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Error Estimates for Multinomial Approximations of American Options in Merton's Model

We derive error estimates for multinomial approximations of American options in a multidimensional jump--diffusion Merton's model. We assume that the payoffs are Markovian and satisfy Lipschitz type conditions. Error estimates for such type of approximations were not obtained before. Our main tool is the strong approximations theorems for i.i.d. random vectors which were obtained [14]. For the multidimensional Black--Scholes model our results can be extended also to a general path dependent payoffs which satisfy Lipschitz type conditions. For the case of multinomial approximations of American options for the Black--Scholes model our estimates are a significant improvement of those which were obtained in [8] (for game options in a more general setup)


Introduction
This paper deals with multinomial approximations of American options arbitragefree prices in the multidimensional jump-diffusion Merton's model with finite horizon. The Merton model is a generalization of the Black-Scholes model and it allows the stock to have jumps of compound Poisson type. We consider a Markovian payoffs which satisfy Lipschitz type conditions and derive error estimates for an appropriate multinomial approximations. In the multidimensional Black-Scholes model these error estimates can be also derived for general path dependent payoffs which satisfy Lipschitz type conditions. For American options in finite horizon Merton's model arbitrage-free prices can not be calculated explicitly. Since multinomial models are active on a discrete set of times and defined on a discrete probability space, then arbitrage-free prices in these models can be calculated efficiently by dynamical programming algorithm. Thus convergence results for multinomial approximations provides an efficient tool to evaluate arbitrage free prices in the Merton model.
Several papers dealt with multinomial approximations of American options in models with jumps (see, [10] and [12]). In both of the papers the authors considered the case with one risky asset and used the weak convergence approach. The main tool that was used in the above papers is the stability results for Snell's envelopes under weak convergence which were obtained in [11]. In [2] the weak convergence approach was used to show convergence results for multinomial approximations of game options (which were introduced in [6]) in the Merton model. The main disadvantage of the weak convergence approach is that this machinery can not provide, in principle, speed of convergence estimates. Thus no error estimates were obtained for multinomial approximations of American options in the Merton model.
Clearly, from practical point of view it is valuable to find estimates of the corresponding errors. In order to obtain error estimates we should consider all the market models on the same probability space, and so methods based on strong approximation theorems come into picture. Strong approximation theorem allows to construct a probability space which contain all the markets models such that the risky assets in the discrete time models will be "close" with respect to the sup norm to the risky assets in the continuous model. Several authors applied strong invariance principles in order to obtain error estimates for American and game options in the one dimensional BS model (see, [5], [7] and [3]). In all of these papers the authors used the Skorokhod embedding tool of i.i.d. random variables into the one dimensional Brownian motion. This tool can not be applied for the multidimensional case. In [8] the author studied discrete time approximations of Dynkin's game values for the multidimensional Brownian motion. The main tool that was used there is strong approximation theorems which were developed in [1] and they work for sequences of random vectors with close characteristic functions.
In [14] the author considered a new approach to strong approximations in the multidimensional case. He showed that for a given sequence of i.i.d. random vectors X (1) , X (2) ...., and a random vector Y which has the same expectation and covariance matrix as X (1) , it is possible to construct a sequence of i.i.d. vectors Y (1) , Y (2) .... such that Y (1) ∼ Y and the normalized sums of the last sequence will be "close" to the normalized sums of the first sequence. Furthermore for any k < m, Y (k) is independent of X (m) . The above approach will be the main tool that we use in order to establish the results in this paper. For Merton's model strong approximation theorems were not used before. An interesting question which is still open, is whether the method from [14] can be applied for game options approximations.
Let us remark, that the estimates for the Brownian motion by means of normalized sums of independent random vectors obtained in [14] are much better than those which were obtained in [1].
Main results of this paper are formulated in the next section where we also introduce the notations that will be used. In Section 3 we derive auxiliary lemmas that we use. In Section 4 we complete the proof of main results of the paper.

