﻿ 随机波动模型的首中时问题
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 浙江大学学报(理学版)  2017, Vol. 44 Issue (3): 296-301  DOI:10.3785/j.issn.1008-9497.2017.03.009 0

引用本文 [复制中英文]

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ZHANG Miao, LIU Hui, ZHANG Feilong. The first hitting time of stochastic volatility models[J]. Journal of Zhejiang University(Science Edition), 2017, 44(3): 296-301. DOI: 10.3785/j.issn.1008-9497.2017.03.009.
[复制英文]

文章历史

1. 西安电子科技大学 数学与统计学院，陕西 西安 710126;
2. 北京大学 地球与空间科学学院，北京 100871;
3. 西安电子科技大学 物理与光电工程学院，陕西 西安 710126

The first hitting time of stochastic volatility models
ZHANG Miao1 , LIU Hui2 , ZHANG Feilong3
1. School of Mathematics and Statistics, Xidian University, Xi'an 710126, China;
2. School of Earth and Space Scienecs, Peking University, Beijing 100871, China;
3. School of Physics and Optoelectronic Engineering, Xidian University, Xi'an 710126, China
Abstract: This paper explores the first passage times of stochastic volatility CEV model. We mainly solve the joint Laplace transform of the first hitting time and volatility. Firstly, we use the It${\rm{\hat o}}$ formula to construct the martingale which can convert the problem into the process of solving a differential equation. Then, we introduce an appropriate second order variable coefficient ordinary differential equation, after a change of variable, it is turned to the Whittaker's equation. It's not difficult to get the general solution of Whittaker's equation. Thus, the explicit expressions for the joint Laplace transformation of the first passage times of stochastic volatility CEV model can be derived. Finally, selecting the parameters γ be 0, 1/2 and 1, let the asset price process covers the O-U process, geometric Brownian motion and square root process. Under different parameters, we obtain explicit expression of the joint Laplace transformation function, and use Matlab to draw the corresponding diagram and analyze the trend of graph.
Key words: stochastic volatility CEV model    first passage times    martingale method    joint Laplace transforms    Whittaker's equation

1 随机波动CEV模型

 ${\rm{d}}{X_t} = {\mu _X}\left( {{X_t}} \right){\rm{d}}t + {\sigma _X}\left( {{X_t},{Y_t}} \right){\rm{d}}W_t^X,$ (1)

 ${\rm{d}}{Y_t} = {\mu _Y}\left( {{Y_t}} \right){\rm{d}}t + {\sigma _Y}\left( {{Y_t}} \right){\rm{d}}W_t^Y.$ (2)

 ${\rm{d}}{Y_t} = \left( {{c_Y} - {b_Y}{Y_t}} \right){\rm{d}}t + \varphi \left( {{Y_t}} \right){\rm{d}}W_t^Y,$ (3)

 $\begin{array}{l} {\rm{d}}{X_t} = {a_X}X_t^\gamma \sqrt {{Y_t}} {\rm{d}}W_t^Y,\\ {\rm{d}}{Y_t} = \left( {{c_Y} - {b_Y}{Y_t}} \right){\rm{d}}t + {a_Y}\sqrt {{Y_t}} {\rm{d}}W_t^Y, \end{array}$ (4)

2 首中时和随机波动因子的联合拉普拉斯变换

 ${\tau _l} = \inf \left( {t \ge 0;{X_{{\tau _l}}} = l} \right),$ (5)

 $\varphi \left( {l;x,y} \right) = {\mathit{\boldsymbol{E}}_{x,y}}\left[ {\exp \left( { - \alpha \tau l - \beta {T_{{\tau _l}}}} \right)} \right],$ (6)

 ${\mathit{\boldsymbol{E}}_{x,y}}\left[ \cdot \right] = \mathit{\boldsymbol{E}}\left[ { \cdot \left| {{X_0} = x,{Y_0} = y} \right.} \right].$

 $\begin{array}{*{20}{c}} {\frac{1}{2}a_X^2f''\left( x \right){x^{2\gamma }} - \rho {a_X}{a_Y}\beta {x^\gamma }f'\left( x \right) + }\\ {\left( {{b_Y}\beta + \frac{1}{2}a_Y^2{\beta ^2}} \right)f\left( x \right) = 0,} \end{array}$ (7)

 $\varphi \left( {l;x,y} \right) = {\mathit{\boldsymbol{E}}_{x,y}}\left[ {\exp \left( { - \alpha \tau l - \beta {T_{{\tau _l}}}} \right)} \right] = \frac{{{{\rm{e}}^{ - \beta y}}f\left( x \right)}}{{f\left( l \right)}}.$ (8)

