Kolmogorov equations and applications to path dependent derivatives - - PowerPoint PPT Presentation

Kolmogorov equations and applications to path dependent derivatives

Andrea Pascucci

University of Bologna, Italy RICAM - Linz, November 2008 – Special Semester on Stochastics with Emphasis on Finance –

The model operator: Kolmogorov (1934)

∂vv + v∂x + ∂t, (x, v, t) ∈ R3

The model operator: Kolmogorov (1934)

∂vv + v∂x + ∂t, (x, v, t) ∈ R3 Kolmogorov Eqs. in Physics

◮ Kinetic theory: Einstein, Langevin (1905)

The model operator: Kolmogorov (1934)

∂vv + v∂x + ∂t, (x, v, t) ∈ R3 Kolmogorov Eqs. in Physics

◮ Kinetic theory: Einstein, Langevin (1905) ◮ Linear Eqs: Ornstein-Uhlenbeck, Vasiˇ

cek

• dXt = Vtdt

dVt = σdWt, related Dynkin (Kolmogorov) operator: σ2 2 ∂vv + v∂x + ∂t

The model operator: Kolmogorov (1934)

∂vv + v∂x + ∂t, (x, v, t) ∈ R3 Kolmogorov Eqs. in Physics

◮ Kinetic theory: Einstein, Langevin (1905) ◮ Linear Eqs: Ornstein-Uhlenbeck, Vasiˇ

cek

• dXt = Vtdt

dVt = σdWt, related Dynkin (Kolmogorov) operator: σ2 2 ∂vv + v∂x + ∂t

◮ Non-linear Eqs: Boltzmann-Landau

a(·, f)∂vvf + v∂xf + ∂tf

Kolmogorov Eqs in Finance

◮ Asian options:

Barucci-Polidoro-Vespri (2001), Di Francesco-P.-Polidoro (2007), P. (2007)

Kolmogorov Eqs in Finance

◮ Asian options:

Barucci-Polidoro-Vespri (2001), Di Francesco-P.-Polidoro (2007), P. (2007)

◮ Path dependent volatility:

Hobson-Rogers (1998), Di Francesco-P. (2004)

Kolmogorov Eqs in Finance

◮ Asian options:

Barucci-Polidoro-Vespri (2001), Di Francesco-P.-Polidoro (2007), P. (2007)

◮ Path dependent volatility:

Hobson-Rogers (1998), Di Francesco-P. (2004)

◮ Pension plans:

Sherris (1995), Friedman and Shen (2002)

Kolmogorov Eqs in Finance

◮ Asian options:

Barucci-Polidoro-Vespri (2001), Di Francesco-P.-Polidoro (2007), P. (2007)

◮ Path dependent volatility:

Hobson-Rogers (1998), Di Francesco-P. (2004)

◮ Pension plans:

Sherris (1995), Friedman and Shen (2002)

◮ Interest rates theory:

Carverhill (1994), Jeffrey (1995), Cheyette (1996), Bhar-Chiarella (1997)

Kolmogorov Eqs in Finance

◮ Asian options:

Barucci-Polidoro-Vespri (2001), Di Francesco-P.-Polidoro (2007), P. (2007)

◮ Path dependent volatility:

Hobson-Rogers (1998), Di Francesco-P. (2004)

◮ Pension plans:

Sherris (1995), Friedman and Shen (2002)

◮ Interest rates theory:

Carverhill (1994), Jeffrey (1995), Cheyette (1996), Bhar-Chiarella (1997)

◮ Stochastic differential utility:

Antonelli-Barucci-Mancino (2001), Antonelli-P. (2002), P.-Polidoro (2003)

Kolmogorov Eqs and linear SDEs

Kolmogorov Eqs and linear SDEs

dXt = BXtdt + σdWt W d-dimensional Brownian motion B constant N × N matrix σ constant N × d matrix

Kolmogorov Eqs and linear SDEs

dXt = BXtdt + σdWt W d-dimensional Brownian motion B constant N × N matrix σ constant N × d matrix Solution: Xt = etB

• x +

t e−sBσdWs

• ,

x ∈ RN

Kolmogorov Eqs and linear SDEs

dXt = BXtdt + σdWt W d-dimensional Brownian motion B constant N × N matrix σ constant N × d matrix Solution: Xt = etB

• x +

t e−sBσdWs

• ,

x ∈ RN Xt is a Gaussian process:

◮ Mean

E(Xt) = etBx

◮ Covariance matrix

C(t) = t esBσ

• esBσ

T ds

Kolmogorov Eqs and linear SDEs

dXt = BXtdt + σdWt C(t) is positive definite ⇔ H¨

• rmander condition

Kolmogorov Eqs and linear SDEs

dXt = BXtdt + σdWt C(t) is positive definite ⇔ H¨

• rmander condition

◮ X has a transition density ◮ Gaussian fundamental solution of the Kolmogorov op.

