Forward Monte-Carlo Scheme for PDEs: Multi-Type Marked Branching Diffusions

Transcription

1 Forward Monte-Carlo Scheme for PDEs: Multi-Type Marked Branching Diffusions Pierre Henry-Labordère 1 1 Global markets Quantitative Research, SOCIÉTÉ GÉNÉRALE

2 Outline 1 Introduction 2 Semi-linear PDEs 3 Non-linear Monte-Carlo algorithms 4 New method: Marked branching diffusions 5 CVA 6 Multi-type Marked branching diffusions

3 Contents Stochastic representation of semi-linear PDEs: Counterparty risk (and American options). Review of Numerical Methods: Brute-force Monte-Carlo of Monte-Carlo" method (with nested simulations). BSDEs. Gradient representation. Branching diffusions. Marked branching diffusions. Numerical results. Multi-type marked branching diffusions: Extensions to fully non-linear PDEs [joint work with X. Tan, N. Touzi].

4 Semi-linear PDEs: CVA examples Two types of PDEs: t u + Lu + ru + r 1 u + = 0, u(t, x) = ψ(x) : PDE1 t u + Lu + ru + r 1 M + r 2 M + + r 3 u + = 0 : PDE2 t M + LM + r 4 M = 0, M(T, x) = ψ(x) Toy example: t u + Lu βu + = 0, u(t, x) = ψ(x)

5 A brut-force algorithm Feynman-Kac s formula: T u(t, x) = E P t [ψ(x T )] βe P t [u + (s, X s )]ds t Approximation (β is small) 1 : u(t, x) E P t [ψ(x T )] n i=1 ( +] ti βe P t [ E P t i [ψ(x T )]) Leads to Monte-Carlo of Monte-Carlo" approach (with nested simulations). Complexity: O(N 2 ). Can we design an algorithm with complexity O(N)? 1 exact for PDE2.

6 1-BSDE [Pardoux-Peng] 1-BSDE: dx t = b(t, X t )dt + σ(t, X t ).dw t dy t = βy + t dt + Z t σ(t, X t ).dw t Y T = ψ(x T ) where (Y, Z ) adapted processes. Unique solution: (Y t = u(t, X t ), Z t = σ(t, X t ) x u(t, X t )). Discretization scheme (Y ti 1 is forced to be F ti 1 -adapted): ( ) Y ti 1 = E P 1 (1 θ)β t i t i 1 [Y ti ] 1 E P ti 1 [Y ti ]> θβ t E P i ti 1 [Y ti ]<0 Needs the computation of E P t i 1 [Y ti ] by regression methods. Quite difficult and time-consuming, specially for multi-asset portfolios.

7 Gradient representation [Talay-al], [Jourdain] Let u be the solution of t u σ2 (t, x) 2 x u + f (u) = 0 u(t, x) = ψ(x) By differentiating w.r.t. x : ( t + (σ x σ) x + 1 ) 2 σ2 (t, x) x 2 + f (u) = 0 Interpreted as a Fokker-Planck PDE: u(t, x) = ψ (a)da E P t [1(X a T S)e T t f (u(t +t s,xs a))ds ] R + dxs a = σ(t + t s, Xs a )db s + (σ 2 σ) (T + t s, Xs a )ds

8 Branching Diffusions [MCKean] Branching diffusions first introduced by McKean for KPP type PDE: ( ) t u(t, x) + Lu + β p k u k u = 0 in R d R + k=1 u(t, x) = ψ(x) in R d Restrictive algebraic non-linearity: f (u) p k u k, p k = 1, 0 p k 1 k=0 k=0 Feynman-Kac s formula: u(t, x) = E t [1 τ>t ψ(x T )] + p k E t [u k (τ, X τ )1 τ<t ] k=0

9 Probability interpretation Let a single particle starts at the origin, performs an Itô diffusion motion on R d, after a mean β exponential time dies and produces k descendants with probability p k. Then, the descendants perform independent Itô diffusion motions on R d from their birth locations, die and produce descendants after a mean β( ) exponential times, etc. This process is called a d-dimensional branching diffusion with a branching rate β > 0. Stochastic representation [strong Markov property]: N T u(t, x) = E t [ ψ(zt i )] i=1

10 Marked branching diffusions [PHL] Algebraic semi-linear PDE: t u + Lu + Φ(u) = 0 with Φ(u) = β(f(u) u) and F(u) = M k=0 a ku k. From Feynman-Kac s formula: u(t, x) = E t [1 τ>t ψ(x T )] + E t [F(u τ )1 τ<t ] Recursively solved in terms of multiple exp. random times τ i : u(t, x) = E t [1 τ0 >T ψ(x T )] +E t [F ( E τ [1 τ0 >T ψ(x T )] + E τ [F(u τ2 )1 τ2 <T ] ) 1 τ<t ]

