Rough PDEs for Local Stochastic Volatility Models

This thesis presents our study on using the hybrid stochastic-local volatility model for option pricing. Many researchers have demonstrated that stochastic volatility models cannot capture the whole volatility surface accurately, although the model parameters have been calibrated to replicate the market implied volatility data for near at-the-money strikes. On the other hand, the local volatility model can reproduce the implied volatility surface, whereas it does not consider the stochastic behaviour of the volatility. To combine the advantages of stochastic volatility (SV) and local volatility (LV) models, a class of stochastic-local volatility (SLV) models has been developed. The SLV model contains a stochastic volatility component represented by a volatility process and a local volatility component represented by a so-called leverage function. The leverage function can be roughly seen as a ratio between local volatility and conditional expectation of stochastic volatility. The difficulty of implementing the SLV model lies in the calibration of the leverage function. In the thesis, we first review the fundamental theories of stochastic differential equations and the classic option pricing models, and study the behaviour of the volatility in the context of FX market. We then introduce the SLV model and illustrate our implementation of the calibration and pricing procedure. We apply the SLV model to exotic option pricing in the FX market and compare pricing results of the SLV model with pure local volatility and pure stochastic volatility models. Numerical results show that the SLV model can match the implied volatility surface very well as well as improve the pricing performance for barrier options. In addition, we further discuss some extensions of the SLV project, such as parallelization potential for accelerating option pricing and pricing techniques for window barrier options. Although the SLV model we use in the thesis is not entirely new, we contribute to the research in the following aspects: 1) we investigate the hybrid volatility modeling thoroughly from theoretical backgrounds to practical implementations; 2) we resolve some critical issues in implementing the SLV model such as developing a fast and stable numerical method to derive the leverage function; and 3) we build a robust calibration and pricing platform under the SLV model, which can be extended for practical uses.

This thesis presents our study on using the hybrid stochastic-local volatility model for option pricing. Many researchers have demonstrated that stochastic volatility models cannot capture the whole volatility surface accurately, although the model parameters have been calibrated to replicate the market implied volatility data for near at-the-money strikes. On the other hand, the local volatility model can reproduce the implied volatility surface, whereas it does not consider the stochastic behaviour of the volatility. To combine the advantages of stochastic volatility (SV) and local volatility (LV) models, a class of stochastic-local volatility (SLV) models has been developed. The SLV model contains a stochastic volatility component represented by a volatility process and a local volatility component represented by a so-called leverage function. The leverage function can be roughly seen as a ratio between local volatility and conditional expectation of stochastic volatility. The difficulty of implementing the SLV model lies in the calibration of the leverage function. In the thesis, we first review the fundamental theories of stochastic differential equations and the classic option pricing models, and study the behaviour of the volatility in the context of FX market. We then introduce the SLV model and illustrate our implementation of the calibration and pricing procedure. We apply the SLV model to exotic option pricing in the FX market and compare pricing results of the SLV model with pure local volatility and pure stochastic volatility models. Numerical results show that the SLV model can match the implied volatility surface very well as well as improve the pricing performance for barrier options. In addition, we further discuss some extensions of the SLV project, such as parallelization potential for accelerating option pricing and pricing techniques for window barrier options. Although the SLV model we use in the thesis is not entirely new, we contribute to the research in the following aspects: 1) we investigate the hybrid volatility modeling thoroughly from theoretical backgrounds to practical implementations; 2) we resolve some critical issues in implementing the SLV model such as developing a fast and stable numerical method to derive the leverage function; and 3) we build a robust calibration and pricing platform under the SLV model, which can be extended for practical uses.

Calibrating and Pricing with Stochastic-Local Volatility Model

Efficient simulation of generalized SABR and stochastic local volatility models based on Markov chain approximations

Stochastic Local Volatility

Calibrating stochastic local volatility (SLV) models to market implied volatility surfaces remains an inherently complex and ill-posed inverse problem, as it involves solving nonlinear PDEs under sparse and noisy data, while simultaneously ensuring consistency with the underlying risk-neutral probability density. We present a probability density-consistent PINNs (PDC-PINNs) that embeds the forward Kolmogorov Fokker Planck equation and the risk-neutral density constraints whose joint validity ensures the absence of static and dynamic arbitrage. A self-adaptive bilevel optimiser is developed to balance the PDE, data-fit, and density losses by rescaling them according to gradient magnitudes for stabilising learning. In synthetic and real foreign exchange datasets, PDC-PINNs match or exceed the accuracy of conventional calibrators while halving computational time through TPU vectorisation and producing reasonable surfaces that remain all constraints. The framework further enables consistent pricing of path-dependent options and generalises to other constrained financial PDE problems.

