On density asymptotics for diffusions, old and new

Transcription

1 On density asymptotics for diffusions, old and new Stefano De Marco (Ecole Polytechnique) Paris, January 2014 S. De Marco (Ecole Polytechnique) 7/01/14 0 / 15

2 d dx = b(x)dt + σ j (X)dW j, X 0 = x 0, t T j=1 a diffusion process in in R n. Under some conditions, P(X t dx) = p t (x 0, x)dx. Y t = Xt 1 models a (log)-price Z t = (Xt 2,..., Xt n ) a stoch. volatility component, or other (correlated) prices S. De Marco (Ecole Polytechnique) 7/01/14 1 / 15

3 d dx = b(x)dt + σ j (X)dW j, X 0 = x 0, t T j=1 a diffusion process in in R n. Under some conditions, P(X t dx) = p t (x 0, x)dx. Y t = Xt 1 models a (log)-price Z t = (Xt 2,..., Xt n ) a stoch. volatility component, or other (correlated) prices Question of the asymptotic behavior of the marginal density p Yt (y) : as t 0, or y Translates into and σ implied (t, y) σ implied (0, y), t 0 σ implied (t, y) 2 t c t y, y Lee 04, Benaim&Friz 09, Gulisashvili 10,(...). S. De Marco (Ecole Polytechnique) 7/01/14 1 / 15

4 Conditional laws Focus on : Law(Z Y t = y) Question of the asymptotic behavior : as t 0, or y. Related to : law of diffusion bridge under partial conditioning, X Y t = y Applications : asymptotics of Allows for : σ 2 loc(t, y) := E[Z 2 t Y t = y] Extrapolation of local volatility in given models. Justification of parameterizations of the local volatility surface : a result such as σloc 2 (t, y) lim = c t > 0 y ± y motivates asymptotically linear (e.g. Gatheral s SVI-type) functional forms. S. De Marco (Ecole Polytechnique) 7/01/14 2 / 15

5 T Local variance logmoneyness log(k/f_0) maturity Local variance logmoneyness log(k/f_0) maturity k S. De Marco (Ecole Polytechnique) 7/01/14 3 / 15

6 Stochastic volatility systems are hypoelliptic Generic SV model (S t = S 0 e Yt ) : dy = 1 2 Z 2 dt + ZdB 1, Y 0 = 0, dz = β(z)dt + α(z)db 2, Z 0 0 (B 1, B 2 ) = C(W 1, W 2 ), C correlation. Diffusion vector fields : ( z C σ 1 (z) = 11 α(z) C 21 ) ( z C σ 2 (z) = 12 α(z) C 22 ). {z = 0, α(z) 0} elliptic region {z = 0} sub-elliptic set S. De Marco (Ecole Polytechnique) 7/01/14 4 / 15

7 Density asymptotics, small time (old) Log-estimate (Varadhan) lim t log p t(x 0, x) = Λ(x 0, x) t 0 with Λ(x 0, x) the energy of the deterministic control system dϕ = j σ j (ϕ)ḣj dt, ϕ 0 = x 0, ḣ L 2 ([0, 1]) 1 } Λ(x 0, x) := inf{ 2 ḣ 2 : ϕ 0 (h) = x 0, ϕ 1 (h) = x Varadhan 67 : σ 1,..., σ d uniformly elliptic Léandre 87,87 : degenerate vector fields : Lie(σ 1,..., σ d ) x = R n for all x Very general : no condition on x 0, x. Note : Λ(x 0, x) = 1 2 d2 (x 0, x) the geodesics distance of x 0 to x. S. De Marco (Ecole Polytechnique) 7/01/14 5 / 15

8 Density asymptotics, small time (old) Exact asymptotics of p t heat kernel expansion p t (x 0, x) = 1 e tn/2 Λ(x 0,x) t ( k j=0 ) c j (x 0, x)t j + R k+1 (x 0, x; t), t 0 Molchanov 75 (elliptic diffusions), Léandre 87, Ben Arous 88, (building on Bismut, Azencott), Watanabe 87,(...) based on the regularity of the energy Λ(x 0, x). holds under non-degeneracy conditions on (x 0, x) : not in cut-locus condition from sub-riemannian geometry. Diagonal behavior (x = x 0 ) is different in general : exactly because the non-degeneracy condition fails if x 0 is a sub-elliptic point. S. De Marco (Ecole Polytechnique) 7/01/14 6 / 15

9 Toolbox : small-noise large deviations dx ε = b ε (X ε )dt + εσ(x ε )dw, X 0 = x0 ε. with b ε b 0 uniformly and x0 ε x 0 as ε 0 Limiting ODE as ε 0 dϕ = b 0 (ϕ)dt + σ(ϕ)dh, ϕ 0 = x 0 Wentzell-Freidlin large deviations (84) : 1 } I(φ) = inf{ 2 ḣ 2 : ϕ(h) = φ the energy of the path φ For fixed t P(X ε Γ) = e 1 ε 2(I(Γ)+o(1)), I(Γ) := inf φ Γ I(φ). } Λ t (x 0, x) = inf{i(φ) : φ 0 = x 0, φ t = x 1 } = inf{ 2 ḣ 2 : ϕ 0 (h) = x 0, ϕ t (h) = x S. De Marco (Ecole Polytechnique) 7/01/14 7 / 15