Preliminaries and main results
Consider a complete probability space (Ω, F , P ) together with a standard ddimensional continuous in time Brownian motion {W (t) = (W 1 (t), ..., W d (t))} ∞ t=0 , a Poisson process {N (t)} ∞ t=0 with intensity λ and independent of W , and a sequence of i.i.d. random vectors independent of W and N . We also assume that for any 1 ≤ j ≤ d, E|U (1) j | 2 < ∞ where E denotes the expectation with respect to P . A d-dimensional Merton's model with horizon T < ∞ consists of a savings account which given by and of d risky stocks {S(t) = (S 1 (t), ..., S d (t))} T t=0 given by where σ = (σ ij ) 1≤i,j≤d is a nonsingular matrix, S i (0) > 0 and without loss of generality we assume that Consider an American option with the payoff process where F : R d + × R + → R + is a function such that for some constant L ≥ 1 and for any t ≥ s ≥ 0 and υ,υ ∈ R d , Let T be the set of stopping times with respect to the natural filtration generated by S (which satisfies the usual conditions) with values not exceeding T . The equality (2.3) guaranties that the probability measure P is a martingale measure. Thus the term gives an arbitrage-free price for the American option. For any n ∈ N define a sequence of i.i.d. random vectors {U n,i = (U n,i 1 , ..., where I Q = 1 if an event Q occurs and = 0 if not, and M (n) ∈ N satisfies Notice that Next we describe the discrete time markets which we use to approximate the Merton model. Let A ∈ M d+1 (R) be an orthogonal matrix such that it last column equals to 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 d+1 , ..., 1 d+1 } ∞ . Define a sequence of i.i.d. random vectors ξ (1) , ξ (2) , ... by The probability space (Ω ξ , P ξ ) was introduced in [4]. For any n we extend (Ω ξ , P ξ ) to a probability space (Ω n , G n , P n ) such that it contains a three independent se- The second sequence is a sequence of Bernoulli random variables such that P n {ρ n,1 = 1} = 1 − exp(−λT /n) and the second sequence satisfies u n,1 ∼ U n,1 .
For any n ∈ N, 0 ≤ k ≤ n and 1 ≤ i ≤ d set N n,k = k m=1 ρ n,m and (2.12) n and E n denotes the expectation with respect to P n . Consider a multinomial n-step market which is active in the moments 0, T n , 2T n , ..., T and consists of a savings account which given by (2.1) and of d risky stocks S (n) = (S (n) 1 , ..., S (n) d ) given by (2.12). Next, we introduce an American option with the payoff process Let T n be the set of stopping times with respect to the filtration {σ(S with values in {0, 1, ..., n}. Observe that for any n, P n is a martingale measure for the n-step market. Thus is an arbitrage-free price of the n-step market. The following theorem says that the arbitrage-free prices of the n-step markets converge to the arbitrage-free price of the Merton model and provides an estimates on the error terms.
Theorem 2.1. For any ǫ > 0 there exists a constant C ǫ such that for any n Remark 2.2. Theorem 2.1 can be extended to a case where we have a finite number of Poisson clocks. Namely, consider a complete probability space (Ω, F , P ) together with a standard d-dimensional continuous in time Brownian motion W , m independent Poisson processes N (1) , ..., N (m) , which are independent of W , and for any We assume that the sequences are independent of each other and independent of W and N (1) , ..., N (m) . We also assume E|U k,1 j | 2 < ∞, for j ≤ d and k ≤ m. The risky assets are given by For an analogical multinomial models (to those that we used for one Poisson process) we can prove a similar result to Theorem 2.1. The proof for this case can be done in a similar way to the proof of Theorem 2.1 and by using the same ideas. For simplicity we provide the proof only for the case with one Poisson process. Remark 2.3. By using the Cauchy-Schwarz inequality and the Chebyshev's inequality it follows that we can set M (n) = [cn 1/4 ] for some constant c ([x] is the integer part of x). Thus in the n-step mutinomial model which is given by (2.12) the number of growth rates is proportional to n 1/4 . If U (1) takes on a finite number of values, then we can construct the multinomial models with a fixed number of growth rates by letting U (n,i) = U (i) for any i, n. In this case the proof of Theorem 2.1 is simpler than for the general case and does not require Lemma 3.3.

