High order Cubature on Wiener space applied to derivative pricing and sensitivity estimations

Transcription

1 1/25 High order applied to derivative pricing and sensitivity estimations University of Oxford 28 June 2010

2 2/25 Outline

3 3/25 Assume that a set of underlying processes satisfies the Stratonovich SDE d dy t = V i (Y t ) dbt i i=0 for some vector fields V 0,...,V d on R N, (Bt 1,..., Bd t ) d-dimensional Brownian motion, with Bt 0 = t. Then the expected payoff of a European derivative can be expressed as an integral on the Wiener (Ω, F, P) - where Ω = C0 0([0, T ], Rd ), F is the Borel σ-field and the Wiener measure P: E[f (Y T )] = f (Y T (ω))p(dω). Ω

4 4/25 (Kusuoka, Lyons, Victoir): replace P with a measure Q on the Wiener supported by finitely many paths {ω 1,..., ω M } C 0 0 ([0, T ], Rd ), i.e. Q = M λ i δ ωi for some positive weights λ i, such that E P [f (Y T )] = f (Y T (ω))p(dω) E Q [f (Y T )] = λ i f (Y T (ω i )) Ω i=1 using the notation ωi 0 (t) = t. Analogy: numerical integration in finite dimensional s: f (x)φ(x)dx λ i f (x i ) D exact for polynomials up to degree n. Questions: 1 conditions for/order of convergence? 2 how to find the weights and paths? (how to interpret Y T (ω i )?)

5 5/25 Criteria for Cubature measure There exist a measure Q 1, supported by finitely many piece-wise linear paths, such that E P [ 0<u 1 < <u k <1 db i 1 u1 db i k uk ] }{{} B I 0,1 := = E Q1 [B I 0,1 ] for all multi-indices (i 1,...) = I A n. Rescaling property Rescaling the paths in the support of Q 1, we can construct a measure Q t = λ j δ ωj,t such that E P [B0,t I ] = E Q t [B0,t I ] for all multi-indices (i 1,...) = I A n.

6 6/25 Local approximation For smooth f with bounded derivatives, (E P E Qt )[f (Y t )] = (E P E Qt )[R0,t n ] Ctγ where C depends on f, V and A n, γ depends on A n. (Proof: based on stochastic-taylor expansion and the rescaling property of BM.) Global approximation Given a partition D = {0 = t 0 < t 1 < < t m = T }, we define the global cubature measure Q D Q D = M j 1,...,j m=1 λ j1 λ jm δ ωj1, t 1 ω jm, t1 supported by the rescaled and concatenated paths.

7 7/25 Classical global cubature measure - smooth payoff T t 2 t 1 t 0 Y t0

8 8/25 Classical global cubature measure - uneven step-size T t 2 t 1 t 0 Y t0

9 9/25 Algebraic approach to constructing cubature formulae Algebraic representation of paths: The truncated signature S (n) s,t (B) of B on [s, t] is an element in the truncated tensor algebra T (n) generated by the letters e 0,... e d : S (n) 0,t ( B) := (i 1,...,i k ) A B (i 1,...,i k ) 0,t e i1 e ik Properties of the signature The truncated log-signature lies in the truncated Lie algebra L (n) generated by the brackets of e 0,...,e d For any Lie element l in L (n) there exists a piece-wise linear path with log-signature l

10 10/25 Algebraic approach to constructing cubature formulae Constructing cubature formulae The moment matching condition is equivalent to E P [ S (n) 0,1 ( B) ] = E Q1 [ S (n) 0,1 ( B) ] Furthermore E P [ S (n) 0,1 ( B) ] = exp (n) e d e i e i 2 i=1 The moment matching problem is to find Lie elements l j and weights λ j such that exp (n) e d e i e i = λ j exp (n) ( ) l j. 2 i=1 j Once the l j s are found, the corresponding piece-wise linear paths can be constructed.

