Numerical probabalistic methods for high-dimensional problems in finance

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1 Numerical probabalisic mehods for high-dimensional problems in finance The American Insiue of Mahemaics This is a hard copy version of a web page available hrough hp:// Inpu on his maerial is welcomed and can be sen o workshops@aimah.org Version: Fri Mar 26 16:54:

2 2 Table of Conens A. Open problems High-dimensional Problems in Finance and exension a. Some sochasic conrol problems in finance b. Exensions and BSDE s 2. Some New Mehodologies a. The case where b and a does no depend on Y and Z b. The case where b or a depends on Y and Z 3. A es problem 4. Reducion of he dimension 5. References

3 3 Chaper A: Open problems A.1 High-dimensional Problems in Finance and exension A.1.a Some sochasic conrol problems in finance. Several examples were considered. Opimal Sopping and free boundary problems. Le s consider he following financial marke conaining a non-risky asse S 0 = e r, and d risky asses (e.g. socks) wih prices modeled by a d-dimensional diffusion process S wih dynamics ds = rs d + diag(s )σ(, S )dw, where W is a sandard Brownian moion. An imporan problem in finance is he pricing of American opions. Given a reward funcion g, an American opion is a conrac ha provides o he owner he righ o receive (from he seller) he amoun g(s ) a any ime if exercised before some fixed mauriy T. This righ can be exercised only once during he period [0, T ]. The price of his opion a ime can be expressed as he value funcion associaed o he opimal sopping problem v(, S ) = sup τ T [,T ] E[e r(τ ) g(s ) S ] (1.1.1) where T [,T ] is he se of all sopping imes wih values in [, T ]. The associaed value funcion v is he soluion of he free boundary problem min{rv v Lv ; v g} = 0 on [0, T ) [0, ) d where L is he generaor of he diffusion S. v(t,.) = g(.) on [0, ) d Opimal Invesmen and Hamilon-Jacobi-Bellman equaions. An oher imporan issue in finance is ha of opimal invesmen. Denoing by ν he number of socks held by a given financial agen a ime, he associaed wealh-process X ν has dynamics given by dx = ν ds + (X ν S )ds 0 where sands for ransposiion, and S may have a general dynamics of he form ds = µ(, S, ν )d + σ(, S, ν )dw, in order o ake ino accoun a possible influence of he agen s financial sraegy on he dynamics of he risky asses. Given a concave (uiliy) funcion V, he agen ries o maximize he expeced uiliy of erminal wealh max E[V (X ν T )] (1.1.2) over a se of admissible financial sraegies (ν ) wih values in some subse U of R d. The associaed value funcion v is he soluion of Hamilon-Jacobi-Bellman equaion v sup L ν v = 0 on [0, T ) [0, ) d R ν U v(t, s, x) = V (x) for (s, x) [0, ) d R where L ν is he generaor of he diffusion (S, X ν ).

4 4 A.1.b Exensions and BSDE s. The value funcion of Problem (1.1.1) can be reformulaed in erms of he Y componen of he soluion (Y, Z, A) of he Backward Sochasic Differenial Equaion Y = g(s T ) Y g(s ) where A is a non-decreasing process saisfying 0 Z dw + A T A (Y g(s ))da = 0, see e.g. [KKPPQ]. This leads us o consider he more general problem of approximaing he soluion of Refleced Forward - Backward Sochasic Differenial Equaions of he form X = X 0 + Y = g(t, X T ) + Y g(, X ) b(, X, Y, Z )d + where A is a non-decreasing process saisfying 0 f(, X, Y, Z )d a(, X, Y, Z )dw (Y g(, X ))da = 0. Z dw + A T A This framework parially includes conrol problems associaed o HJB equaions. I is relaed o non-linear PDE s of he form where θ(, x) solves hrough he relaions See e.g. [MY]. min{ v (, x) 1 2 Trace[v xx(, x)aa (, x, v(, x), θ(, x))] b(, x, v(, x), θ(, x)) v x (, x) f(, x, v(, x), θ(, x)) v(t,.) = g(t,.) A.2 Some New Mehodologies v x (, x)a(, x, v(, x), θ(, x)) = θ(, x), v(, X ) = Y and θ(, X ) = Z. ; v(, x) g(, x)} = 0 A.2.a The case where b and a does no depend on Y and Z. Several mehods were considered. Pure Mone Carlo Mehods. Pure Mone-Carlo mehods are based on a discree ime approximaion of he forwardbackward equaion. Once discreized he forward process can be simulaed. These simulaions are hen used o compue he condiional expecaions involved in he backward

