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

Non Euclidean structure K = ∂ xx + x∂ y + ∂ t ◮ B -Homogeneous norm: 1 1 2 + | x | + | y | � ( t, x, y ) � B = | t | 3 ◮ B -H¨ older continuity: | u ( z ) − u ( ζ ) | ≤ c � ζ − 1 ⊕ z � α B

Kolmogorov equations with H¨ older coefficients d d � � K = a ij ( t, x ) ∂ x i x j + a i ( t, x ) ∂ x i + a ( t, x ) + � Bx, ∇� + ∂ t � �� � i,j =1 i =1 Y ◮ ( t, x ) ∈ R × R N but d ≤ N ◮ ( a ij ) ∼ I R d

Kolmogorov equations with H¨ older coefficients d d � � K = a ij ( t, x ) ∂ x i x j + a i ( t, x ) ∂ x i + a ( t, x ) + � Bx, ∇� + ∂ t � �� � i,j =1 i =1 Y ◮ ( t, x ) ∈ R × R N but d ≤ N ◮ ( a ij ) ∼ I R d ◮ a ij , a i , a are bounded and B -H¨ older continuous ◮ B is constant and K 0 = △ R d + Y verifies the H¨ ormander 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 ) = R N 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 ) = R N 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 Γ 0 ( z, ζ )

Gaussian estimates of the fundamental solution Polidoro (1994-95) Di Francesco - P. (2006) ◮ Gaussian upper bounds: Γ( z, ζ ) ≤ C Γ 0 ( z, ζ ) Polidoro (1997) ◮ Harnack inequality and Gaussian lower bounds Γ( z, ζ ) ≥ C Γ 0 ( z, ζ )

Parametrix method and analytical pricing formulas

Parametrix method (E. E. Levi, 1907) ( t, S ) ∈ R 2 L = a ( t, S ) ∂ SS + ∂ t ,

Parametrix method (E. E. Levi, 1907) ( t, S ) ∈ R 2 L = a ( t, S ) ∂ SS + ∂ t , � Option price C ( t, S ) = ϕ ( ξ ) Γ( t, S ; T, ξ ) dξ R

Parametrix method (E. E. Levi, 1907) ( t, S ) ∈ R 2 L = a ( t, S ) ∂ SS + ∂ t , � Option price C ( t, S ) = ϕ ( ξ ) Γ( t, S ; T, ξ ) dξ R 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) ( t, S ) ∈ R 2 L = a ( t, S ) ∂ SS + ∂ t , � Option price C ( t, S ) = ϕ ( ξ ) Γ( t, S ; T, ξ ) dξ R 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) ( t, S ) ∈ R 2 L = a ( t, S ) ∂ SS + ∂ t , � Option price C ( t, S ) = ϕ ( ξ ) Γ( t, S ; T, ξ ) dξ R 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) ( t, S ) ∈ R 2 L = a ( t, S ) ∂ SS + ∂ t , � Option price C ( t, S ) = ϕ ( ξ ) Γ( t, S ; T, ξ ) dξ R 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 = ϕ ( ξ ) Π( t, S ; T, ξ ) dξ ���� ���� z R ζ

The backward parametrix Π( z ; ζ ) is a Gaussian function as a function of z but � Option price = ϕ ( ξ ) Π( t, S ; T, ξ ) dξ ���� ���� z R ζ 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 = ϕ ( ξ ) Π( t, S ; T, ξ ) dξ ���� ���� z R ζ 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 L Γ = 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 L Γ = 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 = C k ( t, S ) k =1

Option price expansion C option price with payoff ϕ : � ∞ � C ( t, S ) = ϕ ( ξ )Γ( t, S ; T, ξ ) dξ = C k ( t, S ) k =1 ◮ First term: Black&Scholes price with σ = σ ( t, S ) � C 1 ( 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 = C k ( t, S ) k =1 ◮ First term: Black&Scholes price with σ = σ ( t, S ) � C 1 ( t, S ) = ϕ ( ξ ) P ( t, S ; T, ξ ) dξ ◮ k -th term: B&S price with σ = σ ( t, S ) and transaction cost LC k − 1 � C k ( t, S ) = LC k − 1 ( ζ ) P ( t, S ; ζ ) dζ

Global error estimates | Γ( t, x ) − P 2 n ( t, x ) | ≤ δ t n n ! Γ heat ( t, x ) P n = parametrix expansion of order n

Global error estimates | Γ( t, x ) − P 2 n ( t, x ) | ≤ δ t n n ! Γ heat ( t, x ) P n = parametrix expansion of order n δ = explicit constant

Global error estimates | Γ( t, x ) − P 2 n ( t, x ) | ≤ δ t n n ! Γ heat ( t, x ) P n = parametrix expansion of order n δ = explicit constant Γ heat = Black&Scholes density with volatility sup σ

Global error estimates | Γ( t, x ) − P 2 n ( t, x ) | ≤ δ t n n ! Γ heat ( t, x ) P n = parametrix expansion of order n δ = explicit constant Γ heat = 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 ( · ) ∂ SS C + ∂ t C