 $\begin{array}{*{20}{l}} \begin{array}{l} {{\rm{e}}^{ - \beta {Y_t}}}f\left( {{X_t}} \right) = {{\rm{e}}^{ - \beta {Y_0}}}f\left( {{X_0}} \right) + \int_0^t {{\rm{d}}\left( {{{\rm{e}}^{ - \beta {Y_s}}}f\left( {{X_s}} \right)} \right)} {\rm{ = }}\\ {{\rm{e}}^{ - \beta {Y_0}}}f\left( {{X_0}} \right) + \int_0^t {{{\rm{e}}^{ - \beta {Y_s}}}} {\rm{d}}\left( {f\left( {{X_s}} \right)} \right) + \end{array}\\ {\int_0^t {f\left( {{X_s}} \right){\rm{d}}\left( {{{\rm{e}}^{ - \beta {Y_s}}}} \right)} + \int_0^t {{\rm{d}}\left( {f\left( {{X_s}} \right)} \right){\rm{d}}\left( {{{\rm{e}}^{ - \beta {Y_s}}}} \right)} = }\\ {{{\rm{e}}^{ - \beta {Y_0}}}f\left( {{X_0}} \right) + {a_X}\int_0^t {{{\rm{e}}^{ - \beta {Y_s}}}f'\left( {{X_s}} \right)X_s^\gamma \sqrt {{Y_s}} {\rm{d}}W_s^X} + }\\ {\frac{1}{2}a_X^2\int_0^t {{{\rm{e}}^{ - \beta {Y_s}}}f''\left( {{X_s}} \right)X_s^{2\gamma }{Y_s}{\rm{d}}s} - }\\ {\beta \int_0^t {{{\rm{e}}^{ - \beta {Y_s}}}f\left( {{X_s}} \right)\left( {{c_Y} - {b_Y}{T_s}} \right){\rm{d}}s} - }\\ {\beta {a_Y}\int_0^t {{{\rm{e}}^{ - \beta {Y_s}}}f\left( {{X_s}} \right)\sqrt {{Y_s}} {\rm{d}}W_s^X} + }\\ {\frac{1}{2}{\beta ^2}a_Y^2\int_0^t {{{\rm{e}}^{ - \beta {Y_s}}}f\left( {{X_s}} \right){Y_s}{\rm{d}}s} - }\\ {\beta \rho {a_X}{a_Y}\int_0^t {{{\rm{e}}^{ - \beta {Y_s}}}f'\left( {{X_s}} \right)X_s^\gamma {Y_s}{\rm{d}}s} .} \end{array}$

 $\begin{array}{l} {M_t} = {a_X}\int_0^t {{{\rm{e}}^{ - \beta {Y_s}}}f'\left( {{X_s}} \right)X_s^\gamma \sqrt {{Y_s}} {\rm{d}}W_s^X} - \\ \;\;\;\;\;\;\;\;\beta {a_Y}\int_0^t {{{\rm{e}}^{ - \beta {Y_s}}}f\left( {{X_s}} \right)\sqrt {{Y_s}} {\rm{d}}W_s^X} ,t \ge 0. \end{array}$

 $\begin{array}{l} {{\rm{e}}^{ - \alpha t - \beta {Y_t}}}f\left( {{X_t}} \right) = {{\rm{e}}^{ - \beta {Y_0}}}f\left( {{X_0}} \right) + \int_0^t {{\rm{d}}\left( {{{\rm{e}}^{ - \alpha s - \beta {Y_s}}}f\left( {{X_s}} \right)} \right)} = \\ \;\;\;\;\;\;\;\;\;{{\rm{e}}^{ - \beta {Y_0}}}f\left( {{X_0}} \right) + \int_0^t {{{\rm{e}}^{ - \alpha s}}{\rm{d}}\left( {{{\rm{e}}^{ - \beta {Y_s}}}f\left( {{X_s}} \right)} \right)} + \\ \;\;\;\;\;\;\;\;\;\int_0^t {{{\rm{e}}^{ - \beta {Y_s}}}f\left( {{X_s}} \right){\rm{d}}\left( {{{\rm{e}}^{ - \alpha s}}} \right)} = \\ \;\;\;\;\;\;\;\;\;{{\rm{e}}^{ - \beta {Y_0}}}f\left( {{X_0}} \right) + \int_0^t {{{\rm{e}}^{ - \alpha s}}{\rm{d}}{M_s}} - \\ \;\;\;\;\;\;\;\;\;\alpha \int_0^t {{{\rm{e}}^{ - \alpha s - \beta {Y_s}}}f\left( {{X_s}} \right){\rm{d}}s} + \\ \;\;\;\;\;\;\;\;\;\frac{1}{2}a_X^2\int_0^t {{{\rm{e}}^{ - \alpha s - \beta {Y_s}}}f''\left( {{X_s}} \right)X_s^{2\gamma }{Y_s}{\rm{d}}s} - \\ \;\;\;\;\;\;\;\;\;\beta \int_0^t {{{\rm{e}}^{ - \alpha s - \beta {Y_s}}}f\left( {{X_s}} \right)\left( {{c_Y} - {b_Y}{Y_s}} \right){\rm{d}}s} + \\ \;\;\;\;\;\;\;\;\;\frac{1}{2}{\beta ^2}a_Y^2\int_0^t {{{\rm{e}}^{ - \alpha s - \beta {Y_s}}}f\left( {{X_s}} \right){Y_s}{\rm{d}}s} - \\ \;\;\;\;\;\;\;\;\;\beta \rho {a_X}{a_Y}\int_0^t {{{\rm{e}}^{ - \alpha s - \beta {Y_s}}}f'\left( {{X_s}} \right)X_s^\gamma {Y_s}{\rm{d}}s} . \end{array}$

