#### 期刊菜单

European Option Pricing and Hedging Strategy Based on the Finite Moment Log-Stable Process
DOI: 10.12677/AAM.2019.88174, PDF, HTML, XML, 下载: 1,034  浏览: 2,040  科研立项经费支持

Abstract: The empirical test suggests that the log-return series of stock price in US market reject the normal distribution and admit instead a subclass of the asymmetric distribution. Therefore, the European option pricing and variance optimal hedging strategy are investigated based on the finite moment log-stable process (FMLS) in this paper. Firstly, the pricing formula of European option is derived by using the principle of convolution pricing, and we design a mathematical experiment to test this formula. Secondly, according to the principle of option hedging, the variance optimal hedging of the European option obtained under the FMLS frame by solving a nonlinear optimization problem. Finally, the numerical results are simulated and we compare the results between the FMLS and Black-Scholes (BS) models to observe the ability of capture risk of FMLS model.

1. 引言

2. 欧式期权定价公式

${S}_{T}={S}_{t}\mathrm{exp}\left[\left(r-D\right)\tau \right]\mathrm{exp}\left(-\nu \tau +\sigma {L}_{\tau }^{\alpha ,-1}\right),$ (1)

$\begin{array}{c}{\Phi }_{FMLS}\left(\omega ;t,T\right)={\stackrel{˜}{E}}_{t}\left[\mathrm{exp}\left(i\omega \mathrm{ln}\left({S}_{T}/{S}_{t}\right)\right)\right]\\ =\mathrm{exp}\left[i\omega \left(r-\nu \right)\tau -{\left(i\omega \sigma \right)}^{\alpha }\tau \mathrm{sec}\frac{\alpha \text{π}}{2}\right],\end{array}$ (2)

${Y}_{\tau }=\nu \tau +\sigma {L}_{\tau }^{\alpha ,-1}，$ (3)

${S}_{T}=K{\text{e}}^{{X}_{T}},$ (4)

$C\left(t,{X}_{t}\right)$ 表示欧式看涨期权在时刻t的价值，根据欧式期权的定价原理可知

$\begin{array}{c}C\left(t,{X}_{t}\right)={\text{e}}^{-r\left(T-t\right)}{\stackrel{˜}{E}}_{t}\left[{\left({S}_{T}-K\right)}^{+}\right]\\ =K{\stackrel{˜}{E}}_{t}\left[{\left({\text{e}}^{{X}_{T}-r\tau }-{\text{e}}^{-r\tau }\right)}^{+}\right]\\ =K{\int }_{-\infty }^{\infty }{\left({\text{e}}^{y-r\tau }-{\text{e}}^{-r\tau }\right)}^{+}{f}_{{X}_{t}}\left(y\right)\text{d}y,\end{array}$ (5)

${\Phi }_{X}\left(\omega ;t,T\right)={\stackrel{˜}{E}}_{t}\left[\mathrm{exp}\left(i\omega {X}_{T}\right)\right]=\mathrm{exp}\left[i\omega \left(r\tau +{X}_{t}\right)\right]{\stackrel{˜}{E}}_{t}\left[\mathrm{exp}\left(i\omega {Y}_{\tau }\right)\right],$

${\Phi }_{X}\left(\omega ;\text{\hspace{0.17em}}t,\text{\hspace{0.17em}}T\right)=\mathrm{exp}\left[i\omega \left(r\tau +{X}_{t}\right)\right]\mathrm{exp}\left[i\omega \nu \tau -{\left(i\omega \sigma \right)}^{\alpha }\tau \mathrm{sec}\frac{\alpha \text{π}}{2}\right].$ (6)

$C\left(t,{X}_{t}\right)=\frac{K{\text{e}}^{-r\tau }}{\text{π}}{\int }_{0}^{\infty }R\left\{\frac{{\Phi }_{X}\left(\omega -i-i\stackrel{¯}{\alpha };t,T\right)}{\left(i\omega +1+\stackrel{¯}{\alpha }\right)\left(i\omega +\stackrel{¯}{\alpha }\right)}\right\}\text{d}\omega ,$ (7)

