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
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
为了有效规避期权市场风险,BLACK等[1]提出了Black-Scholes期权定价模型(也称B-S模型).经典的B-S模型建立在波动率为常数这一假设基础之上,然而,大量研究表明,隐含波动率常呈现“微笑”的特征,这与B-S模型的基本假设相矛盾.为此,提出了随机波动模型CanKaoWenXian_4.本文讨论的随机波动CEV(亦称常弹性方差)模型也是随机波动模型的一种,可用2个扩散方程描述,即资产价格过程和波动过程[5],其中波动过程是平方根扩散过程,资产价格过程则涵盖了Ornstein-Uhlenbec过程(也称O-U过程)、几何布朗运动、平方根过程等几种常见的扩散过程.
近年来,首中时问题被广泛研究,MARIO[6]总结和推广了CARLSUND[7]的结论,得到布朗运动首中时的生成函数.文献[8-11]对O-U过程和反射O-U过程的首中时问题进行了研究.同时,文献[12]讨论了反射CEV模型的首中时问题,利用鞅方法计算得到首中时的拉普拉斯变换.以上研究均只针对单变量扩散过程,对随机波动CEV模型的双变量扩散过程首中时的研究却非常少.
本文借鉴单变量扩散过程首中时问题的求解方法——鞅方法[13-16],在βcY+α=0这一假定条件下,将随机波动模型的首中时问题转化为求解一类变系数二阶常微分方程,通过变量代换[17-19],将此方程转化为Whittaker方程,最终计算首中时和波动率的联合拉普拉斯变换.
下文安排如下:首先,简要介绍随机波动CEV模型;其次,求解首中时和波动率的联合拉普拉斯变换;接着,讨论γ为0,
尽管B-S期权定价模型的改进方法众多,但因随机波动模型独具优势,能够很好地描述时变波动,被广泛应用于建模利率期限结构和期权定价研究.
通常,随机波动模型描述成如下随机微分方程:
$ {\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) |
其中,Y=(Yt; t≥0) 为随机波动因子,满足:
$ {\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) |
随机微分方程(1)、(2) 分别代表随机波动模型的资产价格过程和波动过程.其中,布朗运动WtX和WtY的相关系数为ρ,ρ∈[-1,1].同时,假定上述随机微分方程的系数函数μX,μY,σX和σY是充分光滑的,从而使随机微分方程(1) 和(2) 有唯一强解(X, Y).
在常方差弹性过程中,方程(2) 应为:
$ {\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) |
其中,φ(x)=aYxν,0≤ν≤1,系数aY为波动率的波动,ν为方差弹性.当ν=0.5时,为HESTON提出的均值回归过程;当ν=1时,为GARCH扩散过程.
本文主要考虑如下形式的随机波动CEV模型:
$ \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) |
其中,γ≥0, aX > 0, cY > 0, bY≥0, aY > 0.
2 首中时和随机波动因子的联合拉普拉斯变换本节的目标是得到随机变量(Yτl, τl)的联合拉普拉斯变换表达式.对一个已知的首中阀值l∈R,定义随机波动模型的首中时如下:
$ {\tau _l} = \inf \left( {t \ge 0;{X_{{\tau _l}}} = l} \right), $ | (5) |
特别地,取infφ=∞.
通常情况下,由于很难得到随机变量(Yτl, τl)的联合分布表达式,因此可用联合拉普拉斯变换来代替.随机变量(Yτl, τl)的联合拉普拉斯变换定义如下:
$ \varphi \left( {l;x,y} \right) = {\mathit{\boldsymbol{E}}_{x,y}}\left[ {\exp \left( { - \alpha \tau l - \beta {T_{{\tau _l}}}} \right)} \right], $ | (6) |
其中,E是期望算子且
$ {\mathit{\boldsymbol{E}}_{x,y}}\left[ \cdot \right] = \mathit{\boldsymbol{E}}\left[ { \cdot \left| {{X_0} = x,{Y_0} = y} \right.} \right]. $ |
定理1 设非零函数f∈C2(R2),R为实数,满足方程:
$ \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) |
那么对任意满足条件βcY+α=0的(α, β),有
$ \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) |
证明 用鞅方法求解式(6) 的解析表达式.首先,对f(x)∈C2(R2)应用It
$ \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} $ |
定义:
$ \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} $ |
那么M=(Mt; t≥0) 是一个局部鞅.由分部积分公式,得:
$ \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} $ |
因为函数f(x)满足方程(7),并且βcY+α=0,则有
$ \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} $ |
这样
$ {{\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}} . $ |
因此,对任意t≥0,有
$ {{\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}} . $ |
注意到Xτl=l,P-a.s.,于是令t→∞,
$ {\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). $ |
可以证明
$ {\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)}}. $ |
此时,只要知道方程(7) 的通解f(x),就可以得到随机变量(Yτl, τl)对应的联合拉普拉斯变换,因此,本文接下来的工作重点就是求解方程(7).
