山本, 有作

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Titleオプション価格評価のための高性能計算技術 : 高速多重極展開法と二重指数型数値積分公式の適用 (数値解析と新しい情報技術)Author(s)山本, 有作Citation数理解析研究所講究録 (2004), 1362: 113-123Issue Date2004-04URL http://hdl.handle.net/2433/25285RightType Departmental Bulletin PaperTextversionpublisherKyoto University113--Yusaku Yamamoto, Department of Computational Science & Engineering, Nagoya University1, ,, , , , 11000 ( ) 1 3 $*$ 1 110 () , ( 1 3 $t$ ),, ,, $t$ t $S_{t}$ , $T$ ,( ) $K$ , 0 $A$ $B$, $S_{T}>K$ $A$ $B$, $S_{t}$ , $S_{T}-K$ ,$s_{T}\leq K$ , $A$ , , $t$ , $S_{t}$$A$ ( ) $h_{t}$ (St) ,$h_{T}(S_{T})= \max(S_{T}-K, 0)$ (1), , $B$, $A$ $B$,,, $S_{t}$ ,$dSt=\mu \mathrm{S}tdt+\sigma StdWt$ , (2)= , , $W_{t}$, $\mu$ ( ) $\sigma$( 0 ) , $\supset-\nearrow\backslash$$Q_{0}(S_{0})=e^{-\mathrm{r}T}$E$0[h_{T}(S_{T})]$ (3)Black&Scholes [5] , I( ) , $e^{-rT}$ $T$0 , $E_{0}$0 ,$dS_{t}=rS_{t}dt+\sigma S_{t}dW_{t}$ (4)(3), $[9][15]$1362 2004 113-123114, (3) (1), = ,, , ,, , [11][4],[16] ,[7][8] , ,$[8]_{0}$22.1, $t$ , $S_{t}$$h_{t}$ (St)( $h_{t}(S_{t})= \max(S_{t}-K,$ $0$)), 0 ,$Q_{0}(S_{0})= \sup_{\tau}\{e^{-rr}E_{0}[h_{\tau}(S_{\tau})]\}$ (5)[9] , $E_{0}$0 , $\tau$( ) ,0$j\backslash \cdot\backslash \backslash \backslash \backslash$ ,, (5) $T$ ,, $T$,$Q_{T}(S_{T})=h_{T}(S_{T})$ (6), $t_{n+1}$ $Q_{t_{\iota+1}},(S_{t_{n+1}})$ ( $S_{t_{n+1}}$ ). 1 $t_{n}$ $Q_{t_{n}}$ (St ) $t_{n}$ ,2 ,( ) $h_{t_{\hslash}}(S_{t_{n}})$$t_{n+1}$. ( ,)$e^{-\mathrm{r}(t_{n+1}-t_{n})}E_{t_{n}}$ [ $Q_{t_{n+1}}$ (St + )][3] , $t_{n}$ 2 ,, , $t_{n}$ , $s_{t_{n}}$, ,$Q_{t_{n}}(S_{t_{n}})= \max\{h_{t_{n}}(S_{t_{\mathfrak{n}}}), e^{-\mathrm{r}(t_{n+1}-t_{n})}E_{t_{n}}[Q_{t_{n+1}}(S_{t_{n+1}})]\}$ (7)( [9] ) $\text{ }$ (7) , $t_{n}$$s_{t_{\mathrm{n}}}$ , $Q_{t_{n}}(S_{t_{n}})$ , $A^{1}F.\cdot J$ $t_{n}$ ,$t_{n-1},$ $t_{n-2},$ $\ldots$ , 0 $Q_{0}$ (S0), (7) , $s_{t_{n}}$ $Q_{t_{n}}(S_{t_{n}})$$E_{t_{n}}[Q_{t_{n+1}}(S_{t_{n+1}})]$ ,1152.2, [1][6], $(t, W)$ , $W_{t}$, $|$ $t=i\Delta$t $W=j\Delta$W , $1$ $t=(i+\mathfrak{y}\Delta t$ } $W$ $\{(j-b)\Delta W$,$(j-b+1)\Delta W,$$\ldots,$$(j+b)\Delta W\}g)$ $p_{-b},p_{-b+1},$ $\ldots,p$b $b=2$(5 ) 1$\ovalbox{\tt\small REJECT}$,$\}1$,$|$$-A^{u}$1: 5 $(b=2)$, $(i,j)$ , $t=(i+1)\Delta t$ $2b+1$ \vdash ^$E[Q_{t}:+ \iota|\mathrm{S}_{j,j}]=.