A profit or loss (P&L) of a dynamically hedged option depends on the implied volatility used to price the option and implement the hedges. Break-even volatility is a method of solving for the volatility which yields no profit or loss based on replicating the hedging procedure of an option on a historical share price time series. This dissertation investigates the traditional break-even volatility method on simulated data, how the break-even formula is derived and details the implementation with reference to MATLAB. We extend the methodology to the Heston model by changing the reference model in the hedging process. Resultantly, the need to employ characteristic function pricing methods arises to calculate the Heston model sensitivities. The break-even volatility solution is then found by means of an optimisation of the continuously delta hedged P&L over the Heston model parameters.

In 2009, Trolle and Schwartz (2008) produced an instantaneous forward interest rate model with several stylised facts such as stochastic volatility. They derived pricing formulae in order to price bonds and bond options, which can be altered to price interest rate options such as caplets, caps and swaptions. These formulae involve implementing numerical methods for solving an ordinary differential equation (ODE). Schumann (2016) confirmed the accuracy of the pricing formulae in the Trolle and Schwartz (2008) model using Monte-Carlo simulation. Both authors used a numerical ODE solver to estimate the ODE. In this dissertation, a closed-form solution for this ODE is presented. Two solutions were found. However, these solutions rely on a simplification of the instantaneous volatility function originally proposed in the Trolle and Schwartz (2008) model. This case happens to be the stochastic volatility version of the Hull and White (1990) model. The two solutions are compared to an ODE solver for one stochastic volatility term and then extended to three stochastic volatility terms.

The valuation of employee stock options has become a key requirement due to the rapid growth in the use of these options as a means of employee compensation. IFRS 2 Share-based Payment stipulates that these instruments must be valued and expensed on the date the awards are issued. This dissertation aims to value an employee stock option, in a case where both the equity and vesting (performance) condition are based on a reported earnings process. The equity dependency on earnings stems from the fact that we are primarily concerned with the valuation of employee stock options that are issued by a private firm. We implement a capital structure framework provided by Goldstein, Ju and Leland (2001). In this framework, equity and debt are derived from an underlying EBIT process that is governed by a geometric Brownian motion. The model also accounts for taxation and bankruptcy. The research aim is addressed by incorporating the capital structure model into our employee stock option pricing framework.

Financial practitioners use models in order to price, hedge and measure risk. These models are reliant on assumptions and are prone to ”model risk”. Increased innovation in complex financial products has lead to increased risk exposure and has spurred research into understanding model risk and its underlying factors. This dissertation quantifies model risk inherent in Value-at-Risk (VaR) on a variety of portfolios comprised of European options written on the ALSI futures index across various maturities. The European options under consideration will be modelled using the Black-Scholes, Heston and Variance-Gamma models.

The post 2007-financial crisis era has led to renewed zeal in quantifying market incompleteness when pricing contingent claims. This quantification exercise is necessary in maintaining a stable and sustainable banking operation and thus the XVAs have emerged as the metrics for market incompleteness. This dissertation focuses solely on the capital valuation adjustment (KVA) and aims to use the definition of KVA as set out by Albanese et al. (2016) in an investigation of different numerical techniques for calculating KVA. A single equity forward is considered first, followed by an equity option and then portfolios of options on two underlying assets, with the dissertation ending by considering a practical example on discrete delta and vega-delta hedging an index option. The numerical approaches explored are the binomial tree method and a combination of the crude and quasi-Monte Carlo method.

This dissertation details the performance of two specific trading strategies which are based on the Johansen-Ledoit-Sornette (JLS) model. Both positive and negative bubbles are modelled as a log-periodic power law (LPPL) ending in a finite time singularity. The stock prices of the constituents of the FTSE/JSE Top40 index are taken as inputs to the JLS model from 3 June 2003 to 31 August 2015. It is shown that for certain time horizons into the past, the JLS based trading strategies significantly outperform random trading strategies. However this result is highly dependent on how far the model looks into the past, and if the model is calibrating to positive or negative bubbles. The lack of research with regards to the "stylized facts" of the JLS model, specifically relating to the time horizon and type of bubble, poses a significant hurdle in correctly identifying a LPPL structure in stock prices. These core features of the JLS model were developed from a number of positive bubbles that built up over many years. The results suggest that these features may not apply over all time horizons, and for both types of bubbles.