Auxiliary lemmas
We start with a standard result, but since we could not find a direct reference its proof is given here for readers' convenience.
where M is the set of stopping times with respect to the above filtration with values is a sequence of i.i.d. random vectors such that for any k, ( Let p(·) be the probability density function of ( Then where ψ 0 is given by the following dynamical programming relations It is well known (see [13]) that A = A(0) can be calculated by the following dynamical programming relations A(n) = Y (n) and for 0 ≤ k < n, A(k) = max(Y (k), E(A(k + 1)|G k )). Next, we derive several estimates that will be used in this Section. For any n ∈ N and 1 ≤ i ≤ d denote, By using the inequality | exp(x) − exp(y)| ≤ |x − y| exp(max(x, y)) we obtain that for any From the scaling property of the Brownian motion it follows E max for some constant c 1 . This together with (3.7) and the Holder inequality gives for some constant c 2 . Let T 1,n ⊂ T be the set of stopping times with values in Lemma 3.2. There exists a constant C 1 such that for any n is obvious. Thus it remains to prove that V − V (1) n ≤ C 1 n −1/4 . Fix n ∈ N and choose ǫ > 0. There exist τ ∈ T such that (3.11) V < ǫ + E(exp(−rτ )F (S(τ ), τ )).

Proof. ¿From Lemma 3.1 it follows that V
(1) n = sup τ ∈T 2,n E(exp(−rτ )F (S(τ ), τ )). This together with (2.5) gives i . The sequences U n,j , U (j) are independent of the Poisson process N . Thus i ) m−j = from (2.9) and the inequality |x m − y m | ≤ m|x − y|(max(x, y)) m−1 (for x, y ≥ 0, and m ∈ N) we obtain i ))n −1/8 . By combining (3.19)-(3.21) we obtain (3.18). For any n ∈ N, 0 ≤ k ≤ n and 1 ≤ i ≤ d set Notice that Z n,k ∼ N n,k (the last term was defined before (2.12)). ¿From (2.3) and (2.9) it follows that there exists a constant c 5 such that Lemma 3.4. There exists a constant C 3 such that for any n ¿From Lemma 3.1 it follows that V (2) n = sup τ ∈T 3,n E(exp(−rτ )F (S(τ ), τ )). It is easy to verify that the terms E(sup 0≤t≤T (S 1,n i (t) + S 2,n i (t)) 2 ) are uniformly (with respect to n) bounded. Thus (3.26) and the Cauchy-Schwarz inequality gives that there exist constants c 6 , c 7 such that and the result follows.
Next, we derive estimates on a discrete probability spaces. For any n and 0 ≤ k ≤ n define (1−exp(−λT /n))Enu n,1 i k , for kT /n ≤ t < (k + 1)T /n and V (4) n = sup τ ∈Tn E n (exp(−τ T /n)F (τ, S 3,n (τ )), where S 3,n = (S 3,n 1 , ..., S 3,n d ). Lemma 3.5. There exists a constant C 4 such that for any n Proof. For any n ∈ N, 0 ≤ m ≤ n and 1 ≤ i ≤ d set, Observe that for any 1 ≤ j ≤ d, |ξ This together with the Doob-Kolmogorov inequality (see [9]) gives for some constant c 9 . It is easy to verify that the terms E n ( i (t)) 2 ), n ∈ N are uniformly bounded. From (3.31)-(3.32) and the Cauchy-Schwarz inequality we obtain that there exists a constant C 4 such that

Proof of main results
The following result which we state without proof was established in [14] under more general assumptions (see Theorem 2.1, Corollary 3.1 and Lemma 5.1 there). This result is the main tool that we use in order to complete the proof of Theorem 2.1.
