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 Université du Maine, Laboratoire Statistiques et Processus, BP. 535, 72017 Le Mans Cedex, France; e-mail: [email protected] 2 Université de Franche-Comté,LaboratoiredeMathématiques 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. The important fact is the knowledge of the law of σ . Indeed, we have (see p. 233 in [4]), q bε 1 b2 ε2 2cosh a2 sinh a 2 λ + a2 E(exp(−λσε)) = q · 2 b2 ε2 sinh a 2λ + a2 4 V. BALLY AND B. SAUSSEREAU

E ε 2 E ε − 2 So we can compute αε := σ and βε := (σ αε) and show that − − 2|b|ε 1 exp a2 1 αε = −−−→ | | − 2|b|ε ε→0 a2 ε b 1 + exp a2 2 −−−→ 2 · βε ε→0 3a4

Anyway, it is easy to see that sup(αε + βε) ≤ C<∞. ε≤1 For each time t we denote t k(t):= 2 , αεε where [ ] is the integer part. We prove that τk(t) −−−→ t.Wewrite ε→0

2 Xk(t) 2 2 2 E|t − τk(t)| = E − t − αεε k(t) + (σk − αεε )

k=0 Xk(t) 2 2 2 2 = t − αεε k(t) + E(σk − αεε ) k=0 Xk(t) 2 2 4 ε 2 = t − αεε k(t) + ε (Eσ − αε) k=0 − 2 2 4 2 = t αεε k(t) + k(t)ε βε .

So we ﬁnally obtain that

2 2 E| − |2 ≤ 2 4 tβε ε ≤ 2 t τk(t) αεε + Cε . (1.1) αε The approximation scheme is deﬁned by ε,x E x F Qk := sup h(τν,Xτν ) τk , ν∈Sk,k(T )

S F − { } where k,k(T ) is the family of all the ( τi )0≤i≤k(T ) stopping times with values in the set k,...,k(T ) . F ∈S { ≤ } S Note that τν is a stopping time with respect to the filtration ( t)t≥0. Indeed if ν k,k(T ),then τν t = { ≤ } { }∈F { ≤ }∈F k τk t,ν = k and since ν = k τk and τk is a stopping time, we have τk t,ν = k t.Weshall ε,x give later an algorithm which permits to compute Q0 . x − ε,x For the moment, we want to evaluate the error Y0 Q0 . Except for Lipschitz continuous functions h,we shall also consider the following more special class of functions. H.1. For any t ∈ R+ and x ∈ R, h(t, x)=ϕ(t)f(x) with 1 R+ i) ϕ belongs to Cb ( ); Z x ii) there exists f˜ with finite variation such that f(x)=f(0) + f˜(u)du; 0 iii) we denote by µ the signed measure associated to f˜ via its representation as a difference of two increasing functions and we assume that µ has finite total variation, denoted by kµk. APPROXIMATION OF THE SNELL ENVELOPE AND AMERICAN OPTIONS PRICES 5

∈ 2 R The simplest example of a function satisfying H.1 is h(t, x)=ϕ(t)f(x) with f Cb ( ). In this case, µ(dx)= ∂2f(x) dx. Another interesting example is h(t, x) = exp(−rt)(K − exp(x)) which corresponds to a put. Then, ∂x2 + ˜ f(x)=− exp(x)1x≤ln(K) and µ(dx)=− exp(x)1x≤ln(K)dx + Kδln(K)(dx). Remark that we have kfk∞ = kf˜k∞ = K and kµk =2K. More generally, one may prove that a function f satisﬁes H.1 ii) and iii) if and only if it is a linear combination of convex functions (see [13], p. 23). Forafunctionh satisfying H.1, we have the following extension of Itˆo’s rule: Z Z Z t t x x 0 x ˜ x x x,y h(t, Xt )=h(0,X0 )+ ϕ (s)f(Xs )ds + ϕ(s)f(Xs )dXs + ϕ(t) Lt µ(dy), (1.2) 0 0 R

x,y x ∈ R where Lt denotes the local time of (Xt )t≥0 at the point y (see for example [4] for the case of h depending only on x and apply the classical integration by parts formula with the function ϕ(.) for the general case). We are now able to give the main result of this section. Theorem 1.2. Suppose that h is Lipshitz continuous. Then √ | ε,x − x|≤ Q0 Y0 C1(h) ε, (1.3) where C1(h)=C [h]1 is the Lipschitz constant of h. Suppose that h satisﬁes H.1,then √ | ε,x − x|≤ − Q0 Y0 C2(h, x) ε ln ε, (1.4) where 0 C2(h, x)=C kϕ k∞ |f(x)| +[f]1 + kµk + kϕk∞(kµk + |f˜(0)|) . (1.5)

