A smooth estimator for MC/QMC methods in finance

In this paper, we consider volatility swap and variance swap when the underlying asset is described by a process with multiple stochastic volatility models. The model considered in this paper is the multi-factor Heston stochastic volatility model. We obtain pricing formulas for the weighted variance swap and approximate expressions for the weighted volatility swap. The bounds of the arbitrage-free variance swap price are also found. The proposed pricing formulas are easy to compute in real time and can be applied efficiently for practical applications. We study the problem of hedging volatility swap with variance swap. We also determined the optimal amount of the underlying asset that has to be held for minimizing the hedging error by taking positions in options and weighted variance swap. From the numerical analysis, a couple of important features of the usefulness of the multi-factor Heston stochastic volatility model are discussed.

Efficient 'Black-Scholes' Approximations of the Heston and Multi-Factor Heston Option Prices for a Correct Spot Price

This paper builds and implements multifactor stochastic volatility models where the main objective is step ahead volatility prediction and to describe its relevance for the equity markets. The paper outlines stylised facts from the volatility literature showing density tails, persistence, mean reversion, asymmetry and long memory, all contributing to systematic data dependencies. As a by-product of the multifactor stochastic volatility model estimation, a long-simulated realization of the state vectors is available. The realization establishes a functional form of the conditional distribution, which is evaluated on observed data convenient for step ahead predictions. The paper uses European equity for relevance arguments and illustrational prediction purposes. Multifactor SV models empower volatility visibility and predictability enriching the amount of information available for equity market participants.

The conventional control variate method proposed by Kemna and Vorst for evaluating Asian options using the Black-Scholes model utilizes a constant control parameter.We generalize this method, applying it to a stochastic control process through the martingale representation of the conventional control. This generalized control variate has zero variance in the optimal case, whereas the conventional control can only reduce its variance by a finite factor. By means of option price approximations, the generalized control is reduced to a linear martingale control. It is straightforward to extend this martingale control to a non-linear situation such as the American Asian option problem. From the variance analysis of martingales, the performance of control variate methods depends on the distance between the approximate martingale and the optimal martingale. This measure becomes helpful for the design of control variate methods for complex problems such as Asian options using stochastic volatility models.We demonstrate multiple choices of controls and test them with Monte Carlo and quasi-Monte Carlo simulations. Quasi-Monte Carlo methods perform significantly better after adding a control when using the two-step control variate method, while the variance reduction ratios increase to 320 times for randomized quasi-Monte Carlo methods, compared with 60 times for Monte Carlo simulations with a control.

This paper is mainly about the research object of early exercise of American options by two-period binomial models. In this work, we use the two-period binomial model to analyze the price of American options and European options, decide whether to exercise early or not, and compare the differences between American options and European options. What is more, we use Python to help us with our calculations. Options are a kind of financial derivatives that help one to arbitrage, speculate, and hedge. To make more profits, people can use the binomial model to calculate the price of options. Through this analysis, we got to know each price of options in different situations, and we know that the price of American options is always equal to or larger than the price of European options. This work can help us to calculate the price of the option and decide if it is exercised early, and make sure to get more profits and apply it to daily life.

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.

The SABR stochastic volatility model with β-volatility β ? (0,1) and an absorbing barrier in zero imposed to the forward prices/rates stochastic process is studied. The presence of (possibly) nonzero correlation between the stochastic differentials that appear on the right hand side of the model equations is considered. A series expansion of the transition probability density function of the model in powers of the correlation coefficient of these stochastic differentials is presented. Explicit formulae for the first three terms of this expansion are derived. These formulae are integrals of known integrands. The zero-th order term of the expansion is a new integral formula containing only elementary functions of the transition probability density function of the SABR model when the correlation coefficient is zero. The expansion is deduced from the final value problem for the backward Kolmogorov equation satisfied by the transition probability density function. Each term of the expansion is defined as the solution of a final value problem for a partial differential equation. The integral formulae that give the solutions of these final value problems are based on the Hankel and on the Kontorovich-Lebedev transforms. From the series expansion of the probability density function we deduce the corresponding expansions of the European call and put option prices. Moreover we deduce closed form formulae for the moments of the forward prices/rates variable. The moment formulae obtained do not involve integrals or series expansions and are expressed using only elementary functions. The option pricing formulae are used to study synthetic and real data. In particular we study a time series (of real data) of futures prices of the EUR/USD currency's exchange rate and of the corresponding option prices. The website: http://www.econ.univpm.it/recchioni/finance/w18 contains material including animations, an interactive application and an app that helps the understanding of the paper. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website:http://www.econ.univpm.it/recchioni/finance.

