Topics on discrete-time stochastic volatility models with applications in finance and insurance

二十多年来，金融保险产品的定价和套期保值一直是研究的主要对象。尽管巨大的研究努力已经解决了重要的方面，但仍有“谜题”有待解决。本研究项目的重点是从金融计量经济学和数学金融文献中选择的主题。波动率交易最近变得几乎和期权交易一样重要，因为波动率交易的日交易量最近已经超过了标准普尔500指数期权的日交易量。方差掉期合约是波动性衍生品的基石。尽管绝大多数数学金融文献研究了连续时间金融资产的建模和波动性衍生品的定价（主要是由于这种设置提供的可追溯性），但方差掉期在实践中是在固定日期采样的，因此，离散时间设置可能更合适。以下是我今后的研究方向。在本提案的第一部分，我计划通过研究几个流行的仿射和非仿射模型（如广义自回归条件异方差（GARCH）家族和自回归随机波动（SV）模型）的弱收敛性来研究离散和连续时间定价模型之间的关系。这些结果将在单变量和多变量设置下得出，并将讨论欧洲和美国期权定价的应用。本练习的目标之一是确定新的定价模型和策略，这些模型和策略在拟合金融资产数据时利用了非仿射结构的优势，在为衍生品定价时利用了仿射属性。在提案的第二部分中，我打算研究在离散时间点进行抽样时方差互换的新定价方法。使用第一部分计算的收敛结果，我的目标是推导出离散抽样方差互换的新公式，当基础资产在连续时间内建模时，这在直接计算后是不可能的。使用真实的市场报价进行方差互换，我计划构建适合其期限结构的模型。例如，在期权定价理论中，一个众所周知的事实是，在随机波动率模型中加入跳跃只有助于拟合短期现金外合约，因此测试方差掉期是否也是如此将是很有趣的。我相信，拟议的研究计划将为金融衍生品（尤其是波动性衍生品）的建模、定价和对冲带来几项重要贡献。

The pricing and hedging of financial and insurance products have been key objects of study in over two decades. Although tremendous research efforts have addressed important aspects, there are still 'puzzles' yet to be solved. This research project focuses on topics selected from the financial econometrics and mathematical finance literature.Volatility trading has recently become almost as important as option trading, as daily volumes of volatility trading have recently become larger than daily volumes of S&P 500 option trading. Variance swap contracts are the building blocks of volatility derivatives. Although a vast majority of the mathematical finance literature examines the modelling of financial assets and pricing of volatility derivatives in continuous-time (mainly due to the tractability offered by this setup) variance swaps are in practice sampled at fixed dates and therefore, a discrete-time setting might be more appropriate. The following summarizes my future research directions. In the first part of this proposal, I plan to study the relationship between discrete and continuous time pricing models, by investigating the weak convergence of several popular affine and non-affine models such as the Generalized Autoregressive Conditional Heteroskedastic (GARCH) family and autoregressive Stochastic Volatility (SV) models. These results will be derived in both a univariate and multivariate setting, and applications to pricing European and American options will be discussed. One of the goals of this exercise, is to identify new pricing models and strategies which make use of the advantages of the non-affine structure, when fitting financial asset data, and the affine properties when pricing derivatives. In the second part of the proposal, I intend to look at novel pricing methodologies for variance swaps when the sampling is performed at discrete time points. Using the convergence results computed in the first part, I aim to derive new formulas for the discretely sampled variance swaps, when the underlying asset is modelled in continuous time, which is not possible following a direct calculation. Using real market quotes for variance swaps, I plan to construct models which fit well their term structure. For example, in the option pricing theory it is a well-known fact that adding jumps to a stochastic volatility models only helps in fitting the short-term out-of-money contracts, and therefore it will be interesting to test if that is also the case for variance swaps. I believe the proposed research plan will bring several important contributions to the modelling and pricing and hedging of financial derivatives, in particular volatility derivatives.