用变量代换法求解常微分方程(7).
情况1 0≤γ < 1,首先定义
$ \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} $ |
因为
$ \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}}}, $ |
代入方程(7) 得
$ \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} $ |
令
$ \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) |
其中,
$ \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} $ |
易知,常微分方程(9) 是Whittaker方程,有2个线性无关的解Mk, m(z)和Wk, m(z)(称M(·)和W(·)是Whittaker函数),因此,方程(7) 的通解为
$ \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} $ |
情况2 γ=1,此时常微分方程(7) 是欧拉方程,其通解为
$ f\left( x \right) = {C_1}{x^{{\lambda _1}}} + {C_2}{x^{{\lambda _2}}}, $ |
其中λ1和λ2是常微分方程(7) 的2个不相等的特征值.
将求出的通解f(x)代入定理1的式(8),即有:
定理2 令(x, y)∈R×R+,R为实数,对满足条件βcY+α=0的(α, β),随机变量(Yτl, τl)对应的联合拉普拉斯变换为
$ \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) |
其中超几何函数φ(x)和ψ(x)定义为
$ \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} $ |
这里Mk, m(·)和Wk, m(·)称为Whittaker函数,由Kummer合流超几何函数M和U给出Whittaker函数的定义:
$ \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} $ |
同时,
$ \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} $ |
对定理2,参数γ分别取为0,
推论1 当γ=0时,令(x, y)∈R×R+,对任意满足条件βcY+α=0的(α, β),随机变量(Yτl, τl)对应的联合拉普拉斯变换为
$ \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} $ |
其中,
$ \varepsilon = \frac{{\rho {a_Y}\beta }}{{\sqrt {\left( {{\rho ^2} - 1} \right)a_Y^2{\beta ^2} - 2{b_Y}\beta } }}. $ |
推论2 当γ=
$ \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} $ |
其中,
$ \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} $ |
推论3 当γ=1,令(x, y)∈R×R+,对任意满足条件βcY+α=0的(α, β),随机变量(Yτl, τl)对应的联合拉普拉斯变换为
$ \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}}}}}, $ |
其中,λ1和λ2是常微分方程(7) 的2个不相等的特征值.
利用这3个推论,分别画出当资产价格过程是O-U过程、平方根过程以及几何布朗运动时,联合拉普拉斯变化函数(x, y)→φ(l; x, y)的图像,并分析其变化趋势.
这里,取系数aX=1, aY=1, β=0.5以及首中阀值l =1,首先画出当γ=0,ρ分别取-1,0,0.5和1时,联合拉普拉斯变化函数(x, y)→φ(l; x, y).
从图 1可看出相关程度ρ和(Xt, Yt)初值(x, y)的选取对φ值的影响.首先,在不同相关程度ρ下,x取值越大,y取值越小, φ值就越大.随着相关程度的降低,φ值整体呈下降趋势.并且在不同相关程度ρ下,函数(x, y)→φ(l; x, y)的整体变化趋势基本相同.
接着画出当γ=
从图 2和3中可以看出,当γ=
研究了一类波动率是平方根过程的随机波动CEV模型的首中时问题.利用鞅方法求解得到首中时和波动率的联合拉普拉斯变换表达式,并分析函数(x, y)→φ(l; x, y)在不同参数下的变化趋势.不足之处是,在给出拉普拉斯变换的显式表达式时,需有限定条件βcY+α=0.去除或弱化这一条件,使求解的拉普拉斯变换更为严格,这一问题有待进一步研究.
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