论文标题
在随机波动率模型下定价欧洲选择：五参数差异过程的案例
Pricing European Options under Stochastic Volatility Models: Case of five-Parameter Variance-Gamma Process
论文作者
论文摘要
该论文构建了具有五个参数的方差 - 伽马（VG）模型：位置（$μ$），对称性（$δ$），波动性（$σ$），Shape（$α$）和比例尺（$θ$）;并研究其在欧洲选择定价中的应用。我们的分析结果表明，五参数VG模型是一个随机波动率模型，具有$γ（α，θ）$ ornstein-uhlenbeck类型过程； VG模型的相关lévy密度是订单$ν= 0 $，强度$α$和陡度参数的Kobol家族$ \fracδ{σ^2} - \ sqrt {\ frac {Δ^frac {δ^2} $ \fracδ{σ^2}+ \ sqrt {\ frac {δ^2} {σ^4}+ \ frac {2} {θσ^2}} $; VG过程在分布中渐近地收敛到由平均$ $（μ +αθδ）$的正态分布和方差$α（θ^2δ^2δ^2 +σ^2θ）$驱动的Lévy过程。通过将五参数方差 - 伽马（VG）模型拟合到每日间谍ETF数据的基础分布中，获得了用于经验分析的数据。关于五参数VG模型的应用，实施了十二点规则综合牛顿 - 纽顿 - cots Quadrature和分数快速傅立叶（FRFT）算法以计算欧洲期权价格。与黑色choles（BS）模型相比，经验证据表明，VG期权价格的价格低估（OTM）选项价格低估，并且价格过高，用于货币（ITM）期权。这两种模型几乎都可以为Deep Out-Out-Out-Out-Ofnoy（OTM）和Deep-In-Itm（ITM）选项产生相同的选项定价结果
The paper builds a Variance-Gamma (VG) model with five parameters: location ($μ$), symmetry ($δ$), volatility ($σ$), shape ($α$), and scale ($θ$); and studies its application to the pricing of European options. The results of our analysis show that the five-parameter VG model is a stochastic volatility model with a $Γ(α, θ)$ Ornstein-Uhlenbeck type process; the associated Lévy density of the VG model is a KoBoL family of order $ν=0$, intensity $α$, and steepness parameters $\fracδ{σ^2} - \sqrt{\frac{δ^2}{σ^4}+\frac{2}{θσ^2}}$ and $\fracδ{σ^2}+ \sqrt{\frac{δ^2}{σ^4}+\frac{2}{θσ^2}}$; and the VG process converges asymptotically in distribution to a Lévy process driven by a normal distribution with mean $(μ+ αθδ)$ and variance $α(θ^2δ^2 + σ^2θ)$. The data used for empirical analysis were obtained by fitting the five-parameter Variance-Gamma (VG) model to the underlying distribution of the daily SPY ETF data. Regarding the application of the five-parameter VG model, the twelve-point rule Composite Newton-Cotes Quadrature and Fractional Fast Fourier (FRFT) algorithms were implemented to compute the European option price. Compared to the Black-Scholes (BS) model, empirical evidence shows that the VG option price is underpriced for out-of-the-money (OTM) options and overpriced for in-the-money (ITM) options. Both models produce almost the same option pricing results for deep out-of-the-money (OTM) and deep-in-the-money (ITM) options
