Tail asymptotics for diffusion processes, with applications to local volatility and CEV-Heston models

虚假平稳粗糙Heston波动率：微结构启发的基础

Fake stationary rough Heston volatility: Microstructure-inspired foundations

本文提出了一种基于时间非齐性粗糙分数CIR方程的虚假平稳粗糙Heston波动率模型，研究其在重尾核函数下的渐进行为及性质。

Heston-Hawkes随机波动率模型下投连险的Thiele偏微分积分方程

Thiele's PIDE for unit-linked policies in the Heston-Hawkes stochastic volatility model

本文在Heston-Hawkes随机波动率模型下推导了投连险的Thiele微分方程，通过研究方差和随机指数矩找到合适的风险中性测度族，为无套利模型下的寿险准备金计算和保证偿付能力提供了实用方法。

Sensitivity analysis in the infinite dimensional Heston model

We consider the infinite dimensional Heston stochastic volatility model proposed in \arXiv:1706:03500. The price of a forward contract on a non-storable commodity is modelled by a generalized Ornstein-Uhlenbeck process in the Filipović space with this volatility. We prove different representation formulas for the forward price. Then we consider prices of options written on these forward contracts and we study sensitivity analysis with computation of the Greeks with respect to different parameters in the model. Since these parameters are infinite dimensional, we need to reinterpret the meaning …

Realized volatility and parametric estimation of Heston SDEs

We present a detailed analysis of \emph{observable} moments based parameter estimators for the Heston SDEs jointly driving the rate of returns $R_t$ and the squared volatilities $V_t$. Since volatilities are not directly observable, our parameter estimators are constructed from empirical moments of realized volatilities $Y_t$, which are of course observable. Realized volatilities are computed over sliding windows of size $\varepsilon$, partitioned into $J(\varepsilon)$ intervals. We establish criteria for the joint selection of $J(\varepsilon)$ and of the sub-sampling frequency of return rates…

The rough Hawkes Heston stochastic volatility model

We study an extension of the Heston stochastic volatility model that incorporates rough volatility and jump clustering phenomena. In our model, named the rough Hawkes Heston stochastic volatility model, the spot variance is a rough Hawkes-type process proportional to the intensity process of the jump component appearing in the dynamics of the spot variance itself and the log returns. The model belongs to the class of affine Volterra models. In particular, the Fourier-Laplace transform of the log returns and the square of the volatility index can be computed explicitly in terms of solutions of …

On refined volatility smile expansion in the Heston model

It is known that Heston's stochastic volatility model exhibits moment explosion, and that the critical moment $s_+$ can be obtained by solving (numerically) a simple equation. This yields a leading order expansion for the implied volatility at large strikes: $σ_{BS}( k,T)^{2}T\sim Ψ(s_+-1) \times k$ (Roger Lee's moment formula). Motivated by recent "tail-wing" refinements of this moment formula, we first derive a novel tail expansion for the Heston density, sharpening previous work of Dragulescu and Yakovenko [Quant. Finance 2, 6 (2002), 443--453], and then show the validity of a refined expan…

Large deviations for the extended Heston model: the large-time case

We study here the large-time behaviour of all continuous affine stochastic volatility models (in the sense of Keller-Ressel) and deduce a closed-form formula for the large-maturity implied volatility smile. Based on refinements of the Gartner-Ellis theorem on the real line, our proof reveals pathological behaviours of the asymptotic smile. In particular, we show that the condition assumed in Gatheral and Jacquier under which the Heston implied volatility converges to the SVI parameterisation is necessary and sufficient.

Stationary Heston model: Calibration and Pricing of exotics using Product Recursive Quantization

A major drawback of the Standard Heston model is that its implied volatility surface does not produce a steep enough smile when looking at short maturities. For that reason, we introduce the Stationary Heston model where we replace the deterministic initial condition of the volatility by its invariant measure and show, based on calibrated parameters, that this model produce a steeper smile for short maturities than the Standard Heston model. We also present numerical solution based on Product Recursive Quantization for the evaluation of exotic options (Bermudan and Barrier options).

对Heston随机波动率模型的分析：使用Matlab进行实现与校准

An Analysis of the Heston Stochastic Volatility Model: Implementation and Calibration using Matlab

本文分析了Heston模型的实现与校准，通过特征函数估计期权价格，并展示简单方法可快速准确计算欧式期权价格，同时通过局部和全局优化进行校准测试，证明简单设置能获得良好校准结果。

A decomposition formula for fractional Heston jump diffusion models

We present an option pricing formula for European options in a stochastic volatility model. In particular, the volatility process is defined using a fractional integral of a diffusion process and both the stock price and the volatility processes have jumps in order to capture the market effect known as leverage effect. We show how to compute a martingale representation for the volatility process. Finally, using Itô calculus for processes with discontinuous trajectories, we develop a first order approximation formula for option prices. There are two main advantages in the usage of such approxim…

具有 Zumbach 效应的非对称 super-Heston-rough 波动率模型作为二次 Hawkes 过程的标度极限

Asymmetric super-Heston-rough volatility model with Zumbach effect as scaling limit of quadratic Hawkes processes

本文提出二元 QHawkes 过程，推导出保留 Zumbach 效应的非对称 super-rough-Heston 模型，并发现近不稳定状态下驱动布朗运动具有依赖即期波动率的随机协变。

