Error Estimates for Multinomial Approximations of American Options - PowerPoint PPT Presentation

Outline American options in continuous time models Discrete time approximations Results Sketch of the proof Remarks References Error Estimates for Multinomial Approximations of American Options in a Class of Jump Diffusion Models Yan Dolinsky ETH Zurich AnStAp 2010 15.07.2010 Yan Dolinsky Error Estimates for Multinomial Approximations of American Options

Outline American options in continuous time models Discrete time approximations Results Sketch of the proof Remarks References The setup ◮ (Ω , F , P )–complete probability space. ◮ { W ( t ) = ( W 1 ( t ) , ..., W d ( t )) } ∞ t =0 – standard d dimensional Brownian motion. ◮ { N ( t ) } ∞ t =0 – Poisson process with intensity λ and independent of W . ◮ { U ( i ) = ( U ( i ) 1 , ..., U ( i ) d ) } ∞ i =1 –sequence of i.i.d. random vectors with values in ( − 1 , ∞ ) d , independent from W and N . Assume that the random vector U (1) takes on a finite number of values and denote u j = EU (1) , 1 ≤ j ≤ d . j Yan Dolinsky Error Estimates for Multinomial Approximations of American Options

Outline American options in continuous time models Discrete time approximations Results Sketch of the proof Remarks References The market ◮ B ( t ) = B (0) exp( rt ). ◮ For 1 ≤ i ≤ d , N ( t ) d (1 + U ( j ) � � S i ( t ) = S i (0) exp( µ i t + σ ij W j ( t )) ) i j =1 j =1 where σ = ( σ ij ) 1 ≤ i , j ≤ d is a nonsingular matrix. ◮ Without loss of generality we assume that � d j =1 σ 2 ij µ i = r − λ − u i − , 1 ≤ i ≤ d . 2 Yan Dolinsky Error Estimates for Multinomial Approximations of American Options

Outline American options in continuous time models Discrete time approximations Results Sketch of the proof Remarks References American options ◮ American option with the payoff process Y ( t ) = F ( S ( t ) , t ) , 0 ≤ t ≤ T where F : R d + × R + → R + . ◮ The term V = sup E (exp( − r τ ) Y ( τ )) τ ∈T gives an arbitrage–free price for the American option. ◮ We assume that for some constant L ≥ 1 d d � � | F ( υ, t ) − F (˜ υ, s ) | ≤ L | υ i − ˜ υ i | + L ( t − s )(1 + | υ i | ) i =1 i =1 υ ∈ R d . for any t ≥ s ≥ 0 and υ, ˜ Yan Dolinsky Error Estimates for Multinomial Approximations of American Options

Outline American options in continuous time models Discrete time approximations Results Sketch of the proof Remarks References Construction of the discrete probability spaces ◮ Let A ∈ M d +1 ( R ) be an orthogonal matrix such that it last 1 1 column equals to ( d +1 , ..., d +1 ). √ √ ◮ Let Ω ξ = { 1 , 2 , ..., d + 1 } ∞ be the space of infinite sequences ω = ( ω 1 , ω 2 , ... ); ω i ∈ { 1 , 2 , ..., d + 1 } with the product probability P ξ = { 1 1 d +1 } ∞ . d +1 , ..., ◮ Define a sequence of i.i.d. random vectors ξ (1) , ξ (2) , ... by √ ξ ( i ) ( ω ) = d + 1( A ω i 1 , A ω i 2 , ..., A ω i d ) , i ∈ N . We have E ξ (1) = 0 and E ξ (1) ξ (1) = δ ij . i j Yan Dolinsky Error Estimates for Multinomial Approximations of American Options

Outline American options in continuous time models Discrete time approximations Results Sketch of the proof Remarks References Discrete probability spaces For any n we extend (Ω ξ , P ξ ) to a probability space (Ω n , P n ) such that it contains a three independent sequences of i.i.d. random vectors ◮ { ξ ( k ) } ∞ k =1 –as above. ◮ { ˜ U ( k ) } ∞ k =1 ∼ { U ( k ) } ∞ k =1 . ◮ { ρ n , k } n k =1 – sequence of Bernoulli random variables such that P n { ρ n , 1 = 1 } = 1 − exp( − λ T / n ). Yan Dolinsky Error Estimates for Multinomial Approximations of American Options

Outline American options in continuous time models Discrete time approximations Results Sketch of the proof Remarks References Multinomial n –step market n , 2 T ◮ The market is active in the moments 0 , T n , ..., T . ◮ B ( t ) = B (0) exp( rt ). � ◮ S ( n ) j =1 σ ij ξ ( m ) n ) = S i (0) exp( rkT / n ) � k � d ( kT T m =1 (1 + ) i j n � ˜ Nn , k U ( j ) j =1 (1+˜ ) × i � k � 1+(1 − exp( − λ T / n )) u i N n , k = � k where ˜ m =1 ρ n , m . ◮ T n –the set of all stopping times σ : Ω n → { 0 , T n , ..., T } with respect to the filtration generated by S ( n ) . Yan Dolinsky Error Estimates for Multinomial Approximations of American Options

