### Browsing by Author "McWalter, Thomas"

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- ItemOpen AccessA Review of Multilevel Monte Carlo Methods(2020) Jain, Rohin; McWalter, ThomasThe Monte Carlo method (MC) is a common numerical technique used to approximate an expectation that does not have an analytical solution. For certain problems, MC can be inefficient. Many techniques exist to improve the efficiency of MC methods. The Multilevel Monte Carlo (ML) technique developed Giles (2008) is one such method. It relies on approximating the payoff at different levels of accuracy and using a telescoping sum of these approximations to compute the ML estimator. This dissertation summarises the ML technique and its implementation. To start with, the framework is applied to a European call option. Results show that the efficiency of the method is up to 13 times faster than crude MC. Then an American put option is priced within the ML framework using two pricing methods. The Least Squares Monte Carlo method (LSM) estimates an optimal exercise strategy at finitely many instances, and consequently a lower bound price for the option. The dual method finds an optimal martingale, and consequently an upper bound for the price. Although the pricing results are quite close to the corresponding crude MC method, the efficiency produces mixed results. The LSM method performs poorly within an ML framework, while the dual approach is enhanced.
- ItemOpen AccessAdjoint Venture: Fast Greeks with Adjoint Algorithmic Differentiation(2017) McPetrie, Christopher Lindsay; McWalter, ThomasThis dissertation seeks to discuss the adjoint approach to solving affine recursion problems (ARPs) in the context of computing sensitivities of financial instruments. It is shown how, by moving from an intuitive 'forward' approach to solving a recursion to an 'adjoint' approach, one might dramatically increase the computational efficiency of algorithms employed to compute sensitivities via the pathwise derivatives approach in a Monte Carlo setting. Examples are illustrated within the context of the Libor Market Model. Furthermore, these ideas are extended to the paradigm of Adjoint Algorithmic Differentiation, and it is illustrated how the use of sophisticated techniques within this space can further improve the ease of use and efficiency of sensitivity calculations.
- ItemOpen AccessAnalytical Solution of the Characteristic Function in the Trolle-Schwartz Model(2019) Van Gysen, Richard John; McWalter, Thomas; Kienitz, JoergIn 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.
- ItemOpen AccessCharacteristic function pricing with the Heston-LIBOR hybrid model(2019) Sterley, Christopher; Ouwehand, Peter; McWalter, ThomasWe derive an approximate characteristic function for a simplified version of the Heston-LIBOR model, which assumes a constant instantaneous volatility structure in the underlying LIBOR market model. We also implement measures to improve the numerical stability of the characteristic function derived in this dissertation as well as the one derived by Grzelak and Oosterlee. The ultimate aim of the dissertation is to prevent these characteristic functions from exploding for given parameter values.
- ItemOpen AccessConcurrence Between the Displaced Libor Market and Hull-White Models(2021) Thantsha, Kgothatso; McWalter, ThomasThe 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.
- ItemOpen AccessThe cost of using misspecified models to exercise and hedge American options on coupon bearing bonds(2016) Welihockyj, Alexander; Silverman, Searle; McWalter, ThomasThis dissertation investigates the cost of using single-factor models to exercise and hedge American options on South African coupon bearing bonds, when the simulated market term structure is driven by a two-factor model. Even if the single factor models are re-calibrated on a daily basis to the term structure, we find that the exercise and hedge strategies can be suboptimal and incur large losses. There is a vast body of research suggesting that real market term structures are in actual fact driven by multiple factors, so suboptimal losses can be largely reduced by simply employing a well-specified multi-factor model.
- ItemOpen AccessEmployee Stock Option Valuation with Earnings-Based Vesting Condition(2018) Patel, Kavir; McWalter, Thomas; Musvosv, ChiedzaThe 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.
