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Approximation of the Snell Envelope and American Options Prices in Dimension One

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Approximation of the Snell Envelope and American Options Prices in Dimension One ESAIM: Probability and Statistics March 2002, Vol. 6, 1–19 URL: http://www.emath.fr/ps/ DOI: 10.1051/ps:2002001 APPROXIMATION OF THE SNELL ENVELOPE AND AMERICAN OPTIONS PRICES IN DIMENSION ONE Vlad Bally1 and Bruno Saussereau2 Abstract. We establish some error estimates for the approximation of an optimal stopping problem along the paths of the Black–Scholes model. This approximation is based on a tree method. Moreover, we give a global approximation result for the related obstacle problem. Mathematics Subject Classification. 49L20, 60G40, 65M15, 91B28. Received February 15, 2001. Revised November 13, 2001. Introduction The purpose of this article is to estimate the rate of convergence of the approximation scheme of an optimal stopping problem along Brownian paths, when the Brownian Motion is approximated by a random walk. F F P We consider a standard Brownian Motion (Bt)t≥0 defined on a filtered probability space (Ω, , ( t)t≥0, ) and we denote x ∈ R Xt = x + bt+ aBt, a,b,x ,a>0. Given a function h which is assumed to be (at least) Lipschitz-continuous, our aim is to compute x E x |F Yt =sup (h(τ,Xτ ) t) , τ∈Tt,T where Tt,T is the set of all the stopping times taking values in [t,T ]. This is the Snell Envelope of the process x (h(t, Xt ))t∈[0,T ]. Our results apply to standard American Options. Recall, for example, that an American Put on a stock is the right to sell one share at a specified price K (called the exercise price) at any instant until a given future date T (called the expiration date or date of maturity). In the Black–Scholes model, the value at time t (0 ≤ t ≤ T ) x −rt − x + of such an option is given by the Yt defined above with h(t, x):=e (K e ) . The constant r is the interest rate (assumed to be positive), the diffusion coefficient a is the so-called volatility (assumed to be positive also) and the drift coefficient b, in this case, is equal to r − a2/2. We give here a discretization scheme based on a tree method, which is a variant of the tree method (see (1.20)) studied in [8–11]. These authors study a binomial approximation based on the discretization of the Brownian Keywords and phrases: Dynamic programming, snell envelope, optimal stopping. 1 Universit´e du Maine, Laboratoire Statistiques et Processus, BP. 535, 72017 Le Mans Cedex, France; e-mail: [email protected] 2 Universit´e de Franche-Comt´e,LaboratoiredeMath´ematiques de Besan¸con, 16 route de Gray, 25000 Besan¸con, France; e-mail: [email protected] c EDP Sciences, SMAI 2002 2 V. BALLY AND B. SAUSSEREAU −rt x+(r−a2/2)t+ay motion (Bt)t∈R+ .Sotheytake(t, y) 7→ h(t, y)=e (K − e )+ which is slightly different from the function h considered here. An alternative to the tree method is to approach the numerical solution of the optimal stopping problem based on the discretization of the variational inequality satisfied by the value function. We refer to [1] for the error estimates for finite difference schemes. A comparison of existing methods forAmericanOptionsisgivenin[5]. Let us present our algorithm. We assume that the function h is of the special form h(t, x)=e−rtf(x) (which covers the applications in mathematical finance). We fix ε>0 and we define the sequence of functionsu ˆk by uˆk(T )(x + iε)=f(x + iε), −k(T ) ≤ i ≤ k(T ), . + − − − ≤ ≤ uˆk(x + iε)=max f(x + iε),γε uˆk+1(x + iε + ε)+γε uˆk+1(x + iε ε) , k i k, ∼ 2 2 + − where k(T ) is an integer depending on ε such that k(T ) a T/ε . The constants γε and γε appear from exact computations related to the first exit time for the Brownian motion with drift. In Section 1 we give their interpretations and their exact values. Let us give here just an expansion in powers of ε which is sufficient for concrete calculations: 1 b r b3 br 5r2 rb γ+ = + ε − ε2 − + ε3 + + ε4 + O(ε5) ε 2 2a2 2a2 6a6 2a4 12a4 12a6 1 b r b3 br 5r2 rb γ− = − ε − ε2 + + ε3 + + ε4 + O(ε5). ε 2 2a2 2a2 6a6 2a4 12a4 12a6 We prove that | x − |≤ Y0 uˆ0(x) cδε. (0.1) The values√ of c and δε depend on the regularity of the function f (see Th.p 1.2). If f is Lipschitz continuous