K = div(A∇) + Bx, ∇ + ∂t, (t, x) ∈ RN+1 where A = 1 2σσ∗ = Id

An example

In R3 ∂xx + x∂y + ∂t A = 1

• B =

1

An example

In R3 ∂xx + x∂y + ∂t A = 1

• B =

1

• Covariance matrix

C(t) = t esBσ

• esBσ

T ds =

• t

t2 2 t2 2 t3 3

• > 0

An example

In R3 ∂xx + x∂y + ∂t A = 1

• B =

1

• Covariance matrix

C(t) = t esBσ

• esBσ

T ds =

• t

t2 2 t2 2 t3 3

• > 0

Fundamental solution Γ(t, x, y) = √ 3 πt2 exp

• −x2

t − 3xy t2 − 3y2 t3

Examples

Example 1: Geometric Asian options

◮ Log-price:

dXt = µ(t, Xt)dt + σ(t, Xt)dWt

◮ Geometric average:

Yt = t Xsds Pricing PDE σ2 2 ∂xxu + x∂yu + ∂tu = 0 (t, x, y) ∈ R3

Example 2: Arithmetic Asian options

◮ Asset price:

dSt = µ(t, St)Stdt + σ(t, St)StdWt

◮ Arithmetic average:

Yt = t Sτdτ Pricing PDE (locally, for s > 0, a Kolmogorov PDE) σ2s2 2 ∂ssu + s∂yu + ∂tu = 0

Example 3: path dependent volatility

Hobson-Rogers (1998) Foschi-P.(2007)

Example 3: path dependent volatility

Hobson-Rogers (1998) Foschi-P.(2007) Xt = log-price dXt = µ(Dt)dt + σ(Dt)dWt Dt = deviation from the normal trend Dt = Xt − +∞ e−s Xt−s ds

Example 3: path dependent volatility

Hobson-Rogers (1998) Foschi-P.(2007) Xt = log-price dXt = µ(Dt)dt + σ(Dt)dWt Dt = deviation from the normal trend Dt = Xt − +∞ e−s Xt−s ds Pricing PDE: a(x, y, t)∂xx + x∂y + ∂t, (x, y, t) ∈ R3

Path dependent volatility: PROs

◮ market completeness ◮ with the simple specification of the volatility function

σ2(x) = a1 + a2(x − a3)2, it reproduces observed volatility surfaces

◮ information on the past: better out of sample and P&L

performance

◮ not more difficult than local volatility

Kolmogorov PDEs and control theory: the Harnack inequality

Kolmogorov PDEs and control theory

K =

d

• i=1

∂xixi + Bx, ∇ + ∂t

• Y

d ≤ N

◮ the cov. matrix C(t) is positive definite in RN:

Kolmogorov PDEs and control theory

K =

d

• i=1

∂xixi + Bx, ∇ + ∂t

• Y

d ≤ N

◮ the cov. matrix C(t) is positive definite in RN: ◮ given x0, y0, an integral curve γ : [t0, t1] −

→ RN+1 exists ˙ γ =

d

• j=1

ξjej + Y (γ)

• Y

x0 x1 t1 t0

An example

∂vv + v∂x + ∂t

• Y

An example

∂vv + v∂x + ∂t

• Y
• x

v t

˙ γ = ξ1e1 + Y (γ) = ξ1   1   +   γ1 1  

Harnack inequality and arbitrage

V positive − → admissible solution − → self-financing strategy then V (t1, x1) ≤ c V (t0, x0) with c = c(t0, t1, x0, x1)

Harnack inequality and arbitrage

V positive − → admissible solution − → self-financing strategy then V (t1, x1) ≤ c V (t0, x0) with c = c(t0, t1, x0, x1) = ⇒ absence of arbitrage opportunities

• Y

x0 x1 t1 t0

Harnack inequality (constant coefficients)

◮ Heat equation: Pini, Hadamard (1954), Li-Yau (1987)

V (t1, x1) ≤ t1 t0 N

2

exp 1 4 |x1 − x0|2 (t1 − t0)

• V (t0, x0)

Harnack inequality (constant coefficients)

◮ Heat equation: Pini, Hadamard (1954), Li-Yau (1987)

V (t1, x1) ≤ t1 t0 N

2

exp 1 4 |x1 − x0|2 (t1 − t0)

• V (t0, x0)

◮ Kolmogorov eqs (constant coeff.):

Kuptsov (1968), Garofalo-Lanconelli (1990), Lanconelli-Polidoro (1994)

• ptimal Harnack constant: P.-Polidoro (2004)

Harnack inequality (variable coefficients)

◮ Parabolic operators with H¨

• lder coefficients:

Moser (1963), Aronson, Serrin (1967)

Harnack inequality (variable coefficients)

◮ Parabolic operators with H¨

• lder coefficients:

Moser (1963), Aronson, Serrin (1967)

◮ Kolmogorov eqs with H¨

• lder coefficients:

Polidoro (1997)

Harnack inequality (variable coefficients)

◮ Parabolic operators with H¨

• lder coefficients:

Moser (1963), Aronson, Serrin (1967)

◮ Kolmogorov eqs with H¨

• lder coefficients:

Polidoro (1997)

◮ Parabolic eq. with measurable coeff.

