Stock returns are fat tailed. Model with Stochastic volatility? Jumps? Assume stock return process (under the EMM) is S t = S 0 exp( rt + λ t - wt) where λ t is a Lévy process, wt term makes S t e -rt a martingale, e w = E[exp(λ 1 )]. Lots of literature: Empirical fits to time series, volatility smile. Formulae for (European) options....some on pricing methods. Interest Rates? Very little literature as yet. Problem: In general, hard to price non-vanilla derivatives. How to value Bermudan/American puts? 11

12 Numerical Methods Need for numerical solutions: Few formulae for non-vanilla options. Valuing American and Bermudan options? Existing methods for option pricing: Explicit solutions Fourier transform methods Monte Carlo: Directly. As subordinated Brownian motion, Mean-variance mixture. As compound Poisson approximation. Method of Lines Lattice methods: Can price American and Bermudan options. Flexible for different payoffs. Can get high accuracy cheaply. 1

13 Modelling Interest Rates with Lévy Processes Lévy Processes Grace Kuan and 13

14 Lévy Processes: Definition and Examples Lévy Processes: Stationary with independent increments, Continuous in probability, (starts at zero, càdlàg). Lévy processes applied in finance: Generalised hyperbolic processes (GH), Normal inverse Gaussian (NIG), Variance Gamma (VG). VG and NIG are special cases of GH, but are unique subclasses closed under convolution, hence worth considering separately. Have explicit functions for the densities of these processes. Lévy processes are infinitely divisible. 14

15 Infinitely Divisible Measures and Processes A measure µ on R d is infinitely divisible if n there exists µ n such that µ = (µ n ) * = µ n * *µ n (n-fold convolution) If µ is infinitely divisible then µˆ (z) = (µˆ n(z)) n Infinitely divisible distributions: Lévy processes. GIG processes. On R d : Gaussian, Cauchy On R: Poisson, exponential, Γ. Can t be infinitely divisible if µ has bounded support, µˆ has zeros. Uniform distribution is not infinitely divisible. Every infinitely divisible distribution is the limit of a sequence of compound Poisson distributions. 15

16 Infinitely Divisible Measures µ is infinitely divisible on R d iff there exists a Lévy process (in law) with F X1 = X 1 X 0 = µ X t is unique up to identity in law. It has a càglàg modification. 16

17 The Lévy-Khintchine representation If X t is infinitely divisible then µˆ (z) = exp( φ(z) ), z R, with φ(z) = -½z Az + iz γ + d R (e iz x iz x.1 D (x))ν(dx). where A is a symmetric non-negative definite matrix, γ R d, ν is a measure on R d, such that ν{0} = 0, d R ( x 1)ν(dx) ) <, D = { x x 1} is the unit ball in R d, ν is not necessarily a probability measure. Need not be integrable. i) (A,ν,γ) is unique ii) All (A,ν,γ) give infinitely divisible distributions (A,ν,γ) is the generating triplet of µ. 17

18 Notes: If µ has generating triplet (A,ν,γ) then µ 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 + d R (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 + d R x( c(x) - 1 D (x) )ν(dx). Write (A, ν, γ c ) c. 18

19 Centre and Drift Suppose that x 1 x ν(dx) <, then can set c(x) = 0 and φ(z) = -½z Az + iz γ 0 + d R (e iz x - 1)ν(dx). This γ 0 is the drift of µ. Suppose that x > 1 x ν(dx) <, then can set set c(x) = 1 and φ(z) = -½z Az + iz γ 1 + d R (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. 19

20 Observations µ compound Poisson then A = 0, ν = cσ, γ 0 = 0. Jump times are exponential, mean c, each jump size is distributed as cν. 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. 0

21 Connections X t additive process (in law) then, for all t, F Xt is infinitely divisible. (A t,ν t,γ t ) triple for µ t = F Xt then X t is additive (in law) iff (A 0,ν 0,γ 0 ) = (0,0,0) for 0 s t <, z A s z z A t z, ν s (B) ν t (B), as s t, z A s z z A t z, ν s (B) ν t (B), γ s γ t. Effectively: Infinitely divisible distribution Lévy process Generating triple (A,ν,γ) 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) =. 1