11 11/25 Cubature formulae constructions degree=2γ 1 dimension support size ref. 3 n 2 n Lyons & Victoir Lyons & Victoir 5 2 8, 10 Gyurkó & Lyons Lyons& Victoir 5 n 2 GaussCub(n, 5) Lyons & Victoir Gyurkó & Lyons Litterer Gyurkó & Lyons Gyurkó & Lyons

12 12/25 Properties High order weak approximation Cubature measure corresponding to dimension d and order γ is re-scalable to any time-step Cubature measure is supported by piece-wise linear paths, Y T (ω i ) is interpreted as an ODE If the support of the global cubature measure is small enough, the exact Q D -expectation can be worked out - there is no approximation error due to variance The support of Q D grows exponentially with the levels - improvements: MC-sampling, other partial re-sampling (Ninomiya) recombination - deterministic support reduction, polynomial support growth (Litterer & Lyons) repeated cubature - computational cost optimized for a given error tolerance (Lyons & Gyurkó)

13 13/25 Outline

14 14/25 - smooth terminal condition Technical detour: Given an ODE dy s = d V i (Y s)dω i (s), Y 0 = y, s [0, t] i=0 driven by a piece-wise linear path ω = (ω 0, ω 1,..., ω d ). Then there exists a vector field W spanned by the (high order) brackets of V 0,...,V d, such that Y t Z 1 where dz u = W (Z u)du, Z 0 = y. (*) W can be directly derived from the log-signature of the paths supporting Q. Test SDE: ds t = as t dt + bs t db t da t = S t dt Note: the corresponding W is always of the form W (x, y) = (c 1 x, c 2 x) where the constants c 1 and c 2 depend on the cubature path. I.e. the exact solution to (*) is known.

15 15/25 Numerical examples - smooth terminal condition 1D problem with smooth payoff 1 Degree Degree 5 Degree 7 Degree Degree 11 (1/n) 10-5 (1/n) 2 (1/n) (1/n) 4 (1/n) Error Error Number of steps Number of ODEs Terminal condition: f (s, a) = a 3

16 16/25 Numerical examples - smooth terminal condition 1D problem with smooth payoff 2 Degree Degree 5 Degree Degree 9 Degree 11 (1/n) 10-5 (1/n) 2 (1/n) (1/n) 4 (1/n) Error Error Number of steps Number of ODEs Terminal condition: f (s, a) = a 2 s

17 17/25 Numerical examples - non-smooth payoff 1D problem with non-smooth payoff 10 0 Degree 5, γ=2 Degree 7, γ= Degree 9, γ=3 Degree 11, γ= Degree 5, γ=2 Degree 7, γ= Degree 9, γ=3 Degree 11, γ= Error Error Number of steps Number of steps Process: BM, Terminal conditions: f (s) = (s k) + and g(s) = 1 s>k

18 18/25 Outline

19 19/25 Extension Aims: Improving the convergence when the payoff is only Lipschitz. Extending the method to piece-wise smooth payoffs, not excluding discontinuous payoffs. Key ideas: If we are far enough in from the discontinuity, the payoff looks smooth, we can exploit the accuracy of a single step high degree cubature by jumping straight to maturity If we can see the discontinuity, we refine the scale approaching the maturity with geometrical speed Detecting the closeness of discontinuity: compare a coarse-scale estimate to a finer-scale estimate Key ingredients: High degree cubature formulae, Recombination.

20 20/25 T t t 6 7 t 5 t 4 t 3 t 2 t 1 t 0 Y t0

21 21/25 Numerical examples Kusuoka s uneven stepsize Degree 5, γ=2 Degree 7, γ=2.5 Degree 9, γ=3 Degree 11, γ= Degree 5, γ=2 Degree 7, γ=2.5 Degree 9, γ=3 Degree 11, γ= Error 10-4 Error Number of steps Number of steps

22 22/25 Numerical examples Error 10-8 Error Degree 9, θ=0.25 Degree 11, θ=0.25 Degree 23, θ=0.3 Degree 23, θ= Degree 9, θ=0.25 Degree 11, θ=0.25 Degree 23, θ=0.3 Degree 23, θ= Number of levels Number of levels

23 23/25 Boundary value problems T t 0 Y t0 K δ K

24 24/25 Outline

25 25/25 weak approximations (Kusuoka, Lyons, Victoir, Nonomiya, Litterer, etc.) sensitivity estimations (Teichmann) SPDEs (Crisan, Lyons, Teichmann) BSDEs (Manolarakis, Crisan) Data compression - representing data by its expected signature (Lyons)