∈ 1,2 R+ × R Suppose that h Cb ( ) then

| ε,x − x|≤ Q0 Y0 C3(h) ε, (1.6)

a2 where C (h)=C k∂ h + b∂ h + ∂ 2 hk∞. 3 s x 2 x

Remark 1.3. By (1.5), one remarks that C2(h, x) is independent of x if f is bounded. This is the case when −rt x we apply this result to an American Put. Then h(t, x)=e f(x) with f(x)=(K − e )+ which satisfies kfk∞ = kf˜k∞ = K. Remark 1.4. Notice that the convergence rate in (1.6) is the same as in [11] because our time discretization is of order n = ε−2. Note also that in [8], the author proved that in the√ case of a put option (which corresponds to the case (1.4) 4 in our theorem), the speed of convergence is better: ( ln n/n) 5 (see also the results in [9]). In fact numerical experiments (see [5]) show that for a put, the speed of convergence is 1/n. So for some particular payoff functions, one may obtain better results. Proof of Theorem 1.2. Step 1: The first part of this proof is devoted to state estimates analogous to those in Theorem 1.2 for the following intermediary quantity. 6 V. BALLY AND B. SAUSSEREAU

For a stopping time τ, we denote τ T = τ ∧ T and we deﬁne ε,x T x Q¯ =supE h(τ ,X T ) . 0 ν τν ν∈S0,k(T )

∈S | T − |≤| − | Note ﬁrst that for every ν 0,k(T ) one has τν τν T τk(T ) , so that by (1.1)

k − T k ≤ τν τν 2 Cε. (1.7)

• We prove that, if h is Lipschitz continuous √ ¯ε,x − ε,x ≤ Q0 Q0 C1,1(h) ε with C1,1(h)=C [h]1. (1.8)

For every ν ∈S0,k(T ) x T x T x x E − T ≤ E| − | E| T − | h τν ,Xτν h τν ,Xτ [h]1 τν τν + Xτ Xτν ν ν q ≤ | | E| T − | E| T − | [h]1 (1 + b ) τν τν + a τν τν .

By (1.7), it follows that (1.8) holds true.

• ∈ 1,2 R+ × R We prove now that if h Cb ( )then

2 ¯ε,x − ε,x ≤ a Q0 Q0 C3,1(h) ε with C3,1(h)=C ∂sh + b∂xh + ∂x2 h . (1.9) 2 ∞

Using Itˆo’s formula one gets Z Z τν 2 τν x T x a x x E − T E 2 E h τν ,Xτν h τν ,Xτ = ∂sh + b∂xh + ∂x h (s, Xs )ds + ∂xh(s, Xs )σdBs. ν T 2 T τν τν

Since ∂xh is bounded and τν is integrable, the integral with respect to B is a stochastic integral (not only a local martingale), so its expectation vanishes. It follows that

2 x T x a T E − T ≤ 2 E| − | h(τν,Xτν ) h(τν ,Xτ ) ∂sh + b∂xh + ∂x h τν τν , ν 2 ∞ so (1.9) follows from (1.7).

• Suppose now that h satisﬁes H.1. We prove that √ ¯ε,x − ε,x ≤ − Q0 Q0 ε ln εC2,1(h, x) with (1.10) 0 | | 00 C2,1(h, x)=C2,1(h) f(x) + C2,1(h)where 0 0 C (h)=C kϕ k∞ and 2,1 00 k 0k k k k k k k | ˜ | C2,1(h)=C ϕ ∞ ([f]1 + µ )+C ϕ ∞ µ + f(0) +1 . APPROXIMATION OF THE SNELL ENVELOPE AND AMERICAN OPTIONS PRICES 7

T We apply the formula (1.2) between 0 and τν , and between 0 and τν . Subtracting the two formulas we obtain

E x − T x ≤ h(τν ,Xτ ) h(τν ,Xτ T ) A1 + A2 with ν Z ν τν E 0 x ˜ x A1 = 1[τν ≥T ](ω) ϕ (s)f(Xs )+bϕ(s)f(Xs ) ds Z T A = E ϕ(τ )Lx,y − ϕ(τ T )Lx,y µ(dy) . 2 ν τν ν τ T R ν

We ﬁrst estimate A1.Wewrite !

≤k − k k 0k | x | | |k k | ˜ x | A1 τk(T ) T L2 ϕ ∞ sup f(Xs ) + b ϕ ∞ sup f(Xs ) . ∈ ∈ s [0,τk(T )] L2 s [0,τk(T )] L2

By H.1 |f˜(y) − f˜(z)| = |µ({u : y ≤ u ≤ z})|≤kµk. Hence

kf˜k∞ ≤kµk + |f˜(0)|.

Note that | x |≤| | | x − |≤| | | | | | · f(Xs ) f(x) +[f]1 Xs x f(x) +[f]1 b s + σ BS So, using (1.1), we obtain 0 A1 ≤ εC kϕ k∞ |f(x)| +[f]1 + |b|kϕk∞ kµk + |f˜(0)| .