5 discree ime approximaion of (Y, Z). Two differen mehods can be used o compue hese condiional expecaions. a. The Longsaff-Schwarz (Carriere) approach consiss in approximaing he condiional expecaions by regressions on a given basis of funcions. This provides a very powerful, and, easy o implemen, algorihm for which convergence has been shown o hold in he case of he American opion pricing problem. However, no rae of convergence is given and he choice of he basis is a quie difficul problem. I has been used successfully in up o 20 facors models. See [C], [LS], [CLP]. b. The Malliavin approach consiss in re-wriing he condiional expecaions as he raio of wo uncondiional expecaions ha can be esimaed by sandard Mone-Carlo mehods. Upper bounds for he rae of convergence are proved. Conrary o he previous approach, his algorihm also provides good approximaions for he greeks, i.e. he gradien of he associaed value funcion (which is relaed o he Z, see above). So far, he exising algorihm has a oo imporan complexiy, which explains why i has only been esed in small dimensions (up o 5). Some numerical improvemens have been proposed, reducing he complexiy from N 2 o N ln(n) d, where N is he number of simulaed pahs. This work is sill in progress, he aim being o develop an algorihm such ha he major par of he work is done before a reward funcion is specified, so as o reduce as much as possible he effecive compuaion ime once a paricular payoff is defined. See [BET], [BT]. Grid approximaions. The Quanizaion approach consiss in approximaing he original forward process by a discree ime process which evolves on some finie grid. The grid is consruced so has o provide he bes L p approximaion. I was firs applied o he pricing of American opions in dimension up o 10. The consrucion of he opimal grid (and he compuaion of he associaed ransiion probabiliies) is very ime consuming, bu i can be done once for all. Once he grid, which is independen of he payoff funcion, is consruced, i provides a very quick algorihm for pricing American opions on differen payoffs, whenever hey are wrien on he same asses. As in he Malliavin and Longsaff-Schwarz approach, he use of a good conrol variable is required. Exensions o non-linear filering, opimal conrol and Asian-ype opions have also been sudied. See [BP1], [BP2], [PP1], [PP2]. Dual formulaion. This algorihm is based on a dual formulaion for problem (1.1.1): [ ] v(0, S 0 ) = inf E sup(e r g(s ) M ) T where he inf is aken over a well suied se of maringales M. This algorihm consiss in providing a upper bound for v by choosing some maringale M and compuing [ ] E sup(e r g(s ) M ) T In cases where a good maringale ˆM can be found, ypically when 5 E[e r(t ) g(s T ) S ] =: ˆM

6 6 is known as a funcion of S, his provides a quie sharp upper bound for he price of he American opion. Numerical experimens, up o dimension 15, have been performed. See [R]. Cubaure on Wiener spaces. Cubaure formula on Wiener spaces have been developed in [VL] in order o consruc probabiliy measures wih finie suppor which approximae he Wiener measure in he sense ha he expecaion of ieraed Sraonovich inegrals under he approximaing measure and he Wiener measure are close. In a sense, his approach is similar o ha of he quanizaion since i allows o reduce o a finie dimensional seing. So far, i has been used o develop high order numerical schemes for high dimensional SDE s and semi-ellipic PDE s. A.2.b The case where b or a depends on Y and Z. In ha case, i is no more possible o approximae (or simulae) he forward process X since is dynamic depends on he soluion. In such siuaions, wo differen soluions have been proposed: a - Consruc an a priori grid for X, possibly based on some a priori on he dynamics of Y and Z. b - Given an a priori soluion Y 0 and Z 0, approximae (or simulae) he corresponding forward process X 0 and use he above mehodologies o consruc he corresponding soluion (Y 1, Z 1 ) of he BSDE. Then, use his soluion (Y 1, Z 1 ) o approximae (or simulae) he corresponding forward process X 1 and go on ieraing his procedure. Under some mild assumpions, his algorihm should be convergen. However, i seems o be quie heavy o implemen. Soluion a. has already been applied in he quanizaion approach bu, so far, does no provide very good resuls. See [PP1]. A.3 A es problem A es problem has been proposed o compare he performance of he differen mehods. I corresponds o he compuaion of a d dimensional a-he-money American min-pu opion in he Black-Scholes model. The asses are uncorrelaed, have he same volailiy and he same iniial value : S i = 100 exp((r σ 2 /2) σw i ) for each i = 1,..., d where W = (W 1,..., W d ) is a sandard Brownian moion. The payoff funcion is g(x) = [100 min{x 1,..., x d }] +. Numerical ess will be performed for differen dimensions wih σ = 20% and r = 5%. Resuls will be colleced and compared by J. Cvianic. A.4 Reducion of he dimension Differen ways of reducing he dimension were proposed. a- The firs consiss in using Principal Componen Analysis in order o aggregae a large number of random variables in a small number of principal direcions. Also i is used in

