Supervised deep learning in computational finance

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.

A multiscale extension of the Margrabe formula under stochastic volatility

Stochastic volatility (SV) models take a very important role in financial market, while there still exist some difficulties in estimating parameters in SV models. In this paper, a unified parameter estimation algorithm is proposed to estimate continuous-time SV model. Parameters in equity prices and volatilities stochastic processes are estimated separately after orthogonalization of Brownian motion in SV models. A closed-form of Maximum Likelihood Estimation (MLE) and moment estimation results for four parameters in SV models are deducted. Five typical SV models are simulated to reveal the characteristics in empirical volatilities data. Parameter estimation results are presented to prove the convergence of estimation algorithm. We also compare our algorithm with Least Square (LS) estimations and numerical solutions of all the parameters in with MLE method, respectively. LS can only work in a specific form of SV models and the convergence speed is limited. Numerical solutions of the likelihood function with four parameters are unstable and can not converge to true values of all the parameters. Thus our algorithm works more efficiently than other methods.

This study performs a comparative analysis of various GARCH and stochastic volatility (SV) models using the BIST 100 index. The models we examine include traditional GARCH (1,1) models and SV models with an AR (1) log-volatility process. Additionally, we consider more flexible models that incorporate jump components, volatility-in-mean, leverage effects, and innovations following a t-distribution or a moving average. The empirical findings reveal the following findings: (1) Stochastic Volatility (SV) models generally demonstrate better performance when compared to their GARCH counterparts. (2) The inclusion of a jump component and innovations following a tdistribution notably enhances the performance of the standard GARCH model while having less impact on the SV model. (3) The presence of a

Stochastic Local Volatility

This article introduces a discrete‐time fractional stochastic volatility model (FSV) based on fractional Gaussian noise. The new model includes the standard stochastic volatility model as a special case and has the same limit as the fractional integrated stochastic volatility (FISV) model, which is the continuous‐time fractional Ornstein–Uhlenbeck process. A simulated maximum likelihood method, which maximizes the time‐domain log‐likelihood function calculated by the importance sampling technique, and a frequency‐domain quasi maximum likelihood method (or quasi Whittle) are employed to estimate the model parameters. Simulation studies suggest that, while both estimation methods can accurately estimate the model, the simulated maximum likelihood method outperforms the quasi Whittle method. As an illustration, we fit the FSV and FISV models with the proposed estimation techniques to the S&P 500 composite index over a sample period spanning 45 years.

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.

In this paper, we propose a fractional stochastic volatility jump-diffusion model which extends the Bates(1996) model. Where we model the volatility as a fractional process. Extensive empirical studies show that the distributions of the logarithmic returns of financial asset usually exhibit properties of self-similarity and long-range dependence and since the fractional Brownian motion has these two important properties, it has the ability to capture the behavior of underlying asset price. Further incorporating jumps into the stochastic volatility framework gives further freedom to financial mathematicians to fit both the short and long end of the implied volatility surface. We propose a stochastic model which contains both fractional and jump process. Then we price options using Monte Carlo simulations along with a variance reduction technique(antithetic variates). We use market data from the S&P 500 index and we compare our results with the Heston and Bates model using error measures. The results show our model greatly outperforms previous models in terms of estimation accuracy.

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

ABSTRACTIn this work, we introduce a novel pricing methodology in general, possibly non‐Markovian local stochastic volatility (LSV) models. We observe that by conditioning the LSV dynamics on the Brownian motion that drives the volatility, one obtains a time‐inhomogeneous Markov process. Using tools from rough path theory, we describe how to precisely understand the conditional LSV dynamics and reveal their Markovian nature. The latter allows us to connect the conditional dynamics to so‐called rough partial differential equations (RPDEs), through a Feynman–Kac type of formula. In terms of European pricing, conditional on realizations of one Brownian motion, we can compute conditional option prices by solving the corresponding linear RPDEs, and then average over all samples to find unconditional prices. Our approach depends only minimally on the specification of the volatility, making it applicable for a wide range of classical and rough LSV models, and it establishes a PDE pricing method for non‐Markovian models. Finally, we present a first glimpse at numerical methods for RPDEs and apply them to price European options in several rough LSV models.

Modeling Energy Price Dynamics: GARCH Versus Stochastic Volatility

Modeling energy price dynamics: GARCH versus stochastic volatility

Portfolio Benchmarking under Drawdown Constraint and Stochastic Sharpe Ratio

This paper introduces a general class of stochastic volatility models that can serve as a basis for modeling and estimating simultaneous equation systems involving multivariate volatility processes. The class consists of processes obtained as monotone polynomial transformations of so-called stochastic autoregressive volatility (SARV) models. The class permits a flexible modeling of volatility and avoids strong distributional assumptions. Most of the standard stochastic volatility models are incorporated in the framework, including the lognormal autoregressive model and stochastic volatility generalizations of GARCH and EGARCH. General conditions for strict statinarity, ergodicity, and the existence of finite lower-order unconditional moments for SARV are established. Closed-form expressions for the lowerorder moments constitute the basis for the proposed generalized method of moments estimation procedure. As an illustration we consider an informationdriven model of the return volatility–trading volume system for financial assets. Parameters with structural interpretation are estimated and an extended Kalmanfilter procedure allows for the construction of volatility forecasts based on the bivariate return-volume series. Empirical results are based on daily NYSE data for IBM over 1973–1991.