Stochastic processes and applications to mathematical finance

Proceedings of the 6th Ritsumelkan International Symposium STOCHASTIC PROCESSES AND APPLICATIONS TO MATHEMATICAL FINANCE This page intentionally left blank Proceedings of the 6the Ritsumeikan International Symposium STOCHASTIC PROCESSES AND APPLICATIONS TO MATHEMATICAL FINANCE Ritsumeikan University,, Japan 6–10 March 2006 Editors Joro Akahori Shigeyoshi Ogawa Shinzo Watanabe Ritsumeikan University,, Japan World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI Published by World Scientific Publishing Co Pte Ltd Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library STOCHASTIC PROCESSES AND APPLICATIONS TO MATHEMATICAL FINANCE Proceedings of the 6th Ritsumeikan International Symposium Copyright © 2007 by World Scientific Publishing Co Pte Ltd All rights reserved This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA In this case permission to photocopy is not required from the publisher ISBN-13 978-981-270-413-9 ISBN-10 981-270-413-2 Printed in Singapore Chelsea - Stochastic Processes (6th).pmd 1/18/2007, 2:37 PM December 26, 2006 14:28 Proceedings Trim Size: 9in x 6in PREF-06-2+ PREFACE The 6th Ritsumeikan international conference on Stochastic Processes and Applications to Mathematical Finance was held at Biwako-Kusatsu Campus (BKC) of Ritsumeikan University, March 6–10, 2006 The conference was organized under the joint auspices of Research Center for Finance and Department of Mathematical Sciences of Ritsumeikan University, and financially supported by MEXT (Ministry of Education, Culture, Sports, Science and Technology) of Japan, the Research Organization of Social Sciences, Ritsumeikan University, and Department of Mathematical Sciences, Ritsumeikan University The series of the Ritsumeikan conferences has been aimed to hold assemblies of those interested in the applications of theory of stochastic processes and stochastic analysis to financial problems The Conference, counted as the 6th one, was also organized in this line: there several eminent specialists as well as active young researchers were jointly invited to give their lectures (see the program cited below) and as a whole we had about hundred participants The present volume is the proceedings of this conference based on those invited lectures We, members of the editorial committee listed below, would express our deep gratitude to those who contributed their works in this proceedings and to those who kindly helped us in refereeing them We would express our cordial thanks to Professors Toshio Yamada, Keisuke Hara and Kenji Yasutomi at the Department of Mathematical Sciences, of Ritsumeikan University, for their kind assistance in our editing this volume We would thank also Mr Satoshi Kanai for his works in editing TeX files and Ms Chelsea Chin of World Scientific Publishing Co for her kind and generous assistance in publishing this proceedings December, 2006, Ritsumeikan University (BKC) Jiroˆ Akahori Shigeyoshi Ogawa Shinzo Watanabe v December 26, 2006 14:28 Proceedings Trim Size: 9in x 6in PREF-06-2+ vi The 6th Ritsumeikan International Conference on STOCHASTIC PROCESSES AND APPLICATIONS TO MATHEMATICAL FINANCE Date March 6–10, 2006 Place Rohm Memorial Hall/Epoch21, in BKC, Ritsumeikan University 1-1-1 Nojihigashi, Kusatsu, Shiga, 525-8577, Japan Program March, (Monday): at Rohm Memorial Hall 10:00–10:10 Opening Speech, by Shigeyoshi Ogawa (Ritsumeikan University) 10:10–11:00 T Lyons (Oxford University) Recombination and cubature on Wiener space 11:10–12:00 S Ninomiya (Tokyo Institute of Technology) Kusuoka approximation and its application to finance 12:00–13:30 Lunch time 13:30–14:20 T Fujita (Hitotsubashi University, Tokyo) Some results of local time, excursion in random walk and Brownian motion 14:30–15:20 K Hara (Ritsumeikan University, Shiga) Smooth rough paths and the applications 15:20–15:50 Break 15:50–16:40 X-Y Zhou (Chinese University of Hong-Kong) Behavioral portfolio selection in continuous time 17:30– Welcome party March, (Tuesday): at Rohm Memorial Hall 10:00–10:50 M Schweizer (ETH, Zurich) Aspects