De Giorgi (1956), Nash (1957) Krylov, Safonov (1979)

Harnack inequality (variable coefficients)

◮ Parabolic operators with H¨

• lder coefficients:

Moser (1963), Aronson, Serrin (1967)

◮ Kolmogorov eqs with H¨

• lder coefficients:

Polidoro (1997)

◮ Parabolic eq. with measurable coeff.

De Giorgi (1956), Nash (1957) Krylov, Safonov (1979)

◮ Kolmogorov eq. with measurable coeff.

still an open problem partial results with S. Polidoro

Fundamental solution of Kolmogorov PDEs with H¨

• lder coefficients

Non Euclidean structure

K =

d

• i=1

∂xixi + Bx, ∇ + ∂t, (t, x) ∈ RN+1

◮ B-Translations: z = (t, x), ζ = (τ, ξ)

ζ ⊕ z = (t + τ, x + etBξ)

◮ B-Dilations:

K = ∂xx + x∂y + ∂t is homogeneous w.r.t. (t, x, y)

δλ

− − − − → (λ2t, λx, λ3y)

Non Euclidean structure

K =

d

• i=1

∂xixi + Bx, ∇ + ∂t, (t, x) ∈ RN+1

◮ B-Translations: z = (t, x), ζ = (τ, ξ)

ζ ⊕ z = (t + τ, x + etBξ)

◮ B-Dilations:

K = ∂xx + x∂y + ∂t is homogeneous w.r.t. (t, x, y)

δλ

− − − − → (λ2t, λx, λ3y) All depends only on the matrix B!

Non Euclidean structure

K = ∂xx + x∂y + ∂t

◮ B-Homogeneous norm:

(t, x, y)B = |t|

1 2 + |x| + |y| 1 3

◮ B-H¨

• lder continuity:

|u(z) − u(ζ)| ≤ cζ−1 ⊕ zα

B

Kolmogorov equations with H¨

• lder coefficients

K =

d

• i,j=1

aij(t, x)∂xixj +

d

• i=1

ai(t, x)∂xi + a(t, x) + Bx, ∇ + ∂t

• Y

◮ (t, x) ∈ R × RN but d ≤ N ◮ (aij) ∼ IRd

Kolmogorov equations with H¨

• lder coefficients

K =

d

• i,j=1

aij(t, x)∂xixj +

d

• i=1

ai(t, x)∂xi + a(t, x) + Bx, ∇ + ∂t

• Y

◮ (t, x) ∈ R × RN but d ≤ N ◮ (aij) ∼ IRd ◮ aij, ai, a are bounded and B-H¨

• lder continuous

◮ B is constant and

K0 = △Rd + Y verifies the H¨

• rmander condition

Existence and uniqueness results

Polidoro (1994-95) Di Francesco - P. (2006) By the parametrix method:

◮ K has a fundamental solution (transition density)

Γ(t, x; T, y): u(t, x) =

• RN Γ(t, x; T, y)ϕ(y)dy

is the solution to the Cauchy problem for K with final datum (payoff) ϕ

Existence and uniqueness results

Polidoro (1994-95) Di Francesco - P. (2006) By the parametrix method:

◮ K has a fundamental solution (transition density)

Γ(t, x; T, y): u(t, x) =

• RN Γ(t, x; T, y)ϕ(y)dy

is the solution to the Cauchy problem for K with final datum (payoff) ϕ

◮ Uniqueness of non-negative or non-rapidly increasing

solutions to the Cauchy problem

Gaussian estimates of the fundamental solution

Polidoro (1994-95) Di Francesco - P. (2006)

◮ Gaussian upper bounds:

Γ(z, ζ) ≤ C Γ0(z, ζ)

Gaussian estimates of the fundamental solution

Polidoro (1994-95) Di Francesco - P. (2006)

◮ Gaussian upper bounds:

Γ(z, ζ) ≤ C Γ0(z, ζ) Polidoro (1997)

◮ Harnack inequality and Gaussian lower bounds

Γ(z, ζ) ≥ C Γ0(z, ζ)

Parametrix method and analytical pricing formulas

Parametrix method (E. E. Levi, 1907)

L = a(t, S)∂SS + ∂t, (t, S) ∈ R2

Parametrix method (E. E. Levi, 1907)

L = a(t, S)∂SS + ∂t, (t, S) ∈ R2 Option price C(t, S) =

• R

ϕ(ξ) Γ(t, S; T, ξ)dξ

Parametrix method (E. E. Levi, 1907)

L = a(t, S)∂SS + ∂t, (t, S) ∈ R2 Option price C(t, S) =

• R

ϕ(ξ) Γ(t, S; T, ξ)dξ First idea: Γ(t, S; T, ξ) = Π(t, S; T, ξ) + “correction term” Π(t, S; T, ξ) is the parametrix:

Parametrix method (E. E. Levi, 1907)