 $\begin{array}{l} \frac{1}{2}a_X^2f''\left( x \right){x^{2\gamma }}y - \beta f\left( x \right)\left( {{c_Y} - {b_Y}y} \right) + \frac{1}{2}a_Y^2{\beta ^2}f\left( x \right)y - \\ \;\;\;\;\;\;\;\rho {a_X}{a_Y}\beta {x^\gamma }f'\left( x \right)y - \alpha f\left( x \right) = \\ \;\;\;\;\;\;\;\left[ {\frac{1}{2}a_X^2f''\left( x \right){x^{2\gamma }} - \rho {a_X}{a_Y}\beta {x^\gamma }f'\left( x \right) + } \right.\\ \;\;\;\;\;\;\;\left. {\left( {{b_Y}\beta + \frac{1}{2}a_\mathit{Y}^2{\beta ^2}} \right)f\left( x \right)} \right]y - \left( {\beta {c_Y} + \alpha } \right)f\left( x \right) = 0. \end{array}$

 ${{\rm{e}}^{ - \alpha t - \beta {Y_t}}}f\left( {{X_t}} \right) = {{\rm{e}}^{ - \beta {Y_0}}}f\left( {{X_0}} \right) + \int_0^t {{{\rm{e}}^{ - \alpha s}}{\rm{d}}{M_s}} .$

 ${{\rm{e}}^{ - \alpha \left( {t \wedge {\tau _l}} \right) - \beta {Y_{t \wedge {\tau _l}}}}}f\left( {{X_{t \wedge {\tau _l}}}} \right) = {{\rm{e}}^{ - \beta {Y_0}}}f\left( {{X_0}} \right) + \int_0^{t \wedge {\tau _l}} {{{\rm{e}}^{ - \alpha s}}{\rm{d}}{M_s}} .$

 ${\mathit{\boldsymbol{E}}_{x,y}} = \left[ {{{\rm{e}}^{ - \alpha {\tau _l} - \beta {Y_{{\tau _l}}}}}f\left( l \right)} \right] = {{\rm{e}}^{ - \beta y}}f\left( x \right).$

 ${\mathit{\boldsymbol{E}}_{x,y}}\left[ {{{\rm{e}}^{ - \alpha \tau \_\beta {Y_{{\tau _l}}}}}} \right] = \frac{{{{\rm{e}}^{ - \beta y}}f\left( x \right)}}{{f\left( l \right)}}.$

 $\begin{array}{l} z\left( x \right) = \sqrt {4\left[ {\left( {{\rho ^2} - 1} \right)a_Y^2{\beta ^2} - 2{b_Y}\beta } \right]/a_X^2{{\left( {1 - \gamma } \right)}^2}} {x^{1 - \gamma }} = \\ \;\;\;\;\;\;\;\;\;A{x^{1 - \gamma }},x \in {\bf{R}}. \end{array}$

 $\frac{{{\rm{d}}f}}{{{\rm{d}}x}} = \frac{{{\rm{d}}f}}{{{\rm{d}}z}}\frac{{{\rm{d}}z}}{{{\rm{d}}x}},\frac{{{{\rm{d}}^2}f}}{{{\rm{d}}{x^2}}} = \frac{{{{\rm{d}}^2}f}}{{{\rm{d}}{z^2}}}{\left( {\frac{{{\rm{d}}z}}{{{\rm{d}}x}}} \right)^2} + \frac{{{\rm{d}}f}}{{{\rm{d}}z}}\frac{{{{\rm{d}}^2}z}}{{{\rm{d}}{x^2}}},$

