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Outline American options in continuous time models Discrete time approximations Results Sketch of the proof Remarks References

Error Estimates for Multinomial Approximations

• f 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) = (W1(t), ..., Wd(t))}∞ t=0– standard d dimensional

Brownian motion.

◮ {N(t)}∞ t=0– Poisson process with intensity λ and independent

• f 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

• f values and denote uj = EU(1)

j

, 1 ≤ j ≤ d.

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,

Si(t) = Si(0) exp(µit +

d

• j=1

σijWj(t))

N(t)

• j=1

(1 + U(j)

i

) where σ = (σij)1≤i,j≤d is a nonsingular matrix.

◮ Without loss of generality we assume that

µi = r − λ − ui − d

j=1 σ2 ij

2 , 1 ≤ i ≤ d.

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 : Rd

+ × R+ → R+. ◮ The term

V = sup

τ∈T

E(exp(−rτ)Y (τ)) gives an arbitrage–free price for the American option.

◮ We assume that for some constant L ≥ 1

|F(υ, t) − F(˜ υ, s)| ≤ L

d

• i=1

|υi − ˜ υi| + L(t − s)(1 +

d

• i=1

|υi|) for any t ≥ s ≥ 0 and υ, ˜ υ ∈ Rd.

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 ∈ Md+1(R) be an orthogonal matrix such that it last

column equals to (

1 √ d+1, ..., 1 √ d+1). ◮ Let Ωξ = {1, 2, ..., d + 1}∞ be the space of infinite sequences

ω = (ω1, ω2, ...); ωi ∈ {1, 2, ..., d + 1} with the product probability Pξ = {

1 d+1, ..., 1 d+1}∞. ◮ Define a sequence of i.i.d. random vectors ξ(1), ξ(2), ... by

ξ(i)(ω) = √ d + 1(Aωi1, Aωi2, ..., Aωid), i ∈ N. We have Eξ(1) = 0 and Eξ(1)

i

ξ(1)

j

= δij.

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 (Ωξ, Pξ) to a probability space (Ωn, Pn) 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. ◮ {ρn,k}n k=1– sequence of Bernoulli random variables such that

Pn{ρn,1 = 1} = 1 − exp(−λT/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

◮ The market is active in the moments 0, T n , 2T n , ..., T. ◮ B(t) = B(0) exp(rt). ◮ S(n) i

(kT

n ) = Si(0) exp(rkT/n) k m=1(1 +

• T

n

d

j=1 σijξ(m) j

) ×

˜

Nn,k j=1 (1+˜

U(j)

i

)

• 1+(1−exp(−λT/n))ui

k where ˜ Nn,k = k

m=1 ρn,m. ◮ Tn–the set of all stopping times σ : Ωn → {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 n

• = F
• S(n)kT

n

• , kT

n

• , 0 ≤ k ≤ n.

◮ The term

Vn = sup

τ∈Tn

En(exp(−rτ)Y (n)(τ)) 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 Vn

For any 0 ≤ k ≤ n define F (n)

k

: Rd

+ → R+ by

F (n)

n (x) = F(x, T)

and for k < n , F (n)

k

(x) = max

• F(x, kT/n), EF (n)

k+1(x1Γ(n) 1 , ..., xdΓ(n) d )

• where for any 1 ≤ i ≤ d

Γ(n)

i

= exp(rT/n)

• 1 +
• T

n

d

• j=1

σijξ(1)

j

• 1 + ˜

U(j)

i

Iρn,1=1 1 + (1 − exp(−λT/n))ui . Then Vn = F (n) (S(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 Cǫ such that for any n |V − Vn| < Cǫnǫ− 1

8.

If F is bounded then there exists a constant C such that for any n |V − Vn| < Cn− 1

8.

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 Technical preparations Strong approximation theorems

Preparations

Fix n. For 1 ≤ i ≤ d and 0 ≤ k ≤ n set

◮ βi = r − d

j=1 σ2 ij

2

.