$p\left(x\right)={\text{e}}^{\left(1+\stackrel{¯}{\alpha }\right)x}{\left({\text{e}}^{x}-{\text{e}}^{-r\tau }\right)}^{+},$

$\begin{array}{c}{\left({\text{e}}^{x-r\tau }-{\text{e}}^{-r\tau }\right)}^{+}={\text{e}}^{\left(1+\stackrel{¯}{\alpha }\right)\left(\tau r-x\right)}{\text{e}}^{\left(1+\stackrel{¯}{\alpha }\right)\left(x-\tau r\right)}{\left({\text{e}}^{x-r\tau }-{\text{e}}^{-r\tau }\right)}^{+}\\ ={\text{e}}^{\left(1+\stackrel{¯}{\alpha }\right)\left(\tau r-x\right)}{\text{e}}^{\left(1+\stackrel{¯}{\alpha }\right)\left(x-\tau r\right)}{\left({\text{e}}^{-\left(r\tau -x\right)}-{\text{e}}^{-r\tau }\right)}^{+}\\ =p\left(r\tau -x\right){\text{e}}^{\left(1+\stackrel{¯}{\alpha }\right)\left(y-r\tau \right)},\end{array}$

$h\left(x\right)={\text{e}}^{\left(1+\stackrel{¯}{\alpha }\right)x}{f}_{{X}_{t}}\left(x\right),$

$\begin{array}{c}C\left(t,{X}_{t}\right)=K{\int }_{-\infty }^{\infty }p\left(r\tau -y\right){\text{e}}^{\left(1+\stackrel{¯}{\alpha }\right)\left(y-r\tau \right)}{f}_{{X}_{t}}\left(y\right)\text{d}y\\ =K{\text{e}}^{-r\tau \left(1+\stackrel{¯}{\alpha }\right)}{\int }_{-\infty }^{\infty }p\left(r\tau -y\right)h\left(y\right)\text{d}y\\ =K{\text{e}}^{-r\tau \left(1+\stackrel{¯}{\alpha }\right)}\left(p\ast h\right)\left(r\tau \right),\end{array}$

$F\left(p\ast h\right)=F\left(p\right)\ast F\left(h\right)$

$\begin{array}{l}F\left[\frac{C\left(t,{X}_{t}\right)}{K}{\text{e}}^{r\tau \left(1+\stackrel{¯}{\alpha }\right)}\right]\left(\omega \right)\\ =F\left(p\right)\left(\omega \right)F\left(h\right)\left(\omega \right)\\ ={\int }_{-\infty }^{\infty }{\text{e}}^{i\omega x}p\left(x\right)\text{d}x{\int }_{-\infty }^{\infty }{\text{e}}^{i\omega x}h\left(x\right)\text{d}x\\ ={\int }_{-\infty }^{\infty }{\text{e}}^{i\omega x}{\text{e}}^{\left(1+\stackrel{¯}{\alpha }\right)x}{\left({\text{e}}^{-x}-{\text{e}}^{-r\tau }\right)}^{+}\text{d}x{\int }_{-\infty }^{\infty }{\text{e}}^{i\omega x}{\text{e}}^{\left(1+\stackrel{¯}{\alpha }\right)x}{f}_{{X}_{t}}\left(x\right)\text{d}x\\ ={\int }_{-\infty }^{r\tau }{\text{e}}^{i\omega x}{\text{e}}^{\left(1+\stackrel{¯}{\alpha }\right)x}\left({\text{e}}^{-x}-{\text{e}}^{-r\tau }\right)\text{d}x{\int }_{-\infty }^{\infty }{\text{e}}^{i\omega x}{\text{e}}^{\left(1+\stackrel{¯}{\alpha }\right)x}{f}_{{X}_{t}}\left(x\right)\text{d}x\\ =\frac{{\text{e}}^{r\tau \left(1+\stackrel{¯}{\alpha }\right)x}{\Phi }_{X}\left(\omega -i-i\stackrel{¯}{\alpha };t,T\right)}{\left(i\omega +1+i\stackrel{¯}{\alpha }\right)\left(i\omega +i\stackrel{¯}{\alpha }\right)},\end{array}$