 $\begin{array}{*{20}{c}} {z\frac{{{{\rm{d}}^2}f}}{{{\rm{d}}{z^2}}} - \left[ {\frac{\gamma }{{1 - \gamma }} + \frac{{2\rho {a_Y}\beta }}{{{a_X}\left( {1 - \gamma } \right)A}}z} \right]\frac{{{\rm{d}}f}}{{{\rm{d}}z}} + }\\ {\frac{{a_Y^2{\beta ^2} + 2{b_Y}\beta }}{{a_X^2{{\left( {1 - \gamma } \right)}^2}{A^2}}}zf\left( z \right) = 0.} \end{array}$

$f(z) = {z^{\frac{\gamma }{{^2(1 - \gamma )}}}}{\rm{e}^{\frac{{\rho aY\beta }}{{^a{X^{(1 - \gamma )A}}}}z}}g(z)$，得

 $\frac{{{{\rm{d}}^2}g}}{{{\rm{d}}{z^2}}} + \left[ {\frac{{1/4 - {m^2}}}{{{z^2}}} + \frac{k}{z} - \frac{1}{4}} \right]g\left( z \right) = 0,$ (9)

 $\begin{array}{*{20}{c}} {m = \frac{1}{{2\left( {1 - \gamma } \right)}},}\\ {k = - \frac{{\gamma \rho {a_Y}\beta }}{{2\left( {1 - \gamma } \right)}}{{\left[ {\left( {{\rho ^2} - 1} \right)a_Y^2{\beta ^2} - 2{b_Y}\beta } \right]}^{ - \frac{1}{2}}}.} \end{array}$

 $\begin{array}{l} f\left( x \right) = {C_1}z{\left( x \right)^{\frac{\gamma }{{2\left( {1 - \gamma } \right)}}}}{\rm{e}}\left\{ {\frac{{\rho {a_Y}\beta z\left( x \right)}}{{\left[ {2\sqrt {\left( {{\rho ^2} - 1} \right)a_Y^2{\beta ^2} - 2{b_Y}\beta } } \right]}}} \right\}{\mathit{\boldsymbol{M}}_{k,m}}\left( {z\left( x \right)} \right) + \\ \;\;\;\;\;\;\;\;\;{C_2}z{\left( x \right)^{\frac{\gamma }{{2\left( {1 - \gamma } \right)}}}}{\rm{e}}\left\{ {\frac{{\rho {a_Y}\beta z\left( x \right)}}{{\left[ {2\sqrt {\left( {{\rho ^2} - 1} \right)a_Y^2{\beta ^2} - 2{b_Y}\beta } } \right]}}} \right\}{\mathit{\boldsymbol{W}}_{k,m}}\left( {z\left( x \right)} \right) = \\ \;\;\;\;\;\;\;\;\;{C_1}\varphi \left( x \right) + {C_2}\psi \left( x \right). \end{array}$

 $f\left( x \right) = {C_1}{x^{{\lambda _1}}} + {C_2}{x^{{\lambda _2}}},$

 $\varphi \left( {l;x,y} \right) = \frac{{{{\rm{e}}^{ - \beta y}}\left( {{C_1}\varphi \left( x \right) + {C_2}\psi \left( x \right)} \right)}}{{{C_1}\varphi \left( l \right) + {C_2}\psi \left( l \right)}},$ (10)

 $\begin{array}{l} \varphi \left( x \right) = z{\left( x \right)^{\frac{\gamma }{{2\left( {1 - \gamma } \right)}}}}{{\rm{e}}^{\frac{{\varepsilon z\left( x \right)}}{2}}}{\mathit{\boldsymbol{M}}_{k,m}}\left( {z\left( x \right)} \right),\\ \psi \left( x \right) = z{\left( x \right)^{\frac{\gamma }{{2\left( {1 - \gamma } \right)}}}}{{\rm{e}}^{\frac{{\varepsilon z\left( x \right)}}{2}}}{\mathit{\boldsymbol{W}}_{k,m}}\left( {z\left( x \right)} \right), \end{array}$

 $\begin{array}{l} {\mathit{\boldsymbol{M}}_{k,m}}\left( x \right) = {{\rm{e}}^{ - \frac{x}{2}}}{x^{m + \frac{1}{2}}}M\left( {m - k + \frac{1}{2},1 + 2m;x} \right),\\ {\mathit{\boldsymbol{W}}_{k,m}}\left( x \right) = {{\rm{e}}^{ - \frac{x}{2}}}{x^{m + \frac{1}{2}}}U\left( {m - k + \frac{1}{2},1 + 2m;x} \right). \end{array}$