Outline American options in continuous time models Discrete time approximations Results Sketch of the proof Remarks References American options ◮ American option with the payoff process Y ( n ) � kT S ( n ) � kT , kT � � � � = F , 0 ≤ k ≤ n . n n n ◮ The term E n (exp( − r τ ) Y ( n ) ( τ )) V n = sup τ ∈T n is an arbitrage–free price of the n –step market. Yan Dolinsky Error Estimates for Multinomial Approximations of American Options

Outline American options in continuous time models Discrete time approximations Results Sketch of the proof Remarks References Dynamical programming algorithm for V n For any 0 ≤ k ≤ n define F ( n ) : R d + → R + by k F ( n ) n ( x ) = F ( x , T ) and for k < n , F ( n ) F ( x , kT / n ) , EF ( n ) k +1 ( x 1 Γ ( n ) 1 , ..., x d Γ ( n ) � � ( x ) = max d ) k where for any 1 ≤ i ≤ d U ( j ) d 1 + ˜ � T I ρ n , 1 =1 Γ ( n ) σ ij ξ (1) � i � � = exp( rT / n ) 1 + . i j n 1 + (1 − exp( − λ T / n )) u i j =1 Then V n = F ( n ) ( S (0)) . 0 Yan Dolinsky Error Estimates for Multinomial Approximations of American Options

Outline American options in continuous time models Discrete time approximations Results Sketch of the proof Remarks References The main result Theorem: For any ǫ > 0 there exists a constant C ǫ such that for any n | V − V n | < C ǫ n ǫ − 1 8 . If F is bounded then there exists a constant C such that for any n | V − V n | < Cn − 1 8 . Yan Dolinsky Error Estimates for Multinomial Approximations of American Options

Outline American options in continuous time models Discrete time approximations Technical preparations Results Strong approximation theorems Sketch of the proof Remarks References Preparations Fix n . For 1 ≤ i ≤ d and 0 ≤ k ≤ n set � d j =1 σ 2 ◮ β i = r − ij . 2 ◮ N n , k = � k j =1 I { N ( jT / n ) − N (( j − 1) T / n ) ≥ 1 } . ◮ S C , n ( kT / n ) = S i (0) exp( β i kT / n + � d j =1 σ ij W j ( kT / n )) i � Nn , k j =1 (1+ U ( j ) ) × i � k . � 1+(1 − exp( − λ T / n )) u i Yan Dolinsky Error Estimates for Multinomial Approximations of American Options

Outline American options in continuous time models Discrete time approximations Technical preparations Results Strong approximation theorems Sketch of the proof Remarks References First step Set Y C , n ( t ) = F ( t , S C , n ( t )). Let T ( n ) ⊂ T be the set of all stopping times (in the continuous model) with values in the set { 0 , T n , ..., T } . Define the map φ n : T → T ( n ) by φ n ( τ ) = T n min { k : kT n ≥ τ } . By using the regularity properties of F and the fact that 0 ≤ φ n ( τ ) − τ ≤ 1 n we obtain that for V C τ ∈T ( n ) E (exp( − r τ ) Y C , n ( τ )) , n := sup | V C n − V | ≤ C 1 n − 1 / 4 . Yan Dolinsky Error Estimates for Multinomial Approximations of American Options

Outline American options in continuous time models Discrete time approximations Technical preparations Results Strong approximation theorems Sketch of the proof Remarks References Second step Fix n . For 1 ≤ i ≤ d and 0 ≤ k ≤ n set ◮ S D , n ( kT / n ) = i j =1 σ ij ξ ( m ) T / n � k � d � S i (0) exp( β i kT / n + ) m =1 j � ˜ Nn , k U ( j ) j =1 (1+˜ ) × i � k . � 1+(1 − exp( − λ T / n )) u i ◮ Y D , n ( kT / n ) = F ( kT / n , S D , n ( kT / n )). By using the regularity properties of F we obtain that for V D E (exp( − r τ ) Y D , n ( τ )) , n := sup τ ∈T n | V D n − V n | ≤ C 2 n − 1 / 4 . Yan Dolinsky Error Estimates for Multinomial Approximations of American Options

Outline American options in continuous time models Discrete time approximations Technical preparations Results Strong approximation theorems Sketch of the proof Remarks References Final step We need to estimate | V C n − V D � sup exp( − r τ ) F ( τ, S C , n ( τ )) � � � n | := τ ∈T ( n ) E exp( − r τ ) F ( τ, S D , n ( τ )) �� � − sup E n � τ ∈T n where S C , n ( kT / n ) = S i (0) exp( β i kT / n + � d j =1 σ ij W j ( kT / n )) × i � Nn , k j =1 (1+ U ( j ) ) i � k . � 1+(1 − exp( − λ T / n )) u i S D , n m =1 ξ ( m ) ( kT / n ) = S i (0) exp( β i kT / n + � d T / n � k � j =1 σ ij ) × i j � ˜ Nn , k U ( j ) j =1 (1+˜ ) i � k . � 1+(1 − exp( − λ T / n )) u i Yan Dolinsky Error Estimates for Multinomial Approximations of American Options