Variational inequalities and the pricing of American options

Acta Applicandae Mathematicae 21: 263-289, 1990. 263 ,~'~ 1990 Kluwer Academic Publishers. Printed in the Netherlands. Variational Inequalities and the Pricing of American Options* PATRICK JA ILLET, DAMIEN LAMBERTON, and BERNARD LAPEYRE LAMM, CERMA-ENPC La Courtine, 93167 Noisy le Grand, France (Received: 28 February 1989; revised: 22 August 1990) Abstract. This paper is devoted to the derivation of some regularity properties of pricing functions for American options and to the discussion of numerical methods, based on the Bensoussan-Lions methods of variational inequalities. In particular, we provide a complete justification of the so-called Brennan- Schwartz algorithm for the valuation of American put options. AMS subject classifications (1950). 90A09, 60G40, 60J60, 65K10, 65M 10. Key words. Option pricing, variational inequalities, optimal stopping problem. I. Introduction In their celebrated paper E4], Black and Scholes derived explicit pricing formulas for European call and put options on stocks which do not pay dividends. The absence of closed-form expressions for the pricing of American options caused an extensive literature on numerical methods with, in general, little mathematical support. Actually, the rigorous theory of American options is rather recent (see [2], [22- 24]). In [2], [22], Bensoussan and Karatzas derive pricing formulas involving Snell envelopes. The aim of the present paper is to study and exploit these formulas within the framework of diffusion models, including the classical Black-Scholes [4] and Garman-Kohlagen [17] models. In this setting, we can rely on the connection between optimal stopping and variational inequalities as established by Bensoussan and Lions (see [3], [14], chapter 16) in order to investigate regularity properties of pricing functions and discuss the accuracy of numerical methods. In particular, we will examine an algorithm due to Brennan and Schwartz [5]. In Section 2, we set the basic assumptions of our model and show how the price of an American option can be obtained as a function of the current stock prices. Note that in order to deal with nondegenerate operators, we choose the logarithms of stock prices as state variables. Section 3.1 contains general results on v~ariational inequalities which are, for the most part, included in [3]. We have tried to weaken some regularity assumptions, so that the results can be applied to put and call options. In Section 3.2, we concentrate on one-dimensional models. We show, among other things, that the American put price, in the Black-Scholes model, is a continuously differentiable *Research supported in part by a contract from Banque INDOSUEZ. 264 PATRICK JAILLET ET AL. function of the stock price, before maturity (note that this property can also be derived from [32]). In Section 4, we localize the inequalities and review some results of [18] on the approximation of variational inequalities. Section 5 contains a detailed discussion of an algorithm which, except for the logarithmic change of variable, is the Brennan- Schwartz method of valuation of American put options (cf. [5]). It turns out that, although the formulation of the boundary problem in [5] was mathematically incorrect, this algorithm can be completely justified. However, it must be emphasized that this justification relies