Traditional option pricing methods like Monte Carlo simulation can be time consuming when pricing and hedging exotic options under stochastic volatility models like the Heston model. The purpose of this research is to apply the Gaussian Process Regression (GPR) method to the pricing and hedging of exotic options under the Black-Scholes and Heston model. GPR is a supervised machine learning technique which makes use of a training set to train an algorithm so that it makes predictions. The training set is composed of the input vector X which is a n × p matrix and Y an n×1 vector of targets, where n is the number of training input vectors and p is the number of inputs. Using a GPR with a squared-exponential kernel tuned by maximising the log-likelihood, we established that this GPR works reasonably for pricing Barrier options and Asian options under the Heston model. As compared to the traditional method of Monte Carlo simulation, GPR technique is 2 000 times faster when pricing barrier option portfolios of 100 assets and 1 000 times faster computing a portfolio of Asian options. However, the squared-exponential GPR does not compute reliable hedging ratios under Heston model, the delta is reasonably accurate, but the vega is off.

The recent explosion of work on rough volatility and fractional Brownian motion has led to the development of a new generation of stochastic volatility models. Such models are able to capture a wide range of stylised facts that classical models simply do not. While these models have sound mathematical underpinnings, they are difficult to implement, largely due to the fact that fractional Brownian motion is neither Markovian nor a semimartingale. One idea is to investigate the behaviour of these models as maturities become very small (or very large) and consider asymptotic estimates for quantities of interest. Here we investigate the performance of small-time asymptotic formulae for the cumulant generating function of the Fractional Heston model as presented in Guennoun et al. (2018). These formulae and their effectiveness for small-time pricing are interrogated and compared against the Rough Heston model proposed in El Euch and Rosenbaum (2019).

The concurrence between the displaced lognormal forward-Libor model (DLFM), Gaussian Heath-Jarrow-Morton (GHJM) model and Hull-White (HW) model is explored. We briefly present the theory underpinning these models, specifically focusing on single factors. A useful volatility relation result adapted from Andersen and Piterbarg (2010) is derived. It relates the instantaneous volatility functions of the GHJM model and the DLFM model. The volatility relation allows us to state a specific GHJM model and derive a corresponding DLFM model that it is concurrent with. We take the Hull-White model and derive its corresponding GHJM model, the volatility of the GHJM model is then fed into the volatility relation in order to derive the corresponding DLFM model. This was sufficient mathematical proof of the concurrence, but numerical confirmation is also essential. The HW, GHJM and DLFM models were implemented, with applications to pricing European swaptions. Numerical results show that swaption prices are consistent across the three models. This provides good numerical evidence to support the concurrence between the DLFM and HW models.

Includes bibliographical references. / This dissertation is an investigation into realised volatility (RV) estimators. Here, RV is defined as the sum-of-squared-returns (SSR) and is a proxy for integrated volatility (IV), which is unobservable. The study focuses on a subset of the universe of RV estimators. We examine three categories of estimators: historical, high-frequency (HF) and implied. The need to estimate RV is predominantly in the hedging of options and is not concerned with speculation or forecasting. The main research questions are; (1) what is the best RV estimator in a historical study of S&P 500 data? (2) What is the best RV estimator in a Monte Carlo simulation when delta hedging synthetic options? (3) Do our findings support the stylized fact of `Asymmetry in time scales' (Cont, 2001)? In the answering of these questions, further avenues of investigation are explored. Firstly, the VIX is used as the implied volatility. Secondly, the Monte Carlo simulation generates stock price paths with random components in the stock price and the volatility at each time point. The distribution of the input volatility is varied. The question of asymmetry in time scales is addressed by varying the term and frequency of historical data. The results of the historical and Monte Carlo simulation are compared. The SSR and two of the HF estimators perform best in both cases. Accuracy of estimators using long term data is shown to perform very poorly.