22 Sample path properties 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.

23 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)), ( 0 + )dy is the modified Bessel function of the third kind. ν 1 where K ν (z) = ½ y exp 1 z( 1 y y ) 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. 3

24 λ = 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 λ = -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). 4

25 Representation of GH distributions Represent as mixtures with GIG distributions. Generalised inverse Gaussian distribution λ ( δ γ) 1 f GIG (x λ, δ, γ) = 1 exp δ x + γ x, K λ ( γδ) x > 0, if λ > 0 then δ 0, γ > 0 if λ = 0 then δ > 0, γ > 0 if λ > 0 then δ > 0, γ 0 x λ-1 ( ( ) Reciprocal inverse Gaussian f RIP (x δ, γ) = f GIG (x ½, δ, γ) Inverse Gaussian distribution (λ = -½) f IG (x δ, γ) = f GIG (x -½, δ, γ) = γ ( ) x-3/ exp δ x x γ δ π Gamma (δ = 0) f Γ (x λ, γ) = f GIG (x λ,0,γ) = Γ(λ,½γ ) ν x ν 1 Γ(ν,a) = a Γ( ν) exp(-ax), ν > 0. Reciprocal Gamma (γ = 0), x > 0. f Γ -1(x λ, δ) = f GIG (x -λ,δ,0) = Γ -1 (λ,½δ ), λ > 0. 5

26 Relationship to Mean-Variance Mixtures Mean-Variance mixture: location µ, correlation matrix, drift β, mixing distribution F. µ,β R d, symmetric positive definite, = 1. Let u ~ F. Write N for normal distribution function. Then mean-variance mixture M(µ,β, ; F) has M X u ~ N(µ + uβ, u ). Characteristic function of M: Mˆ (z) = e iz µ Fˆ (z β + ½z z) Convolutions: M(µ,β,,F) n = M(nµ,β, ; F n ) eg, Generalised hyperbolic distribution: F GH (λ,α,β,δ,µ, ) = M(µ,β, ; N - (λ,δ,γ)) γ = α - β β, N - (λ,δ,γ) is generalised inverse Gaussian. 6

27 Mixtures of Distributions Write n(x µ, ) for the normal density function. Then (d = 1) f GH (x λ,α,β,δ,µ) = 0 n(x µ+βu,u)f GIG(u λ,δ, α β )du f H (x α,β,δ,µ) = 0 n(x µ+βu,u)f IG(u δ, α β )du f NIG (x α,β,δ,µ) = 0 n(x µδ u,δ u)f IG (u α,β)du f VG (x) = 0 n(x θt,σ s)f GIG (s t,0,ν)ds Student-t distribution Student-t: mixture of normal and inverse gamma ( 1 Γ ( 1+ f )) f t (x f) = x 1 + f 1+ fπγ ( ) ( ) ( ) 1 f f = 0 n(x 0,u)f GIG(u -½f, f)du 7

28 The NIG Lévy process NIG Lévy density: ( ) k NIG (x) = βx K1 α x e π δα x NIG Characteristic function: ( ) α β φ NIG (z α,β,tδ) = exp( -tδ ( β + iz) α ) ) No Gaussian component: is pure jump (plus drift) Has finite moments: No problems with big jumps Set ρ = β/α, then δρ κ 1 = µ + ( ) 1 κ = κ 3 = α 1 ρ δ, ( 1 ρ ) 3 3δρ,, skew = 3ρ α ( 1 ρ ) 5 αδ( 1 ρ ) 1 4 3δ κ 3 ( 1+ 4ρ ) 3( 1+ 4ρ ) =, kurtosis = α3( 1 ρ ) 7 αδ( 1 ρ ) 1. 8

29 Time changed Brownian motion X t a 1-dimensional semimartingale: Representable as a time-changed Brownian motion. X t = z A(t), z t a Brownian motion A(t) a time change. z t is subordinated to A(t) NIG: Brownian motion subordinated to IG. VG: Brownian motion subordinated to Γ. 9