Time-Changed Bessel Processes and Credit Risk

The Constant Elasticity of Variance (CEV) model is mathematically presented and then used in a Credit-Equity hybrid framework. Next, we propose extensions to the CEV model with default: firstly by adding a stochastic volatility diffusion uncorrelated from the stock price process, then by more generally time changing Bessel processes and finally by correlating stochastic volatility moves to the stock ones. Properties about strict local and true martingales in this study are discussed. Analytical formulas are provided and Fourier and Laplace transform techniques can then be used to compute optio…

Large-Maturity Regimes of the Heston Forward Smile

We provide a full characterisation of the large-maturity forward implied volatility smile in the Heston model. Although the leading decay is provided by a fairly classical large deviations behaviour, the algebraic expansion providing the higher-order terms highly depends on the parameters, and different powers of the maturity come into play. As a by-product of the analysis we provide new implied volatility asymptotics, both in the forward case and in the spot case, as well as extended SVI-type formulae. The proofs are based on extensions and refinements of sharp large deviations theory, in par…

粗糙Hawkes-Heston模型的微观结构基础

Microstructural Foundation for the Rough Hawkes--Heston Model

本文构建泊松嵌入标记Hawkes订单流模型，为含共同价格-波动率跳跃的粗糙Hawkes-Heston模型提供微观结构基础，推导其收敛性并经数值实验验证。

From Hawkes-type processes to stochastic volatility

We introduce a Hawkes-like process and study its scaling limit as the system becomes increasingly endogenous. We derive functional limit theorems for intensity and fluctuations. Then, we introduce a high-frequency model for a price of a liquid traded financial instrument in which the nearly unstable regime leads to a Heston-type process where the negative correlation between the noise driving the proce of the instrument and the volatility can be viewed as a result of high variance of the sell-side order arrivals.

Option Pricing in a Regime Switching Stochastic Volatility Model

In the classical model of stock prices which is assumed to be Geometric Brownian motion, the drift and the volatility of the prices are held constant. However, in reality, the volatility does vary. In quantitative finance, the Heston model has been successfully used where the volatility is expressed as a stochastic differential equation. In addition, we consider a regime switching model where the stock volatility dynamics depends on an underlying process which is possibly a non-Markov pure jump process. Under this model assumption, we find the locally risk minimizing pricing of European type v…

局部随机粗糙波动率：路径滤波与条件密度方程

Local Stochastic Rough Volatility: Pathwise Filtering and the Conditional Density Equation

该研究以粗糙Heston模型为示例，证明局部随机粗糙波动率下条件密度SPDE的Itô-Wentzell约化仍成立，得到路径式Fokker-Planck公式，纯粗糙Heston情形下条件密度为显式对数正态形式，关联Rao-Blackwell化校准。

Moment explosions, implied volatility and local volatility at extreme strikes

We consider a stochastic volatility model where the moment generating function of the logarithmic price is finite only on part of the real line. Using a new Tauberian result obtained in [1] and [2], we show that the knowledge of the moment generating function near its critical moment gives a sharp asymptotic expansion (with an error of order o(1)) of the local volatility and implied volatility for small and large strikes. We apply our theoretical estimates to Gatheral's SVI parametrization of the implied volatility and Heston's model.

The Small-Maturity Heston Forward Smile

In this paper we investigate the asymptotics of forward-start options and the forward implied volatility smile in the Heston model as the maturity approaches zero. We prove that the forward smile for out-of-the-money options explodes and compute a closed-form high-order expansion detailing the rate of the explosion. Furthermore the result shows that the square-root behaviour of the variance process induces a singularity such that for certain parameter configurations one cannot obtain high-order out-of-the-money forward smile asymptotics. In the at-the-money case a separate model-independent an…

粗糙Heston模型作为双变量累积重尾INAR过程的标度极限：弱误差界和期权定价

Rough Heston model as the scaling limit of bivariate cumulative heavy-tailed INAR processes: Weak-error bounds and option pricing

研究双变量累积重尾INAR(∞)过程的近不稳定特性，证明其在特定参数化和标度下收敛于粗糙Heston模型，推导出INAR不对称参数与极限波动过程杠杆相关性和扩散尺度的联系，并通过离散方程与连续分数Riccati方程比较，获得弱误差界和期权定价方法。

Pricing timer options and variance derivatives with closed-form partial transform under the 3/2 model

Most of the empirical studies on stochastic volatility dynamics favor the 3/2 specification over the square-root (CIR) process in the Heston model. In the context of option pricing, the 3/2 stochastic volatility model is reported to be able to capture the volatility skew evolution better than the Heston model. In this article, we make a thorough investigation on the analytic tractability of the 3/2 stochastic volatility model by proposing a closed-form formula for the partial transform of the triple joint transition density $(X,I,V)$ which stand for the log asset price, the quadratic variation…

Asymptotic behavior of maximum likelihood estimators for a jump-type Heston model

We study asymptotic properties of maximum likelihood estimators of drift parameters for a jump-type Heston model based on continuous time observations, where the jump process can be any purely non-Gaussian Lévy process of not necessarily bounded variation with a Lévy measure concentrated on $(-1,\infty)$. We prove strong consistency and asymptotic normality for all admissible parameter values except one, where we show only weak consistency and mixed normal (but non-normal) asymptotic behavior. It turns out that the volatility of the price process is a measurable function of the price process. …

Heston 模型及相关过程的逼近与正则性结果

Approximation and regularity results for the Heston model and related processes

本文通过三项研究探讨了 Heston 模型的逼近与正则性，利用随机网格技术为 CIR 和 log-Heston 过程实现了任意阶弱逼近，并证明了相关 PDE 在不依赖 Feller 条件下的粘性解存在唯一性及混合数值格式的收敛性。