$\begin{array}{c}C\left(t,{X}_{t}\right)=\frac{K{\text{e}}^{-r\tau }}{\text{2π}}{\int }_{-\infty }^{\infty }\frac{{\Phi }_{X}\left(\omega -i-i\stackrel{¯}{\alpha };t,T\right)}{\left(i\omega +1+i\stackrel{¯}{\alpha }\right)\left(i\omega +i\stackrel{¯}{\alpha }\right)}\text{d}\omega \\ =\frac{K{\text{e}}^{-r\tau }}{\text{2π}}\left[{\int }_{0}^{\infty }\frac{{\Phi }_{X}\left(\omega -i-i\stackrel{¯}{\alpha };t,T\right)}{\left(i\omega +1+i\stackrel{¯}{\alpha }\right)\left(i\omega +i\stackrel{¯}{\alpha }\right)}\text{d}\omega +{\int }_{-\infty }^{0}\frac{{\Phi }_{X}\left(\omega -i-i\stackrel{¯}{\alpha };t,T\right)}{\left(i\omega +1+i\stackrel{¯}{\alpha }\right)\left(i\omega +i\stackrel{¯}{\alpha }\right)}\text{d}\omega \right]\\ =\frac{K{\text{e}}^{-r\tau }}{\text{2π}}\left[{\int }_{0}^{\infty }\frac{{\Phi }_{X}\left(\omega -i-i\stackrel{¯}{\alpha };t,T\right)}{\left(i\omega +1+i\stackrel{¯}{\alpha }\right)\left(i\omega +i\stackrel{¯}{\alpha }\right)}\text{d}\omega -{\int }_{\infty }^{0}\frac{{\Phi }_{X}\left(-\omega -i-i\stackrel{¯}{\alpha };t,T\right)}{\left(-i\omega +1+i\stackrel{¯}{\alpha }\right)\left(-i\omega +i\stackrel{¯}{\alpha }\right)}\text{d}\omega \right]\\ =\frac{K{\text{e}}^{-r\tau }}{\text{2π}}{\int }_{0}^{\infty }\left[\frac{{\Phi }_{X}\left(\omega -i-i\stackrel{¯}{\alpha };t,T\right)}{\left(i\omega +1+i\stackrel{¯}{\alpha }\right)\left(i\omega +i\stackrel{¯}{\alpha }\right)}+\frac{{\Phi }_{X}\left(-\omega -i-i\stackrel{¯}{\alpha };t,T\right)}{\left(-i\omega +1+i\stackrel{¯}{\alpha }\right)\left(-i\omega +i\stackrel{¯}{\alpha }\right)}\right]\text{d}\omega ,\end{array}$

$\begin{array}{c}C\left(t,{X}_{t}\right)=\frac{K{\text{e}}^{-r\tau }}{\text{2π}}{\int }_{0}^{\infty }\left[\frac{{\Phi }_{X}\left(\omega -i-i\stackrel{¯}{\alpha };t,T\right)}{\left(i\omega +1+i\stackrel{¯}{\alpha }\right)\left(i\omega +i\stackrel{¯}{\alpha }\right)}+\frac{{\Phi }_{X}\left(-\omega -i-i\stackrel{¯}{\alpha };t,T\right)}{\left(-i\omega +1+i\stackrel{¯}{\alpha }\right)\left(-i\omega +i\stackrel{¯}{\alpha }\right)}\right]\text{d}\omega \\ =\frac{K{\text{e}}^{-r\tau }}{\text{π}}{\int }_{0}^{\infty }R\left\{\frac{{\Phi }_{X}\left(\omega -i-i\stackrel{¯}{\alpha };t,T\right)}{\left(i\omega +1+i\stackrel{¯}{\alpha }\right)\left(i\omega +i\stackrel{¯}{\alpha }\right)}\right\}\text{d}\omega .\end{array}$