10 Classical : small-noise small-time dx = b(x)dt + Brownian scaling d σ j (X)dW j, X 0 = x 0 j=1 (X tε 2, t 0) = L (Xt ε, t > 0) d dx ε = ε 2 b(x ε )dt + ε σ j (X ε )dw j, X0 ε = x 0 lim ε 0 ε 2 b = b 0 0 no drift in the limiting ODE : dϕ = σ(ϕ)dh. Small-noise small-time : t logp(x t B) = ε 2 logp(x1 ε B) ε 2 =t Λ 1 (B) as t 0. j=1 S. De Marco (Ecole Polytechnique) 7/01/14 8 / 15

11 Small-noise spatial asymptotics If θ-scaling : Y ε L = ε θ Y then p Y (y) = ε θ p Y ε(1) ε= y 1/θ : that is, y translates into ε 0. S. De Marco (Ecole Polytechnique) 7/01/14 9 / 15

12 Small-noise spatial asymptotics If θ-scaling : Y ε L = ε θ Y then p Y (y) = ε θ p Y ε(1) ε= y 1/θ : that is, y translates into ε 0. Example : a model with Ornstein-Uhlenbeck stochastic volatility dy = 1 2 Z 2 dt + ZdB 1, Y 0 = 0 dz = (a+bz)d + cdb 2, Z 0 = z 0 > 0 has 2-scaling for Y : with Y ε = ε 2 Y, Z ε = εz, dy ε = 1 2 (Z ε ) 2 dt + εz ε db 1, Y ε 0 = 0, dz ε = (aε+bz ε )dt + ε c db 2, Z ε 0 = εz 0. note : (Y0 ε, Z 0 ε ) (0, 0) (hypoelliptic point) Also have : lim y ± E[Z 2 t Yt=y] y = lim ε 0 E [ ] (Zt ε ) 2 Yt ε = ±1 S. De Marco (Ecole Polytechnique) 7/01/14 9 / 15

13 Density asymptotics, small-noise (X ε p ε (x)dx) Log-estimates (still old!) Ben Arous & Léandre 91 lim sup ε 2 log p ε (x) Λ(x) ε 0 lim inf ε 0 ε2 log p ε (x) Λ R (x) The upper bound : the classical energy (seen before) 1 } Λ(x) = inf{ 2 ḣ 2 : ϕ 0 (h) = x 0, ϕ t (h) = x The lower bound : minimum energy over the regular controls steering x 0 to x 1 } Λ R := inf{ 2 ḣ 2 : ϕ 0 (h) = x 0, ϕ t (h) = x, rk(d h ϕ(h)) is max In other words : invertibility of the deterministic Malliavin matrix D h ϕ(h), D h ϕ(h) S. De Marco (Ecole Polytechnique) 7/01/14 10 / 15

14 Asymptotics of conditional densities, small-noise Under some technical conditions (Hörmander + existence of regular minimizing controls from x 0 to hyperplane x = (y, )) : Theorem (DM&Friz 13) (i) Log-estimate of the density of Y ε : lim ǫ 0 ε2 log p Y ε(y) = minλ(y, z). z (ii) Law of large numbers for the conditional law : provided z := argmin z Λ(y, z) is unique. Law (Z ε t Y ε t = y) = δ z ε 0 Checkable hypotheses in terms of the SDE coefficients. Limits characterized in terms of Hamiltonian equations (explicit solution in some cases, see below) Estimates of log p ε X are enough to establish the result. S. De Marco (Ecole Polytechnique) 7/01/14 11 / 15

15 Asymptotics of local volatilities Corollary (i) [Local volatility, short time behavior] In a generic stochastic volatility model (Y, Z) (where Y denotes log-price and Z stochastic volatility) σ 2 loc(t, y) = E[Z 2 t Y t = y] z (y) 2 as t 0. removes the ellipticity assumptions in Berestycki et al. 04 (ii) [Local volatility wings ] In models with OU stochastic volatility (Stein Stein, Schöbel Zhu) : lim y ± σloc 2 (t, y) = lim y y ± 1 y E[Z 2 t Y t = y] = z (±1) 2 > 0 z (±1) known explicitly in terms of the model parameters. S. De Marco (Ecole Polytechnique) 7/01/14 12 / 15

16 Convergence of y σ2 (y,t) y, y (a) t = 0.25 (b) t = (c) t = 3 (d) t = 10 Blue line : y σ2 (y,t) y 1 (z ) 2. Red line : 1. S. De Marco (Ecole Polytechnique) 7/01/14 13 / 15

17 Small-noise equations with degenerate limit ODE Several financial models rely on SDEs of the form dx t = (a+bx t )dt + σxt γ db t, γ < 1 (CEV, Sabr, Heston). Spatial scaling holds for X ε := ε 1/(1 γ) X : dxt ε = (ε 1/(1 γ) a+bxt ε )dt + εσ(xt ε ) γ db t, X0 ε = ε 1/(1 γ) x The limit ODE ϕ t = βϕ t + σϕ γ t ḣtdt, ϕ 0 = 0 admits infinitely many solutions, for any h new situation in W F theory. S. De Marco (Ecole Polytechnique) 7/01/14 14 / 15

