CEV及其他可解扩散模型下的阶梯期权定价
Pricing Step Options under the CEV and other Solvable Diffusion Models
摘要：
本文考虑一类特殊的占有时间导数，即Linetsky在[Math. Finance，9，55- 96（1999）]中引入的比例步长期权。我们开发了新的封闭形式的谱展开定价的一类非线性波动扩散过程，其中包括常数方差弹性（CEV）模型作为一个例子。特别是，我们推导出一个一般的解析精确表达式的预解核（即格林函数），这样的过程与杀害在指数停时（独立的过程）的占领以上或以下一个固定的水平。此外，我们成功地在拉普拉斯反演的预解核，从而获得新的封闭形式的谱展开公式的转移概率密度的过程与杀戮。谱展开公式是快速收敛和易于实现的，因为它们是简单地基于知识的一对基本解的基础上可解扩散过程。我们将谱展开公式应用于四个特定的可解非线性扩散资产价格模型族的比例阶跃期权定价，这些模型包括CEV扩散模型和其他三个多参数状态依赖局部波动汇合超几何扩散过程。
英文摘要：
We consider a special family of occupation-time derivatives, namely proportional step options introduced by Linetsky in [Math. Finance, 9, 55--96 (1999)]. We develop new closed-form spectral expansions for pricing such options under a class of nonlinear volatility diffusion processes which includes the constant-elasticity-of-variance (CEV) model as an example. In particular, we derive a general analytically exact expression for the resolvent kernel (i.e. Green's function) of such processes with killing at an exponential stopping time (independent of the process) of occupation above or below a fixed level. Moreover, we succeed in Laplace inverting the resolvent kernel and thereby derive newly closed-form spectral expansion formulae for the transition probability density of such processes with killing. The spectral expansion formulae are rapidly convergent and easy-to-implement as they are based simply on knowledge of a pair of fundamental solutions for an underlying solvable diffusion process. We apply the spectral expansion formulae to the pricing of proportional step options for four specific families of solvable nonlinear diffusion asset price models that include the CEV diffusion model and three other multi-parameter state-dependent local volatility confluent hypergeometric diffusion processes.
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