on special properties of put and call options and that the method fails for the pricing of other types of options (see Remark 5.14 below). Therefore, we have included some remarks on alternate methods in the last section, In order to complete this introduction, we wish to mention van Moerbeke's paper [32], on optimal stopping and free-boundary problems, whose results on warrant pricing extend readily to American options. The main advantage of our approach is that the techniques of variational inequalities provide the adequate framework for the study of numerical methods. For regularity results on European options, we refer the reader to [16]. This paper is a detailed version of [20]. 2. Assumptions and Notations Let (~, ~ , P) be a probability space and (W~)t>~o a standard Brownian motion with values in R". We denote by (~)t~>0 the P-completion of the natural filtration of tw,)t~o. Consider a financial market with n risky assets, with prices S~ . . . . . S~' at time t, and let Xt be the n-dimensional vector with components: X /= log S~, forj = 1 . . . . . n. We will assume that (Xt) satisfies the following stochastic differential equation dX t = fl(t, X,)dt + a(t, X t )dW t (2.1) on a finite interval [0, T], where T is the horizon (namely the date of maturity of the option). We impose the following conditions on fl and a, and on the interest rate: (HI) fl(t,x) is a bounded C t function from [0, T] x R" into R ", with bounded derivatives. (H2) a(t,x) is a bounded C 1 function from [0, T] x R" into the space of n x n matrices, with bounded derivatives. Also a admits bounded continuous second partial derivatives with respect to x, c32aij/c3xic3x j, satisfying a H61der condition in x, uniformly with respect to (t, x) in [0, T] x R ~ (H3) The entries aij of a(t, x). a*(t, x)/2 (where * denotes transposition) satisfy the following coercivity property: 3q>OV(t ,x)e[o,r] x R"V~eR ~ ~, ai.~(t,x)~,~>~r I ~ ~2 t <~i.j<<.n i= I THE PRICING OF AMERICAN OPTIONS 265 (H4) The instantaneous interest rate r(t) is a C ~ function from [0, T] into [0, co). Remark 2.1. Note that condition (H3) can be interpreted in terms of completeness of the market (cf. [19, 2, 23, 24]) An American contingent claim is defined by an adapted process (h(t)) o ~t a continuous version of the flow of the stochastic differential Equation (2.1). Therefore, (s, t, x )~ Xt;X(~o) is continuous for almost all ~o, Xtff = x, and X t'~ satisfies (2.1) on It, T]. PROPOSIT ION 2.2. Assume ¢/is continuous and satisfies [qJ(x)l ~< m e MIxl for some M > O, and define a function u* on [0, T] x R", by u*(t, x) = sup E(e -f:~(~)ds ~(X~ff)), (2.2) rE-Yt.T where J-t,T is the set of all stopping times with values in [t, 7"]. Then u* is continuous and, for any solution (X,) of (2.1) u*(t, Xt) = ess sup E(e-g ~~)d'*qj(X~){~ O. (2.3) Note that if, under probability P, the discounted vector price is a martingale, Equation (2.3) means that u*(t, Xt) is the 'fair price' of the American option defined by ~, at time t (cf. [2, 22, 23, 24]). Proof of Proposition 2.2. Using the equality X'; ~ = x + fl(v, Xlff)dv + a(v, X~'~)dW,, (2.4) it is easy to prove that E ( sup (eM'X:~l)) ~< C e MIx', (2.5) \ t <~s<~ T / where C depends only on T, M, and on the bounds on fl, a. Now, observe that if (ta, xx), (t2, x2)~ [0, T] × R", with t I < t 2, then u*(t 2, X2) - - u*(tl, Xl) = sup E(e -~:~m)d~ $(Xt~ .... )) -- sup E(e-~;, m)ds ~,(X'~ .... )) + + sup E(e-J'g'r(s)ds~k(X'~ .... ))-- sup E(e-J';,m)dsd/(Xt~ .... )). rE .