- ItemOpen AccessEstimating dynamic affine term structure models(2015) Pitsillis, Zachry Steven; Ouwehand, Peter; McWalter, ThomasDuffee and Stanton (2012) demonstrated some pointed problems in estimating affine term structure models when the price of risk is dynamic, that is, risk factor dependent. The risk neutral parameters are estimated with precision, while the price of risk parameters are not. For the Gaussian models they investigated, these problems are replicated and are shown to stem from a lack of curvature in the log-likelihood function. This geometric issue for identifying the maximum of an essentially horizontal log-likelihood has statistical meaning. The Fisher information for the price of risk parameters is multiple orders of magnitude smaller than that of the risk neutral parameters. Prompted by the recent results of Christoffersen et al. (2014) a remedy to the lack of curvature is attempted. An unscented Kalman filter is used to estimate models where the observations are portfolios of FRAs, Swaps and Zero Coupon Bond Options. While the unscented Kalman filter performs admirably in identifying the unobserved risk factor processes, there is little improvement in the Fisher information.
- ItemOpen AccessExposure modelling under change of measure(2017) Roberts, Christopher; Kienitz, JÃ¶rg; McWalter, ThomasThe credit risk of a portfolio is often managed via measures of counter-party exposure, such as potential future exposure (PFE) and expected exposure (EE), with these measures playing an important role in setting economic and regulatory capital levels. For the sake of risk measurement and risk management these exposure measures should be computed under the real-world probability measure. However, due to the similarity of these exposure calculations to those used in calculating credit valuation adjustments, some have begun to compute them under the risk-neutral measure instead. This is problematic, as the magnitudes of PFEs and EEs differ under different equivalent martingale measures and their associated numÃ©raires. Working with the Hull-White (HW) model of the short rate, the effect of a change of measure on the PFE and EE profiles of vanilla interest rate swaps and European swaptions is shown under three common measures: the money-market account measure, the T-forward measure and the Linear Gaussian Markovian (LGM) measure. A modified Least Squares Monte Carlo (LSM) algorithm, which allows for substantial computational savings, is then introduced in order to approximate contract level exposures under each of the aforementioned probability measures. Finally, a change of measure is implemented within the modified LSM algorithm in order to approximate exposure profiles under the real-world measure. The modified LSM algorithm is particularly useful for computing exposure profiles of contracts without closed-form valuation formulae, which would otherwise take significantly longer to compute via a standard Monte Carlo approach.
- ItemOpen AccessFunctional quantization-based stratified sampling(2017) Platts, Alexander; McWalter, ThomasFunctional quantization-based stratified sampling is a method for variance reduction proposed by Corlay and PagÃ¨s (2015). This method requires the ability to both create functional quantizers and to sample Brownian paths from the strata defined by the quantizers. We show that product quantizers are a suitable approximation of an optimal quantizer for the formation of functional quantizers. The notion of functional stratification is then extended to options written on multiple stocks and American options priced using the Longstaff-Schwartz method. To illustrate the gains in performance we focus on geometric brownian motion (GBM), constant elasticity of variance (CEV) and constant elasticity of variance with stochastic volatility (CEV-SV) models. The pricing algorithm is used to price knock-in, knockout, autocall, call on the max and path dependent call on the max options.
- ItemOpen AccessImplementation of numerical Fourier method for second order Taylor schemes(2019) Mashalaba, Qaphela; McWalter, ThomasThe problem of pricing contingent claims in a complete market has received a significant amount of attention in literature since the seminal work of Black, Fischer and Scholes, Myron (1973). It was also in 1973 that the theory of backward stochastic differential equations (BSDEs) was developed by Bismut, Jean-Michel (1973), but it was much later in the literature that BSDEs developed links to contingent claim pricing. This dissertation is a thorough exposition of the survey paper Ruijter, Marjon J and Oosterlee, Cornelis W (2016) in which a highly accurate and efficient Fourier pricing technique compatible with BSDEs is developed and implemented. We prove our understanding of this technique by reproducing some of the numerical experiments and results in Ruijter, Marjon J and Oosterlee, Cornelis W (2016), and outlining some key implementationl considerations.