$C\left(t,{X}_{t}\right)+K{\text{e}}^{-r\tau }=P\left(t,{X}_{t}\right)+{S}_{t}.$

$P\left(t,{X}_{t}\right)=C\left(t,{X}_{t}\right)+K{\text{e}}^{-r\tau }-K\text{exp}\left({X}_{t}\right).$ (8)

$\begin{array}{c}|C\left(t,{X}_{t}\right)-\stackrel{^}{C}\left(t,{X}_{t}\right)|\le \frac{K{\text{e}}^{-r\tau }}{\text{π}}{\int }_{0}^{\infty }|R\left\{\frac{{\Phi }_{X}\left(\omega -i-i\stackrel{¯}{\alpha };t,T\right)}{\left(i\omega +1+i\stackrel{¯}{\alpha }\right)\left(i\omega +i\stackrel{¯}{\alpha }\right)}\right\}|\text{d}\omega \\ \le \frac{K{\text{e}}^{-r\tau }}{\text{π}}{\int }_{A}^{\infty }\frac{{\stackrel{˜}{E}}_{t}\left[{\text{e}}^{\left(1+\stackrel{¯}{\alpha }\right){Y}_{T}}\right]}{\sqrt{\left({\stackrel{¯}{\alpha }}^{2}+\stackrel{¯}{\alpha }-\omega \right)+\left(1+2{\stackrel{¯}{\alpha }}^{2}\right){\omega }^{2}}}\text{d}\omega \\ \le \frac{K{\text{e}}^{-r\tau }{\stackrel{˜}{E}}_{t}\left[{\text{e}}^{\left(1+\stackrel{¯}{\alpha }\right){Y}_{T}}\right]}{\text{π}}{\int }_{A}^{\infty }\frac{1}{{\omega }^{2}}\text{d}\omega \\ =\frac{K{\text{e}}^{-r\tau }{\stackrel{˜}{E}}_{t}\left[{\text{e}}^{\left(1+\stackrel{¯}{\alpha }\right){Y}_{T}}\right]}{A\text{π}}<\epsilon ,\end{array}$

$A>\frac{K{\text{e}}^{-r\tau }{\stackrel{˜}{E}}_{t}\left[{\text{e}}^{\left(1+\stackrel{¯}{\alpha }\right){Y}_{T}}\right]}{\epsilon \text{π}},$

$C\left(t,{X}_{t}\right)=\frac{K{\text{e}}^{-r\tau }}{\text{π}}{\int }_{0}^{A}R\left\{\frac{{\Phi }_{X}\left(\omega -i-i\stackrel{¯}{\alpha };t,T\right)}{\left(i\omega +1+\stackrel{¯}{\alpha }\right)\left(i\omega +\stackrel{¯}{\alpha }\right)}\right\}\text{d}\omega$ (9)

$C\left(t,{X}_{t}\right)\approx \frac{K{\text{e}}^{-r\tau }}{\text{π}}\underset{j=1}{\overset{M+1}{\sum }}R\left\{\frac{{\Phi }_{X}\left({\omega }_{j}-i-i\stackrel{¯}{\alpha };t,T\right)}{\left(i{\omega }_{j}+1+\stackrel{¯}{\alpha }\right)\left(i{\omega }_{j}+\stackrel{¯}{\alpha }\right)}\right\}\eta .$ (10)

3. 方差最优对冲策略

${M}_{0}={\stackrel{˜}{E}}_{0}{\left[{\stackrel{˜}{C}}_{0}+\underset{j=1}{\overset{N}{\sum }}{D}_{j}\Delta {\stackrel{˜}{S}}_{j}-{\stackrel{˜}{C}}_{T}\right]}^{2},$ (11)

${D}^{\ast }=\frac{{\stackrel{˜}{E}}_{j-1}\left[{\stackrel{˜}{C}}_{T}{\stackrel{˜}{S}}_{j}\right]}{{\stackrel{˜}{E}}_{j-1}\left[\Delta {\stackrel{˜}{S}}_{j}\right]}\approx \frac{{\text{e}}^{-r\left(T-{t}_{j}\right)}{v}_{j}-{\stackrel{˜}{C}}_{j-1}{\stackrel{˜}{S}}_{j-1}}{K{\Phi }_{X}\left(-2i;{t}_{j-1},{t}_{j}\right)-{\stackrel{˜}{S}}_{j-1}^{2}},$ (12)