The term A2 can be treated as follows. We write

A ≤ A + A with 2 2 ,1 Z2,2

E x,y − x,y A2,1 = ϕ(τν ) Lτ L T µ(dy) and ν τν R Z E − T x,y · A2,2 = ϕ(τν ) ϕ(τν ) Lτ T µ(dy) R ν

x,y y x We begin with the simplest term. Since L. does not grow up to τx := the ﬁrst hitting time of y by X. ,we x,y y,y y,y have L = L − x 1τ x≤t ≤ L (1 denotes the indicator function). Then using (1.7) t t τy y t Z 0 T x,y ≤k k∞k − k 2 k k 2 | | A2,2 ϕ τν τν L Lτ T L µ (dy) RZ ν 0 y,y ≤k k∞k − k 2 k k 2 | | ϕ T τk(T ) L Lτ T L µ (dy) R ν ≤ k 0k k k k y,yk εC ϕ ∞ µ sup LT L2 . y∈R

Using the Tanaka’s formula, one gets easily that

k y,yk2 ≤ 2 2 2 E| y − |2 ≤ 2 2 2 LT L2 4b T +4a T +4sup XT y 12(b T + a T ), y∈R and this ends the estimation of A2,2. For A2,1, we will need the following lemma (the proof is given in the Appendix 1). 8 V. BALLY AND B. SAUSSEREAU

Lemma 1.5. With the above notations we have, √ x,y x,y E1 ≥ L − L ≤ Cε − ln ε. (1.11) [τk(T ) T ] τk(T ) T

We assume (1.11). For ν ∈S0,k(T ) we have x,y x,y x,y x,y E L − L = E1 ≥ L − L τν τ T [τν T ] τν T ν √ x,y x,y ≤ E1 ≥ L − L ≤ Cε − ln ε. [τk(T ) T ] τk(T ) T Z √ x,y x,y It follows that A ≤kϕk∞ E L − L µ(dy) ≤ Cε − ln ε. 2,1 τν τ T R ν

¯ε,x − x Step 2: In order to prove Theorem 1.2, it remains to estimate QT0 Y0 . ∈T { ≥ } { } { ≤ } { ≤ }∈F ⊆F Let τ 0,T and let ν =inf i : τi τ .Wehave ν>i = j≤i τj τ and since τj τ τj τi ≥ F F − for i j, ν is a stopping time with respect to the ﬁltration ( τk )k.So,τν is also a ( t)t stopping time, and T ∧ τν also. Due to these remarks, one gets that

ε,x T x x Q¯ =supEh(τ ,X T ) ≤ Y . (1.12) 0 ν τν 0 ν∈S0,k(T )

0 { | x − x|≥ } Given a stopping time τ, we denote τ =inf t>τ: Xt Xτ 2ε and using the strong Markov property we get

6 E(τ ∧ T − τ) ≤ E(τ − τ) ≤ E(τ 0 − τ)=4α ε2 ≤ ε2. (1.13) ν ν 2ε a2

The above inequality will play the same role in the sequel that (1.7) played in the first step. E x We fix τ and we denote by ν the corresponding stopping time defined as above. We have to compare h(τ,Xτ ) T x and Eh(τ ,X T ). The same computations as for (1.8) and (1.9) (using (1.13) instead of (1.7)) yield ν τν • if h is Lipschitz continuous

x T x Eh(τ,X ) − Eh(τ ,X T ) ≤ εC1,2(h)with C1,2(h)= C[h]1 τ ν τν

• ∈ 1,2 R+ × R if h Cb ( ) then the same reasoning (based on Itˆo’s formula) as for (1.7) gives

a2 E x − E T x ≤ 2 h(τ,Xτ ) h(τν ,Xτ T ) ε C3,2(h)with C3,2(h)=C ∂sh + b∂xh + ∂x2 h . ν 2 ∞

So we have proved that

x x T x ¯ε,x Y =supEh(τ,X ) ≤ sup Eh(τ ,X T )+εCi,2(h)=Q + εCi,2(h) 0 τ ν τν 0 τ∈T0,T ν∈S0,k(T )

1,2 with i = 1 (respectively 3) if h is Lipschitz continuous (respectively Cb ). This, together with (1.12) proves that • if h is Lipschitz continuous

¯ε,x − x ≤ Q0 Y0 εC1,2(h) with C1,2(h)=C [h]1 APPROXIMATION OF THE SNELL ENVELOPE AND AMERICAN OPTIONS PRICES 9

• ∈ 1,2 R+ × R if h Cb ( )

2 ¯ε,x − x ≤ 2 a Q0 Y0 ε C3,2(h) with C3,2(h)=C ∂sh + b∂xh + ∂x2 h . 2 ∞

If h satisﬁes H.1 we have to give more details. E 0 − 2 4E 2 Using the strong Markov property and the scaling property we get (τ τ) =16ε σ2ε and recall that E 2 ≤ σε C.Sowehave

E(τ 0 − τ)2 ≤ Cε4. (1.14)

Arguing as in the proof of (1.10), one may prove that

E x − E T x ≤ ˜ 2 ˜ h(τ,Xτ ) h(τν ,Xτ T ) Cε + A with Z ν A˜ =supE ϕ(T ∧ τ ) Lx,y − Lx,y µ(dy) and ν T ∧τν τ τ∈T0,T R 0 0 C˜ = C kϕ k∞ |f(x)| +[f]1 + kϕk∞(kµk + |f˜(0)|)+kϕ k∞kµk .