\sum_{k=-b}^{b}p\iota.Q_{i+1_{:}j+k}$ (8)[1] $p_{k}$ , $2b$, ,$\{$1222$\backslash$$0$ 2 $\cdot\Delta$W2 2 $\cdot(2\Delta W)^{2}$ 2 $\cdot(b\Delta W)^{2}$$0$ 2 $\cdot\Delta$W4 2 $\cdot(2\Delta W)^{4}$ 2 $\cdot$ (b$\Delta$W)4... ... .$\cdot$. .$\cdot$.02 $\cdot\Delta$W$2b$ 2 $\cdot(2\Delta W)^{2b}$ 2 $\cdot(b\Delta W)^{2b}$ ,$\{$ $\mathrm{p}_{1}p_{0}^{\backslash }p_{b}p_{2}.\cdot.,$ $=($1$\Delta$t$3\Delta t^{2}$.$\cdot$.$\Leftrightarrow_{2b!}^{(2b!}(\Delta t)^{b}$ ,(9)$p_{-k}=p_{k}$ (10), $M$ $b$, ,$O$ (M$(-b\oplus 1)$ ) [13] ,, $O$ (M) ,$O$ (Mb) , $M$ $b$162.3, , $(t, W_{t})$ $N$ , $t_{n}$$i$ $s_{t,.,i}$. , $t_{n+1}$$E[Q_{t_{n+1}}(S_{t_{n+1}})|S_{t_{n\prime}i}]= \sum_{j=1}^{N}Q_{t_{n+1\prime}j}w_{t_{n},ij}$ $(i=1, . .. , N)$ (11)[6] $wt_{n},\dot{t}j$ , $s_{t_{n}}$ $s_{t_{n+\mathrm{l}}}$ $f_{t_{1}}^{S},(S_{t_{n+1}}|S_{t_{n}})$,$w_{t_{\hslash},\dot{\iota}j}= \frac{f_{t_{\mathfrak{n}}}^{S}(S_{t_{n+1},j}|S_{t_{n}.i})}{\sum_{\dot{\iota}’=1}^{b}f_{t_{\mathrm{r}\iota}}^{S}(S_{t_{n+1},j}|S_{t_{n},i’})}$. (12)[6], $N$ ,, (11) $O$ (N2) ,3, ,3.1 $b$, , $b$ 1\sim 3, , $b$ ( $b>1000$), $\Delta W$ 0 $b$ , (9), (10) $p_{k}$$\frac{1}{\sqrt{2\pi\Delta t}}\exp(-_{22}^{\underline{k}\Delta W}\frac{2}{t})$ [7] , (8)$E[Q_{t}:+1|S_{\dot{\iota},j}]= \frac{\Delta W}{\sqrt{2\pi\Delta t}}\sum_{k=-b}^{b}Q_{i+1.j+k}\exp\{-\frac{1}{2\Delta t}$ $(W_{j}-Wj+k)2\}$ . (13)(13) $O$ (M) $j$ , $O$ (Mb) ,,3.2, Greengard&Strain [11][4],$G(x_{1}.)= \sum_{j=1}^{N}q_{j}\exp$ $\{\frac{(x_{j}-y_{j}\rangle^{2}}{\delta}\}$ $(i=1,2, \ldots, N’)$ (14), $O$ (NN’)$O(N+N’)$117, . $x_{0},$ $y_{0}$ $i,$ $j$$|x_{i}.-x_{0}|<\delta^{1/2},$ $|y_{j}-y_{0}|<\delta^{1/2}$ , (14)[11]$e^{-(_{1}:-y_{j}}$ 2/6$= \sum_{\beta=0}^{\infty}\sum_{\alpha=0}^{\infty}\frac{1}{\beta!}\frac{1}{\alpha!}(\frac{y_{j}-y_{0}}{\sqrt{\delta}})^{\alpha}h_{\alpha+\beta}(\frac{x_{0}-y_{0}}{\sqrt{\delta}})(\frac{x_{i}-x_{0}}{\sqrt{\delta}})^{\beta}$ (15), $h_{\alpha}(x)$$h_{\alpha}(x)=(-1$Y $( \frac{d}{dx})^{\alpha}e^{-oe^{2}}$ (16)(15) , ,$e^{-}(x-y)2= \sum\frac{y^{\alpha}}{\alpha!