Theorem 4.1. Consider a probability space together with a sequence X (1) , X (2) , ... of i.i.d. d-dimensional random vectors and a d-dimensional random vector Y .
¿From Lemmas 3.2-3.5 it follows that there exists a constant C 5 such that for any n ∈ N, |V − V n | ≤ C 5 n −1/8 + |V (2) n − V (3) n |. Thus in order to complete the proof of Theorem 2.1 it remains to establish the following lemma.
Lemma 4.2. For any ǫ > 0 there exists a constantC ǫ such that for any n Proof. Fix n ∈ N and ǫ > 0. We start with proving the inequality ¿From (2.11) and Theorem 4.1 it follows that we can construct on the same probability space (Ω,F ,P ) two sequence of i.i.d. random vectors X (1) , ..., for some constant C 6 . Furthermore, for any k > 1 the random vectors X (1) , ..., X (k−1) , Y (k) , ..., Y (n) are independent. We can extend the constructed probability space such that it will contain also two independent sequences of i.i.d. random vectors F (1) , ..., F (n) and G (1) , ..., G (n) which are independent of X (1) , ..., X (n) , Y (1) , ..., Y (n) and satisfy F (1) ∼ ρ n,1 , G (1) ∼ U (n,1) (ρ n,1 , U n,1 were defined in Section 2). For any p ≤ n and i ≤ d set, It is easy to verify that there exists a constantĈ and for any p ≥ 1 there exists a constantĈ p (the above constants does not depend on n) such that and Observe thatS ∼ S 2,n andŜ ∼ S 3,n . ¿From Lemma 3.1 it follows thatṼ = V (2) n andV = V (3) n , in the last equality we used the fact that X (1) , ..., X (k−1) , Y (k) , ..., Y (n) are independent for any k > 1. SinceT ⊂T then from (4.5), (4.7)-(4.9), the inequality | exp(x) − exp(y)| ≤ |x − y| max(exp(x), exp(y)) and the Holder inequality we obtain 1−8ǫ and q = 1 8ǫ . This completes the proof of the inequality (4.4). The inequality V 1 8 can be proved in a similar way, just take Consider an American option with the payoff process Y (t) = F t (S) where S = S(ω) ∈ M [0, ∞) is a random function taking the value S(ω, t) at t ∈ [0, ∞). When considering F t (S B ) for t < ∞ we take the restriction of S to the interval [0, t]. The term V = sup τ ∈T E(exp(−rτ )Y (τ )) gives an arbitrage-free price for our model. For the multinomial models we consider the payoffs Y (n) (k) = F kT n (S (n) ) and the arbitrage-free prices V n = sup τ ∈Tn E n (exp(−τ T /n)Y (n) (τ )). Lemmas 3.2, 3.5 and 4.2 can be extended to this setup in a way that does not ruin the estimates of Theorem 2.1. The problem is with Lemmas 3.3-3.4. For path dependent options the equality before (3.19) does not follows from Lemma 3.2 (although is correct) and the the first inequality in (3.27) should be replaced by |V The way to fix it is to consider a piecewise constant approximations of S,Ṡ (n) (t) = S kT /n for kT /n ≤ t < (k + 1)T /n, 0 ≤ k ≤ n. If we could provide an estimates of the term E max 0≤k≤n |F kT n (S) −    n that were defined in (3.6) and the events Q k which were defined before (3.26). For Russian options we have the following inequality max 0≤k≤n |F kT n (S) − F kT n (Ṡ (n) )| ≤ LI {∪ n j=1 Qj } sup 0≤t≤T F t (S) + sup 0≤t≤T By using estimates that were derived in Section 3 we obtain that E(max 0≤k≤n |F kT n (S)− F kT n (Ṡ (n) )|) is of order n −1/4 and so the estimates from Theorem 2.1 are also valid for Russian options in Merton's model.