We write

x,y x,y ˜ ≤k k∞k k E − A ϕ µ sup sup (Lτν Lτ ). τ∈T0,T y∈R

Using Tanaka’s formula, one gets Z Z τν τν 1 x,y x − + − − + − Lτν =(Xτν y) (x y) + a 1[Xs>y]dBs b 1[Xs>y]ds 2 0 0 and the analogous formula for τ.Since|(z − y)+ − (z0 − y)+|≤|z − z0|, it yields E x,y − x,y 2 ≤ 2E| − |2 2E| − | (Lτν Lτ ) 36 b τν τ + a τν τ , so by (1.13) and (1.14)

A˜ ≤ εC kϕk∞kµk.

Then the same reasoning as above yields

¯ε,x − x ≤ Q0 Y0 εC2,2(h, x) with 0 0 C2,2(h, x)=C kϕ k∞ |f(x)| +[f]1 + kϕk∞(kµk + |f˜(0)|)+kϕ k∞kµk . (1.15)

The Algorithm ε,x In order to give an explicit algorithm for computing Q0 , we have to consider functions h satisfying H.1 which depend on the time in a special way, namely h(t, x) = exp(−rt)f(x). This is suﬃcient for applications in Mathematical Finance. 10 V. BALLY AND B. SAUSSEREAU

From the dynamic programming principle, we already know that

Qε,x = exp(−rτ )f(Xx ) k(T ) k(T ) τk(T ) . . Qε,x =max exp(−rτ )f(Xx ), E(Qε,x |F ) . (1.16) k k τk k+1 τk

ε,x x x Our aim is to compute Qk . To this end, we ﬁrst write τk+1 = τk + σk and Xτk+1 = Xτk + δk. The important F fact is that (δk ,σk)k∈N are independent of τk (see Prop. 1.1) and we know that (see [4], p. 233) q bε ε b2 exp a2 sinh a 2r + a2 E − + exp( rσk)1δk =ε = q := γε and (1.17) 2ε b2 sinh a 2r + a2 q − bε ε b2 exp a2 sinh a 2r + a2 E − − exp( rσk)1δk =−ε = q := γε . (1.18) 2ε b2 sinh a 2r + a2

We deﬁne now ˆε,x ε,x Qk = exp(rτk)Qk , and we multiply with exp(rτk) in (1.16) in order to obtain the recurrence relation

Qˆε,x = f(Xx ) k(T ) τk(T ) . . n o ˆε,x x E − ˆε,x |F · Qk =max f(Xτk ), exp( rσk)Qk+1 τk

We want to construct a sequence of functionsu ˆ such that Qˆε,x =ˆu (Xx ). Suppose that this is true. Then k k k τk the above recurrence relation yields x x x uˆk(X )=max f(X ), E(exp(−rσk)ˆuk+1(X + δk)|Fτ ) τk τk τk k x −rσk x −rσk x =max f(X ), E e 1δ =ε uˆk+1(X + ε)E e 1δ =−ε uˆk+1(X − ε) τk k τk k τk =max f(Xx ),γ+uˆ (Xx + ε)+γ−uˆ (Xx − ε) · τk ε k+1 τk ε k+1 τk

This leads us to deﬁne

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. (1.19)

ˆε,x x Then Qk =ˆuk(Xτk ) and in particular for k =0

ˆε,x ε,x uˆ0(x)=Q0 = Q0 . APPROXIMATION OF THE SNELL ENVELOPE AND AMERICAN OPTIONS PRICES 11

Remarks. Notice that the algorithm is analogous to the one proposed by [11] except for the contribution of + − γε and γε . In [11], the authors proceed as follows. Given a terminal time T (T = 1 for simplicity), they approximate the Brownian motion by a standard binomial random walk which can be embedded in the paths x of Brownian motion. The approximation of Y0 , x =0,isthengivenbyU0(0) where the sequence (Uk(.))0≤k≤n is deﬁned by i i U √ = h 1, √ , −n ≤ i ≤ n, n n n . . i k i i 1 i 1 U √ =max h , √ , 0.5 U √ + √ +0.5 U √ − √ , (1.20) k n n n k+1 n n k+1 n n − k ≤ i ≤ k,

−rt where h(t, u)=e f(x + bt + au). √ ∈ 1,2 × R | − 0|≤ In [11], the authors prove that for f C ([0,T ] ), one√ has U0 Y0 C/ n, and in [8], one considers 4 | − 0|≤ 5 the particular case of a put and proves that U0 Y0 C( ln n/n) . In our framework, we do not give a binomial approximation of the Brownian motion but of the diﬀusion itself (including the drift part). This leads us to use the function h(t, x)=ϕ(t)f(x). So we do not work with the same function h as Lamberton–Rogers in [11]. We can compute explicitly the coeﬃcient of our binomial + − approximation. Whereas Lamberton and Rogers have γε = γε =1/2, in our frame we have 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