of large investor models 11:10–12:00 J Imai (Tohoku University, Sendai) A numerical approach for real option values and equilibrium strategies in duopoly 12:00–13:30 Lunch time December 26, 2006 14:28 Proceedings Trim Size: 9in x 6in PREF-06-2+ vii 13:30–14:20 H Pham (Univ Paris VII) An optimal consumption model with random trading times and liquidity risk and its coupled system of integrodifferential equations 14:30–15:20 K Hori (Ritsumeikan University, Shiga) Promoting competition with open access under uncertainty 15:20–15:50 Break 15:50–16:40 K Nishioka (Chuo University, Tokyo) Stochastic growth models of an isolated economy March, (Wednesday): at Rohm Memorial Hall 10:00–10:50 H Kunita (Nanzan University, Nagoya) Perpetual game options for jump diffusion processes 11:10–11:50 E Gobet (Univ Grenoble) A robust Monte Carlo approach for the simulation of generalized backward stochastic differential equations 12:00– Excursion March, (Thursday): at Epoch21 10:00–10:50 P Imkeller (Humbold University, Berlin) Financial markets with asymmetric information: utility and entropy 11:00–12:00 M Pontier (Univ Toulouse III) Risky debt and optimal coupon policy 12:00–13:30 Lunch time 13:30–14:20 H Nagai (Osaka University) Risk-sensitive quasi-variational inequalities for optimal investment with general transaction costs 14:30–15:20 W Runggaldier (Univ Padova) On filtering in a model for credit risk 15:20–15:50 Break 15:50–16:40 D A To (Univ Natural Sciences, HCM city) A mixed-stable process and applications to option pricing 16:50– Short Communications Y Miyahara (Nagoya City University) T Tsuchiya (Ritsumeikan University, Shiga) K Yasutomi (Ritsumeikan University, Shiga) December 26, 2006 14:28 Proceedings Trim Size: 9in x 6in PREF-06-2+ viii March, 10 (Friday): Epoch21 10:00–10:50 R Cont (Ecole Polytechnique, France) Parameter selection in option pricing models: a statistical approach 11:10–12:00 T V Nguyen (Hanoi Institute of Mathematics) Multivariate Bessel processes and stochastic integrals 12:00–13:30 Lunch time 13:30–14:20 J-A, Yan (Academia Sinica, China) A functional approach to interest rate modelling 14:30–15:20 M Arisawa (Tohoku University, Sendai) A localization of the L´evy operators arising in mathematical finances 15:20–15:50 Break 15:50–16:40 A N Shiryaev (Steklov Mathem Institute, Moscow) Some explicit stochastic integral representation for Brownian functionals 18:30– Reception at Kusatsu Estopia Hotel December 26, 2006 14:28 Proceedings Trim Size: 9in x 6in PREF-06-2+ ix December 26, 2006 14:56 Proceedings Trim Size: 9in x 6in quantifpartial-rits 282 First, notice by definition of the closest neighbour projection that the L2 quantization error is the minimum of L2 -error Z − Y among all random variables Y taking values in the grid z Then, two questions arise naturally : for fixed N, is there an optimal grid z∗ which minimizes the L2 -quantization error (or equivalently the distorsion), and how does this minimum behave when N goes to infinity? The latter question is answered by the so-called Zador theorem : Theorem 4.1 (see [7]) Assume that Z ∈ L2+ε (P, Rq ) for some ε > Then, lim N q Z − Zˆ z N z 2 = Jq ( q | f | q+2 dλq ) q+2 q , where PZ (dξ) = f (ξ)λq (dξ) + ν(dξ) is the Lebesgue decomposition of P Z with respect to the Lebesgue measure λq on Rq , and Jq is a constant depending on q, corresponding to the uniform distribution on [0, 1] q Remark 4.1 In dimensions q = and 2, J = ∼ q 2πe 12 and J2 = 5√ 18 For q ≥ 3, Jq as q goes to infinity The optimal N-quantization problem that consists in determining a grid z , which minimizes the L2 -quantization error, relies on the property that the distorsion is continuously differentiable at any N-tuple having pairwise distinct components, with a gradient obtained by formal differentiation in (4.1) : ∗ (4.2) ∇DZN (z) = 2E KN (z, Z)], where KN : (Rq )N × Rq → (Rq )N is defined by KN (z, ξ) = ((zi − ξ)1ξ∈Ci(z) )1≤i≤N A quantizer Zˆ = Zˆ z is said stationary if the associated N-tuple z satisfies ∇DZN (z) = An optimal quantizer is a stationary quantizer The integral representation (4.2) of ∇DZN suggests, as