30 Relationship to Mean-Variance Mixtures Mean-Variance mixture: location µ, correlation matrix, drift β, mixing distribution F. µ,β R d, symmetric positive definite, = 1. Let u ~ F. Write N for normal distribution function. Then mean-variance mixture M(µ,β, ; F) has M X u ~ N(µ + uβ, u ). Characteristic function of M: Mˆ (z) = e iz µ Fˆ (z β + ½z z) Convolutions: M(µ,β,,F) n = M(nµ,β, ; F n ) eg, Generalised hyperbolic distribution: F GH (λ,α,β,δ,µ, ) = M(µ,β, ; N - (λ,δ,γ)) γ = α - β β, N - (λ,δ,γ) is generalised inverse Gaussian. 30

31 Mixtures of Distributions Write n(x µ, ) for the normal density function. Then (d = 1) f GH (x λ,α,β,δ,µ) = 0 n(x µ+βu,u)f GIG(u λ,δ, α β )du f H (x α,β,δ,µ) = 0 n(x µ+βu,u)f IG(u δ, α β )du f NIG (x α,β,δ,µ) = 0 n(x µδ u,δ u)f IG (u α,β)du f VG (x) = 0 n(x θt,σ s)f GIG (s t,0,ν)ds Student-t distribution Student-t: mixture of normal and inverse gamma ( 1 Γ ( 1+ f )) f t (x f) = x 1 + f 1+ fπγ ( ) ( ) ( ) 1 f f = 0 n(x 0,u)f GIG(u -½f, f)du 31

32 The Variance-Gamma Process: (Madan etc) Γ ν t a Γ-distribution, mean rate t, variance rate νt. Its density is f Γ (x t,ν) = ( t ν) 1 x e t ν ν Γ x ( t ν) Laplace transform E[ exp(-λγ ν t ) ] = (1 + λν) -t/ν. Let z t be a Wiener process. Define: X VG (t σ,ν,µ) = µγ ν ν t + σz. Γt Subordinated Brownian motion. The characteristic function of X VG (t) is: E[ exp(-izx VG t ) ] = (1 - iµνz + ½σ νz ) -t/ν. Lévy density is: ( ) k VG (x) = exp µ x σ µ exp 1 + x ν x σ σ ν ν The density function of X t VG is f t VG (x σ, ν, µ) = ( µ x σ ) exp t ν ν πσγ t 1, ( ) ( ) ν 4 x 1 K t 1 x t () σ ν + µ σ ν+µ ν σ ν. 3

33 Modelling Interest Rates with Lévy Processes Interest Rate Models and Lévy Processes Grace Kuan and 33

34 Interest Rate Modelling Use framework of Eberlein and Raible. Denote bond prices by P(t,T), accumulator account numeraire, p t = exp( (0,t] r s ds ). Let L t be a Lévy process, ν its Lévy measure, such that x > 1 e λx ν(dx) <, λ < M, some M. E & R: under the equivalent martingale measure exp ( s,t) dls P(t,T) = P(0,T)p 0 t T E exp σ( s,t) dl T σ where 0 t T T *, a maximum time horizon, σ > 0, deterministic, σ(t,t) = 0, twice-differentiable, bounded. (Need to impose mild restrictions on L t ) The process for L t is specified under risk-neutrality. 0 s 34

35 The process for P(t,T) Laplace exponent, e κ(u) = E[e ul 1 ], κ(u) = bu + ½cu + R (e ux ux)ν(dx). Fact: E[ exp( - (0,t] f(s)dl s ) ] = exp( - (0,t] κ(f(s))ds). (mild conditions on f). Then E[ exp( - (0,t] σ s,t dl s ) ] = exp( - (0,t] κ(σ s,t )ds). Find that P(t,T) = P(0,T)exp( (0,t] (r s - κ(σ s,t ))ds + (0,t] σ s,t dl s ). κ gives the compensator for L t. The bond process is dp( t,t) P( t,t) = r r dt + ( ½cσ (t,t) - κ(σ(t,t) )dt + σ(t,t)dl t + e σ(t,t) L t σ(t,t) L t. If L t = W t, a Wiener process, then κ(u) = ½u, c = 1. Reduces to usual HJM case. 35