◮ Nn,k = k j=1 I{N(jT/n)−N((j−1)T/n)≥1}. ◮ SC,n i

(kT/n) = Si(0) exp(βikT/n + d

j=1 σijWj(kT/n))

×

Nn,k

j=1 (1+U(j) i

)

• 1+(1−exp(−λT/n))ui

k .

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 Technical preparations Strong approximation theorems

First step

Set Y C,n(t) = F(t, SC,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 φn : T → T (n) by φn(τ) = T

n min{k : kT n ≥ τ}.

By using the regularity properties of F and the fact that 0 ≤ φn(τ) − τ ≤ 1

n we obtain that for

V C

n := sup τ∈T (n) E(exp(−rτ)Y C,n(τ)),

|V C

n − V | ≤ C1n−1/4.

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 Technical preparations Strong approximation theorems

Second step

Fix n. For 1 ≤ i ≤ d and 0 ≤ k ≤ n set

◮ SD,n i

(kT/n) = Si(0) exp(βikT/n +

• T/n k

m=1

d

j=1 σijξ(m) j

) ×

˜

Nn,k j=1 (1+˜

U(j)

i

)

• 1+(1−exp(−λT/n))ui

k .

◮ Y D,n(kT/n) = F(kT/n, SD,n(kT/n)).

By using the regularity properties of F we obtain that for V D

n := sup τ∈Tn

E(exp(−rτ)Y D,n(τ)), |V D

n − Vn| ≤ C2n−1/4.

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 Technical preparations Strong approximation theorems

Final step

We need to estimate |V C

n − V D n | :=

• sup

τ∈T (n) E

• exp(−rτ)F(τ, SC,n(τ))
• − sup

τ∈Tn

En

• exp(−rτ)F(τ, SD,n(τ))
• where

SC,n

i

(kT/n) = Si(0) exp(βikT/n + d

j=1 σijWj(kT/n))× Nn,k

j=1 (1+U(j) i

)

• 1+(1−exp(−λT/n))ui

k . SD,n

i

(kT/n) = Si(0) exp(βikT/n + d

j=1 σij

• T/n k

m=1 ξ(m) j

)×

˜

Nn,k j=1 (1+˜

U(j)

i

)

• 1+(1−exp(−λT/n))ui

k .

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 Technical preparations Strong approximation theorems

The main tool

Theorem:(Sakhanenko 2002). Let X, Y be a d–dimensional random vectors. Assume that EX = EY and E(XiXj) = E(YiYj) ∀i, j ∈ {1, ..., d}. For any z > 0 and n ∈ N it is possible to construct a probability space which contains two sequence of i.i.d. random vectors X (1), ..., X (n), Y (1), ..., Y (n) such that X (1) ∼ X, Y (1) ∼ Y and P( max

1≤k≤n || k

• m=1

X (m) − Y (m)|| > z) ≤ ˜ Cn z3 E(||X||3 + ||Y ||3). Furthermore, for any k the random vectors X (1), ..., X (k−1), Y (k), Y (k+1), ..., Y (n), are independent.

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 Technical preparations Strong approximation theorems

The inequality V D

n − V C n < Cǫnǫ−1/8

From the Sakhanenko theorem it follows that we can construct a probability space which contains {W (kT/n)}n

k=0, ξ(1), ξ(2), ..., ξ(n)

such that

◮

P( max

1≤k≤n ||W (kT/n) −

• T/n

k

• m=1

ξ(m)|| > n−1/8) ≤ ˆ Cn−1/8.

◮ For any k the random vectors ξ(1), ..., ξ(k−1), W (kT/n) −

W ((k − 1)T/n), ..., W (T) − W ((n − 1)T/n) are independent. We extend the probability space such that it will also contain two independent sequences of i.i.d. random vectors {U(k)}∞

k=1 and

{ρn,k}n

k=1 which are independent of W and ξ.