${v}_{N}={K}^{2}\underset{i=1}{\overset{M}{\sum }}\left[\mathrm{exp}\left(2{z}_{i}+2{x}_{N-1}\right)-\mathrm{exp}\left({z}_{i}+{x}_{N-1}\right)f\left({z}_{i}\right)\right]\Delta$

${v}_{j}=\frac{{K}^{2}}{\text{π}}\underset{i=1}{\overset{N}{\sum }}\left[\mathrm{exp}\left({y}_{i}+{x}_{j}\right)C\left({y}_{i}+{x}_{j}\right)f\left({y}_{i}\right)\right]{\Delta }^{\prime },$

$C\left(t,{X}_{t}\right)\approx \underset{j=1}{\overset{M+1}{\sum }}R\left\{\frac{{\Phi }_{X}\left({\omega }_{j}-i-i\stackrel{¯}{\alpha };t,T\right)}{\left(i{\omega }_{j}+1+\stackrel{¯}{\alpha }\right)\left(i{\omega }_{j}+\stackrel{¯}{\alpha }\right)}\right\}\eta ,$

$f\left(•\right)$ 是随机变量 $X~{L}_{\alpha }\left(\left(r+\nu \right)\left({t}_{j}-{t}_{j-1}\right),\sigma {\left({t}_{j}-{t}_{j-1}\right)}^{\alpha },-1\right)$ 的概率密度函数。

${M}_{0}={\stackrel{˜}{E}}_{0}\left[{\stackrel{˜}{C}}_{T}^{2}\right]-{\stackrel{˜}{C}}_{0}+{\stackrel{˜}{E}}_{0}{\left[\underset{j=1}{\overset{N}{\sum }}{D}_{j}^{2}{\stackrel{˜}{E}}_{j-1}\left[\Delta {\stackrel{˜}{S}}_{j}^{2}\right]-2{D}_{j}{\stackrel{˜}{E}}_{j-1}\left[{\stackrel{˜}{C}}_{T}^{2}\Delta {\stackrel{˜}{S}}_{j}^{2}\right]\right]}^{2},$

${D}^{\ast }=\frac{{\stackrel{˜}{E}}_{j-1}\left[{\stackrel{˜}{C}}_{T}^{2}\Delta {\stackrel{˜}{S}}_{j}^{2}\right]}{{\stackrel{˜}{E}}_{j-1}\left[\Delta {\stackrel{˜}{S}}_{j}^{2}\right]}.$ (13)

$\begin{array}{c}{\stackrel{˜}{E}}_{j-1}\left[\Delta {\stackrel{˜}{S}}_{j}^{2}\right]={\stackrel{˜}{E}}_{j-1}\left[{\stackrel{˜}{S}}_{j}^{2}-{\stackrel{˜}{S}}_{j-1}^{2}\right]={\stackrel{˜}{E}}_{j-1}\left[{\stackrel{˜}{S}}_{j}^{2}\right]-{\stackrel{˜}{S}}_{j-1}^{2}\\ ={K}^{2}{\stackrel{˜}{E}}_{j-1}\left[\mathrm{exp}\left({X}_{j}^{2}\right)\right]-{\stackrel{˜}{S}}_{j-1}^{2}\\ ={K}^{2}{\Phi }_{X}\left(-2i;{t}_{j-1},{t}_{j}\right)-{\stackrel{˜}{S}}_{j-1}^{2}.\end{array}$