36 The extended generalised Vasicek process E & R show that if r t is Markov σ is time homogenous (stationary) then σ(t,t) σ(t-t) = σˆ. 1 e a ( T t ) a Vasicek volatility is case a > 0., a > 0 = σˆ.(t - t), a = 0, Given initial term structure, forward rates f(0,t), then dr t = a(θ(t) - r t )dt + σˆ dl t, where ρ(t) = a 1 f (0,t) - κ (σ(t-t)) σˆ. e at a + κ(σ(t-t)) + f(0,t). E & R investigate the case where L t is hyperbolic. 36

37 NIG case Kuan and Webber investigate pricing when L t is NIG. Construct a lattice method, calibrating to the initial term structure. L t represented as time-changed Brownian motion, L t = β.h t + z ht, where h t ~ IG[ δt, α β ] is inverse Gaussian with density f IG (x δ,γ), x > 0, f IG (x δ,γ) = δ π x-3/ exp( δγ - ½(δ x -1 + γ x). where γ = α β. The short rate process is dr t = a(θ(t) - r t )dt + σdl t, where a and σ are constants and θ(t) is deterministic. Can choose θ(t) to fit initial term structure. 37

38 Numerical methods of option valuation. Can use Monte Carlo for European options. Evolve L t, L t = β.h t + z h t, where h t ~ IG[ δ t, α β ], and use Euler discretisation to evolve r t, r r = a(θ(t) - r t ) t + σ L t. Problems i) Hard to use Monte Carlo for American options ii) Can t easily match initial term structure iii) Hard to ensure arbitrage-free pricing, (eg that put-call parity is satisfied). Alternative: use a lattice method. Can do American options easily. Can easily match to an initial term structure. Get arbitrage-free pricing. 38

39 Lattice Methods Option value : expected discounted payoff (accumulator numeraire), V 0 = E 0 [ exp( T 0 r s ds ).V T ]. At time T, V T = H T is just the option payoff. r t and H T : functions of the state variables. Approximate continuous time state variables λ t by discrete time and space state variables, λ t, time step t, space step λ. Require convergence to continuous time as t 0. λ t takes values λ i,j at time t i = i t, level λ j = j λ. Require λ i,j+1 - λ i,j = λ, for all i,j λ i+1,j = λ i,j λ j, for all i,j λ t0 = λ 0,0 = 0. 39

40 Branching probabilities: Process is stationary with independent increments. Set p k = Pr[ λ ti+1 = λ i+1,j+k λ ti = λ i,j ], -K < k < K. Choose p k so that λ t approximates λ t. Obtaining prices on a lattice: Calculate payoff at terminal time. Iterate back step by step. Continuous time: V t = E t [ V t+ t.exp( - t+ t t r s ds ) ]. On the lattice: (eg trinomial case, K = 1). For T = t N, set c N,j = H T (j λ) c i-1,j = e -r t (p 1 c i,j+1 + p 0 c i,j + p -1 c i,j-1 ) Option value is c 0,0. 40

41 Problems Evolving forward: (obtaining probabilities). Base on true moments, or good approximations. (Avoid using Euler discretisation). Discounting back. If r t not constant: Must use Brownian Bridge discounting. Sample low probability regions? Prune to get rid of wasted branches. Non-uniform convergence? Use terminal correction. 41

42 A Trinomial Lattice State variable evolved on a trinomial lattice. Discretise calendar time as t i = t 0 + i t, i = 0,...,N. Nodes on the lattice are labelled (i,j), where i labels time, j labels level, j = -N,...,N t 0 t 1 t t 3 t 4 t 5 t 6 Branching from node (i,j) is to { (i+1, j-1), (i+1, j), (i+1, j+1) } with probabilities p d, p m, p u. 4