11 Marked branching diffusions (2) Stochastic representation: N T M u(t, x) = E t [ ψ(zt i ) i=1 k=0 ( ak p k ) ωk ]

12 Marked branching Brownian motion (2) Algebraic PDE type 2: t u(t, x) + Lu(t, x) + β(f (E t [ψ(x T )]) u(t, x)) = 0 Feynman-Kac s formula: u(t, x) = E t [1 τ>t ψ(x T )] + E t [F (E τ [ψ(x T )]) 1 τ<t ] As compared to the previous section, we have the term F (E τ [ψ(x T )]) 1 τ<t. This term can be computed using the previous algorithm by imposing that the particle can default only once. This corresponds to the first three diagrams in Fig. (1).

13 Convergence Proposition 1 Let us assume that ψ L (R d ). Set q(s) := M k=0 a k ψ k 1 sk. 1 Case q(1) > 1: We have u L ([0, T ] R d ) if there exists X R + such that X 1 ds q(s) s = βt In the particular case of one branching type k, the sufficient condition for convergence reads as a k ψ k 1 ( 1 e βt (k 1)) < 1 2 Case q(1) 1: u L ([0, T ] R d ) for all T.

14 Optimal probabilities By assuming that ψ L (R d ), the expectation in (1) can then be bounded by ( ) M ωk ( ak û(0, x) E 0,x [ ψ N(ω) p ] = ψ ˆP T, ln a ) k ln ψ k 1 k=0 k p k p k = a k ψ k M i=0 a i ψ i

15 Bias Proposition 2 Let us assume that F(v) and F(v) are two polynomials satisfying (Comp), the sufficient condition in Prop. 1 for a maturity T and F (x) x + F (x) We denote v and v the corresponding solutions of (PDE(F, F )) and v the solution of (PDE(v + )). Then v v v

16 Numerical Experiments We have implemented our algorithm for the two PDE types t u + Lu + β(f(u) u) = 0, u(t, x) = 1 x>1 : PDE1 and t u + Lu + β(f(e t [1 XT >1]) u) = 0, u(t, x) = 1 x>1 : PDE2 L is the Itô generator of a geometric Brownian motion with a volatility σ BS = 0.2 and the Poisson intensity is β = 0.05.

17 Numerical Experiment 1 N Fair(PDE2) Stdev(PDE2) Fair(PDE1) Stdev(PDE1) Table: MC price quoted in percent as a function of the number of MC paths 2 N. PDE pricer(pde1) ( = PDE pricer(pde2) = Non-linearity F(u) = 1 2 u 3 u 2).

18 Numerical Experiment 2 N Fair(PDE2) Stdev(PDE2) Fair(PDE1) Stdev(PDE1) Table: MC price quoted in percent as a function of the number of MC paths 2 N. PDE pricer(pde1) ( = PDE pricer(pde2) = Non-linearity F(u) = 1 3 u 3 u 2 u 4).

19 Numerical Experiment 3 The semi-linear PDE in R d t u + Lu + u 2 = 0 blows up in finite-time if and only if d 2 for any bounded positive payoff [Sugitani]. Maturity(Year) BBM alg.(stdev) PDE (0.09) (0.49) ( ) Table: MC price quoted in percent as a function of the maturity for the non-linearity F (u) = u 2 + u. ψ(x) 1 x>1.

20 Polynomial approximation Figure: u + versus its polynomial approximation.

21 Algorithm: Final recipe 1 Simulate the assets and the Poisson default time 2. 2 At each default time, produce k descendants with probability p k. For PDE type 2, the particles are not allowed to die anymore. 3 Evaluate for each particle alive the payoff N T M ψ(zt i ) i=1 k=0 ( ) ωk ak where ω k denotes the number of branching type k. p k 2 The intensity β can stochastic (Cox process).

22 Two PDE types We have implemented our algorithm for the two PDE types t u x 2 σbs 2 x 2 u βu + = 0, u(t, x) = 2.1 x>1 1 : PDE1 and t u x 2 σbs 2 x 2 u βe t [2.1 x>1 1] + = 0 : PDE2 with Poisson intensities β = 1% and β = 3%. σ BS = 20%.

23 Numerical example 1 Maturity(Year) PDE with poly. BBM alg. PDE (0.00) (0.00) (0.00) (0.00) (0.00) Table: MC price quoted in percent as a function of the maturity for PDE 1 with β = 1%. Maturity(Year) PDE with poly. BBM alg.(stdev) PDE (0.00) (0.02) (0.00) (0.00) (0.00) Table: MC price quoted in percent as a function of the maturity for PDE 2 with β = 1%.