}h_{a}(x)\infty$ (17)$a=0$(18), (15) , $\alpha=\beta=8$(15) \mbox{\boldmath $\alpha$}=\beta =\mbox{\boldmath $\alpha$}m , (14)$G(x_{\dot{\iota}})$ $\cong$ $\sum_{j=1}^{N}q_{j}\sum_{\beta=0}^{a_{\max}}\sum_{\alpha=0}^{\alpha_{\mathrm{m}\cdot \mathrm{r}}}\frac{1}{\beta!}\frac{1}{\alpha!}(\frac{y_{j}-y_{0}}{\sqrt{\delta}}.)^{\alpha}h_{\alpha+\beta}(\frac{x_{0}-y_{0}}{\sqrt{\delta}})(.\frac{x_{*}-x_{0}}{\sqrt{\delta}})^{\beta}$$=$ $\sum_{\beta=0}^{\alpha_{\mathrm{m}*\mathrm{x}}}\frac{1}{\beta!}(\frac{x_{\dot{\iota}}-x_{0}}{\sqrt{\delta}})^{\beta}$$\cross\{\sum_{\alpha=0}^{\alpha_{\mathrm{m}\mathrm{n}\mathrm{x}}}h_{\alpha+\beta}(\frac{x_{0}-y_{0}}{\sqrt{\delta}})\{\frac{1}{\alpha!}\sum_{j=1}^{N}q_{j}(\frac{y_{j}-y_{0}}{\sqrt{\delta}})^{\alpha}\}\}$, $G$ (xj) $(i=1,2, . . . , N’)$ 31. $A \text{ }\equiv\frac{1}{a!}\sum_{j=1}^{N}q_{j}(\frac{y\cdot-y0}{\sqrt{\delta}})^{\alpha}$ ( $\alpha=0,$$\ldots$ , \mbox{\boldmath $\alpha$}2. $B_{\beta} \equiv\sum_{\alpha}^{\alpha};A_{\alpha}$h +\beta $(_{\sqrt{\delta}}^{\underline{x}\underline{0-}L^{\mathrm{Q}}})$ $(\beta=0, \ldots, \alpha_{111\mathrm{a}\mathrm{x}})$3. $G(x_{i})= \sum_{\beta=0}^{\alpha_{\mathrm{m}*\mathrm{x}}}B$. $\beta^{\frac{1}{\beta!}}(\frac{\varphi-}{\sqrt{\delta}}\mathrm{A}\mu)^{\beta}$ ($i=1,$ $\ldots$ , N$\alpha_{\max}$ , 1, 3 $O$ (N), $O$ (N’) ,2 $N$ $N’$ ,(14) $O(N+N’)$, $|x_{i}-x_{0}|<\delta^{1/2}$ . $|$yj $-y_{0}|<\delta^{1/2}$ $i,$ $j$ $\mathrm{A}\mathrm{a}$ $x_{0},$ $yj$, 2 $x,$ $y$ $\delta^{1/2}$ ,3 , $O(N+N’)$[11], ,, ,[11][4](13) , $O$ (Mb)$O(M+b)$ ,3.3, = ,= , [15]1182: $\delta^{1/2}$, ( ) $S_{t}$$S_{t+\Delta t}=St\exp\{$ $(r- \frac{1}{2}\sigma^{2})\Delta t+\sigma\sqrt{\Delta t}z0+\sum_{i=1}^{N_{t}^{P}(\Delta t)}(\delta z_{\dot{\mathrm{t}}}-\kappa)\}$ (19), $N_{t}^{P}$ (\Delta t) $t$ $t+\Delta t$ , $\lambda$, $z_{i}$ $(i=0,1, . . .)