Complexity of the algorithm. It is clear that the complexity of the algorithm depends on the number of points k = k(T ) that we consider at the ﬁrst stage. So we have to take advantage of the fact that the diﬀusion process does not go too far away from its starting point x. More precisely, we have " # " # | | P | x − |≥ ≤ P | |≥cε + b T sup Xs x cε sup Bs , s∈[0,T ] s∈[0,T ] a and by Bernstein’s inequality (see [12], Eq. (3.16), p. 145), " # | | 2 P | x − |≥ ≤ −(cε + b T ) sup Xs x cε exp 2 s∈[0,T ] 2a T |b|2 c2 ≤ exp − × exp − ε · 2a2 2a2T

So if we take p 2 cε = −2a T ln ε, (1.21)

P | x − |≥ ≤ ˜ √ we get [sups∈[0,T ] Xs x cε] ε. This permits to use the function h(t, y)=h(t, y)1|y−x|≤ −2a2T ln ε in cε the above algorithm. Then the number of operations to be performed is of the order ε k(T ) which is of the order 12 V. BALLY AND B. SAUSSEREAU √ ε−3 − ln ε (instead of ε−4 in the initial algorithm) corresponding to an error of: √ • √ε for a Lipschitz continuous function, • ε − ln ε for a function satisfying H.1 and • ε for a regular function.

2. Global approximation The aim of this section is to prove that the functionu ˆ constructed in the previous section provides an approximation of the solution u of the PDE associated to our obstacle problem. Let us be more precise. We ﬁx the time horizon T and the obstacle of special form h(t, x) = exp(−rt)f(x) and we denote by uT,h the unique solution of the PDE   (∂t + L) uT,h(t, x)=0, on uT,h >h (∂ + L) u (t, x) ≤ 0, on u = h (2.1)  t T,h T,h uT,h(T,x)=h(T,x),

2 where L is the inﬁnitesimal generator of the diﬀusion process X,namely,L =(a /2)∂x2 +b∂x. The equation (2.1) can be rewritten as

min (uT,h(t, x) − h(t, x), −∂tuT,h(t, x) − LuT,h(t, x)) = 0, (t, x) ∈ [0,T ] × R

uT,h(T,x)=h(T,x).

Our aim is to give a uniform approximation of uT,h(., .) coming from the previous algorithm constructed for x { − ≤ ≤ } Y0 = uT,h(0,x). We will prove that the values computed on the grid x + iε, tk i tk will provide an 2 2 approximation of u(tk,y)for0≤ k ≤ a T/ε and y in the above grid. The solution of the above PDE has to be understood in a weak sense. In [6] one considers solutions in viscosity sense and in [2], the variational sense is considered and it corresponds to variational inequalities (see [3] for the analytical approach to optimal stopping problems by means of variational inequalities). We will precise later x x x the sense given to (2.1). The relation between the process (Yt )0≤t≤T and u is given by Yt = u(t, Xt ). We x ≤ ≤ shall obtain an algorithm which permits to compute Yt for 0 t T , by means of the previous algorithm x constructed for Y0 . T We fix now ε>0 such that 2 is an integer (of course we may find ε as small as we want, having this ε αε property). We denote T · N := 2 ε αε 2 For n ∈{0,... ,N} we define tn := n(ε αε) (remark that tN = T ) and we note that

k(tn)=n and k(T − tn)=N − n.

Let x be the starting point of the diﬀusion process for which we have already constructed the functionu ˆn on a triangular grid. For y ∈ R, we deﬁney ¯ the projection of y on the grid {x + iε, − n ≤ i ≤ n,0 ≤ n ≤ N},thatis h 1 1 y¯ = iε if y ∈ i − ε, i + ε . 2 2

We are now able to give our result. Theorem 2.1. Let uˆ be deﬁned in (1.19). For every n =0,... ,N and every y ∈ [x − nε, x + nε]

|uT,h(tn,y) − exp(−rtn)ûn(¯y)|≤Kδε APPROXIMATION OF THE SNELL ENVELOPE AND AMERICAN OPTIONS PRICES 13 where √ 4[h] δ = ε and K = C (h)+ √ 1 if h is Lipschitz continuous, ε 1 ε √ 4[h]1 δε = ε − ln ε and K = C2(h, y)+√ if h satisfies H.1, − ln ε 1,2 δε = ε and K =(C3(h)+4[h]1) if h is C , and C1(h), C2(h, y) and C3(h) are the constants appearing in Theorem 1.2. In particular, under H.1,iff is bounded then the evaluation is uniform with respect to y. Before proving this, we need some preliminary results about the solution of the PDE (2.1). We recall the definition of viscosity solution.