$\begin{array}{c}{\stackrel{˜}{E}}_{j-1}\left[{\stackrel{˜}{C}}_{T}^{2}\Delta {\stackrel{˜}{S}}_{j}^{2}\right]={\stackrel{˜}{E}}_{j-1}\left[{\stackrel{˜}{C}}_{T}\left({\stackrel{˜}{S}}_{j}-{\stackrel{˜}{S}}_{j-1}\right)\right]\\ ={\stackrel{˜}{E}}_{j-1}\left[{\stackrel{˜}{C}}_{T}{\stackrel{˜}{S}}_{j}\right]-{\stackrel{˜}{S}}_{j-1}{\stackrel{˜}{E}}_{j-1}\left[{\stackrel{˜}{C}}_{T}\right]\\ ={\stackrel{˜}{E}}_{j-1}\left[{\stackrel{˜}{C}}_{T}{\stackrel{˜}{S}}_{j}\right]-{\stackrel{˜}{S}}_{j-1}{\stackrel{˜}{C}}_{j-1},\end{array}$

${\stackrel{˜}{E}}_{j-1}\left[{\stackrel{˜}{C}}_{T}{\stackrel{˜}{S}}_{j}\right]=\mathrm{exp}\left[-r\left(T+{t}_{j}\right)\right]{\stackrel{˜}{E}}_{j-1}\left[{C}_{T}{S}_{j}\right],$

1) 先讨论 $j=N$ 的时候，亦即 ${t}_{j}=T$ ，那么此时

$\begin{array}{c}{v}_{N}={\stackrel{˜}{E}}_{N-1}\left[{C}_{T}{S}_{T}\right]={\stackrel{˜}{E}}_{N-1}\left[{\left(S-K\right)}^{+}{S}_{T}\right]\\ ={K}^{2}{\stackrel{˜}{E}}_{N-1}\left[{\left({\text{e}}^{{X}_{T}}-1\right)}^{+}{\text{e}}^{{X}_{T}}\right]\\ ={K}^{2}{\int }_{-\infty }^{+\infty }{\left({\text{e}}^{y}-1\right)}^{+}{\text{e}}^{y}{f}_{{X}_{N-1}}\left(y\right)\text{d}y,\end{array}$ (14)

${f}_{{X}_{N-1}}\left({X}_{N}\right)=f\left({X}_{N}-{X}_{N-1}\right),$ (15)

$\begin{array}{c}{v}_{N}={K}^{2}{\int }_{-\infty }^{+\infty }{\left({\text{e}}^{y}-1\right)}^{+}{\text{e}}^{y}{f}_{{X}_{N-1}}\left(y\right)\text{d}y\\ ={K}^{2}{\int }_{-\infty }^{+\infty }{\left(\mathrm{exp}\left(z+{x}_{N-1}\right)-1\right)}^{+}\mathrm{exp}\left(z+{x}_{N-1}\right)f\left(z\right)\text{d}z\\ ={K}^{2}{\int }_{-{x}_{N-1}}^{+\infty }\left[\mathrm{exp}\left(2z+2{x}_{N-1}\right)-\mathrm{exp}\left(z+{x}_{N-1}\right)\right]f\left(z\right)\text{d}z.\end{array}$

$\begin{array}{c}{v}_{N}\approx {K}^{2}{\int }_{-{x}_{N-1}}^{d}\left[\mathrm{exp}\left(2z+2{x}_{N-1}\right)-\mathrm{exp}\left(z+{x}_{N-1}\right)\right]f\left(z\right)\text{d}z\\ \approx {K}^{2}\underset{j=1}{\overset{M}{\sum }}\left\{\left[\mathrm{exp}\left(2{z}_{j}+2{x}_{N-1}\right)-\mathrm{exp}\left({z}_{j}+{x}_{N-1}\right)\right]f\left({z}_{j}\right)\right\}\Delta ,\end{array}$ (16)

(16)中的最后一步是利用梯形公式而得到的，其中 $M\Delta -{x}_{N-1}=d$ 是截断无穷积分的上限， ${z}_{j}=\left(j-1\right)\Delta -{x}_{N-1}$ 。为了保证计算的精度，在数值仿真中本文取 $M={10}^{6},\Delta =0.001$