$ $N(0,1)$ , $\kappa$ $\delta$ 1(19) $\{\}$ 2 =(4) , $\sum_{i=1}^{N_{t}^{P}(\Delta t)}(\delta z_{-}-\kappa)$, Amin [2] , , (8)$p_{k}$ ,[7], $Kou$ [14],$\sigma$ , Heston [12],[10], ,[7]3.4, 2, $w_{t_{n},.j}$. (12) ,St ’ $i$ $yt_{n},t$ ,$w_{t_{n}},ij= \frac{1}{\sqrt{2\pi\Delta t}}\exp\{-\frac{1}{2\Delta t}(y_{t_{n+1},j}-y_{t_{n},\mathrm{i}})^{2}\}c_{j}$ (20), $c_{j}$ $j$ [7](11) ,1184(7) $E$ [Q $t_{n+1}(S_{t_{n+1}})$ ] , ,, –$W_{t}$ ,,, $E[Q_{t_{n+1}}(S_{t_{n+1}})]$ ,, [16][8] , ( ) () ,, (7) $t_{n+1}$ , $Q_{t+1}(S_{t+1})$ $\max$, 1 ,, ,$h_{t_{\mathfrak{n}+1}}(St_{n+1})=Et_{n+1}[Qt_{n+2}(St_{n+2})]$ (21)2 ( 11 ) : [8], $N$$t_{n}$$N$ $O$ (N) ,$O$ (N2) , , $o(N)$, $O$ (N) , $N$55.1, $\mathrm{C}$ ,Pentium $\mathrm{I}\mathrm{I}$ (266MHz) $\sigma)\mathrm{P}\mathrm{C}$ 4 $\mathrm{a}$ , Red-Hat Linux GNU $\mathrm{C}++$= .= ,, (1) ( 2 ,). (2) (3.2 ). (3)(4 ) 3$S_{0}=100$ , $T=0.5$ ( ), $r=0.05$ (/ ) $j$$\sigma=0.2$ (/ ), $q=0.07$ (/ ) $n=10$, $K$ 90 110 5$Qo$ 1 $b=409,6$00, 6 [7]1 =$K$ 90 95 100 105 110$Q\mathrm{o}$ 10.73001013 7.32288562 4.75727741 2.94105489 1.73255637120( ) ,3 [8] , (1), (2), (3) binomial,mulfinomial-FGT, DE-FGT ,2 5 ,, (2) (1) |’ $\text{ },$ (3), , (2), (3),, , (1)Amin [2], (2) Amin (3.3 ) :(3)3$S_{0}=40$ , $T=1.0$ ( ) 1 $r=0.08$ (/ ) -$\sigma=\sqrt{0.05}$ (/ ), $q=0$ (/ ) $n=10$ ,$\lambda=5.0$ (/ ) $l\delta=\sqrt{0.05},$ $\kappa=-\frac{1}{2}\delta^{2}$ , $K$ 3050 5 $Q_{0}$ 2 [8]4 [8] , (1), (2), (3) Amin,Amin-FGT, DE-FGT =, (2) (1)$\mathrm{y}\text{ }$.,(3) (2) ,$\mathrm{F}x$ tm$p^{\infty}\_{\Rightarrow}\omega\not\in|$.$- 9-8\triangleleft-5- 4-2\ovalbox{\tt\small REJECT}-a-\mathrm{u}\mathrm{l}\mathrm{h}\mathrm{n}\mathrm{m}\mathrm{a}\mathrm{l}arrow-\mathrm{m}\mathrm{l}l$$\mathrm{x}$ cun $\mathrm{m}$ ec$\mapsto_{1}^{1}\infty \mathfrak{m}|$$\mathrm{p}^{\infty}\prec=\Phi\not\in \mathrm{q})$$-\theta- 6*\ovalbox{\tt\small REJECT}_{\wedge\sim\cdot\mapsto}^{\mathrm{A}\mathrm{m}\mathrm{m}\sigma \mathrm{r}_{\vee\bigvee_{\mathfrak{l}}}}-\triangleleft 441_{\vee \mathrm{w}\wedge}-arrow \mathrm{A}\mathrm{m}1\mathrm{n}H\mathrm{G}\mathrm{T}\mathrm{s}_{1}\mathrm{a}1||e_{\mathrm{I}}\mathrm{s}_{1}|\mathrm{t}|\mathrm{I}\iota_{1}1|l|,.