This paper builds and implements a multifactor stochastic volatility model for the latent (and observable) volatility from the front month future contracts at the Intercontinental Commodity Exchange (ICE), London, applying Bayesian Markov chain Monte Carlo simulation methodologies for estimation, inference and model adequacy assessment. Stochastic volatility is the main way time‐varying volatility is modelled in financial markets. An appropriate scientific model description, specifying volatility as having its own stochastic process, broadens the applications into derivative pricing purposes, risk assessment and asset allocation and portfolio management. From an estimated optimal and appropriate stochastic volatility model, the paper reports risk and portfolio measures, extracts conditional one‐step‐ahead moments (smoothing), forecasts one‐step‐ahead conditional volatility (filtering), evaluates shocks from conditional variance functions, analyses multistep‐ahead dynamics and calculates conditional persistence measures. (Exotic) option prices can be calculated using the re‐projected conditional volatility. Observed market prices and implied volatilities establish market risk premiums. The analysis adds insight and enables forecasts to be made, building up the methodology for developing valid scientific commodity market models.

We present option pricing under the double stochastic volatility model with stochastic interest rates and double exponential jumps with stochastic intensity in this article. We make two contributions based on the existing literature. First, we add double stochastic volatility to the option pricing model combining stochastic interest rates and jumps with stochastic intensity, and we are the first to fill this gap. Second, the stochastic interest rate process is presented in the Hull–White model. Some authors have concentrated on hybrid models based on various asset classes in recent years. Therefore, we build a multifactor model with the term structure of stochastic interest rates. We also approximated the pricing formula for European call options by applying the COS method and fast Fourier transform (FFT). Numerical results display that FFT and the COS method are much faster than the numerical integration approach used for obtaining the semi-closed form prices. The COS method shows higher accuracy, efficiency, and stability than FFT. Therefore, we use the COS method to investigate the impact of the parameters in the stochastic jump intensity process and the existence of the process on the call option prices. We also use it to examine the impact of the parameters in the interest rate process on the call option prices.

I address the dichotomy between American put option pricing theory and the numerical algorithms designed to estimate American put option prices. The literature has focused only on pricing error. However, early exercise is the essence of American option pricing (exercising) and with it comes the possibility of early exercise error and its opportunity costs. I introduce an economically viable metric that identifies all the errors of American put option pricing algorithms. The accuracy of such algorithms can thereby be fully assessed. A rational option pricing result generalises the usual integral equation and motivates pure pricing error. This provides new intuition for the optimality condition for early exercise. Early exercise error is motivated by comparing discounted expected profits generated by the estimated model vs. the optimal early exercise model. The error measure applies to any put pricing algorithm and any benchmark. We illustrate our total error measure with a new algorithm.

In this article, we develop an identifiable multi-factor stochastic volatility model for international portfolio choice problems in complete and incomplete markets. Allowing for stochastic covariance between financial asset returns and foreign exchange rates, optimal investment strategies are derived in closed form and welfare losses arising from suboptimal investment strategies are analysed. Moreover, we provide a two-step procedure for estimating as well as calibrating the model parameters and use this ansatz to illustrate optimal investment decisions for the S&P 500, the German blue chip index DAX, and the USD/EUR foreign exchange rate. We find, both theoretically and empirically, that the model satisfies various well-known stylized facts of equity and foreign exchange rate markets and that investors who invest myopically or ignore derivative assets can incur substantial welfare losses implying strong evidence for significant welfare benefits from international diversification across different asset classes.

The paper describes an alternative options pricing method which uses a binomial tree linked to an innovative stochastic volatility model. The volatility model is based on wavelets and artificial neural networks. Wavelets provide a convenient signal/noise decomposition of the volatility in the nonlinear feature space. Neural networks are used to infer future volatility from the wavelets feature space in an iterative manner. The bootstrap method provides the 95% confidence intervals for the options prices. Market options prices as quoted on the Chicago Board Options Exchange are used for performance comparison between the Black‐Scholes model and a new options pricing scheme. The proposed dynamic volatility model produces as good as and often better options prices than the conventional Black‐Scholes formulae.

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.

We introduce a hybrid stochastic volatility model where the asset price process follows the Heston model and interest rates are governed by a two-factor stochastic model. Two cases are considered. First, it is assumed that interest rates and asset prices are uncorrelated. The characteristic function method is used to derive semi-analytical pricing formulae for plain vanilla options. In the second case we introduce a correlation between the asset price process and the short rate process and use Monte Carlo simulations for pricing options. To reduce the stochastic error, we implement the control variate method where an estimator of the option value for the uncorrelated case is used as a control variate. The options are priced with a varying correlation coefficient. We observe that the control variate method allows us to speed up Monte Carlo computations by a factor with the magnitude of several hundreds. The efficiency of the method is higher for smaller values of the correlation coefficient. We then study the impact a correlation between the two processes has on option prices. It has been noticed that the call option price is an increasing function of the correlation coefficient.