~- f 2, r ~E'~-t t, T 266 PATRICK JAILLET ET AL. Therefore, lu*(t~, x~) - u*(t~, xOI / \ E( sup .... tl] + / \ t z <~s<~ T ,/ + E sup te-I;'r(O't~b(X'~ .... ) - - ~-v*,~ ,,] ti ~s~t2 and the continuity of u* follows from the continuity of 0 and of the flow and from (2.5) (which, applied with 2M instead of M, ensures uniform integrability). Equality (2.3) is essentially the expression of the Snell envelope in terms of the 'reduite' (see [12]). It can bc proved directly, by arguing that the sup in (2.2) and the essential sup in (2.3) are the same when ~-t.r is replaced by the set of stopping times with respect to the filtration (~t.~)~t of the increments W~ - Wt, s/> t (see [213 for details). [] 3. Variational Inequalities 3.1. EXISTENCE, UNIQUENESS AND REGULARITY RESULTS Before stating the variation inequalities related to the computation of u*, we introduce some function spaces. Let m be a nonnegative integer and let 1 ~< p ~< c~, 0 < # < 0o. W"'P'U(R ") will denote the space of all functions u in LP(R ", e ~,Ixl dx), whose weak derivatives of all orders ~< m exist and belong to LP(R ", e ulxl dx). We will write Hu (resp. V~) instead of W°'2'U(R") (resp. W l'2'u(R")) and the inner product on Hu (resp. Vu) will be denoted by (., .)u (resp. ((., .))u). We define a bilinear form on Vu, for each t ~ [0, T], in the following way: '~U, V ~ V u aU( t;u,v)= z I aij(t,x) OT ~-u ov e-~lxldx - i,j ,JRn ' CX i ~Xj fR ( "--- Xi \ aU -- j=,~ " ai(t'x) + #,~1 ai'g(t'x)~l)~xjV e-UlXldx + + fa" r(t)uv e-~lxl dx, where 0aij. x), for j 1, n. aj(t, x) = flj(t, x) - i~1 ~ (t, = .... THE PRICING OF AMERICAN OPTIONS 267 THEOREM 3.1. Under hypotheses (H1)-(H4), /f 0~ Vu, there is one and only one function u, defined on [0, T] x R" such that: (I) • OU 2 u~L2([0, T]; Vu), -~-eL ([0, T]; Hu) u >1 O a.e. in [0, T] x R", u(T) = O and v. >/0 =, _(Ou, v - u] + a"(t; u, Vve U) ~> 0 a.e . in [o, T] \~t /,, For this theorem, we refer the reader to [3], chapter 3, section 2 (see [21] for a detailed exposition of the Bensoussan-Lions methods in this setting). Note that the solution of system (I) is also the solution of the system obtained by changing # into any number v>#. THEOREM 3.2. Let O ~ WX'P'U(R"), with p > n, and let u*(t, x) = sup E(e-I;r(s)ds 0(X,;X)). "~E~'-t, T Then u* is the solution of system (I). Note that if 0eWI'P'U(R"), exp(--(#/p)lxl)O(x) is in WI"P'°(R"), therefore, exp(-(#/p)lxl)O(x) has a bounded continuous version by the Sobolev imbedding theorem (see, e.g., [1]) and Proposition 2.2 can be applied. The proof of Theorem 3.2 when 0 e Wz'P'"(R"), with p > n/2 + 1 follows from the Bensoussan-Lions methods (see [3], chapter III, No. 4 or [21] for details). For 0 ~ WI'p'U(R"), with p > n, we need the following lemma. LEMMA 3.3. Let 0 < # < v. I f O e WI'P'U(R"), with p > n, there exists a sequence (Or.), with Om ~ W2'p'~(R") for all m, such that (Or.) converges uniformly to O. Before proving this lemma, we complete the proof of Theorem 3.2. Let 0 and (0r") be as in Lemma 3.3. Denote by u the solution of system (I) and ur" the solution of system (I) when 0 is replaced by Or.. With obvious notations, we have u* = u,, and u* ~ u* uniformly as rn ~ ~. On the other hand, it follows from [3] (chapter 3, paragraphs 1.8 and 2.13) that um converges uniformly to u as