24 Numerical example 2 Maturity(Year) PDE with poly. BBM alg. PDE (0.00) (0.00) (0.00) (0.00) (0.00) Table: MC price quoted in percent as a function of the maturity for PDE 1 with β = 3%. Maturity(Year) PDE with poly. BBM alg.(stdev) PDE (0.00) (0.00) (0.01) (0.01) (0.01) Table: MC price quoted in percent as a function of the maturity for PDE 2 with β = 3%.

25 Multi-type Marked branching diffusions Joint work with X. Tan and N. Touzi. Semi-linear PDE system with polynomial non-linearities: t u i (t, x) + Lu i + β i (F i (u 0,..., u N ) u i ) = 0, u i (T, x) = ψ i (x), i = 0, N where N F i (u 1,..., u N ) = M ij u µi p (j) p j=0 p=1 Formula: [ N N j T N û i (t, x) = E ψ j (zt i ) j=0 i=1 j=0 k=1 M ω j (k) jk z i t = x, N j t = δ ji ]

26 Fully non-linear PDE - toy example Burgers: t u + σ2 2 2 x u + β 2 ( xu) 2 = 0, u(t, x) = ψ(x) C (R) Solution: u(t, x) = σ2 β ln E t,x[e β σ 2 ψ(x T ) ] Bootstrapping method (set u 0 = u and u i = i xu): t u 0 + σ2 2 2 x u 0 + β 2 u2 1 = 0, u 0(T, x) = ψ(x) t u 1 + σ2 2 2 x u 1 + βu 1 u 2 = 0, u 1 (T, x) = x ψ(x) t u 2 + σ2 2 2 x u 2 + β (u 2 ) 2 + u 1u 3 = 0, u 2 (T, x) = 2 x ψ(x)... t u K x u K = 0, u K (T, x) = K x ψ(x) Semi-linear PDE system with polynomial non-linearities!

27 Numerical example 3 species: N Fair Stdev Table: MC price quoted in percent as a function of the number of MC paths 2 N. T = 1 year. Exact price ( σ2 2 ln ( T ) ) = Non-linearity β = 1, σ = 0.2, ψ(x) = x 2 /3. Blow-up for T 1.5 as expected.

28 Fully non-linear PDE - toy example One-dimensional UVM: t u σ2 2 x u ( σ 2 σ 2) ( 2 x u ) + = 0, u(t, x) = ψ(x) Set u = e β(t t) v with β = 1 2 ( σ 2 σ 2) : t v σ2 2 x v (σ 2 σ 2) (( 2 x v ) + v ) = 0, v(t, x) = ψ(x) We approximate Γ + by a polynomial P(Γ) 3 : t v σ2 2 x v ( σ 2 σ 2) ( P ( 2 ) ) x v v = 0 3 This is not really an approximation. In practise, rather than taking σ = σθ(γ) + σ(1 θ(γ)), we can use some smoother functions of Γ, for example requiring more comfortable break-even levels as the gamma notional increases.

29 Bootstrap+ truncation t v σ2 2 x v t v σ2 2 x v t v σ2 2 x v t v σ2 2 x v ( σ 2 σ 2) (P (v 2 ) v 0 ) = 0, v 0 (T, x) = ψ(x) (σ 2 σ 2) ( P (v 2 ) v 3 v 1 ) = 0, v 1 (T, x) = ψ (x) (σ 2 σ 2) ( P (2) (v 2 ) v P (v 2 ) v 4 v 2 ) = 0, v 2 (T, x) = ψ (2) (x) (σ 2 σ 2) ( P (3) (v 2 )v P(2) (v 2 )v 3 v 4 + P (v 2 )v 5 v 3 ) = 0, t v K σ2 2 x v K = 0, v K (T, x) = ψ (K ) (x) In practise, 1 2 ( σ 2 σ 2) 1 (i.e. small perturbation).

30 Numerical example 5 species: N Fair Stdev Table: MC price quoted in percent as a function of the number of MC paths 2 N. T = 10 year. Exact price = 20. Non-linearity" P (Γ) = Γ, σ = 0.2, ψ(x) = x 2 /2. N Fair Stdev Table: MC price quoted in percent as a function of the number of MC paths 2 N. T = 10 year. Exact price = Non-linearity P (Γ) = Γ 2 /2, σ = 0.2, ψ(x) = x 2 /2.

31 Conclusions 1 Forward MC scheme for fully non-linear parabolic PDEs. 2 Applicable in higher dimensions (no grid space). 3 No regressions and finite elements required. 4 Algorithm fully parallelizable (independent particles - no interaction).

32 Some references PHL: Counterparty Risk Valuation: A Marked Branching Diffusion Approach, ssrn(2012), submitted. PHL, Tan, X., Touzi, N. : A numerical algorithm for a class of BSDEs via branching processes, in preparation.