soon as independent copies of Z can be simulated, to implement a stochastic gradient algorithm (descent), in order to get numerically a stationary quantizer By denoting, z(s) = (zs,1 , , zs,N ) the grid (or N-tuple in Rq ) at step s, the stochastic gradient descent procedure is recursively defined by : z(s+1) = z(s) − δs+1 KN (z(s) , ξs+1 ), December 26, 2006 14:56 Proceedings Trim Size: 9in x 6in quantifpartial-rits 283 where (ξs )s are independent copies of Z, and (δs )s is a positive sequence of step parameters satisfying the usual conditions : δ2s < ∞ δs = ∞ and s s In our context, this leads to the Kohonen algorithm or competitive learning vector quantization (CLVQ) algorithm, which also provides as a byproduct an estimation of the weights pˆ i of the Voronoi tesselations associated to the stationary quantizer We refer to [10] for a complete description and discussion of the convergence of algorithm Optimal grids and their companion parameters, i.e weights of the Voronoi tesselation and distorsion, for the normal distribution are available and downloadable on the webpages of Gilles Pag`es or Jacques Printems Quantization of the Filter Process In view of solving dynamic optimization problems under partial observation, we need an approximation of the filter process (Πk )k Recall the dependence of the random filter on the observation : Πk = Πk (Y1 , , Yk ) An usual approach, suggested e.g in [3], consists of approximating Πk (Y1 , , Yk ) by Πk (Yˆ , , Yˆ k ) where Yˆ k is a quantizer of Yk The main problem in effective implementation is the growing dimension of this approximating filter : indeed, for instance, if each Yˆ k takes M values, then at time n, the random filter Πn (Yˆ , , Yˆ n ) would take Mn values in Km , which is not realistically implementable for a long horizon n In order to overcome this numerical difficulty, we present a quantization approach introduced in [11] and based on the Markov property of the pair filter-observation (Πk , Yk ) with respect to the observation filtration (FkY ) In other words, the conditional law of Xk+1 given FkY is summarized by the sufficient statistic (Πk , Yk ), and we shall approximate the pair Markov chain (Πk , Yk ) by an approximation of their successive probability transitions One first proves that the probability transition Rk (from time k − to k) of the Markov chain (Zk ) = (Πk , Yk ) in Km × Rd is given by : Rk ϕ(π, y) = ¯ k (π, y, y ), y )Qk (π, y, dy ), ϕ(H where Qk (π, y, dy ) is the law of Yk conditional on (Πk−1 , Yk−1 ) = (π, y) with density : m (5.1) ij gk (xi , y, x j , y )Pk πi y −→ i, j=1 December 26, 2006 14:56 Proceedings Trim Size: 9in x 6in quantifpartial-rits 284 This shows in particular that Zk may be simulated through the following simulation procedure of its probability transition (Rk ) : for k = 0, Z0 is a known deterministic vector equal to z0 = (µ, y0 ), and for k ≥ 1, starting from (Πk−1 , Yk−1 ), • we simulate Yk according to the law Qk (Πk−1 , Yk−1 , dy ) given in (5.1) • we compute Πk by the forward filtering equation Πk = H¯ k (Πk−1 Yk−1 , Yk ) Once we are able to simulate independent copies of (Z0 , , Zn ), we apply an optimal quantization to each Zk in Km × Rd , for k = 0, , n, following the vector quantization method described in the previous section For each k = 0, , n, we denote by Zˆ k the zk -Voronoi quantizer of Zk , valued in the k grid zk = (z1k , , zN ) consisting of Nk points in Km × Rd associated to the k Voronoi tesselations Ci (zk ), i = 1, , Nk As a byproduct, we approximate the probability transitions (Rk ) of the Markov chain (Zk ) by the probability transition matrices (ˆrk ) defined by : ij j rˆk = P Zˆ k = zk Zˆ k−1 = zik−1 = P Zk ∈ C j (zk ), Zk−1 ∈ Ci (zk−1 ) P [Zk−1 ∈ Ci (zk−1 )] =: ij βˆk pˆik−1 , for all k ≥ 1, i = 1, , Nk−1 , j = 1, , Nk The process (Zˆ k ) obtained by this method, is called a marginal quantization of the process (Zk ) : it is characterized for each k by its grid space z k , and by the probability ij transition matrix rˆk = (ˆrk ) Denoting by ξs = (ξs0 , , ξsn )s , independent copies of (Z0 , , Zn ), the optimal grids zk that