m goes to infinity. Therefore, u = u*. Proof of Lemma 3.3. We can assume that 0 vanishes in a neighborhood of the origin. Let ¢(x) = ' Ixl ' i f l x l > 1, 10, if Ix l ~< 1, where y is a positive number to be chosen later. Note that q5 vanishes in a 268 PATRICK JAILLET ET AL. neighborhood of the unit ball and that ]V~b(x)] ~ #, then ~R" IV~(x)lPdx < o% which implies that q~ is uniformly continuous on R". From now on, we assume 7(P - n) >/~. Now, let p be a nonnegative, C ~ function, with compact support on R", satisfying ~R" p(x)dx = I. For any integer m, let p.,(x) = m"p(mx), (~,. = p,. * tk, and O,.(x) = ~,. (e ylxl x] \ Ixl)" Clearly, ~'m vanishes in a neighborhood of the origin for m large enough and is C ~. Moreover, (~,,) converges uniformly to ~ as m goes to infinity, since th is uniformly continuous. On the other hand, if ~ is a multiindex of length I~1 ~< 2, we have / ID~<~,.(x)l ~< C el~( z _ y)l p =~ zl~l Izl "-I f _ I dx(log Ix+yl)"- Ix + yl -I~l~v I~ip~)14~(x)lp. -~ ~+vl~>l Ix+yl ~ Now, assume v > 2p7 and lYl ~< 1/3. Then (log Ix + yl)" Ix + yl "+tlly~v Npy)Iq~(x)lP ftx+yt~ l dx f l (log(1 + Ixl)) " - I ~< xl 1> 213 dx (lxT ---- 113) ~ +(ID')4~- I-elm914>(x)l" (3.1) THE PRICING OF AMERICAN OPTIONS 269 Observe that f (log( [xl )" - x -~1~>1 dx ixl.+~l/~,~_l~le~, ) kb(x)[ p = v" f dz e Izl(v-I~lP~'ll~b(z)[P dR n Therefore, ifv > 2pT+p and m is large enough (so that supp p,, c {yeR"l FYl ~< 1/3}), the right-hand side in (3.1) is finite for I~1 ~< 2, and, consequently, ~,, is in Wz'P'~(R"), which completes the proof. [] Remark 3.4. One can construct a function in W~'z'"(R) which cannot be approxi- mated uniformly by functions in W2'2'~(R). Therefore, the number v in Lemma 3.3 has to be greater than p. 3.2. ONE-DIMENSIONAL MODELS In this section, we assume that n = I and that the drift term fl and the diffusion term a do not depend on X t, so that (2.1) becomes: dX, = fl(t)dt + a(t)dW,, (3.2) where fl and a are C I functions on [0, T], and inf~[o.r]aZ(t) >~ q > O. Recall that in the classical Biack-Scholes and Garman-Kohlhagen models, a, fl, r are constants, with f i+ a2/2 = r in the Black-Scholes case and fl + a2/2 equals the differential interest rate in the Garman-Kohlhagen case. We will also assume that ~k satisfies the following assumptions: (H5) There exists #> 0 such that ~b(x)exp(-plxl) is a bounded, continuous function and admits a bounded (weak) derivative on R. (H6) The function x ~ ¢(log(x)) is convex over (0, ~) Note that (H5) and (H6) are satisfied by the put and call functions ~Op(x) = (K - eX)+, ffc = ( ex - K)+. We first observe that the flow of SDE (3.2) is given by Xt; ~ = x + fl(v) dv + a(v) dWv. (3.3) Therefore, assuming (H5) and using the results and notations of subsection 3.1, we have u(t, x) = u*(t, x) = sup E(e ~:m~d~(Xt;~)), and, by (3.3) ( o(f: f: )) u(t, x) = sup E e -I:~s)ds x + fl(v)dv + a(v)dW, . (3.4) '~E.ff-t. T With this equality, the following proposition is easy to prove. 