 $\begin{array}{*{20}{c}} {\varepsilon = \frac{{\rho {a_Y}\beta }}{{\sqrt {\left( {{\rho ^2} - 1} \right)a_Y^2{\beta ^2} - 2{b_Y}\beta } }},\;\;\;\;m = \frac{1}{{2\left( {1 - \gamma } \right)}},}\\ {k = - \frac{{\gamma \rho {a_Y}\beta }}{{2\left( {1 - \gamma } \right)}}{{\left[ {\left( {{\rho ^2} - 1} \right)a_Y^2{\beta ^2} - 2{b_Y}\beta } \right]}^{ - \frac{1}{2}}}.} \end{array}$
3 推论和结果分析

 $\begin{array}{l} \varphi \left( {l;x,y} \right) = \\ \;\;\;\;\;\;\;\frac{{{{\rm{e}}^{ - \beta \gamma }}{{\rm{e}}^{\frac{{\varepsilon z\left( x \right)}}{2}}}\left[ {{C_1}{\mathit{\boldsymbol{M}}_{0,\frac{1}{2}}}\left( {z\left( x \right)} \right) + {C_2}{\mathit{\boldsymbol{W}}_{0,\frac{1}{2}}}\left( {z\left( x \right)} \right)} \right]}}{{{{\rm{e}}^{\frac{{\varepsilon z\left( l \right)}}{2}}}\left[ {{C_1}{\mathit{\boldsymbol{M}}_{0,\frac{1}{2}}}\left( {z\left( l \right)} \right) + {C_2}{\mathit{\boldsymbol{W}}_{0,\frac{1}{2}}}\left( {z\left( l \right)} \right)} \right]}}, \end{array}$

 $\varepsilon = \frac{{\rho {a_Y}\beta }}{{\sqrt {\left( {{\rho ^2} - 1} \right)a_Y^2{\beta ^2} - 2{b_Y}\beta } }}.$

 $\begin{array}{l} \varphi \left( {l;x,y} \right) = \\ \;\;\;\;\;\;\;\frac{{{{\rm{e}}^{ - \beta \gamma }}{\rm{z}}{{\left( x \right)}^{\frac{1}{2}}}{{\rm{e}}^{\frac{{\varepsilon z\left( x \right)}}{2}}}\left[ {{C_1}{\mathit{\boldsymbol{M}}_{k,1}}\left( {z\left( x \right)} \right) + {C_2}{\mathit{\boldsymbol{W}}_{k,1}}\left( {z\left( x \right)} \right)} \right]}}{{{\rm{z}}{{\left( l \right)}^{\frac{1}{2}}}{{\rm{e}}^{\frac{{\varepsilon z\left( l \right)}}{2}}}\left[ {{C_1}{\mathit{\boldsymbol{M}}_{k,1}}\left( {z\left( l \right)} \right) + {C_2}{\mathit{\boldsymbol{W}}_{k,1}}\left( {z\left( l \right)} \right)} \right]}}, \end{array}$

 $\begin{array}{*{20}{c}} {\varepsilon = \frac{{\rho {a_Y}\beta }}{{\sqrt {\left( {{\rho ^2} - 1} \right)a_Y^2{\beta ^2} - 2{b_Y}\beta } }}.}\\ {k = - \frac{{\rho {a_Y}\beta }}{2}{{\left[ {\left( {{\rho ^2} - 1} \right)a_Y^2{\beta ^2} - 2{b_Y}\beta } \right]}^{ - 1/2}}.} \end{array}$

 $\varphi \left( {l;x,y} \right) = \frac{{{{\rm{e}}^{ - \beta y}}\left( {{C_1}{x^{{\lambda _1}}} + {C_2}{x^{{\lambda _2}}}} \right)}}{{{C_1}{l^{{\lambda _1}}} + {C_2}{l^{{\lambda _2}}}}},$

 图 1 当γ=0，联合拉普拉斯变化函数(x, y)→φ(l; x, y) Fig. 1 The joint Laplace transform function (x, y)→φ(l; x, y) when γ=0

 图 2 当γ= $\frac{1}{2}$，联合拉普拉斯变化函数(x, y)→φ(l; x, y) Fig. 2 The joint Laplace transform function (x, y)→φ(l; x, y) when γ= $\frac{1}{2}$
 图 3 当γ=1，联合拉普拉斯变化函数(x, y)→φ(l; x, y) Fig. 3 The joint Laplace transform function (x, y)→φ(l; x, y) when γ=1
4 结论

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