## Pricing Asian Options Under Time-Changed Mixed Fractional Brownian Motion with Transactions Costs

Ding Yi,, Guo Jingjun,

 基金资助: 国家自然科学基金.  71961013国家自然科学基金.  72061020甘肃省飞天学者项目和兰州财经大学科研创新团队支持计划

 Fund supported: the NSFC.  71961013the NSFC.  72061020the Feitian Scholars Project in Gansu Province and the Research Innovation Team Support Plan of Lanzhou University of Finance and Economics

Abstract

Considering that the classical Black-Scholes(B-S) option pricing model can not describe the characteristics of constant value periodicity and long-term dependence of financial asset prices, the time-changed mixed fractional Brownian motion is used to describe the changes of financial asset prices. By using self-financing delta hedging strategy, the partial differential equation of geometric average Asian call option price in discrete case and the pricing formula of geometric average Asian call and put option are obtained, and the influence of parameters in pricing model on option price is analyzed. Based on the daily closing price of Vanke stock, this article makes an empirical analysis on the established pricing model, and verifies the effectiveness of the pricing model.

Keywords： Asian option ; Time-changed process ; Mixed fractional Brownian motion ; Transaction cost

Ding Yi, Guo Jingjun. Pricing Asian Options Under Time-Changed Mixed Fractional Brownian Motion with Transactions Costs. Acta Mathematica Scientia[J], 2021, 41(4): 1135-1146 doi:

## 2 预备知识

$$$\Delta \left( {\frac{{\partial C}}{{\partial S_t }}} \right) = \frac{{\partial ^2 C}}{{\partial S_t \partial t}}\Delta t + \frac{{\partial ^2 C}}{{\partial S_t^2 }}\Delta S_t + \frac{1}{2}\frac{{\partial ^3 C}}{{\partial S_t^3 }} \Delta S_t^2+ \frac{{\partial ^2 C}}{{\partial S_t \partial J_t }}\Delta J_t + o\left( {\Delta t} \right),$$$

$$$\left| {\Delta \left( {\frac{{\partial C}}{{\partial S_t }}} \right)} \right|S_{t + \Delta t} = S_t^2 \left| {\frac{{\partial ^2 C}}{{\partial S_t^2 }}} \right|\left| {\sigma \left( {\Delta Z_\alpha (t) + \Delta Z_{\alpha , H} (t)} \right)} \right| + o\left( {\Delta t} \right),$$$

$$$C\left( {t, S_t , J_t } \right) = U_t S_t + F_t ,$$$

$U_t = \frac{{\partial C}}{{\partial S_t }}$, 由(3.12)–(3.14)式得

$\begin{eqnarray} \Delta \Pi& = & \frac{{\partial C}}{{\partial S_t }}\left[ {\mu S_t \Delta T_\alpha (t) + \sigma S_t \left( {\Delta Z_\alpha (t) + \Delta Z_{\alpha , H} (t)} \right) + \frac{1}{2}\sigma ^2 S_t \left( {\left( {\Delta Z_\alpha (t)} \right)^2 + \left( {\Delta Z_{\alpha , H} (t)} \right)^2 } \right)} \right]{}\\ & & + rF_t \Delta t - \frac{k}{2}\left| {\Delta \frac{{\partial C}}{{\partial S_t }}} \right|S_{t + \Delta t} + o\left( {\Delta t} \right){}\\ & = & \frac{{\partial C}}{{\partial S_t }}\left[ {\mu S_t \Delta T_\alpha (t) + \sigma S_t \left( {\Delta Z_\alpha (t) + \Delta Z_{\alpha , H} (t)} \right) + \frac{1}{2}\sigma ^2 S_t \left( {\left( {\Delta Z_\alpha (t)} \right)^2 + \left( {\Delta Z_{\alpha , H} (t)} \right)^2 } \right)} \right]{}\\ & &+ rF_t \Delta t - \frac{k}{2}S_t^2 \left| {\frac{{\partial ^2 C}}{{\partial S_t^2 }}} \right|\left| {\sigma \left( {\Delta Z_\alpha (t) + \Delta Z_{\alpha , H} (t)} \right)} \right| + o\left( {\Delta t} \right), \end{eqnarray}$

$\begin{eqnarray} \Delta \Pi - \Delta C& = & \left( {rC - rS_t \frac{{\partial C}}{{\partial S_t }} - \frac{{\partial C}}{{\partial t}} - \frac{{\partial C}}{{\partial J_t }} \cdot \frac{{\rm d}J_t }{{\rm d}t}} \right)\Delta t{}\\ & &- \frac{1}{2}\frac{{\partial ^2 C}}{{\partial S_t^2 }}\left( {\sigma ^2 S_t^2 \left( {\Delta Z_\alpha (t)^2 + \Delta Z_{\alpha , H} (t)^2 } \right)} \right){}\\ & &- \frac{k}{2}S_t^2 \left| {\frac{{\partial ^2 C}}{{\partial S_t^2 }}} \right|\left| {\sigma \left( {\Delta Z_\alpha (t) + \Delta Z_{\alpha , H} (t)} \right)} \right| + o\left( {\Delta t} \right), \end{eqnarray}$