Deﬁnition 2.2. a) uT,h ∈ C([0,T ] × R ; R) is called a viscosity sub-solution of (2.1) if uT,h(T,x) ≤ h(T,x),x∈ R and moreover for any ϕ ∈ C1,2([0,T ] × R) and every (t, x) ∈ [0,T [×R which is a local maximum of uT,h − ϕ,

min (uT,h(t, x) − h(t, x), −∂tϕ(t, x) − Lϕ(t, x)) ≤ 0

b) uT,h ∈ C([0,T ] × R ; R) is called a viscosity super-solution of (2.1) if uT,h(T,x) ≥ h(T,x), x ∈ R and 1,2 moreover for any ϕ ∈ C ([0,T ] × R) and every (t, x) ∈ [0,T [×R which is a local minimum of uT,h − ϕ,

min (uT,h(t, x) − h(t, x), −∂tϕ(t, x) − Lϕ(t, x)) ≥ 0

c) uT,h ∈ C([0,T ]×R ; R) is called a viscosity solution of (2.1) if it is both a viscosity sub- and super-solution. It is proved in [6] that the equation (2.1) has a unique viscosity solution. Given t>0, we introduce the function (s, x) 7→ ht(s, x) := exp(−rt) × h(s, x)=h(t + s, x).

Lemma 2.3. For any t

Proof. We denote v(s, x)=uT,h(t + s, x)for0≤ s ≤ T − t and w(s, x)=uT −t,ht (s, x)for0≤ s ≤ T − t.We shall prove that ∀ 0 ≤ s ≤ T − t, v(s, x)=w(s, x) and in particular s = 0 gives the result. In order to do this, we prove that these two functions solve the same following PDE in viscosity sense:  t  (∂s + L) z(s, x)=0, 0 ≤ s ≤ T − t, on z>h (∂ + L) z(s, x) ≤ 0, 0 ≤ s ≤ T − t, on z = ht (2.2)  s z(T − t, x)=h(T,x), with z ∈ C([0,T − t] × R ; R). We prove that both v and w are viscosity sub-solution of (2.2). By deﬁnition, w solves (2.2) so in particular it is a sub-solution. It remains to deal with v.Firstwehavev(T − t, x)=uT,h(t + T − t, x)=h(T,x). Let ϕ ∈ C1,2([0,T −t]×R)and(s, x) ∈ [0,T −t[×R which is a local maximum of v−ϕ. We deﬁne ϕt ∈ C1,2([t, T ]×R) t t by ϕ (r, x)=ϕ(r − t, x)for(r, x) ∈ [t, T ] × R.Nowsince(t + s, x) is a local maximum of uT,h − ϕ and uT,h solves (2.1) in the viscosity sense, we have t t min uT,h(t + s, x) − h(t + s, x), −∂tϕ (t + s, x) − Lϕ (t + s, x) ≤ 0, which reads t min v(s, x) − h (s, x), −∂tϕ(s, x) − Lϕ(s, x) ≤ 0. So v is a viscosity sub-solution of (2.2). The same arguments show that v is also a viscosity super-solution of (2.2) so the lemma is proved. 14 V. BALLY AND B. SAUSSEREAU

We shall ﬁnally need the following lemma: Lemma 2.4. Suppose h Lipschitz continuous. Then for any 0 ≤ t ≤ T and any x ∈ R, we have

|uT,h(t, x) − uT,h(t, x¯)|≤4[h]1ε.

Proof. We use the following expression of the solution of the obstacle problem as the value function of an optimal stopping problem: E x uT,h(t, x)= sup h(τ,Xτ ). τ∈Tt,T Inthisframewehave

≤ [h]1|x − y|, and the result follows.

Now we can turn to the proof of Theorem 2.1.

T Proof. Remind that we have denoted tn = n N . For n =0,... ,N and y ∈ [x − nε, x + nε], we introduce

− − − ≤ ≤ − u¯k(tN −tn)(N n, y + iε)=f(y + iε), (N n) i N n, . . + u¯ − (k, y + iε)=max f(y + iε),γ u¯ − (k +1,y+ iε + ε) k(tN tn) ε k(tN tn) − − − ≤ ≤ +γε u¯k(tN −tn)(k +1,y+ iε ε) , k i k. (2.3)

Clearly

uˆn(y)=¯uk(tN −tn)(0,y). (2.4)

Using Lemma 2.3 and (2.4)

−rtn uT,h(tn,y) − e uˆn(¯y)=uT,h(tn,y) − uT,h(tn, y¯)

− t + uT,h(tn, y¯) uT −tn,h n (0, y¯)

−rtn t − + uT −tn,h n (0, y¯) e u¯k(T −tn)(0, y¯) − − rtn − rtn +e u¯k(T −tn)(0, y¯) e uˆn(¯y)

−rtn − t − = uT,h(tn,y) uT,h(tn, y¯)+uT −tn,h n (0, y¯) e u¯k(T −tn)(0, y¯)

= a1 + a2.