L = a(t, S)∂SS + ∂t, (t, S) ∈ R2 Option price C(t, S) =

• R

ϕ(ξ) Γ(t, S; T, ξ)dξ First idea: Γ(t, S; T, ξ) = Π(t, S; T, ξ) + “correction term” Π(t, S; T, ξ) is the parametrix: fundamental solution to L(T,ξ) = a(T, ξ)∂SS + ∂t

Parametrix method (E. E. Levi, 1907)

L = a(t, S)∂SS + ∂t, (t, S) ∈ R2 Option price C(t, S) =

• R

ϕ(ξ) Γ(t, S; T, ξ)dξ First idea: Γ(t, S; T, ξ) = Π(t, S; T, ξ) + “correction term” Π(t, S; T, ξ) is the parametrix: fundamental solution to L(T,ξ) = a(T, ξ)∂SS + ∂t

◮ L(T,ξ) is a heat operator (Black&Scholes)

Parametrix method (E. E. Levi, 1907)

L = a(t, S)∂SS + ∂t, (t, S) ∈ R2 Option price C(t, S) =

• R

ϕ(ξ) Γ(t, S; T, ξ)dξ First idea: Γ(t, S; T, ξ) = Π(t, S; T, ξ) + “correction term” Π(t, S; T, ξ) is the parametrix: fundamental solution to L(T,ξ) = a(T, ξ)∂SS + ∂t

◮ L(T,ξ) is a heat operator (Black&Scholes) ◮ Π(t, S; T, ξ) is a Gaussian function in (t, S)

The backward parametrix

Π(z; ζ) is a Gaussian function as a function of z but Option price =

• R

ϕ(ξ) Π( t, S

• z

; T, ξ

• ζ

)dξ

The backward parametrix

Π(z; ζ) is a Gaussian function as a function of z but Option price =

• R

ϕ(ξ) Π( t, S

• z

; T, ξ

• ζ

)dξ Corielli-P.(2008): define a parametrix P using the backward (adjoint) PDE

The backward parametrix

Π(z; ζ) is a Gaussian function as a function of z but Option price =

• R

ϕ(ξ) Π( t, S

• z

; T, ξ

• ζ

)dξ Corielli-P.(2008): define a parametrix P using the backward (adjoint) PDE Γ(z; ζ) = P(z; ζ) + “correction term” P(z; ζ) is a Gaussian function in ζ

The backward parametrix, II

Second idea: look for Γ in the form Γ(z; ζ) = P(z; ζ) +

• P(z; ·)Φ(· ; ζ)

here z = (t, S) and ζ = (T, ξ)

The backward parametrix, II

Second idea: look for Γ in the form Γ(z; ζ) = P(z; ζ) +

• P(z; ·)Φ(· ; ζ)

here z = (t, S) and ζ = (T, ξ)

◮ being LΓ = 0, the unknown function Φ satisfies

0 = LP(z; ζ) − Φ(z; ζ) +

• LP(z; ·)Φ(· ; ζ)

The backward parametrix, II

Second idea: look for Γ in the form Γ(z; ζ) = P(z; ζ) +

• P(z; ·)Φ(· ; ζ)

here z = (t, S) and ζ = (T, ξ)

◮ being LΓ = 0, the unknown function Φ satisfies

0 = LP(z; ζ) − Φ(z; ζ) +

• LP(z; ·)Φ(· ; ζ)

◮ recursive formula

Φ(z; ζ) = LP(z; ζ) +

• LP(z; ·)LP(· ; ζ) + . . .

Option price expansion

C option price with payoff ϕ: C(t, S) =

• ϕ(ξ)Γ(t, S; T, ξ)dξ =

∞

• k=1

Ck(t, S)

Option price expansion

C option price with payoff ϕ: C(t, S) =

• ϕ(ξ)Γ(t, S; T, ξ)dξ =

∞

• k=1

Ck(t, S)

◮ First term: Black&Scholes price with σ = σ(t, S)

C1(t, S) =

• ϕ(ξ)P(t, S; T, ξ)dξ

Option price expansion

C option price with payoff ϕ: C(t, S) =

• ϕ(ξ)Γ(t, S; T, ξ)dξ =

∞

• k=1

Ck(t, S)

◮ First term: Black&Scholes price with σ = σ(t, S)

C1(t, S) =

• ϕ(ξ)P(t, S; T, ξ)dξ

◮ k-th term:

B&S price with σ = σ(t, S) and transaction cost LCk−1 Ck(t, S) =

• LCk−1(ζ)P(t, S; ζ)dζ

Global error estimates

|Γ(t, x) − P2n(t, x)| ≤ δ tn n! Γheat(t, x) Pn = parametrix expansion of order n

Global error estimates

|Γ(t, x) − P2n(t, x)| ≤ δ tn n! Γheat(t, x) Pn = parametrix expansion of order n δ = explicit constant

Global error estimates

|Γ(t, x) − P2n(t, x)| ≤ δ tn n! Γheat(t, x) Pn = parametrix expansion of order n δ = explicit constant Γheat = Black&Scholes density with volatility sup σ

Global error estimates

|Γ(t, x) − P2n(t, x)| ≤ δ tn n! Γheat(t, x) Pn = parametrix expansion of order n δ = explicit constant Γheat = Black&Scholes density with volatility sup σ

◮ asymptotically exact as t → 0 ◮ rate of convergence independent on dimension

Example 1: explicit 2-parametrix for loc. vol.