$’3: = $/\backslash \backslash ^{\backslash }$ $\backslash \backslash \backslash$ 4:1215.2, ,$=$ $S_{0}=100$ , $T=3.0$ ( ), $r=0.05$ (/ )$\sigma=0.20$ (/ ) $q=\mathrm{O}\mathrm{J}\mathrm{O}$ $(/\dotplus/)$ $K=100$, $n=2$,3 , 5 [7]. ,direct , FGT10( )00100001000100$\hat{.\not\in\dot{\S}v.}$1010.10.011NO 3000 $10\alpha n$ 30000. $1\mathrm{B}+0S$monber of sample paths5:122, , 6, ,$\overline{\mathrm{a}}$ $10,000$ 1000 , 5, $N$ $O$ (N2), $o$ (N) $|arrow(\llcorner$.6, ,,, $H$ ,$S_{t}$ $H$ ( $H$) , $\text{ }\grave{\vee}$ , $[0, T]$ $s_{t}$ () $S_{\max}(S_{\mathrm{n}1\mathrm{i}\mathrm{n}})$ , $T$ $S_{\max}$ ( $S_{\min}$) ,, ( ) ,,, ,,, ,[8], \rightarrow ,[, [17]7,,,, Kou [14]Heston [12], [10] ,,Mark Broadie , $\text{ }$, ,( )123[1] J. Alford and N. Webber: Very High Order Lattice Methods for Orte Factor Models, Jam. 2001.[2] K. Amin: Jump Diffusion Option Valuation in Discrete Time, Journal of Finance, Vol. 48, No. 5,pp. 1833-1863 (1993).[3] L. Andersen and M. Broadie: Practical Primal-Dual Simulation Algorithms for Pricing Multidimen-sional American Options, Working paper, Columbia University, March 2001.[4] B. Baxter and G. Roussos: A New Error Estimate of the Fast Gauss Transform, SIAM Journal onScientific Computing, Vol. 24, No. 1, pp. 257-259 (2002).[5] F. Black and M. Scholes: The Pricing of Options and Corporate Liabilities, Journal of PoliticalEconomy, Vol. 81, pp. 637-654 (1973).[6] M. Broadie and P. Glasserman: A Stochastic Mesh Method for Pricing High-Dimensional AmericanOptions, Working paper, Columbia University, 1997.-[7] M. Broadie and Y. Yamamoto: Application of the Fast Gauss Transform to option Pricing, Man-agement Science, Vol. 49, No. 8, pp. 1071-1088 (2003).[8] M. Broadie and Y. 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Merton: Continuous-Time Finance, Basil Blackwell, 1992.[16] H. Takahashi and M. Mori: Double Exponential FormuIas for Numerical Integration, Publ. RIMSKyoto Univ., Vol. 9, pp. 721-741 (1974).[17] , : ,ACS .

オプション価格評価のための高性能計算技術 : 高速多重極展開法と二重指数型数値積分公式の適用 (数値解析と新しい情報技術)

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オプション価格評価のための高性能計算技術 : 高速多重極展開法と二重指数型数値積分公式の適用 (数値解析と新しい情報技術)

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