7 pracice, i does no really solve he problem of he dimension, bu only surrounds i, replacing he iniial model by a more racable, bu differen, one. b- Insead of relying on PCA, dimension could be reduced by considering models where a few number of economic fundamenals could explain he behavior of many financial asses. c- In some cases, i is possible o reduce he dimension wihou changing he iniial model. This is he case in P. Carr s Canadizaion approach for he compuaion of American opions, where he randomizaion of he mauriy allows o reduce o a ime homogeneous problem. Exensions o Markovian conrol problems have been discussed. See [Carr] and [BKT]. A.5 References [BP1] Bally V. and G. Pages: A quanizaion algorihm for solving discree ime mulidimensional opimal sopping problems, Bernoulli, (9) 6, 1-47, [BP2] Bally V. and G. Pages: Error analysis of he quanizaion algorihm for obsacle problems, Sochasic Processes and heir Applicaions,106 (1), 1-40, [BT] Bouchard B. and N. Touzi: Discree Time Approximaion and Mone-Carlo Simulaion of Backward Sochasic Differenial Equaions, Sochasic Processes and heir Applicaions, o appear. [BET] Bouchard B., I. Ekeland and N. Touzi: On he Malliavin approach o Mone Carlo approximaion of condiional expecaions, Finance and Sochasics, 8 (1), [BKT] Bouchard B., N. El Karoui and N. Touzi: Mauriy randomizaion for sochasic conrol problems, in preparaion. [Carr] Carr P.: Randomizaion and he American Pu, The Review of Financial Sudies, 11, , [C] Carriere E.: Valuaion of he Early-Exercise Price for Opions using Simulaions and Nonparameric Regression, Insurance: mahemaics and Economics, 19, 19-30, [CLP] Clémen E., D. Lamberon, and P. Proer: An analysis of a leas squares regression mehod for American opion pricing, Finance and Sochasics, 6, , [KKPPQ] El Karoui N., C. Kapoudjan, E. Pardoux, S. Peng and M.C. Quenez: Refleced soluions of backward sochasic differenial equaions and relaed obsacle problems for PDE s, Annals of Probabiliy, 25, , [LS] Longsaff F.A. and R.S. Schwarz: Valuing American Opions By Simulaion: A simple Leas-Square Approach, Review of Financial Sudies, 14, , [MY] Ma J. and J. Yong: Forward-Backward Sochasic Differenial Equaions and Their Applicaions, Lecure Noes in Mah., 1702, Springer, [PP1] Pages G. and H. Pham: A quanizaion algorihm for mulidimensional sochasic conrol problems, preprin LPMA n 697, [PP2] Pages G. and H. Pham: Opimal quanizaion mehods for nonlinear filering wih discree-ime observaions, preprin LPMA n 778,

8 8 [R] Rogers L.C.G.: Mone Carlo valuaion of American opions, Mahemaical Finance, 12, , [VL] Vicoir N. and T. Lyons: Cubaure on Wiener space, Proc. R. Soc. Lond. A, 460, , 2004.