18 Small-noise equations with degenerate limit ODE Several financial models rely on SDEs of the form dx t = (a+bx t )dt + σx γ t db t, γ < 1 (CEV, Sabr, Heston). Spatial scaling holds for X ε := ε 1/(1 γ) X : The limit ODE dx ε t = (ε 1/(1 γ) a+bx ε t )dt + εσ(x ε t ) γ db t, ϕ t = βϕ t + σϕ γ t ḣtdt, ϕ 0 = 0 X ε 0 = ε 1/(1 γ) x admits infinitely many solutions, for any h new situation in W F theory. Theorem (Conforti,DM&Deuschel 14) X ε satisfies a large deviation principle on C([0, T],R + ) with rate function : I T (ϕ) := 1 T 2σ 2 0 ( ) 2 ϕt bϕ t 1 ϕt>0dt. Solution of the variational problem restores exact log-asymptotics of densities known for CEV process Applies to diffusions with more general drift term S. De Marco (Ecole Polytechnique) 7/01/14 14 / 15 ϕ γ t

19 Special regimes A Black-Scholes basket (St 1, St), 2 the index J t = St 1 + St 2, then (Bayer, Friz & Laurence 13) : 1 p Jt (K) c e Λ(K) t, t 0 t p Jt (K ) c e Λ(K ) t 1 t 3/4, t 0 Heston model (Y t, V t), S t = S 0e Yt, stochastic variance V t follows a CIR process : p Yt (y = 0 Y 0 = 0, V 0 > 0) c V t 0, t 0 p Yt (y = 0 Y 0 = 0, V 0 = 0) c0 t, t 0 Explicit examples from finance (with non-lipschitz coefficients) motivate research on heat kernels IHP Trimester - Geometry, Analysis and Dynamics on Sub-Riemannian Manifolds, September 1 December 12, S. De Marco (Ecole Polytechnique) 7/01/14 15 / 15

20 References A. Agrachev and P. W. Lee, Continuity of optimal control costs and its application to weak KAM theory., Calc. Var. Partial Differ. Equ., 39 (2010), pp I. Bailleul, Large deviation principle for bridges of degenerate diffusion processes, ArXiv. C. Bayer, P. Friz & P. Laurence, On the probability density function of baskets, ArXiv. G. Ben Arous, Développement asymptotique du noyau de la chaleur hypoelliptique hors du cut-locus, Annales scientifiques de l Ecole Normale Supérieure, 4 (1988), pp G. Ben Arous and R. Léandre, Décroissance exponentielle du noyau de la chaleur sur la diagonale (II), Probab. Theory Relat. Fields, 90 (1991), pp H. Berestycki, J. Busca, and I. Florent, Computing the implied volatility in stochastic volatility models, Comm. on Pure and App. Mathematics, LVII (2004), pp S. De Marco (Ecole Polytechnique) 7/01/14 15 / 15

21 S. De Marco, P. Friz, and S. Gerhold, Rational Shapes of local volatility, Risk, February 2013, pp S. De Marco and P. Friz, Varadhan s formula, conditioned diffusions, and local volatilities. ArXiv J.-D. Deuschel, P. Friz, A. Jacquier, and S. Violante, Marginal density expansion for diffusions and stochastic volatility, I and II. To appear in Comm. on Pure and App. Mathematics. P. Henry-Labordère, Analysis, geometry, and modeling in finance : Advanced methods in option pricing. Chapman & Hall/CRC, Financial Mathematics Series, R. Léandre, Majoration en temps petit de la densité d une diffusion dégénérée, Probability Theory and Related Fields, 74 (1987), pp R. Léandre, Minoration en temps petit de la densité d une diffusion dégénérée, Journal of functional analysis, 74 (1987), pp A. Reghai, Model evolution. Presentation at the Parisian Model Validation seminar, February J.-M. Bismut, Large deviations and the Malliavin calculus, Birkhäuser, Boston, S. De Marco (Ecole Polytechnique) 7/01/14 15 / 15

22 Some key points Ensure the two energy functions Λ and Λ R coincide around a minimizer z For the marginal f ε (y) = p ε (y, z)dz LB : lim inf ε 0 ε2 log p ε (y, z) Λ(y, z) UB : lim sup ε 2 log p ε (y, z) Λ(y, z) ε 0 UB uniform over z in a compact n hood of z around the minimizer. Tail estimate : for every q (0, 1), every R < 1 and x R n can integrate the limit p ε t(x) C q,t (1+R N(q) ) ε N(q) P( X ε t x R) q Study the conditional expectation : E[φ(Z ε ) Y ε = y] = φ(z) pε (y, z) f ε (y) dz φ(z ) with the same upper, lower and tail estimates. S. De Marco (Ecole Polytechnique) 7/01/14 15 / 15