minimize the L2 -quantization error Zk − Zˆ k for ij each k, and the companion parameters rˆk , are practically implemented according to the Kohonen algorithm as follows : Initialisation phase : (0) k , , z0,N ) ∈ (Km × Rd )Nk for k = 0, , n, • Initialize the n grids zk = (z0,1 k k with Γ(0) = z0 reduced to N0 = point for k = 0 0,i j • Initialize the weights vectors : p0,i = 1/Nk , βk+1 = 0, i = 1, , Nk , j = k 1, , Nk+1 , and the distorsion D0N = 0, for k = 0, , n k (s) k , , zs,N ), the weights Updating s → s + : At step s, the n grids z k = (zs,1 k k s,i j vectors ps,i , βk+1 , i = 1, , Nk , j = 1, , Nk+1 , have been obtained and we k December 26, 2006 14:56 Proceedings Trim Size: 9in x 6in quantifpartial-rits 285 use the sample ξs+1 of (Z0 , , Zn ) to update them as follows : for all k = 0, , n, • Competitive phase : select ik (s + 1) ∈ {1, , Nk } such that ξs+1 ∈ Cik (s+1) (z(s) ), i.e ik (s + 1) ∈ argmin1≤i≤Nk |zs,i − ξs+1 |2 k k • Learning phase : Updating of the grid : = zs,i − δs+1 1i=ik (s+1) zs,i − ξs+1 , zs+1,i k k k i = 1, , Nk Updating of the weights vectors and of the probability transition = ps,i − δs+1 ps,i − 1i=ik (s+1) , ps+1,i k k k s+1,i j s,i j s,i j βk+1 = βk+1 − δs+1 βk+1 − 1i=ik (s+1), j=ik+1 (s+1) , s+1,i j s+1,i j rk+1 = βk+1 ps+1,i k , for all i = 1, , Nk , j = 1, , Nk+1 Numerical Approximation to Optimization Problems under Partial Observation 6.1 Quantization of optimal stopping We turn back to the optimal stopping problem under partial observation considered in paragraph 3.1, and we define the corresponding values : (6.1) Uk = ess sup E h(τ, Xτ , Yτ )| FkY , k = 0, , n, Y τ∈Tk,n Y where Tk,n is the set of (FkY )-stopping times valued in {k, , n} By using the law of iterated conditional expectation and the definition of the filter, we notice that problem (6.1) may be reduced to a complete observation model with state variable the (FkY )-adapted process (Zk ) : n 1τ= j E[h(j, X j , Y j )|F jY ] FkY Uk = ess sup E Y τ∈Tk,n j=k n 1τ= j Π j h(j, , Y j ) FkY = ess sup E Y τ∈Tk,n j=k ˜ Zτ ) F Y , = ess sup E Πτ h(τ, , Yτ )| FkY = ess sup E h(τ, k Y τ∈Tk,n Y τ∈Tk,n December 26, 2006 14:56 Proceedings Trim Size: 9in x 6in quantifpartial-rits 286 with the notation : ˜ z) = πh(., y) = h(k, m h(k, xi , y)πi , ∀z = (π, y), π = (πi )i ∈ Km , y ∈ Rd i=1 By the (FkY )-Markov property of (Zk ) and the dynamic programming principle, we have Uk = uk (Zk ) where functions uk are defined in backward induction by : ˜ z) un (z) = h(n, ˜ z) , E [ uk+1 (Zk+1 )| Zk = z] uk (z) = max h(k, Following [1], we provide a quantization approximation of Uk = uk (Zk ) ˆ k = uˆ k (Zˆ k ), for k = 0, , n, where (Zˆ k ) is a marginal quantization of by U (Zk ) on grids (zk ) with corresponding probability transition matrices (ˆrk ), as described in the previous section, and functions uˆ k are explicitly computed in recursive form by : ˜ z) uˆ n (z) = h(n, ˜ z) , E uˆ k+1 (Zˆ k+1 ) Zˆ k = z uˆ k (z) = max h(k, From an algorithmic viewpoint, this reads as : ˜ zi ), i = 1, , Nn uˆ n (zin ) = h(n, ⎧n ⎫ ⎪ ⎪ Nk+1 ⎪ ⎪ ⎪ ⎪ ⎨˜ ⎬ ij j i ˆ ˆ h(k, z , ) , (z ) uˆ k (zik ) = max ⎪ r u ⎪ k+1 k+1 ⎪ k ⎪ k+1 ⎪ ⎪ ⎩ ⎭ j=1 i = 1, , Nk , k = 0, , n − ˆ k in terms of quantization error Zk − Zˆ k L1 -error estimation Uk − U is stated in [11] By combining with Zador’s theorem, we obtain a rate of C(n) convergence of order , where C(n) is a constant depending essentially N m−1+d on the boundedness and Lipschitz conditions on gk and h, and the horizon n Numerical illustration : Bermudean options in a partially observed stochastic volatility model We consider an observable stock (logarithm) price Yk = ln Sk , with dynamics given by : (6.2) √ Yk+1 = Yk + r − Xk2 δ + Xk δεk+1 , December 26, 2006 14:56 Proceedings Trim Size: 9in x 6in quantifpartial-rits 287 Table Comparison of quantized filter value to its Monte Carlo estimation Monte Carlo Quant with N¯ = 300 Quant with N¯ = 600 Quant with N¯ = 900 Quant with N¯ = 1200 Quant with N¯ = 1500 E[Π1n ] 0.287608 0.301651 0.301604 0.301598 0.301618 0.301605 E[Π2n ] 0.422833 0.421725 0.421458 0.421316 0.42122 0.421205 E[Π3n ] 0.289558 0.276624 