Pricing operator: LC = a(·)∂SSC + ∂tC

Example 1: explicit 2-parametrix for loc. vol.

Pricing operator: LC = a(·)∂SSC + ∂tC Γ(t, S; T, ξ) ≃ P(t, S; T, ξ) + T − t 2 LP(t, S; T, ξ)

Example 1: explicit 2-parametrix for loc. vol.

Pricing operator: LC = a(·)∂SSC + ∂tC Γ(t, S; T, ξ) ≃ P(t, S; T, ξ) + T − t 2 LP(t, S; T, ξ) Call option with strike K: C(t, S) ≃ CBS(t, S) + K(T − t) 2 (a(T, K) − a(t, S))P(t, S; T, K)

Numerical test: CEV model

dSt St = rdt + σ0S−α

t

dWt, α ∈ [0, 1]

Numerical test: CEV model

dSt St = rdt + σ0S−α

t

dWt, α ∈ [0, 1] We compare I and II order parametrix expansions with the analytic approximation formulas by Cox and Ross (1976) (local approx.) for call options

Numerical test: CEV model

dSt St = rdt + σ0S−α

t

dWt, α ∈ [0, 1] We compare I and II order parametrix expansions with the analytic approximation formulas by Cox and Ross (1976) (local approx.) for call options Parameters:

◮ α = 1 2, 3 4 ◮ T = 2, 3, 6, 9, 12 months ◮ K = 1 ◮ σ0 = 30%, r = 5%

Call in CEV: absolute errors of II parametrix

2m 3m 6m 9m 0.6 0.8 1.0 1.2 1.4 S 0.00004 0.00002 0.00002 0.00004 0.00006 0.00008 Absolute Error

Call in CEV: relative errors of II parametrix

2m 3m 6m 9m 0.8 0.9 1.0 1.1 1.2 1.3 S 0.030 0.025 0.020 0.015 0.010 0.005 0.005 Relative Error

Call in CEV: relative errors of II parametrix

2m 3m 6m 9m 0.8 0.9 1.0 1.1 1.2 1.3 S 0.030 0.025 0.020 0.015 0.010 0.005 0.005 Relative Error

Option price: C(t, S) ≃ CBS(t, S) + K(T − t) 2 (a(T, K) − a(t, S))P(t, S; T, K)

Call in CEV: Cox-Ross approximations for n = 200, 300, 400

0.9 1.0 1.1 1.2 S 0.05 0.10 0.15 0.20 0.25 Option Price

Example 2: path dependent volatility (2-dim)

Hobson-Rogers (1998) Foschi-P.(2007)

Example 2: path dependent volatility (2-dim)

Hobson-Rogers (1998) Foschi-P.(2007) Zt = log-price dZt = µ(Dt)dt + σ(Dt)dWt Dt = deviation from the normal trend Dt = Zt − +∞ e−s Zt−s ds

Example 2: path dependent volatility (2-dim)

Hobson-Rogers (1998) Foschi-P.(2007) Zt = log-price dZt = µ(Dt)dt + σ(Dt)dWt Dt = deviation from the normal trend Dt = Zt − +∞ e−s Zt−s ds Pricing PDE (degenerate parabolic): a(x, y, t)∂xx + x∂y + ∂t, (x, y, t) ∈ R3

Path dependent volatility: call option

Monte Carlo vs I (red) and II (blue) parametrix: absolute errors σ(d) = 0.2 ∗ √ 1 + d2, T = 1

4,

0.8 1.0 1.2 1.4 SK 0.0005 0.0005 0.0010 Error

Parametrix for HR model

Path dependent volatility: call option

Monte Carlo vs II parametrix: absolute errors T = 1m (Red), 3m (Green), 9m (Purple), 12m (Cyan)

0.8 1.0 1.2 1.4 SK 0.0001 0.0002 0.0003 0.0004 Error

HR model

The parametrix method: conclusions

The parametrix method: conclusions

◮ analytical global approximation of generic transition

densities

The parametrix method: conclusions

◮ analytical global approximation of generic transition

densities

◮ expansions for prices using as starting point the

Black&Scholes formula

The parametrix method: conclusions

◮ analytical global approximation of generic transition

densities

◮ expansions for prices using as starting point the

Black&Scholes formula

◮ explicit global error estimates

The parametrix method: conclusions

◮ analytical global approximation of generic transition

densities

◮ expansions for prices using as starting point the

Black&Scholes formula

◮ explicit global error estimates ◮ Calibration: analytic formulas for plain vanilla options

(computationally cheap and simple as the Black&Scholes formula)

The parametrix method: conclusions

◮ analytical global approximation of generic transition

densities

◮ expansions for prices using as starting point the

Black&Scholes formula

◮ explicit global error estimates ◮ Calibration: analytic formulas for plain vanilla options

(computationally cheap and simple as the Black&Scholes formula)

◮ Pricing and hedging: potentially useful in high

dimension − → Monte Carlo further investigation and tests needed!