$\begin{eqnarray} E\left( {\Delta \Pi - \Delta C} \right)& = & \left( {rC - rS_t \frac{{\partial C}}{{\partial S_t }} - \frac{{\partial C}}{{\partial t}} - \frac{{\partial C}}{{\partial J_t }} \cdot \frac{{\rm d}J_t }{{\rm d}t}} \right)\Delta t{}\\ & &- \left[ {\frac{1}{2}\frac{{\partial ^2 C}}{{\partial S_t^2 }}\left( {\sigma ^2 S_t^2 E\left( {\Delta Z_\alpha (t)^2 + \Delta Z_{\alpha , H} (t)^2 } \right)} \right)\Delta t^{ - 1} } \right]\Delta t{}\\ & &- \left[ {\frac{k}{2}\Delta t^{ - 1} S_t^2 \left| {\frac{{\partial ^2 C}}{{\partial S_t^2 }}} \right|E\left| {\sigma \left( {\Delta Z_\alpha (t) + \Delta Z_{\alpha , H} (t)} \right)} \right|} \right]\Delta t + o\left( {\Delta t} \right){}\\ & = & 0, \end{eqnarray}$

$\begin{eqnarray} rC & = & rS_t \frac{{\partial C}}{{\partial S_t }} + \frac{{\partial C}}{{\partial t}} + \frac{{\partial C}}{{\partial J_t }} \cdot \frac{{\rm d}J_t }{{\rm d}t}{\kern 1pt} {} \\ &&+ \frac{1}{2}\sigma ^2 S_t^2 \frac{{\partial ^2 C}}{{\partial S_t^2 }}\frac{{t^{\alpha - 1} }}{{\Gamma (\alpha )}} + \frac{1}{2}\sigma ^2 S_t^2 \frac{{\partial ^2 C}}{{\partial S_t^2 }}\left[ {\frac{{t^{\alpha - 1} }} {{\Gamma (\alpha )}}} \right]^{2H} \Delta t^{2H - 1} {\kern 1pt} {}\\ &&+ \frac{k}{2}S_t^2 \left| {\frac{{\partial ^2 C}}{{\partial S_t^2 }}} \right|\sqrt {\frac{2}{\pi }\left( {\sigma ^2 \frac{{t^{\alpha - 1} }}{{\Gamma (\alpha )}}\Delta t^{ - 1} + \sigma ^2 \left( {\frac{{t^{\alpha - 1} }} {{\Gamma (\alpha )}}} \right)^{2H} \Delta t^{2H - 2} } \right)} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} , \end{eqnarray}$

$$$rC = rS_t \frac{{\partial C}}{{\partial S_t }} + \frac{{\partial C}}{{\partial t}} + \frac{{\partial C}} {{\partial J_t }} \cdot \frac{{\rm d}J_t }{{\rm d}t} + \frac{1}{2}\delta (t)^2 S_t^2 \frac{{\partial ^2 C}}{{\partial S_t^2 }},$$$

## 4 模型求解

$$$C\left( {t, S_t , J_t } \right) = \left( {J_t^t S_t^{T - t} } \right)^{\frac{1}{T}} e^{\left( {\theta + \frac{{\omega ^2 }}{2} - r} \right)\left( {T - t} \right)} N\left( {d_1 } \right) - Ke^{ - r\left( {T - t} \right)} N\left( {d_2 } \right),$$$

类似于文献[12]中的方法, 令

$$$\varphi _t = \frac{{t\ln J_t + (T - t)\ln S_t }}{T},$$$

$$$C\left( {t, S_t , J_t } \right) = U(t, \varphi _t ),$$$

$$$\left\{ \begin{array}{l} { } \frac{{\partial U}}{{\partial t}} + \left( {r - \frac{1}{2}\delta (t)^{\rm{2}} } \right)\frac{{(T - t)}}{T}\frac{{\partial U}}{{\partial \varphi _t }} + \frac{1}{2}\delta (t)^{\rm{2}} \left[ {\frac{{T - t}}{T}} \right]^2 \frac{{\partial ^2 U}}{{\partial \varphi _t^2 }} = rU, \\ U(T, \varphi _T ) = \max \left\{ {e^{\varphi _T } - K, 0} \right\}. \end{array} \right.$$$

$$$\tau = \gamma (t), \ \eta _\tau = \varphi _t + \alpha (t), \ X(\tau , \eta _\tau ) = U(t, \varphi _t )e^{\beta (t)} ,$$$