0.276938 0.277086 0.277162 0.27719 Relative error (%) 0.898 0.886 0.881 0.879 0.878 where (εk ) is a sequence of Gaussian white noise, and (Xk ) is the unobservable volatility process δ = n1 is the time step from an Euler scheme over a period [0, 1] We assume that (X k ) is a Markov chain approximation a` la Kushner [8] with spatial step ∆ and with m = states of a mean-reverting process : (6.3) dXt = λ(x0 − Xt )dt + ηdWt In this context of a partially observed stochastic volatility model, we consider a Bermudean put option with payoff y → (κ − e y )+ , and with price : (6.4) u0 = sup E e−rτδ κ − eYτ Y τ∈T0,n + We perform numerical tests with : - Price and put option parameters : r = 0.05, S = 110, κ = 100, - Volatility parameters : λ = 1, η = 0, 1, ∆ = 0, 05, X = 0.15, - Quantization : Grids are of same size N¯ fixed for each time period We first compare in Table the filter expectation at the final date computed with a time step size δ = 1/5 and by using the optimal quantization method with increasing grid size N¯ , and with 106 Monte Carlo iterations of the path observation Y We observe that besides the very low error level, the absolute error (plotted in Fig 1) and the relative error are decreasing as the grid size grows Secondly, in order to illustrate the effect of the time step, we compute the American option price under partial observation when the time step δ decreases to zero (i.e n increases) and compare it with the American option price with complete observation of (Xk , Yk ) Indeed, in the limit for δ → we fully observe the volatility, and so the partial observation price should converge to the complete observation price December 26, 2006 14:56 Proceedings Trim Size: 9in x 6in quantifpartial-rits 288 1.355 x 10 1.35 1.345 1.34 1.335 1.33 1.325 1.32 1.315 200 400 600 800 1000 1200 1400 1600 Figure Filter error convergence as N¯ grows 11 Quadratic risk 10 Total observation Partial observation 2.5 3.5 4.5 Initial capital 5.5 Figure Quadratic hedging of an European put: graph of w → infα∈A E((κ − eYn )+ − Wn )2 ) in the partial and total observation case Size grid for W = 100 points, size grid for (eY , Π) = 1500 points, size grid for (eY , X) = 45 points Moreover, when we have more and more observations, the difference between the two prices should decrease and converge to zero This is shown in figure 6, where we performed option pricing over grids of size N¯ Π,Y = 1500 in case of partial observation The total observation price is given by the same pricing algorithm carried out on N¯ X,Y = 45 points for the product grid of (Xk , Yk ) For fixed n, the rate of convergence for the December 26, 2006 14:56 Proceedings Trim Size: 9in x 6in quantifpartial-rits 289 Table American option price for embedded filtrations—First Example n 16 1.45863 1.75689 1.77642 0.921729 1.13898 1.47089 0.61 0.30 Tot Obs (N¯ X,Y = 30) Part Obs (N¯ = 1000) Variation 0.53 Π,Y Table American option price for embedded filtrations—Second Example n 10 20 Tot Obs (N¯ X,Y = 45) Part Obs (N¯ = 1500) 1.57506 1.72595 1.91208 0.988531 1.30616 1.59632 Variation 0.58 0.42 0.31 Π,Y approximation of the value function under partial observation is of order 1/(m−1+d) where N¯ Π,Y is the number of points used at each time k for the N¯ Π,Y grid of (Πk , Yk ) valued in Km × Rd From results of [1], we also know that the rate of convergence for the approximation of the value function under full observation is of order m × N¯ Y where N¯ X,Y = m × N¯ Y is the number of points at each time k, used for the grid of (Xk , Yk ) valued in E × Rd This explains why, in order to have comparable results, and with m = and d = 1/3 1, we have chosen N¯ Y ∼ N¯ Π,Y In addition, it is possible to observe the effect of information enrichment as the time step decreases In fact, if we consider multiples of n as the time step parameter, we notice that the American option price increases for both total and partial observation models (see Tables and 3) 6.2 Quantization of control problem We turn back to the control problem under partial observation considered in paragraph 3.2 By using the law of iterated conditional