270 PATRICK JAILLET ET AL. PROPOSIT ION 3.5. (1) I f ¢ is nonincreasin9 (resp. nondecreasing) and satisfies (H5), then u(t, .) is nonincreasing (resp. nondecreasing), for all t ~ [0, T]. (2) I f ¢ satisfies (H5) and (H6), then for all t e [0, T], u(t, log(.)) is a convex function on (0, 0o). We now state the main result of this section: THEOREM 3.6. Let ~ satisfy (H5) and (H6), then u admits partial derivatives t of the increments (Ws- Wt), s ~> t. Therefore, since the law o f (W t + ~ - Wt), >~ o is the same as that of (W,) a ~> o, we can state ( fo: )) u(t, x)= sup E e-J'°rtt+a)da~l X .a¢_ fl(t + a)da + a(t + a)dW~ . (3.5) ZE~-O,T t 0 THE PRICING OF AMERICAN OPTIONS 271 Now, observe that z~J-o,r_ t if and only if z can be expressed in the form: r = O(T - t), where 0 is a stopping time with values in [0, 1], with respect to the filtration (Y#~), where acg, is the a-field generated by all random variables W~r- o, a ~< s. Since (Walr-t)).~0 and ( Tx /~- t Wa).~>o are identically distributed, we obtain u(t, x) = sup E (e -I~°'~-''r(t+")d~ × 0eJ-o,t \ ( i )) x 0 x + f l ( t+a)da+ ~r( t+a( r - t))dW. (3.6) do jo and Lemma 3.9 follows from (3.6) by a change of variables in the integrals with respect to da. [] Proof of Theorem 3.6. (1) Choose p > 0 so that ~ ~ V s. From Theorem 3.1, we know that u ~ L2([0, T]; Vs) and &u/&t ~ L2([0, T]; H.) and that for all v ~ V s such that v ~> ff: ) \t3t v -u s+ as(t;u'v-u)>~O' (3.7) Let v(x)= u(t,x)+ ~(x)e ~jxl, where ~b is a nonnegative C ~ function, with compact support. From (3.7), we derive that -(Ou/Ot + A(t)u) defines a nonnegative measure on (0, T) x R. On the other hand, since by Proposition 3.4, u(t, log(.)) is convex for all t e [0, T], O2u/Ox2 - 3u/Ox must be a nonnegative measure on (0, T) x R. Therefore, the following inequalities hold (in the sense of measures on (0, T) × R) OU 2 ~2t/ OU O2U OU ~-~ + (,~ (0 /2 )~x ~ +/~(t) ~ - rIt)u <~ o, &~ ~ > o Hence, 0 ~< (q/2) \Ox2 ~x <~ r(t)u ~ (fl(t) + (tr2(t)/2)) . (3.8) Therefore, O2u/OxZEL2([O, T]; Hs) and in order to prove Theorem 3.6, it suffices to show that Ou/Ot and Ou/Ox are locally bounded on [0, T )x R (derive (Ou/Ot + A(t)u)(~b - u) = 0 by setting v = ~b in (3.7)). (2) In order to prove local boundedness of &u/Ox, fix x, y ~ R 2 and note that, for all z~J t , r qJ(x+ ~,(a)da+ fT,~(a)dW,,)-,l'(Y+ f[B(a)da+ f~"~(a'dW,,) <. lx -y l sup ~b' (z+~r f l (a )da+~ra(a)dW.) ze[x v]~ <~ Cix - Yl exp #({x[ + [Yl) + Ifl(a)ida + I-t tr(a)dW, , where C is some positive constant. Here we have used (H5). Since the stochastic 272 PATRICK JAILLET ET AL. process (exp(#lS~ a(a)d War))s~>t is a submartingale and z ~< T, we have ~< 2 exp (~u2 Sot 22(a) da ) Therefore, using (3.4), we have lu(t, x) -- u(t, Y)I <~ Clx - Yl e~(Ixl+lyl), where C is some positive constant. It then follows that u is locally Lipschitz in x, uniformly with respect to t. Therefore, Ou/Ox is locally bounded on [0, T] x R. (3) It remains to prove local boundedness for Ou/Ot. For that purpose, we will use Lemma 3.9. Fix t, s in R 2 and 0 in 9-0,1 . Then e-I~°~t'a)da~k(x + f : fl(t, a)da + f : ~(t, a)dWa) - -e-I°°r~s'a)daO(x+ff,(s,a, da+f:#(s ,a , dl/V~) ( fo fo ) ~< le-I ~ont'")a~ - e-I~n='o~aOl ¢, x + fi(t, a)da + ?r(t, a)dW, + +e-'°°~(s'a)daO(x+ f]fi(t,a)da+ f~#(t,a)dWa) - ( ;o~ fo ) - 0 x + (s, a)da + ~(s,a)dW. (( ; ;o )) <~ CIt - sl exp /x Ixl + Ifl(t, a)]da + ~r(t, a)dWa + C fo' f j a)) dW~ + (fi(t, a) - fi(s, a))da + (~(t, a) - #(s, x I f: fo ~ ~.1 x exp/~ 21xl + ~(t, a)dW~ + #(s, a)d , where C is some positive constant. Here, we have used the boundedness and regularity of r and (H5). Now, taking expectations of both sides and using the regularity of fl, we obtain lu(t, x) - u(s, x)l ( ( foa)dWafo ' ) ) <~ C] t - - s I e 2#lxl sup E exp t~ ?r(t, + 6(s, a)dW. + OeJ-o,t 273 THE PRICING OF AMERICAN OPTIONS (fo o,