$$$\begin{array}{l} { } \frac{{\partial U}}{{\partial t}} = e^{ - \beta (t)} \left[ {\frac{{\partial X}}{{\partial \tau }}\gamma '(t) - \beta '(t)X + \frac{{\partial X}}{{\partial \eta _\tau }}\alpha '(t)} \right], \\ { } \frac{{\partial U}}{{\partial \varphi _t }} = e^{ - \beta (t)} \frac{{\partial X}}{{\partial \eta _\tau }}, \ {\kern 1pt} \frac{{\partial ^2 U}}{{\partial \varphi _t^2 }} = \frac{\partial }{{\partial \varphi _t }}\left( {e^{ - \beta (t)} \frac{{\partial X}}{{\partial \eta _\tau }}} \right) = e^{ - \beta (t)} \frac{{\partial ^2 X}}{{\partial \eta _\tau ^2 }}, \\ \end{array}$$$

$$$\left\{ \begin{array}{l} { } \frac{{\partial X}}{{\partial \tau }} = \frac{{\sigma ^2 }}{2}\frac{{\partial ^2 X}}{{\partial \eta _\tau ^2 }}, \\ X(0, \eta _0 ) = \max \left\{ {e^{\eta _0 } - K, 0} \right\}. \end{array} \right.$$$

$$$X\left( {\tau , \eta _\tau } \right) = \frac{1}{{\sigma \sqrt {2\pi \tau } }}\int_{ - \infty }^{ + \infty } {\left( {e^y - K} \right)^ + e^{ - \frac{{(y - \eta _\tau )^2 }}{{2\sigma ^2 \tau }}} {\rm d}y}.$$$

$\begin{eqnarray} X\left( {\tau , \eta _\tau } \right) & = & e^{\eta _\tau + \frac{{\sigma ^2 \tau }}{2}} N\left( {\frac{{\eta _\tau + \sigma ^2 \tau - \ln K}}{{\sigma \sqrt \tau }}} \right) - KN\left( {\frac{{\eta _\tau - \ln K}}{{\sigma \sqrt \tau }}} \right) {}\\ & = & \left( {J_t^t S_t^{T - t} } \right)^{\frac{1}{T}} e^{\left( {\theta + \frac{{\omega ^2 }}{2}} \right)\left( {T - t} \right)} N\left( {d_1 } \right) - KN\left( {d_2 } \right), \end{eqnarray}$

$$$C\left( {t, S_t , J_t } \right) = \left( {J_t^t S_t^{T - t} } \right)^{\frac{1}{T}} e^{\left( {\theta + \frac{{\omega ^2 }}{2} - r} \right)\left( {T - t} \right)} N\left( {d_1 } \right) - Ke^{ - r\left( {T - t} \right)} N\left( {d_2 } \right),$$$

$$$P\left( {t, S_t , J_t } \right) = Ke^{ - r\left( {T - t} \right)} N\left( { - d_2 } \right) - \left( {J_t^t S_t^{T - t} } \right)^{\frac{1}{T}} e^{\left( {\theta + \frac{{\omega ^2 }}{2} - r} \right)\left( {T - t} \right)} N\left( { - d_1 } \right),$$$

由边界条件$P\left( {T, S_T , J_T {\kern 1pt} {\kern 1pt} } \right) = \max \left\{ {0, K - J_T {\kern 1pt} {\kern 1pt} } \right\}$, 运用定理4.1中的方法求解方程(3.3)可得看跌期权价格.

## 5 数值试验

 $K$ $C_{GBM}$ $C_{FBM}$ $C_{MFBM}$ $C_{TC-MFBM}$ 90 12.818953 12.689548 13.990609 13.192214 91 12.081619 11.928910 13.335705 12.412176 92 11.366589 11.190936 12.699837 11.636384 93 10.674784 10.476815 12.083259 10.864681 94 10.007019 9.787618 11.486168 10.096917 95 9.363990 9.124284 10.908701 9.332946 96 8.746270 8.487611 10.350937 8.572629 97 8.154301 7.878242 9.812900 7.815831 98 7.588394 7.296662 9.294556 7.062422 99 7.048728 6.743191 8.795820 6.312278 100 6.535349 6.217986 8.316557 5.565279

 $C_{GBM}$ $C_{FBM}$ $C_{MFBM}$ $C_{TC-MFBM}$ 均值 9.489636 9.256528 11.00682 9.351251 最大值 12.81895 12.68955 13.99061 13.19221 最小值 6.535349 6.217986 8.316557 5.565279 中位数 9.36399 9.124284 10.9087 9.332946 标准差 2.089258 2.152659 1.884268 2.529078

### 图 1

$$$\frac{{R_n}}{{S_n}}\sim cn^H,$$$

### 图 3

 均值 最大值 最小值 中位数 标准差 极差 收盘价 17.41 27.57 8.78 14.67 5.44 18.79 模拟价 17.63 27.00 9.44 16.22 4.55 17.56 BSM 17.57 31.47 8.28 15.63 7.56 23.19

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