expectations, we can rewrite the expected cost function as follows: J(α) = E E (Xn , Yn , Wn )|FnY ⎡ m ⎤ ⎢⎢ ⎥⎥ i i (x , Yn , Wn )Πn ⎥⎥⎥⎦ = E ⎢⎢⎢⎣ i=1 = E ˆ(Πn , Yn , Wn ) December 26, 2006 14:56 Proceedings Trim Size: 9in x 6in quantifpartial-rits 290 where m ˆ(π, y, w) := (xi , y, w)πi i=1 The original control problem (3.2) can now be reformulated as a problem under full observation with state variables (Πk , Yk , Wk ), valued in Km × Rd × R, and (FkY )-adapted : Jopt = inf E ˆ(Πn , Yn , Wn ) α∈A Recalling the dynamics (3.1) of (Wk ) and following the dynamic programming principle for discrete-time control problems, we define the sequence of functions on Km × Rd × R : un (π, y, w) = ˆ(π, y, v) uk (π, y, w) = inf E uk+1 (Πk+1 , Yk+1 , F(w, a, y, Yk+1)) (Πk , Yk ) = (π, y) , a∈A for k = 0, , n − 1, so that Jopt = u0 (µ, y0 , w0 ), where w0 is the initial value of W0 at time k = 0, and we recall that (Π0 , Y0 ) = (µ, y0 ) In order to compute this sequence of functions uk , we deal separately with the approximation of the pair filter-observation process (Zk )k = (Πk , Yk )k that does not depend on the control, and the approximation of the controlled process (Wk )k • We apply a marginal quantization of the process (Zk ) = (Πk , Yk ), and we ˆ k , Yˆ k ) the corresponding quantizers on grids (zk ), and denote the (Zˆ k ) = (Π (ˆrk ) the associated probability transition matrices, as described in section The i-th point of the grid zk of size Nk in Km × Rd is denoted zik = (πk (i), yik ) ∈ Km × Rd , i = 1, , Nk • The approximation of Wk is obtained by a classical uniform space discretization similar to the Markov chain method as in Kushner We fix a bounded uniform grid on the state space R for the controlled process (Wk ) Namely, we set Γ = (2ν)Z ∩ [−L, L], where ν is the spatial step and L is the grid size We denote by ProjΓ the projection on the grid Γ according to the closest neighbor rule Recalling the dynamics (3.1) of the controlled process (Wk ), we approximate it as follows : December 26, 2006 14:56 Proceedings Trim Size: 9in x 6in quantifpartial-rits 291 ˆ k ), given a control α ∈ A, we define the discretized controlled process (W valued in Γ, by : ˆ k , αk , Yˆ k , Yˆ k+1 )) ˆ k+1 = Proj (F(W W Γ We then approximate the sequence of functions uk by the sequence of functions uˆ k defined on zk × Γ, k = 0, , n, by a dynamic programming type formula : uˆ n (π, y, w) = ˆ(π, y, w) ˆ k+1 , Yˆ k+1 , Proj (F(w, a, y, Yˆ k+1)) (Π ˆ k , Yˆ k ) = (π, y) uˆ k (π, y, w) = inf E uˆ k+1 Π Γ a∈A From an algorithmic viewpoint, this is computed explicitly as follows : uˆ n (zin , w) = ˆ(zin , w), zin = (πn (i), yin ) ∈ zn , i = 1, , Nn , w ∈ Γ, Nk+1 (6.5) uˆ k (zik , w) = inf a∈A (6.6) zik ij j j rˆk+1 uˆ k+1 zk+1 , ProjΓ (F(w, a, yik, yk+1 )) j=1 = (πk (i), yik ) ∈ zk , i = 1, , Nk , w ∈ Γ, k = 0, , n − For w0 ∈ Γ, the solution Jopt = u(µ, y0 , w0 ) to our control problem is then approximated by Jquant = uˆ (µ, y0 , w0 ) Moreover, this backward dynamic programming scheme allows us to compute at each time k = 0, , n − 1, an approximate control αˆ k (z, w), z ∈ zk , w ∈ Γ, by taking the infimum in (6.5) Error estimation between Jopt and Jquant in terms of the quantization errors Zk − Zˆ k for Zk = (Πk , Yk ), the spatial step ν, and the grid size L for (Wk ) is stated in [5] By combining with Zador’s theorem, this provides a rate of convergence of order C(n) ν + L1 + 11 N m−1+d Numerical illustration : Mean-variance hedging in a partially observed stochastic volatility model In the setting of the stochastic volatility model described in paragraph 6.1, we consider the mean-variance hedging of a put option The logarithm of the observed stock price is Y = ln S, its unobservable volatility is X, and the wealth process W controlled by the number of shares α invested in stock, is governed by : Wk+1 = Wk erδ + αk (eYk+1 − eYk erδ ), where r is the constant interest rate, and δ > is the interval between two trading dates The dynamics of (X, Y) is given by (6.2)-(6.3) Given a put December 26, 2006 14:56 Proceedings Trim Size: 9in x 6in quantifpartial-rits 292 300 points 600 