43 At a typical node Single state variable r t. Trinomial lattice: r t branches up or across or down. r u p u r p m p d r m r d Time t Time t + t Ordinary idea: Choose probabilities p u, p m and p d so that the discrete variable has the same expected value and the same variance as the continuous variable. Doesn t work for Lévy processes. 43

44 Problem Wiener process: defined by its 1st and nd moments. Lévy process: isn t. Define a lattice directly? Requires very high order, very wide branching (Kellezi and Webber) Alternative: (Kuan and Webber) Combine Monte Carlo with lattices. Generate random lattices. On each lattice, compute option value Take the average from many random lattices. 44

45 Modelling Interest Rates with Lévy Processes The Random Lattice Grace Kuan and 45

46 The Random Lattice Evolve a lattice up to time calandar T in N time steps. Calandar time step is t = T/N. Set up a trinomial lattice using random step size. Time-changed time step is τ i with τ i+1 - τ i = τ i ~ IG[ δ t, α β ] Key point: Conditional on τ i, L τi is normal, L τi ~ N[ -β τ i, τ i ] Use a modified Hull and White lattice, with variable time step. Two stage method: First: build a tree for the process Second: dr t = -ar t dt + σdl t Incorporate the drift, θ(t), by offsetting the nodes of the tree. 46

47 The Stage One Tree Use time step t = T/N, base moments on τ i. set r = σ κ τ, where τ = max i { τ i }, κ > 1, (so that probabilities remain positive). Construct a tree with: time steps: 0, t, t, 3 t,, N t = T. space steps: -j m r,, - r, - r, 0, r, r,, j m r. r 4 r 3 r r r - r 0 - r -3 r -4 r 0 t t 3 t 4 t 5 t 5 t t The node for time i t, rate j r, is labelled (i,j). 47

48 Probabilities At each node have three branches. Assign a probability to each branch: p u, p m, p d. At each node fix the probabilities so that the discrete density on the tree matches the continuous density of r. Conditional mean is: m i,j = E[ r (i,j) ] = -a(j r) t + σβ τ i, Conditional variance is: v i,j = var[ r (i,j) ] = σ τ i, Solve to get: p u = 1 (m i,j + v i,j + m i,j r), r 1 r 1 r p m = (m i,j + v i,j ), p d = (m i,j + v i,j - m i,j r). (Have altered branching when j = j max. In practice lattice truncated before hitting level j max.) 48

49 Second Stage: Incorporate the Drift Given initial term structure to match to, P i = value at time 0 of pdb maturing at time i t. R i = the spot rate for time i t, P i = e -i t.r i. Construct lattice for r by offsetting lattice for r. ie, at time i t displace the nodes at j r to α i + j r. Use continuous time α? Doesn t work on lattice. Write Q i,j for value at time 0 of pure security for state (i,j). Pays out 1 at time i t if r has value r i,j = α i + j r, 0 otherwise. 49

50 Forward induction. Iterate forward finding successive α i. Have two recurrence relationships. Replication of pdb value: P m+1 = n m Q m,j.exp( -(α m + j r) t ) j= n m where j = -n m,,n m for the nodes at time m t. Iterated expectation: Q m+1,j = Q m,k.q k,j.exp( -(α m + k r) t ) k where q k,j is the probability of going from node (m,k) to node (m+1,j). Given α m-1 and Q m,j, obtain α m and Q m+1,j. But Q 0,0 = 1 and α 0 = R 1. Now finish the tree. 50

51 Valuation On the lattice, if know payoff V N,j at time T = t N, then option value V at time 0 is N V = j= N Q N,j.V N,j. Using the random lattice. Generate M random lattices, get M option values V k, k = 1,,M. Set option value to be V = 1 V k. M k= 1, Κ, M 51