Obstacle problem for Amerasian options

Obstacle problem for American options

X diffusion in RN (state variables) K related Kolmogorov operator ϕ payoff function

• max{Ku, ϕ − u} = 0,

in ]0, T[×RN u(T, ·) = ϕ(T, ·), in RN

Obstacle problem for American options

X diffusion in RN (state variables) K related Kolmogorov operator ϕ payoff function

• max{Ku, ϕ − u} = 0,

in ]0, T[×RN u(T, ·) = ϕ(T, ·), in RN

T

X

Continuationregion Exerciseregion

Ku=0 u= u>

Optimal stopping and fair price of American

• ptions

u generalized solution to the obstacle problem for the diffusion X u(t, x) = sup

τ∈[t,T]

E

• ϕ(τ, Xt,x

τ )

Optimal stopping and fair price of American

• ptions

u generalized solution to the obstacle problem for the diffusion X u(t, x) = sup

τ∈[t,T]

E

• ϕ(τ, Xt,x

τ )

• u(t, Xt) fair price, i.e. value of a self-financing strategy such

that

◮ u(t, Xt) ≥ ϕ(t, Xt) for any t ∈ [0, T] ◮ u(τ, Xτ) = ϕ(τ, Xτ) for some τ ∈ [0, T]

Classical results for uniformly parabolic PDEs

◮ Variational solutions:

Bensoussan&Lions (1978) Kinderlehrer&Stampacchia (1980)

◮ Obstacle and optimal stopping:

van Moerbeke (1975) Bensoussan&Lions (1978)

◮ American options:

Bensoussan (1984), Karatzas (1988) Jaillet&Lamberton&Lapeyre (1990)

◮ Viscosity solutions:

Fleming&Soner (2006), Barles (1997)

◮ Strong solutions:

Friedman (1975)

Amerasian options: numerical results

Barraquand and Pudet 1996 Barles 1997 Parrott and Clarke 1998 Wu, You and Kwok 1999 Hansen and Jorgensen 2000 Ben-Ameur, Breton and L’Ecuyer 2002 Barone-Adesi, Bermudez and Hatgioannides 2003 Marcozzi 2003 Fu and Wu 2003 Jiang and Dai 2004 d’Halluin, Forsyth and Labahn 2005 Huang and Thulasiram 2005 Bermudez, Nogueiras and Vazquez 2006

Main results

1) Di Francesco, P. and Polidoro (2007) Existence of a strong solution to the obstacle problem for non-uniformly parabolic pricing PDEs for Asian options Ingredients:

◮ a priori estimates (Sobolev and Schauder type) ◮ penalization technique ◮ existence results for quasi-linear Cauchy-Dirichlet problem

Main results

1) Di Francesco, P. and Polidoro (2007) Existence of a strong solution to the obstacle problem for non-uniformly parabolic pricing PDEs for Asian options Ingredients:

◮ a priori estimates (Sobolev and Schauder type) ◮ penalization technique ◮ existence results for quasi-linear Cauchy-Dirichlet problem

2) P. (2007) Solution to the optimal stopping and pricing problems for Asian

• ptions

Ingredients:

◮ upper Gaussian estimates for the transition density of the

state process

◮ generalized Itˆ

• formula

Strong solution

max{Ku, ϕ − u} = 0, a.e. where K =

d

• i,j=1

aij(t, x)∂xixj + Bx, ∇ + ∂t

• Y
• n R × RN

u ∈ S1

loc that is

u, ∂xiu, ∂xixju, Y u ∈ L1

loc

for i, j = 1, . . . , d

A priori estimates

◮ Embedding theorem

uC1,α

B (O) ≤ cuSp(Ω)

for p ≥ Q + 2, α = 1 − Q + 2 p where O ⊂⊂ Ω and c = c(K, O, Ω, p, α) Q is the homogeneous dimension

A priori estimates

◮ Embedding theorem

uC1,α

B (O) ≤ cuSp(Ω)

for p ≥ Q + 2, α = 1 − Q + 2 p where O ⊂⊂ Ω and c = c(K, O, Ω, p, α) Q is the homogeneous dimension

◮ Schauder and Sobolev type a-priori estimates

Bramanti, Cerutti e Manfredini 1996 Di Francesco e Polidoro 2006 Di Francesco, P. e Polidoro 2007

B-H¨

• lder spaces

K =

d

• i,j=1

aij(t, x)∂xixj + Bx, ∇ + ∂t

• Y
• n R × RN

H¨

• lder spaces

uC2,α

B (Ω) = uCα B(Ω) +

d

• i=1

∂xiuCα

B(Ω)

+

d

• i,j=1

∂xixjuCα

B(Ω) + Y uCα B(Ω)