then − δε = ε,iff is a linear combination of convex functions then√ δε = ε ln(ε)andiff is twicep differentiable −3 then δε = ε. The complexity of the algorithm is of order ε − ln ε. The estimation in δε = ε − ln(ε) is new, as far as we know. The idea which leads to our algorithm is the following. We define a sequence of stopping times by τ0 =0 x x x | − |≥ ∈N { ∈ Z} and τk+1 =inf t>τk : Xt Xτk ε .Thechain(Xτk )k lives on the grid x + iε,i and has a very x x simple structure because if we define σ = τ − τ and δ = X − X ,then(σ ,δ ) ∈N are independent k k+1 k k τk+1 τk k k k + − random variables and we know their laws (this permits to compute explicitly γε and γε ). On the other hand, ∼ ∼ 2 2 x using the law of large numbers, τk(T ) T if k(T ) a T/ε . So we replace the Snell envelope (Yt )0≤t≤T of x ε,x x (h(t, Xt ))0≤t≤T by the discrete Snell envelope (Qk )0≤k≤k(T ) of (h(τk,Xτ ))0≤k≤k(T ). The algorithm described ε,x k above represents the computation of Q0 using the dynamical programming principle (see Sect. 1). Note the x analogy of our idea (to produce a chain which approximates (Xt )t≥0) and that of [11] (to embed a chain in the Brownian motion). 2 2 In order to control the error we employ the basic evaluation E|T − τk(T )| ≤ Cε proved in Section 1. This immediately leads to (0.1) in the case of a Lipschitz functions f.Iff is twice differentiable we use Ito’s formula on the intervals [τk,τk+1] in order to improve our evaluation. If f is a linear combination of convex functions (this is the more interesting case) we use the generalization of the Ito’s formula for such functions and then local time comes into play. In this case, we have to prove (see Appendix 1) a more refined inequality concerning the increments of the local time. APPROXIMATION OF THE SNELL ENVELOPE AND AMERICAN OPTIONS PRICES 3 x In the second section, we prove a global approximation result. It is well known that we may write Yt = x ∈ × Rd u(t, Xt )whereu(t, x), (t, x) [0,T ] is the solution of the obstacle problem 2 ∂t +(a /2)∂x2 + b∂x u(t, x)=0, on u>h 2 ∂ +(a /2)∂ 2 + b∂ u(t, x) ≤ 0, on u = h t x x u(T,x)=h(T,x). We prove that the functionu ˆk(x) constructed above provides an uniform approximation for u(t, y). More precisely, we prove that −rt |u(tk,y) − e uˆk(¯y)|≤Cδε 2 2 2 2 where tk = kε /a ,0≤ k ≤ a T/ε ,and¯y is the projection of y on the grid {x + iε, − tk ≤ i ≤ tk} constructed x for the approximation of Y0 . The constant C depends on a, b, T and on the regularity of the obstacle (see Th. 2.1). The above approximation permits also to obtain an approximation of the continuation region for the obstacle problem and of the first and last optimal exercise time for an American Option. A detailed analysis of the constants appearing in our evaluation is rather heavy, so we leave it out here and send the interested reader to [14]. In this paper we will just emphasize the structure of these constants in the following sense: • C will designates constants which depend on a, b and T . It may change from line to line; • Ci(h), i =1, 2,... designates constants which depend on the obstacle h and its structure will be specified in each case. This is interesting because one can see how the regularity of h appears; • Ci(h, x), i =1, 2,... designates constants which depend on the obstacle h and on the starting point x.We are especially interested in the dependence on x because it comes into play in Section 2 when we discuss the uniform approximation of the solution of the variationnal inequality. Approximation of x 1. Y0 x In this section, we give the discretization scheme for computing Y0 . In the next section, we will construct a global approximation scheme for the solution of the corresponding obstacle problem. We fix ε>0 and we define | x − x |≥ τ0 =0,τk+1 =inf t>τk : Xt Xτk ε , and we denote σk = τk+1 − τk. The following proposition is proved in [12] ((3.6), p. 100). ∈ N − x − x F Proposition 1.1. For every k , τk+1 τk,Xτk+1 Xτk is independent of τk . Remark that (σk)k∈N is a sequence of independent identically distributed random variables. Using the scaling 2 ε property and the strong Markov property, σk has the same law that ε σ where ε bε 1 σ := inf t>0: t + Wt ≥ a a ε with (Wt)t≥0 a standard Brownian Motion.
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