We propose a feasible and constructive methodology which allows us to compute pure hedging strategies with respect to arbitrary square-integrable claims in incomplete markets. In contrast to previous works based on PDE and BSDE methods, the main merit of our approach is the flexibility of quadratic hedging in full generality without a priori smoothness assumptions on the payoff. In particular, the methodology can be applied to multidimensional quadratic hedging-type strategies for fully path-dependent options with stochastic volatility and discontinuous payoffs. In order to demonstrate that our methodology is indeed applicable, we provide a Monte Carlo study on generalized Föllmer-Schweizer decompositions, locally risk minimizing, and mean variance hedging strategies for vanilla and path-dependent options written on local volatility and stochastic volatility models.

The averaging principle for non-autonomous slow-fast stochastic differential equations and an application to a local stochastic volatility model

In this paper, we present our research on pricing window barrier options under a hybrid stochastic-local volatility (SLV) model in the foreign exchange (FX) market. Due to the hybrid effect of the local volatility and stochastic volatility components of the model, the SLV model can reproduce the market implied volatility surface, and can improve the pricing accuracy for exotic options at the same time. In this paper, numerical techniques such as Monte Carlo and finite difference methods for standard exotic barrier options under the SLV model are extended to pricing window barrier options and numerical results produced by the SLV model are used to examine the performance and accuracy of the model for pricing window barrier options.

Exact Forward in Monte-Carlo

In this paper, we introduce a new method for pricing CMS derivatives. We utilize Malliavin's calculus to establish a model-free connection between the price of a CMS derivative and a quadratic payoff. Then, we apply Watanabe's expansions to quadratic payoffs under local and stochastic local volatility. The local and stochastic local volatility models are expressed in a general form, providing a generic approximation. To evaluate their accuracy, we will compare the approximations numerically under the normal SABR model against the market standards: Hagan's approximation and Monte Carlo simulation.

Using an expansion of the transition density function of a two dimensional time inhomogeneous diffusion, we obtain the first and second order terms in the short time asymptotics of the local volatility function in a family of time inhomogeneous local-stochastic volatility models. With the local volatility function at our disposal, we show how recent results (Gatheral et al., Math. Financ. 22:591–620, 2012, [28]) for one dimensional diffusions can be applied to also determine expansions for call prices as well as for the implied volatility. The results are worked out in detail in the case of the dynamic Sabr model, thus generalizing earlier work by Hagan et al. (Wilmott Mag. 84–108, 2003, [31]), Hagan and Lesniewski (Springer Proceedings in Mathematics and Statistics, vol. 110, 2015, [32]) and by Henry-Labordere (Springer Proceedings in Mathematics and Statistics, vol. 110, 2015, Geometry, and Modeling in Finance. Chapman & Hall/CRC Financial Mathematics Series, 2008, [39, 40]).

Local Stochastic Volatility with Jumps

Empirical estimates of equilibrium real interest rates are so far mostly limited to advanced economies, since no statistical procedure suitable for a larger set of countries is available. This is surprising, as equilibrium rates have strong policy implications in emerging markets and developing economies as well. Current estimates of the global equilibrium rate rely on only a few countries but estimates for a more diverse set of countries can improve understanding of its dynamics and drivers. This paper uses a model and estimation strategy that decompose ex-ante real interest rates into permanent and transitory components even with short samples and high volatility. This is done with an unobserved component local level stochastic volatility model, which is used to estimate equilibrium rates for 50 countries with Bayesian methods. Equilibrium rates were lower in emerging markets and developing economies than in advanced economies in the 1980s, similar in the 1990s, and have been higher since 2000. In line with economic integration and rising global capital markets, synchronization has been rising over time and is higher among advanced economies. Equilibrium rates of countries with stronger trade linkages and similar demographic and economic trends are more synchronized.

It is extremely challenging to design a model calibrating both SPX and VIX option prices. A long‐standing conjecture due to Julien Guyon is that it may not be possible to calibrate these two quantities with a continuous model. So far, most studied continuous time models are affine, so we investigate the conjecture among 14 well‐known non‐affine local stochastic volatility models in this article. First, we propose a unified efficient willow tree method for S&P500 and VIX option pricing under non‐affine models. Second, we compare the joint calibration performance on these 14 models on the S&P500 and VIX option prices data from 2006 to 2019. We find that the VIX option price data can provide extra information of the variance dynamics of the models, and the non‐affine structure and the volatility with linear, rather than square‐root, diffusion process provide a better fit for the both data sets than the affine counterparts. Among the 14 stochastic models, the SABR model provides the best in‐ and out‐of‐sample performance (in terms of mean square error) regardless of the state of the economy. Nevertheless, even for the best‐fitted SABR model, the relative error on the VIX option is still around 18%, still quite significant. Therefore, we found the non‐affine and local volatility structure improve the joint calibration but are still far from satisfactory.