By Lemma 2.4, it holds that |a1|≤4[h]1ε. The estimation of a2 follows from Theorem 1.2. APPROXIMATION OF THE SNELL ENVELOPE AND AMERICAN OPTIONS PRICES 15

T An Improvement. Remind that we have fixed ε>0 such that 2 := N is an integer. We moreover denote ε αε αN the corresponding value of αε. As in the previous section, we can take advantage√ of the fact that the diffusion process does not go far away 2 from its starting point. Remind that cε = −2a T ln ε (see (1.21)) and we now denote r 2 T CN := −a T ln · NαN ˜ This suggest using h(t, y)=h(t, y)1|y−x|≤cN instead of h.Wedefinetheintervals " # 1/2 1/2 N − T T ∩ − In = x n ,x+ n [x cN ,x+ cN ]. NαN NαN

This ﬁnally leads us to the ∈ N −rt Proposition 2.5. For every n =0,... ,N and every x In we deﬁne uˆn by (1.19) replacing h(t, y)=e f(y) ˜ −rt ∈ N by h(t, y)=e f(y)1|y−x|≤cN . Then we have for any y In

sup |uT,h(tn,y) − exp(−rtn)ûn(¯y)|≤Kδε with n=0,... ,N √ 4[h]1 K = C1(h)+ √ and δε = ε if h is Lipschitz continuous, ε √ 4[h]1 K = C2(h, x)+√ and δε = ε − ln ε if h satisfies H.1, − ln ε 1,2 K =(C3(h)+4[h]1) and δε = ε if h is C , where C1(h), C2(h, x) and C3(h) are defined in Theorem 2.1. In particular if f is bounded under H.1, then all the approximations are uniform with respect to y.

3. The optimal stopping time In the PDE’s language, the complementary of the set

Λ={(t, x) ∈ [0,T ] × R : u(t, x) ≤ h(t, x)} is called the continuation set or the continuation region. The first optimal stopping time is the first time when the diffusion hits Λ. So, as long as we are in Λc we continue (but there may be other optimal stopping times later). Our aim is to give an approximation of this set in order to obtain approximations for the optimal stopping time. We introduce first

− { ∈ × R − ≤ } Λδ = (t, x) [0,T ] : u(t, x) δ h(t, x) , + { ∈ × R ≤ }· Λδ = (t, x) [0,T ] : u(t, x)+δ h(t, x)

+ ⊆ ⊆ − Clearly Λδ Λ Λδ . We consider now the functionu ˆ constructed in the previous section and define Λˆ − = (t ,x),n=0,... ,N,x∈ IN :û(t, x¯) − δ ≤ h(t, x) , δ n n ˆ + ∈ N ≤ · Λδ = (tn,x),n=0,... ,N ,x In :û(t, x¯)+δ h(t, x) 16 V. BALLY AND B. SAUSSEREAU

As an immediate consequence of Corollary 2.5 we get:

Proposition 3.1. Assume that h(t, u)=ϕ(t)f(u).   4[√h]1 2 C1(h)+ ε if h is Lipschitz continuous,  ε 4[h] √ Then for any δ>2 C (h, x)+√ 1 ε − ln ε if h satisﬁes H.1,  2 −  ln ε 1,2 2(C3(h)+4[h]1) ε if h is C ,

ˆ− ⊆ − ⊆ ˆ − ˆ + ⊆ + ⊆ ˆ + one has Λδ/2 Λδ Λ2δ and Λ2δ Λδ Λδ/2. ∗ { ∈ } ∗ { ∈ } Deﬁne now τf =inf t :(t, Xt) Λ the ﬁrst optimal stopping time and τl =sup t :(t, Xt) Λ the last ∗ time when one may exercise in an optimal way (of course τl is no more a stopping time). The discrete versions of these times are given by

− { ∈ ˆ −} + { ∈ ˆ +} τf,δ =inf t :(t, Xt) Λδ ,τf,δ =inf t :(t, Xt) Λδ , − { ∈ ˆ −} + { ∈ ˆ +}· τl,δ =sup t :(t, Xt) Λδ ,τl,δ =sup t :(t, Xt) Λδ

Then, as an immediate consequence of the above Proposition we have:

Proposition 3.2. Under the hypotheses of the above Proposition

− ≤ ∗ ≤ + + ≤ ∗ ≤ − τf,δ τf τf,δ and τl,δ τl τl,δ. Appendix A: Proof of Lemma 1.5

We prove the estimation (1.11). We write E x,y − x,y 1[τ ≥T ] Lτ L = I + J with k(T ) k(T ) T E √ x,y − x,y I = 1[τk(T )≥T ]1[τ

Here 2Tβ K = p ε · k(T )αεε The reason of this choice of K will become clear in the proof. For the moment we note that since k(T ) ≥ 2 T/(αεε ) − 1, we have

2 2 2 ≤ 4T βε ≤ K 2 C. Tαε − αεε

Step 1: We ﬁrst deal with I. Since the local time is an increasing process, we have √ x,y √ x,y x,y √ x,y I = E1 ≥ 1 − L − L ≤ EL − EL . [τk(T ) T ] [τk(T )

7→ We denoteZ (s, x, z) p(s, x, z) the transition density function of the diﬀusion process X and we recall that t E x,y xLt = p(s, x, y)ds (see [4], p. 21). It yields 0

Z √ T +Kε − ln ε I ≤ p(s, x, y)ds T √ ≤ Kε − ln ε sup |p(s, x, y)| s∈[T +∞[ √ 1 ≤ Kε − ln ε sup √ exp −(y − x − bs)2/(2a2s) s∈[T +∞[ 2πa2s √ ≤ Cε − ln ε.

