Pricing European lookback option by a special kind of mixed jump-diffusion model
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摘要: 利用分数Girsanov公式和分数Wick-Itô-Skorohod积分，建立了一个基于标准布朗运动、分数布朗运动、Poisson过程的线性组合的金融市场模型，结合Merton假设条件以及风险资产所满足的随机微分方程的Cauchy初值问题，给出了混合跳-扩散模型下的欧式看跌期权定价的Merton公式，给出了混合跳-扩散分数布朗运动下连续支付红利的欧式固定履约价和浮动履约价的看涨回望期权及看跌回望期权定价公式.数值模拟与仿真结果验证了模型的有效性和准确性.
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关键词:
- 混合跳扩散分数布朗运动 /
- Merton假设条件 /
- 分数Wick-Itô-Skorohod积分 /
- 欧式回望期权
Abstract: By using fractional Girsanov formula and fractional Wick-Itô-Skorohod integral, based on a linear combination of Brownian motion, fractional Brownian motion and Poisson process, a new market pricing model is built. Under the conditions of Merton assumptions, we analyze the Cauchy initial problem of stochastic parabolic partial differential equations. Then the pricing Merton-formula of European option meets the pricing model for the European fixed strike and floating strike price of the lookback option. Finally the pricing formulas of fixed strike and floating strike lookback call option and lookback put option are proved. Numerical simulations illustrate that our model are valid and accurate. -
图 1 不同参数下欧式固定履约回望看跌期权价格
Fig. 1 European lookback fixed strike put option prices for various parameters
图 2 混合跳-扩散分数布朗运动下欧式固定履约回望看跌期权模型的隐含波动率模拟
Fig. 2 Smile surface generated by the mj-dfBm model
表 1 不同模型下选择的参数值
Tab. 1 The valuations of the chosen parameters used in these models
模型类型 $r$ /% $q$ /% $\sigma _1$ /% $K(S_T) $ $H$ $J$ /% $\lambda$ $\mu _j$ /% $\sigma _j$ ( $\delta$ )/% $\sigma _2$ /% $\sigma _3$ /% pmfBm 2.50 3.20 10.73 100 0.558 - - - - - - jmfBm $^{l}$ 2.50 3.20 10.73 100 0.558 0.83 1.68 0.066 5 0.12 - - jmfBm $^{h}$ 2.50 3.20 10.73 100 0.558 0.32 6.68 -0.066 5 0.15 - mj-dfBm $^{l}$ 2. 50 3.20 10.73 100 0.558 0.94 2.68 0.072 1 0.16 2.15 0.071 97 mj-dfBm $^{h}$ 2. 50 3.20 10.73 100 0.558 0.35 7.68 -0.072 1 0.19 2.15 0.072 02 注:其中 $l$ 表示低强度跳环境, $h$ 表示高强度跳环境 
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表 2 不同模型下欧式固定履约回望看跌期权价格
Tab. 2 European lookback fixed strike put option prices for various models
$m/K$ 期权到期时间较短情形 $\left( {T=0.5, t=0} \right)$ 期权到期时间较长情形 $\left( {T=2.0, t=0} \right)$ $P_{p-m}$ $P_{j-m}^{\ast, {l}}$ $P_{j-m}^{\ast , {h}}$ $P_{m-j}^{\ast, {l}}$ $P_{m-j}^{\ast, {h}}$ $P_{p-m}$ $P_{j-m}^{\ast, {l}}$ $P_{j-m}^{\ast, {h}}$ $P_{m-j}^{\ast, {l}}$ $P_{m-j}^{\ast, {h}}$ 0.1 0.900 0 0.883 2 0.821 9 0.859 5 0.7891 0.781 1 0.712 9 0.699 2 0.703 4 0.688 5 0.2 0.800 0 0.746 7 0.680 2 0.701 3 0.6780 0.676 9 0.645 4 0.562 7 0.606 6 0.533 9 0.3 0.700 0 0.683 2 0.658 3 0.621 5 0.564 0 0.541 3 0.538 7 0.485 3 0.568 7 0.480 3 0.4 0.600 0 0.584 1 0.503 5 0.555 1 0.420 8 0.466 0 0.411 7 0.398 4 0.402 1 0.313 5 0.5 0.500 0 0.495 0 0.482 1 0.473 4 0.382 1 0.300 4 0.264 3 0.201 1 0.369 3 0.225 8 0.6 0.400 0 0.375 5 0.320 1 0.371 9 0.300 0 0.288 3 0.175 5 0.114 1 0.226 8 0.106 9 0.7 0.300 0 0.204 5 0.102 0 0.199 1 0.122 8 0.133 5 0.064 3 0.008 1 0.041 7 - 0.8 0.200 0 0.187 3 0.097 8 0.085 2 0.076 0 0.005 1 0.000 3 - - - 0.9 0.100 0 0.119 0 0.076 3 0.044 0 0.021 2 - - - - - 注:其中 $l$ , $h$ 环境下模型参数取值于表 1, 运算的最大循环次数为100
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