Geometric Asian Option Pricing in General Affine Stochastic Volatility Models with Jumps

In this paper we present some results on Geometric Asian option valuation for affine stochastic volatility models with jumps. We shall provide a general framework into which several different valuation problems based on some average process can be cast, and we shall obtain close-form solutions for some relevant affine model classes.Comment: 20 page

General optimized lower and upper bounds for discrete and continuous arithmetic Asian options

We propose an accurate method for pricing arithmetic Asian options on the discrete or continuous average in a general model setting by means of a lower bound approximation. In particular, we derive analytical expressions for the lower bound in the Fourier domain. This is then recovered by a single univariate inversion and sharpened using an optimization technique. In addition, we derive an upper bound to the error from the lower bound price approximation. Our proposed method can be applied to computing the prices and price sensitivities of Asian options with fixed or floating strike price, discrete or continuous averaging, under a wide range of stochastic dynamic models, including exponential Lévy models, stochastic volatility models, and the constant elasticity of variance diffusion. Our extensive numerical experiments highlight the notable performance and robustness of our optimized lower bound for different test cases

A Comparative Predicting Stock Prices using Heston and Geometric Brownian Motion Models

This paper presents a novel approach to predicting stock prices using technical analysis. By utilizing Ito's lemma and Euler-Maruyama methods, the researchers develop Heston and Geometric Brownian Motion models that take into account volatility, interest rate, and historical stock prices to generate predictions. The results of the study demonstrate that these models are effective in accurately predicting stock prices and outperform commonly used statistical indicators. The authors conclude that this technical analysis-based method offers a promising solution for stock market prediction

Multi-scale Volatility in Option Pricing

This PhD thesis investigated the influence of kaolin and bentonite clays in the ore on flotation, filtration and centrifugal concentration. The results showed that the presence of particularly bentonite in the ore had a detrimental effect on flotation and filtration. The information generated from this work will advance our knowledge as well as provide important information for plant metallurgists. The project, therefore, is essential for the mineral industry that process clay-containing ores

Efficient valuation of exotic derivatives with path-dependence and early exercise features

The main objective of this thesis is to provide effective means for the valuation of popular financial derivative contracts with path-dependence and/or early-exercisable provisions. Starting from the risk-neutral valuation formula, the approach we propose is to sequentially compute convolutions of the value function of the contract at a monitoring date with the transition density between two dates, to provide the value function at the previous monitoring date, until the present date. A rigorous computational algorithm for the convolutions is then developed based on transformations to the Fourier domain. In the first part of the thesis, we deal with arithmetic Asian options, which, due to the growing popularity they enjoy in the financial marketplace, have been researched signicantly over the last two decades. Although few remarkable approaches have been proposed so far, these are restricted to the market assumptions imposed by the standard Black-Scholes-Merton paradigm. Others, although in theory applicable to Lévy models, are shown to suffer a non-monotone convergence when implemented numerically. To solve the Asian option pricing problem, we initially propose a flexible framework for independently distributed log-returns on the underlying asset. This allows us to generalize firstly in calculating the price sensitivities. Secondly, we consider an extension to non-Lévy stochastic volatility models. We highlight the benefits of the new scheme and, where relevant, benchmark its performance against an analytical approximation, control variate Monte Carlo strategies and existing forward convolution algorithms for the recovery of the density of the underlying average price. In the second part of the thesis, we carry out an analysis on the rapidly growing market of convertible bonds (CBs). Despite the vast amount of research which has been undertaken yet. This is due to the need for proper modelling of the CBs composite payout structure and the multi factor modelling arising in the CB valuation. Given the dimensional capacity of the convolution algorithm, we are now able to introduce a new jump diffusion structural approach in the CB literature, towards more realistic modelling of the default risk, and further include correlated stochastic interest rates. This aims at fixing dimensionality and convergence limitations which previously have been restricting the range of applicability of popular grid- based, lattice and Monte Carlo methods. The convolution scheme further permits flexible handling of real-world CB specications; this allows us to properly model the call policy and investigate its impact on the computed CB prices. We illustrate the performance of the numerical scheme and highlight the effects originated by the inclusion of jumps

Asymptotic techniques and stochastic volatility in option pricing problems

This thesis investigates the use of asymptotic techniques and stochastic volatility models in option pricing problems. For the Heston stochastic volatility model, a fast mean reverting asymptotic approach, similar to Fouque et al. (2000) is taken. The asymptotic solution derived extends on their most recent work, with the solution presented expanded out to four terms. The worthiness and robustness of the asymptotic solution is then tested by applying it to the theory of locally risk minimizing hedges. The asymptotic approach is then further developed by applying it to a real options framework, allowing for a better understanding of what the asymptotic solution actually reflects under this model, and in particular, how it affects the optimal investment threshold, a key component in real options theory. Asian options with general call type payoffs are then investigated and equivalency theorems derived linking them to Australian options under both a Black-Scholes model and a Heston stochastic volatility model. Examining Asian options from this ‘Australian’ perspective gives a new angle on how one can approach the pricing of Asian options under stochastic volatility. Advances are made in areas such as the PDE pricing equation, and Monte Carlo simulations. Finally, an asymptotic solution under a low volatility assumption in the Black-Scholes model for an Australian call option is derived. This extends the work of Dewynne and Shaw (2008), to cater for Australian options. It is argued that this can be used as an alterative to existing approximations under a low volatility regime, for both pricing general Australian call options and general Asian options through the equivalency theorems. Aside from the over arching theme of asymptotic techniques and stochastic volatility, this thesis looks at how each of the newly presented solutions and/or methods, can be of benefit to the pricing of their respective option types. In particular, focus will be placed on the usage, accuracy and computational efficiency of these techniques. In all cases, the new solutions provide a high level of accuracy compared to the true solution, and/or are much more computationally efficient than existing methodologies. The simplicity and advantages of these solutions make a valuable contribution to current option pricing techniques