Step 2: In order to evaluate J, we ﬁrst use the Cauchy–Schwartz inequality and get

J ≤ J × J with 1 2 x,y − x,y J1 = 1[τ ≥T ] Lτ L k(T ) k(T ) T L2 h √ i 1 2 J2 = P τk(T ) − Kε − ln ε ≥ T .

• We ﬁrst prove that √ J1 ≤ C ε. (3.1)

Using Tanaka’s formula, one gets Z Z 1 τk(T ) τk(T ) Lx,y =(Xx − y)+ − (x − y)+ + a 1 dB − b 1 ds τk(T ) τk(T ) [Xs>y] s [Xs>y] 2 0 0

+ 0 + 0 and one writes down the analog of the above formula with T instead of τk(T ).Since|(z−y) −(z −y) |≤|z−z |, it yields 2 ≤ 2E| − |2 2E| − | J1 36 b τk(T ) T + a τk(T ) T , so using (1.7) we get (3.1). • We end this computations with the estimate √ J2 ≤ C ε.

2 ε We recall that σk has the same law that ε σ for k =0,... ,k(T ) and they are independent. So τk(T ) − T has k(XT )−1 2 − 2 ε − ε the same law as αεε k(T ) T + ε (σk αε)whereσk, k =0,... ,k(T ), is a sequence of independent k=0 18 V. BALLY AND B. SAUSSEREAU random variables which have the same law as σε. So we may write that   k(XT )−1 √ 2 ≤ P  2 − 2 ε − ≥ −  J2 αεε k(T ) T + ε (σk αε) Kε ln ε  k=0  k(XT )−1 √ ≤ P  2 ε − ≥ −  ε (σk αε) Kε ln ε  k=0  k(XT )−1 √ ≤ P  T 1 ε − ≥ −  (σk αε) Kε ln ε αε k(T ) k=0  p  k(T )−1 X σε − α k(T )α √ = P  kp ε ≥ ε Kε − ln ε. Tβε k=0 βε k(T )

ε 3 We denote γε = E|σ − αε| . We apply the Berry–Esseen theorem ([7], p. 542) and get

Z ∞ 2 ≤ √1 − 2 p3γε · J √ √ exp( x /2)dx + 2 k(T )α 3 2π ε Kε − ln ε β k(T ) Tβε ε √ From the choice of K, the lower bound of the above integral is equal to 2 − ln ε so

Z ∞ 1 3γ J 2 ≤ √ exp(−x2/2)dx + p ε · 2 √ 3 2π 2 − ln ε βε k(T ) For any λ ∈ R one may write

Z ∞ 2 2 − λ exp(−x /2)dx ≤ C e 4 . λ √ Applying this inequality with λ =2 − ln ε,weget

3γ J 2 ≤ Cε+ p ε ≤ Cε. 2 3 βε k(T )

Now the proof of (1.11) is complete.

References

[1] C. Baiocchi and G.A. Pozzi, Error estimates and free-boundary convergence for a finite-difference discretization of a parabolic variational inequality. RAIRO Anal. Numér./Numer. Anal. 11 (1977) 315-340. [2] V. Bally, M.E. Caballero and B. Fernandez, Reflected BSDE’s, PDE’s and Variational Inequalities. J. Theoret. Probab. (submitted). [3] A. Bensoussans and J.-L. Lions, Applications of the Variational Inequalities in Stochastic Control. North Holland (1982). [4] A.N. Borodin and P. Salminen, Handbook of Brownian Motion Facts and Formulae. Birkhauser (1996). [5] M. Broadie and J. Detemple, American option valuation: New bounds, approximations, and a comparison of existing methods. Rev. Financial Stud. 9 (1995) 1211-1250. [6] N. El Karoui, C. Kapoudjan, E. Pardoux, S. Peng and M.C. Quenez, Reflected Solutions of Backward Stochastic Differential Equations and related Obstacle Problems for PDE’s. Ann. Probab. 25 (1997) 702-737. APPROXIMATION OF THE SNELL ENVELOPE AND AMERICAN OPTIONS PRICES 19

[7] W. Feller, An Introduction to Probability Theory and its Applications, Vol. II. John Wiley and Sons (1966). [8] D. Lamberton, Error Estimates for the Binomial Approximation of American Put Options. Ann. Appl. Probab. 8 (1998) 206-233. [9] D. Lamberton, Brownian optimal stopping and random walks, Preprint 03/98. UniversitédeMarne-la-Vallée (1998). [10] D. Lamberton and G. Pagès, Sur l’approximation des réduites. Ann. Inst. H. Poincaré Probab. Statist. 26 (1990) 331-335. [11] D. Lamberton and C. Rogers, Optimal Stopping and Embedding, Preprint 17/99. Université de Marne-la-Vallée (1999). [12] D. Revuz and M. Yor, Continuous Martingales and Brownian Motion. Springer Verlag, Berlin Heidelberg (1991). [13] A.W. Roberts and D.E. Varberg, Convex Functions. Academic Press, New York (1973). [14] B. Saussereau, Sur une classe d’équations aux dérivées partielles. Ph.D. Thesis of the University of Le Mans, France (2000).