Sobolev spaces

K =

d

• i,j=1

aij(t, x)∂xixj + Bx, ∇ + ∂t

• Y
• n R × RN

Sobolev spaces uSp(Ω) = uLp(Ω) +

d

• i=1

∂xiuLp(Ω) +

d

• i,j=1

∂xixjuLp(Ω) + Y uLp(Ω)

Obstacle problem on a bounded cylinder

max{Ku, ϕ − u} = 0

• on a bounded cylindrical domain H(T) := ]0, T[×H
• Cauchy-Dirichlet boundary conditions

Obstacle problem on a bounded cylinder

max{Ku, ϕ − u} = 0

• on a bounded cylindrical domain H(T) := ]0, T[×H
• Cauchy-Dirichlet boundary conditions

The cylinder ]0, T[×H is regular any point of the parabolic boundary admits a barrier

Obstacle problem on a bounded cylinder

max{Ku, ϕ − u} = 0

• on a bounded cylindrical domain H(T) := ]0, T[×H
• Cauchy-Dirichlet boundary conditions

The cylinder ]0, T[×H is regular any point of the parabolic boundary admits a barrier w : V ∩ H(T) → R, such that i) Kw ≤ −1 in V ∩ H(T); ii) w(z) > 0 in V ∩ H(T) \ {ζ} and w(ζ) = 0. Max principle = ⇒ uniform bounds at the boundary

Regular cylinder ]0, T[×H: an example

K = ∂xx + x∂y + ∂t

X Y

H

X Y

H

Obstacle / Payoff function

ϕ ∈ Lip(H(T)) and

d

• i,j=1

ξiξj∂xixjϕ ≥ c|ξ|2 in H(T), ξ ∈ Rd in the distributional sense, c ∈ R (possibly c < 0)

Obstacle / Payoff function

ϕ ∈ Lip(H(T)) and

d

• i,j=1

ξiξj∂xixjϕ ≥ c|ξ|2 in H(T), ξ ∈ Rd in the distributional sense, c ∈ R (possibly c < 0) Examples:

◮ C2 functions

Obstacle / Payoff function

ϕ ∈ Lip(H(T)) and

d

• i,j=1

ξiξj∂xixjϕ ≥ c|ξ|2 in H(T), ξ ∈ Rd in the distributional sense, c ∈ R (possibly c < 0) Examples:

◮ C2 functions ◮ Lipschitz continuous functions, convex w.r.t. x1, . . . , xd

Obstacle / Payoff function

ϕ ∈ Lip(H(T)) and

d

• i,j=1

ξiξj∂xixjϕ ≥ c|ξ|2 in H(T), ξ ∈ Rd in the distributional sense, c ∈ R (possibly c < 0) Examples:

◮ C2 functions ◮ Lipschitz continuous functions, convex w.r.t. x1, . . . , xd ◮ call and put options

ϕ(x) = (E − x)+ = ⇒ ϕ′′ = δE ≥ 0

Obstacle problem

Theorem [Di Francesco, P., Polidoro] Problem

• max{Ku, ϕ − u} = 0,

in H(T) u|∂P H(T) = g admits a strong solution u such that uSp

loc ≤ c,

for any p ≥ 1 with c = c(K, p, ϕ∞, g∞)

Obstacle problem

Theorem [Di Francesco, P., Polidoro] Problem

• max{Ku, ϕ − u} = 0,

in H(T) u|∂P H(T) = g admits a strong solution u such that uSp

loc ≤ c,

for any p ≥ 1 with c = c(K, p, ϕ∞, g∞)

• in particular u ∈ C1,α

B

• u is a viscosity solution

Penalization technique

• Ku = βε(u − ϕ),

in H(T) u|∂P H(T) = g

Penalization technique

• Ku = βε(u − ϕ),

in H(T) u|∂P H(T) = g where (βε)ε>0 are in C∞(R), increasing, βε(0) = 0 and

◮ if u > ϕ then βε(u − ϕ) ≤ ε ◮ if u < ϕ then βε(u − ϕ) −

→ −∞ as ε → 0

Quasi-linear problem

If h = h(z, u) ∈ Lip(H(T) × R) then

• Ku = h(·, u),

in H(T), u|∂P H(T) = g has a classical solution u ∈ C2,α

B (H(T)) ∩ C(H(T))

Proof by monotone iteration:

Quasi-linear problem

If h = h(z, u) ∈ Lip(H(T) × R) then

• Ku = h(·, u),

in H(T), u|∂P H(T) = g has a classical solution u ∈ C2,α

B (H(T)) ∩ C(H(T))

Proof by monotone iteration:

◮ λ exists such that

• Kuj − λ uj = h(·, uj−1) − λ uj−1,

in H(T), uj|∂P H(T) = g, if u0 super-solution then −u0 ≤ uj+1 ≤ uj ≤ u0, j ∈ N

Quasi-linear problem

If h = h(z, u) ∈ Lip(H(T) × R) then

• Ku = h(·, u),

in H(T), u|∂P H(T) = g has a classical solution u ∈ C2,α

B (H(T)) ∩ C(H(T))