points 1500 points 8.7 8.6 8.5 8.4 8.3 8.2 8.1 7.9 7.8 2.5 3.5 Figure Quadratic hedging of an European put: graph of w → infα∈A E((κ − eYn )+ − Wn )2 ) for different quantification grid sizes (N = 300, 600, 1500) and a fixed uniform grid size (N W = 400) Table Quadratic hedging of an European put: European put price (defined as the initial capital minimizing the risk) and optimal control strategy calculated for different quantization grid sizes (N = 300, 600, 1500) and a fixed uniform grid size (NW = 400) N 300 600 1500 European put price 3.04132 3.05965 3.07098 Optimal control strategy α0 -0.2813 -0.2813 -0.2813 option of payoff (κ − eYn )+ at maturity n, the investor’s objective is defined by the control problem : inf E (κ − eYn )+ − Wn α∈A We perform numerical tests with : - Price and put option parameters : r = 0.05, S = 110, κ = 110, - Volatility parameters : λ = 1, η = 0, 1, ∆ = 0, 05, X = 0.15, - Quantization of (Zk ) = (Πk , Yk ) : grids are of same size N fixed for each time period with step δ = n1 When it is not precised, we choose n = - Discretization of (Wk ) : we use a N W -point grid defined by Γ = (2ν)Z ∩ [Lin f , lsup ] with Lin f = −10, Lsup = 15 and so ν = 2(N25 W −1) December 26, 2006 14:56 Proceedings Trim Size: 9in x 6in quantifpartial-rits 293 8.6 400 points 200 points 100 points 8.5 8.4 8.3 8.2 8.1 7.9 7.8 7.7 2.4 2.6 2.8 3.2 3.4 3.6 Figure Quadratic hedging of an European put: graph of w → infα∈A E((κ − eYn )+ − Wn )2 ) for different fixed uniform grid sizes (NW = 50, 100, 200, 400) and a fixed quantization grid size (N = 300) - Approximation of the optimal control: golden search method (see [9]) on A = [−1, 1] In order to study the effects of the quantization grid size N and uniform grid size N W , we plot the graph of w0 → infα∈A E((κ − eYn )+ − Wn )2 ) for different values of N and N W (Figs and 4) As expected, the global shape of the graph is parabolic, due to the quadratic hedging criterion that we have used The minimum is reached at wmin which can be considered as the ”quadratic hedging price” of our European put option The corresponding hedging strategies are given in Table 4, and Fig displays the graph of α0 as a function of the initial wealth w0 We can see that the strategy is nearly constant for w0 ∈ [2, 4], where the non constant values may be due to numerical imprecision This is consistent with the theoretical result, which shows that the optimal strategy for the mean-variance hedging problem does not depend on the initial wealth when the (discounted) stock price is a martingale, which is the case here In Fig and in the Table 5, we compare the European put option price under partial and complete observation when we increase the number of observations (i.e the time step δ decreases to zero) Denoting by NΠ,Y the number of grid points used in the partial observation case to make an optimal quantization of the pair (Π, Y), by NX,Y the number of grid points used in the total observation case to make an optimal quantization of the December 26, 2006 14:56 Proceedings Trim Size: 9in x 6in quantifpartial-rits 294 Figure Quadratic hedging of an European put: graph of w → α0 (w0 ) for a quantization grid size of N = 300 and a fixed uniform grid size of NW = 400 0.25 a2 0.15 0.1 0.05 10 15 20 25 Figure Quadratic hedging of an European put: distance between total and partial observation European put prices (defined as the initial capital minimizing the risk) when we increase the number of observations (axis of abscissae) and consequently the time step δ goes to Size grid for W = 30 points, size grid for (eY , Π) = 1500 points, size grid for (eY , X) = 45 points pair (X, Y), and by L the grid size in the discretization of the controlled variable W, we recall that the discretization error is of order −1 d+m−1 NΠ,Y +ν+ L December 26, 2006 14:56 Proceedings Trim Size: 9in x 6in quantifpartial-rits 295 Table Quadratic hedging of an European put: comparison between partial and total observation price (defined as the initial capital minimizing the quadratic risk) and strategies when we increase the number of observations and consequently