52 Results Compute caplet prices on the lattice, benchmark using a Monte Carlo method. Caplet: Maturity time, T = 1, Libor has tenor 0.5. Payoff is H T = max(0, L T - X) where L T is Libor at time T, X is exercise rate. Monte Carlo Method: Benchmark to the term structure. Recovers well (to 4dp) with N = 100, M = For caplet: To get underlying libor, simulate further 0 paths from time 1 to time 1.5. Use Michael, Schucany and Hass (76) algorithm to generate IG variates. 5

53 Caplet Prices on the Lattice Random lattice with N = 40 time steps to 1.5 years. Parameter values: (a little arbitrary...) NIG: α = 10, β =, δ = 10, Short rate: a = 0.01, σ = 0.0, r 0 = Initial term structure flat at Strike, X Monte Carlo M = 5, (7.E-06) (9.9E-06) (5.5E-06) Lattice, M = 4, (9.3E-07) (1.6E-06) (1.1E-06) Lattice, M = 5, (8.7E-07) (1.4E-06) (8.9E-07) Lattice, M = 6, (7.8E-07) (1.3E-06) (8.E-07) Lattice, M = 7, (7.E-07) (1.E-06) (7.8E-07) Lattice, M = 8, (6.8E-07) (1.0E-06) (7.3E-07) Black s implied volatility Run times: Random lattice: 57 secs (M = 8000) Monte Carlo: 47 secs (M = 5000) Monte Carlo used 100 time steps over 1.5 years. Both implemented in C on a Unix Ultra Enterprise

54 Caplet Prices on the Lattice Parameter values: (a little arbitrary...) NIG: α = 36, β =, δ = 10, Short rate: a = 0., σ = 0.0, r 0 = Initial term structure flat at Strike, X Lattice, M = (1.9E-05) (1.7E-05) (1.0E-05) (3.4E-06) (6.E-07) Lattice, M = (1.6E-05) (1.5E-05) (9.4E-06) (3.3E-06) (5.4E-07) Lattice, M = (1.5E-05) (1.4E-05) (8.9E-06) (3.3E-06) (5.E-07) Lattice, M = (1.4E-05) (1.3E-05) (7.9E-06) (.8E-06) (4.7E-07) Lattice, M = (1.3E-05) (1.E-05) (7.5E-06) (.7E-06) (4.5E-07) Blacks vol (M = 8000) Get pronounced volatility skew. 54

55 Caplet Prices on the Lattice Parameter values: (a little arbitrary...) NIG: α = 36, β =, δ = 10, Short rate: a = 0., σ = 0.0, r 0 = Initial term structure flat at Prices for various times to maturity: Strike, X T = (1.6E-06) (5.3E-07) (3.81E-07) (3.4E-07) 5.38E-08 (6.78E-07) T = (3.4E-06) (1.5E-05) (9.37E-06) (1.5E-07) 1.0E-06 (1.06E-08) T = (1.93E-05) (1.86E-05) (1.13E-05) (3.13E-06) 7.0E-06 (4.89E-07) T = (1.7E-05) (1.60E-05) (1.11E-05) (3.37E-06) (6.0E-07) Blacks implied vols. for various times to maturity: Blacks Implied Volatilitiesl Strike, X T = T = T = T =

56 Plot of Blacks Implied Volatilities Black's Implied Volatility Surface Implied volatility Strike index 3 T= T=1 4 T=0.5 Time to maturity 5 T=0.5 56

57 Effect of blipping parameter values Base case Blip alpha Blacks implied volatility Blip beta Blip delta Strike 57

58 Effect of changes in Parameter Values 58

59 Observations Random lattice: More accurate than Monte Carlo. Similar run times at greatest accuracy. Run times for random lattice: M run time (secs) Can get strong volatility skews. Effect of blipping parameter values α 40, Tilt down, short end remains stable β.5, Tilt down, long end remains stable δ 1, Shift upwards Introduction 59

60 Conclusions Demonstrated that i) NIG processes feasible in E & R context ii) Can implement on a lattice Can calibrate to An initial term structure. Potentially, to Blacks implied volatilities. Lattice method: in principle Implementable for any process expressible as time-changed Brownian motion. No reason not to explore use of NIG processes and other Lévy processes in interest rate modelling. Introduction 60