2) 接下来考虑 $0 的情况。首先根据重期望公式

${v}_{j}={\stackrel{˜}{E}}_{j-1}\left[{S}_{j}{C}_{T}\right]={\stackrel{˜}{E}}_{j-1}\left[{\stackrel{˜}{E}}_{j}\left[{S}_{j}{C}_{T}\right]\right]={\stackrel{˜}{E}}_{j-1}\left[{S}_{j}{\stackrel{˜}{E}}_{j}\left[{C}_{T}\right]\right],$ (17)

$C\left(t,{X}_{t}\right)\approx \frac{K}{\text{π}}\underset{j=1}{\overset{M+1}{\sum }}R\left\{\frac{{\Phi }_{X}\left({\omega }_{j}-i-i\stackrel{¯}{\alpha };t,T\right)}{\left(i{\omega }_{j}+1+\stackrel{¯}{\alpha }\right)\left(i{\omega }_{j}+\stackrel{¯}{\alpha }\right)}\right\}\eta ,$

$\begin{array}{l}{\Phi }_{X}\left(\omega ;{t}_{j},T\right)\\ =\mathrm{exp}\left[i\omega \left(r\tau +{X}_{j}\right)\right]\mathrm{exp}\left[i\omega \nu \tau -{\left(i\omega \sigma \right)}^{\alpha }\tau \mathrm{sec}\frac{\alpha \text{π}}{2}\right]\\ =\left[\mathrm{cos}\left(\omega r\left(\tau +{X}_{j}\right)\right)+i\mathrm{sin}\left(\omega r\left(\tau +{X}_{j}\right)\right)\right]\mathrm{exp}\left[i\omega \nu \tau -{\left(i\omega \sigma \right)}^{\alpha }\tau \mathrm{sec}\frac{\alpha \text{π}}{2}\right].\end{array}$

$\begin{array}{c}{v}_{j}=K{\stackrel{˜}{E}}_{j-1}\left[\mathrm{exp}\left({X}_{j}\right)C\left({X}_{j}\right)\right]\\ =K{\int }_{-\infty }^{\infty }\mathrm{exp}\left(x\right)C\left(x\right){f}_{{X}_{j}}\left(x\right)\text{d}x\\ =K{\int }_{-\infty }^{\infty }\mathrm{exp}\left(y+{x}_{j}\right)C\left(y+{x}_{j}\right)f\left(y\right)\text{d}y\\ \approx K\underset{i=1}{\overset{N}{\sum }}\left[\mathrm{exp}\left({y}_{i}+{x}_{j}\right)C\left({y}_{i}+{x}_{j}\right)f\left({y}_{i}\right)\right]{\Delta }^{\prime },\end{array}$

4. 数值分析

${C}_{BS}=SN\left({d}_{1}\right)-{\text{e}}^{-rT}KN\left({d}_{2}\right),$

${d}_{1}=\frac{\mathrm{ln}\left(S/K\right)+\left(r+0.5{\stackrel{^}{\sigma }}^{2}\right)T}{\stackrel{^}{\sigma }\sqrt{T}},\text{\hspace{0.17em}}\text{\hspace{0.17em}}{d}_{2}={d}_{1}-\stackrel{^}{\sigma }\sqrt{T},\text{\hspace{0.17em}}\text{\hspace{0.17em}}\stackrel{^}{\sigma }=\sqrt{2}\sigma .$

$error=|{C}_{FMLS}-{C}_{BS}|,$

(a) CFMLS (α=2) vs CBS (b) 绝对误差图

Figure 1. The strike price $K=10$

5. 参数影响分析

(a) CFMLS vs Monte Carlo (b) 误差图

Figure 2. The strike price $K=10$

Figure 3. European call value, with $r=0.05,T=1,\sigma =0.20,K=30$

Figure 4. European call value, with $r=0.05,T=1,\alpha =1.52,K=35$

(a) σ=0.2 (b) α=1.52

Figure 5. The implied volatility of European call with $r=0.05,T=1,K=30$

Figure 6. Hedging curve with $r=0.05,T=1,K=30,\sigma =0.2,\alpha =1.52$

(a) σ=0.2 (b) α=1.52

6. 结论

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