the time step δ goes to Size grid for W = 30 points, size grid for (eY , Π) = 1500 points, size grid for (eY , X) = 45 points Time step Partial observation Partial observation Total observation Total observation δ price strategy price strategy 1\5 2.9933 −0.2813 3.24459 −0.2734 1\10 3.5255 −0.3013 3.65515 −0.2422 1\20 3.9501 −0.3215 4.02799 −0.3614 2.5 1.5 0.5 Tot Obs Option Price (45 pts) Part Obs Option Price (1500pts) 0 10 15 20 25 30 35 Figure Partial and total observation option prices as δ → for the partial observation case For the total observation case we have: 1 +ν+ NX,Y R where NX,Y = mNY (see [11]) So, in order to obtain comparable results, given the uniform grid discretizing the variable W, we perform an optimal quantization of (Π, Y) and (X, Y) by using grid sizes NΠ,Y and NX,Y = mNY such that: d+m−1 NY NΠ,Y where d = and m = That is why we have chosen NΠ,Y = 1500 and NX,Y = 45 December 26, 2006 14:56 Proceedings Trim Size: 9in x 6in quantifpartial-rits 296 We notice that when the number of observations increases (i.e δ → 0), the partial observation price converges to the complete observation price; this is due to the fact that with observation performed in continuous time we are able to calculate the volatility given by the quadratic variation of the price process (eY ) Figure shows that by working in a total observation setting the quadratic risk associated to a given initial wealth is smaller than the corresponding value obtained in the partial observation case This is consistent with the fact that the filtration generated by the observation price is included in the full information filtration, and consequently the corresponding optimal cost function in the partial information case is larger than the one in the full information case References Bally, V and G Pag`es (2003): “A quantization algorithm for solving discrete time multi-dimensional optimal stopping problems”, Bernoulli, 9, 1003–1049 Bensoussan, A (1992): Stochastic control of partially observable systems, Cambridge university Press Bensoussan A and W Runggaldier (1987): An approximation method for stochastic control problems with partial observation of the state: a method for constructing ε-optimal controls, Acta Appli Math., 10, 145–170 Bouchard, B., I Ekeland, and N Touzi (2004): “On the Malliavin approach to Monte Carlo approximation of conditional expectations”, Finance and Stochastics, 8, 45–71 Corsi, M., H Pham, and W Runggaldier (2006): “Numerical approximation by quantization of control problem in finance under partial observations”, to appear in Mathematical modelling and numerical methods in finance, edited by A Bensoussan and Q Zhang, special volume of Handbook of numerical analysis Di Masi, G B and W J Runggaldier (1987): An Approach to Discrete-Time Stochastic Control Problems under Partial Observation, SIAM J Control & Optimiz 25, pp 38 - 48 Graf, S and H Luschgy (2000): Foundations of quantization for random vectors, Lecture Notes in Mathematics n0 1730, Springer, Berlin, 230 pp Kushner, H J and P Dupuis (1992): Numerical Methods for Stochastic Control Problems in Continuous Time, Springer, New York Luenberger, D (1984): Linear and nonlinear programming, Addison-Wesley 10 Pag`es, G., H Pham, and J Printems (2004): “Optimal quantization methods and applications to numerical problems in finance”, Handbook of computational and numerical methods in finance, ed S Rachev, Birkhauser 11 Pham, H., W Runggaldier, and A Sellami (2005): “Approximation by quanti- ... Symposium STOCHASTIC PROCESSES AND APPLICATIONS TO MATHEMATICAL FINANCE This page intentionally left blank Proceedings of the 6the Ritsumeikan International Symposium STOCHASTIC PROCESSES AND APPLICATIONS. .. catalogue record for this book is available from the British Library STOCHASTIC PROCESSES AND APPLICATIONS TO MATHEMATICAL FINANCE Proceedings of the 6th Ritsumeikan International Symposium Copyright... 6in PREF-06-2+ PREFACE The 6th Ritsumeikan international conference on Stochastic Processes and Applications to Mathematical Finance was held at Biwako-Kusatsu Campus (BKC) of Ritsumeikan University,