- ItemOpen AccessInterpolation of Forward Rates in the LIBOR Market Model(2020) Mbele, Buhlebezwe Bandile Sthombe; McWalter, ThomasSince its development in 1997, the LIBOR market model has gained widespread use in interest rate modelling, largely owing to its consistency with the Black futures formula for pricing interest rate caps and floors. From its original construction(s), the LIBOR market model specifies a discrete set of forward rates that correspond to a fixed tenor structure, e.g. market tenors. This implies the pricing of interest rate contingent claims is restricted to claims with cashflow dates that coincide with the fixed tenor structure. In this light, several interpolation schemes have been suggested to handle the pricing restrictions, however at the cost of introducing possible arbitrage opportunities. The present dissertation studies four such interpolation schemes, paying particular attention to arbitrage-free interpolation schemes: Piterbarg deterministic interpolation, Schlogl deterministic interpolation, Schlogl stochastic interpolation, and Beveridge-Joshi stochastic interpolation.
- ItemOpen AccessLocal Stochastic Volatilityâ€”The Hyp-Hyp Model(2020) Cowen, Nicholas; McWalter, Thomas; Kienitz, JorgVolatility modelling is used predominantly in order to explain the volatility smile observed in the market. Stochastic volatility models are mainly used to capture the curvature of a volatility smile while local volatility models generally model the skew. Jackel and Kahl Â¨ (2008) present a hyperbolic-local hyperbolic-stochastic volatility (Hyp-Hyp) model which aims to improve upon existing local and stochastic volatility models such as the stochastic alpha, beta, rho (SABR) and constant elasticity of variance (CEV) models. The advantageous features of the Hyp-Hyp model are corroborated by implementing the model. Jackel and Kahl Â¨ (2008) investigate the accuracy of a scaled analytical approximation for implied volatility, based on approximations presented by Watanabe (1987) and Fouque et al. (2000), for the Hyp-Hyp model. They use the approximation to derive an expression for the delta of an option. This dissertation analyses the Hyp-Hyp model, as well as the approximation, by deriving expressions for other sensitivities and by investigating the effect of the Hyp-Hyp model parameters on the volatility smile. The accuracy of the analytical approximation for functional forms other than those defined by the Hyp-Hyp model is explored. A derivation of the approximation is undertaken, presenting corrections to the expressions introduced by Kahl (2007) and used by Jackel and Kahl Â¨ (2008).
- ItemOpen AccessMixed Monte Carlo in the foreign exchange market(2017) Baker, Christopher; McWalter, Thomas; Searle Silverman, Searle; Maze, SheldonThe stochastic differential equation (SDE) describing the spot FX rate is of central importance to modelling FX derivatives. A Monte Carlo estimate of the discounted individual payoffs of FX derivatives is taken to arrive at the price, provided there does not exist a closed form solution for the price. One propagates the FX spot rate through time under risk-neutral dynamics to realise the before-mentioned payoffs. A drawback to Monte Carlo becomes evident when the model dynamics become more complicated, such as when more dimensions are added to the dynamics of the model. These additional dimensions can be stochastic volatility and/or stochastic domestic and foreign short rates. This dissertation describes the calibration of such a model using mixed Monte Carlo, as described in Cozma and Reisinger (2015), to both model-generated and market data. Profit and loss analysis of hedging FX derivatives using the mixed Monte Carlo method is conducted when hedging against both model-generated and market data .