Proof by monotone iteration:

◮ λ exists such that

• Kuj − λ uj = h(·, uj−1) − λ uj−1,

in H(T), uj|∂P H(T) = g, if u0 super-solution then −u0 ≤ uj+1 ≤ uj ≤ u0, j ∈ N

◮ boundary datum: barrier functions

Penalized problem as ε → 0+

• Kuε = βε(uε − ϕ),

in H(T) uε|∂P H(T) = g

Penalized problem as ε → 0+

• Kuε = βε(uε − ϕ),

in H(T) uε|∂P H(T) = g Key estimate: |βε(uε − ϕ)| ≤ c (1)

Penalized problem as ε → 0+

• Kuε = βε(uε − ϕ),

in H(T) uε|∂P H(T) = g Key estimate: |βε(uε − ϕ)| ≤ c (1)

Penalized problem as ε → 0+

• Kuε = βε(uε − ϕ),

in H(T) uε|∂P H(T) = g Key estimate: |βε(uε − ϕ)| ≤ c (1) Estimate (1) + barriers + interior estimates in Sp uSp(O) ≤ c

• uLp(Ω) + KuLp(Ω)

Proof of estimate (1)

|βε(uε − ϕ)| ≤ c Proof: ζ interior minimum = ⇒ K(uε − ϕ)(ζ) ≥ 0 βε(uε − ϕ)(ζ) = Kuε(ζ) ≥ Kϕ(ζ) ≥ C

Obstacle problem on the strip

• max{Ku, ϕ − u} = 0,

in ]0, T[×RN u(T, ·) = ϕ(T, ·), in RN Theorem [Di Francesco, P., Polidoro] If a strong super-solution ¯ u exists then there exists a strong solution u ≤ ¯ u and u ∈ Sp

loc for any p ≥ 1

Obstacle problem on the strip

• max{Ku, ϕ − u} = 0,

in ]0, T[×RN u(T, ·) = ϕ(T, ·), in RN Theorem [Di Francesco, P., Polidoro] If a strong super-solution ¯ u exists then there exists a strong solution u ≤ ¯ u and u ∈ Sp

loc for any p ≥ 1

Example: call option, ϕ(x) = (ex − K)+ ¯ u(x) = ex+Ct is a super-solution i.e. max{K¯ u, ϕ − ¯ u} ≤ 0

Optimal stopping

Xt,x diffusion associated to a Kolmogorov PDE

Optimal stopping

Xt,x diffusion associated to a Kolmogorov PDE Theorem [P.] u strong solution to the obstacle problem s.t. |u(t, x)| ≤ cec|x|2 Then u(t, x) = sup

τ∈[t,T]

E

• ϕ(τ, Xt,x

τ )

Optimal stopping

Xt,x diffusion associated to a Kolmogorov PDE Theorem [P.] u strong solution to the obstacle problem s.t. |u(t, x)| ≤ cec|x|2 Then u(t, x) = sup

τ∈[t,T]

E

• ϕ(τ, Xt,x

τ )

• Proof: Γ(t, x; ·, ·) transition density of Xt,x

upper Gaussian estimate = ⇒ Γ(t, x; ·, ·) ∈ Lq(]t, T[×RN) for some q > 1

Itˆ

• formula for strong solutions

Itˆ

• formula for un (regularization of u)

un(τ, Xt,x

τ ) = un(t, x) +

τ

t

Kunds + τ

t

∇unσdWs

Itˆ

• formula for strong solutions

Itˆ

• formula for un (regularization of u)

un(τ, Xt,x

τ ) = un(t, x) +

τ

t

Kunds + τ

t

∇unσdWs Estimate of the deterministic integral E

• τ

t

• Ku(s, Xt,x

s ) − Kun(s, Xt,x s )

• ds
• ≤

Itˆ

• formula for strong solutions

Itˆ

• formula for un (regularization of u)

un(τ, Xt,x

τ ) = un(t, x) +

τ

t

Kunds + τ

t

∇unσdWs Estimate of the deterministic integral E

• τ

t

• Ku(s, Xt,x

s ) − Kun(s, Xt,x s )

• ds
• ≤

≤

• RN

T

t

|Ku(s, y) − Kun(s, y)| Γ(t, x; s, y)dsdy ≤

Itˆ

• formula for strong solutions

Itˆ

• formula for un (regularization of u)

un(τ, Xt,x

τ ) = un(t, x) +

τ

t

Kunds + τ

t

∇unσdWs Estimate of the deterministic integral E

• τ

t

• Ku(s, Xt,x

s ) − Kun(s, Xt,x s )

• ds
• ≤

≤

• RN

T

t

|Ku(s, y) − Kun(s, y)| Γ(t, x; s, y)dsdy ≤ ≤ Ku − KunLpΓ(t, x; ·, ·)Lq − − − →

n→∞ 0

since u ∈ Sp and Γ(t, x; ·, ·) ∈ Lq