Example 1: explicit 2-parametrix for loc. vol. Pricing operator: LC = a ( · ) ∂ SS C + ∂ t C Γ( t, S ; T, ξ ) ≃ P ( t, S ; T, ξ ) + T − t LP ( t, S ; T, ξ ) 2

Example 1: explicit 2-parametrix for loc. vol. Pricing operator: LC = a ( · ) ∂ SS C + ∂ t C Γ( t, S ; T, ξ ) ≃ P ( t, S ; T, ξ ) + T − t LP ( t, S ; T, ξ ) 2 Call option with strike K : C ( t, S ) ≃ C BS ( t, S ) + K ( T − t ) ( a ( T, K ) − a ( t, S )) P ( t, S ; T, K ) 2

Numerical test: CEV model dS t = rdt + σ 0 S − α α ∈ [0 , 1] dW t , t S t

Numerical test: CEV model dS t = rdt + σ 0 S − α α ∈ [0 , 1] dW t , t S t 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 dS t = rdt + σ 0 S − α α ∈ [0 , 1] dW t , t S t 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 ◮ σ 0 = 30% , r = 5%

Call in CEV: absolute errors of II parametrix Absolute Error 0.00008 9m 0.00006 0.00004 6m 0.00002 3m 2m S 0.6 0.8 1.0 1.2 1.4 � 0.00002 � 0.00004

Call in CEV: relative errors of II parametrix Relative Error 9m 0.005 6m S 0.8 0.9 1.0 1.1 1.2 1.3 3m � 0.005 � 0.010 � 0.015 2m � 0.020 � 0.025 � 0.030

Call in CEV: relative errors of II parametrix Relative Error 9m 0.005 6m S 0.8 0.9 1.0 1.1 1.2 1.3 3m � 0.005 � 0.010 � 0.015 2m � 0.020 � 0.025 � 0.030 Option price: C ( t, S ) ≃ C BS ( t, S ) + K ( T − t ) ( a ( T, K ) − a ( t, S )) P ( t, S ; T, K ) 2

Call in CEV: Cox-Ross approximations for n = 200 , 300 , 400 Option Price 0.25 0.20 0.15 0.10 0.05 S 0.9 1.0 1.1 1.2

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) Z t = log-price dZ t = µ ( D t ) dt + σ ( D t ) dW t D t = deviation from the normal trend � + ∞ e − s Z t − s ds D t = Z t − 0

Example 2: path dependent volatility (2-dim) Hobson-Rogers (1998) Foschi-P. (2007) Z t = log-price dZ t = µ ( D t ) dt + σ ( D t ) dW t D t = deviation from the normal trend � + ∞ e − s Z t − s ds D t = Z t − 0 Pricing PDE (degenerate parabolic): ( x, y, t ) ∈ R 3 a ( x, y, t ) ∂ xx + x∂ y + ∂ t ,

Path dependent volatility: call option Monte Carlo vs I (red) and II (blue) parametrix : absolute errors √ T = 1 1 + d 2 , σ ( d ) = 0 . 2 ∗ 4 , Parametrix for HR model Error 0.0010 0.0005 S � K 0.8 1.0 1.2 1.4 � 0.0005

Path dependent volatility: call option Monte Carlo vs II parametrix : absolute errors T = 1m (Red), 3m (Green), 9m (Purple), 12m (Cyan) HR model Error 0.0004 0.0003 0.0002 0.0001 S � K 0.8 1.0 1.2 1.4

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 R N (state variables) K related Kolmogorov operator ϕ payoff function � in ]0 , T [ × R N max { Ku, ϕ − u } = 0 , in R N u ( T, · ) = ϕ ( T, · ) ,

Obstacle problem for American options X diffusion in R N (state variables) K related Kolmogorov operator ϕ payoff function � in ]0 , T [ × R N max { Ku, ϕ − u } = 0 , in R N u ( T, · ) = ϕ ( T, · ) , T Exercise�region Ku=0 u> � u= � Continuation�region X

Optimal stopping and fair price of American options u generalized solution to the obstacle problem for the diffusion X � � ϕ ( τ, X t,x u ( t, x ) = sup E τ ) τ ∈ [ t,T ]

Optimal stopping and fair price of American options u generalized solution to the obstacle problem for the diffusion X � � ϕ ( τ, X t,x u ( t, x ) = sup E τ ) τ ∈ [ t,T ] u ( t, X t ) fair price, i.e. value of a self-financing strategy such that ◮ u ( t, X t ) ≥ ϕ ( t, X t ) for any t ∈ [0 , T ] ◮ u ( τ, X τ ) = ϕ ( τ, X τ ) 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 options Ingredients: ◮ upper Gaussian estimates for the transition density of the state process ◮ generalized Itˆ o formula

Strong solution max { Ku, ϕ − u } = 0 , a.e. where d � on R × R N K = a ij ( t, x ) ∂ x i x j + � Bx, ∇� + ∂ t � �� � i,j =1 Y u ∈ S 1 loc that is Y u ∈ L 1 u, ∂ x i u, ∂ x i x j u, loc for i, j = 1 , . . . , d