金融中的诸多衍生品都涉及复杂期权定价问题,其中大多数可转换为偏微分方程初(终)值问题,但该问题往往难以获得解析解,且面临着“维度诅咒”问题。在单个标的物的期权定价中,可以采用各种方法绕开偏微分方程的求解问题。但是篮子期权以资产组合为标的,其定价难以绕开高维偏微分方程的求解。在这一背景下,本文从倒向随机微分方程(BSDE)的思路出发,提出利用神经网络可以非线性地对任何函数进行拟合的特点,将其引入到一类抛物型偏微分方程数值求解中,将待求解目标作为可更新参数嵌入到深度学习架构中,使得在模型训练结束后便可以获得具有更高精度的目标解。本文的深度BSDE模型避开传统思路中遇到的对数正态分布随机变量的算术平均不再满足对数正态分布的问题,能兼具有效性和准确性对篮子期权定价问题进行求解,且具有可以优化的方向,在未来应用中泛用性较强。
Abstract
Many derivatives in finance involve complex option pricing problems, most of which can be transformed into initial(final) value problem of partial differential equations(PDEs), but this problem is often difficult to obtain analytic solution and we face the“curse of dimensionality”when solving it.In the option pricing of a single underlying, various methods can be used to bypass the problem of solving PDEs. However, basket option is based on asset portfolio, and it is difficult to avoid the solution of high-dimensional PDE in the process of pricing.In this context, following the idea of backward stochastic differential equation(BSDE), we propose to take advantage of the neural network which can fit any function nonlinearly, and introduce it into the numerical solution of parabolic PDEs.The specific idea is to embed the object to be solved as an updateable parameter into the deep learning architecture, so that a more accurate solution can be obtained after model training.The deep BSDE model proposed in this paper avoids the problem that the arithmetic mean of lognormally distributed random variables no longer satisfies the lognormal distribution encountered in traditional thinking, and can solve the basket option pricing problem with both effectiveness and accuracy.Moreover, the model is easy to be optimized and can be widely used in the future.
关键词
深度学习 /
倒向随机微分方程 /
偏微分方程 /
篮子期权 /
期权定价
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Key words
Deep learning /
Backward stochastic differential equation(BSDE) /
Partial differential equation(PDE) /
Basket option /
Option pricing
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中图分类号：
F830.91O211.63TP183
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参考文献
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脚注
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基金
教育部首批新文科研究与改革实践项目(项目编号:2021060011);中央财经大学“科教融合研究生学术新星孵化计划”资助项目(项目编号:202209);中央财经大学新兴交叉学科建设项目(项目编号:2021)。
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