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 Technical preparations Strong approximation theorems

Continue of the inequality V D

n − V C n < Cǫnǫ−1/8

On the above probability space we have V D

n − V C n =

sup

τ∈T (ξ,U,ρ)

E

• exp(−rτ)F(τ, SD,n(τ))
• −

sup

τ∈T (W ,U,ρ,ξ)

E

• exp(−rτ)F(τ, SC,n(τ))
• ≤

sup

T (ξ,U,ρ)

E

• exp(−rτ)F(τ, SD,n(τ))
• −

sup

τ∈T (ξ,U,ρ)

E

• exp(−rτ)F(τ, SC,n(τ))
• ≤

E max

0≤k≤n |F(kT/n, SD,n(kT/n)) − F(kT/n, SC,n(kT/n))|

< Cǫnǫ−1/8.

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 Technical preparations Strong approximation theorems

The inequality V C

n − V D n < Cǫnǫ−1/8

We do it in a similar way. Just change the roles of {W (kT/n)}n

k=1 and {ξ(k)} n k=1.

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

Weak convergence approach

It is well known that S(n) ⇒ S in distribution on the space D[0, T] equipped with the Skorohod topology. By using the stability of

• ptimal stopping values under weak convergence it can be proved

that limn→∞ Vn = V . The main disadvantage of the weak convergence approach is that this machinery can not provide, in principle, speed of convergence estimates.

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

Strong approximation theorems

There are other strong approximation theorems (multidimensional case) which provide a better estimate of max

1≤k≤n ||W (kT/n) −

• T/n

k

• m=1

ξ(m)|| however the construction of the corresponding probability space does not respect filtrations in a very substantial way as they depend on the far away future.

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

Extension to path dependent payoffs

These results are remain valid for path dependent payoffs which satisfy strong regularity conditions. For the Black–Scholes model (no Poisson process) the above results can be extended to path dependent options which satisfy Lipchitz type conditions.

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

Extension to game options

◮ Let F, ∆ : Rd + × R+ → R+. ◮ Consider a game option with the payoff function

H(σ, τ) = F(S(σ ∧ τ), σ ∧ τ) + Iσ<τ∆(S(σ), σ).

◮ V := infσ∈T supτ∈T E

• exp(−r(σ ∧ τ))H(σ, τ)
• is an

arbitrage–free price for the Game option. An interesting question is whether we can apply strong approximation theorems in order to approximate V by similar terms on the discrete probability spaces that were used for American options.

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

References

◮ Ya.Dolinsky and Yu.Kifer, ”Binomial approximations for

barrier options of Israeli style”, Annals of Dynamic Games,

• vol. XI.

◮ B.Hu, J.Liang and L.Liang, ”Optimal convergence rate of the

binomial tree scheme for American options with jump diffusion and their free boundaries”, SIAM Journal on Financial Mathematics. 1 (2010), 30–65.

◮ Yu.Kifer, ”Error estimates for binomial approximations of

game options”, Ann. Appl. Probab. 16 (2006), 984-1033.

◮ D.Lamberton and L.C.G Rogers, ”Optimal stopping and

Embedding”, J. Appl. Probab. 37 (2000), 1143–1148.

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

References

◮ H.He, ”Convergence from discrete to continuous time

contingent claim prices”, Rev. Financial Stud. 3 (1990).

◮ Yu.Kifer, ”Optimal stopping and strong approximation

theorems”, Stochastics 79 (2007), 253–273.

◮ S.Mulinacci, ”American path–dependent options: analysis and

approximations”, Rend. Studi Econ. Quant. 2002 (2003), 93–120.

◮ R.Maller, D.Solomon and A.Szimayer, ”A Multinomial

Approximation of American Option Prices in a Levy Process Model”, Math. Finance 16 (2006), 613–633.

◮ A.I Sakhanenko, ”A New Way to Obtain Estimates in the

Invariance Principle”, High Dimensional Probability II, (2000) 221–243.

Yan Dolinsky Error Estimates for Multinomial Approximations of American Options