- ItemOpen AccessModelling stochastic multi-curve basis(2017) Dalton, Rowan; Kienitz, JÃ¶rg; McWalter, ThomasAs a consequence of the 2007 financial crisis, the market has shifted towards a multi-curve approach in modelling the prevailing interest rate environment. Currently, there is a reliance on the assumption of deterministic- or constant-basis spreads. This assumption is too simplistic to describe the modern multi-curve environment and serves as the motivation for this work. A stochastic-basis framework, presented by Mercurio and Xie (2012), with one- and two-factor OIS short-rate models is reviewed and implemented in order to analyse the effect of the inclusion of stochastic-basis in the pricing of interest rate derivatives. In order to preclude the existence of negative spreads in the model, a constraint on the spread model parameters is necessary. The inclusion of stochastic-basis results in a clear shift in the terminal distributions of FRA and swap rates. In spite of this, stochastic-basis is found to have a negligible effect on cap/floor and swaption prices for the admissible spread model parameters. To overcome challenges surrounding parameter estimation under the framework, a rudimentary calibration procedure is developed, where the spread model parameters are estimated from historical data; and the OIS rate model parameters are calibrated to a market swaption volatility surface.
- ItemOpen AccessOptimal tree methods(2014) Rudd, Ralph; McWalter, Thomas; Taylor, DavidAlthough traditional tree methods are the simplest numerical methods for option pricing, much work remains to be done regarding their optimal parameterization and construction. This work examines the parameterization of traditional tree methods as well as the techniques commonly used to accelerate their convergence. The performance of selected, accelerated binomial and trinomial trees is then compared to an advanced tree method, Figlewski and Gao's Adaptive Mesh Model, when pricing an American put and a Down-And-Out barrier option.
- ItemOpen AccessPotential Future Exposure in the Presence of Initial Margin(2019) Nevin, James; McWalter, ThomasThis dissertation considers the concept of potential future exposure, and how initial margin can be used to mitigate it. In addition to this, the cost of implementing initial margin is estimated, and some of the difficulties associated with it are addressed. The two primary techniques for calculating initial margin considered are nested Monte Carlo, and Gaussian Least Squares Monte Carlo. These two techniques are compared for effectiveness. It is shown that the nested Monte Carlo technique performs well under numerous conditions, and that the Gaussian Least Squares Monte Carlo relies on particular model and instrument characteristics.
- ItemOpen AccessPricing a Bermudan option under the constant elasticity of variance model(2017) Rwexana, Kwaku; McWalter, Thomas; Rudd, RalphThis dissertation investigates the computational efficiency and accuracy of three methodologies in the pricing of a Bermudan option, under the constant elasticity of variance (CEV) model. The pricing methods considered are the finite difference method, least squares Monte Carlo method and recursive marginal quantization (RMQ) method. Specific emphasis will be on RMQ, as it is the most recent method. A plain vanilla European option is initially priced using the above mentioned methods, and the results obtained are compared to the Black-Scholes option pricing formula to determine their viability as pricing methods. Once the methods have been validated for the European option, a Bermudan option is then priced for these methods. Instead of using the Black-Scholes option pricing formula for comparison of the prices obtained, a high-resolution finite difference scheme is used as a proxy in the absence of an analytical solution. One of the main advantages of the recursive marginal quantization (RMQ) method is that the continuation value of the option is computed at almost no additional computational cost, this with other contributing factors leads to a computationally efficient and accurate method for pricing.
- ItemOpen AccessPricing stochastic volatility models using random grids(2022) Rajkumar, Rishay; McWalter, ThomasAssets can be priced using a variety of numerical methods. In some instances, a particular numerical method may be more appropriate than others. If one method is used to calibrate the model to market conditions, but another method is used to price the asset, the results obtained may be inconsistent. This dissertation addresses the fundamental problem of this bias that is introduced when calibrating and pricing options using inconsistent methods. The random grids approach, developed by Andreasen and Huge (2011), is a pricing method that guarantees discrete consistency between calibration, finite difference solution and Markov-chain MonteCarlo simulation based on the random grids approach. This dissertation provides a review and implementation of this random grids approach for pricing under the Heston model as well as the stochastic local volatility model. Consistent results are obtained for a call option under the various pricing methods using similar parameters as those used in the random grids paper. More specifically, when using a Heston model, consistent prices are obtained for the characteristic function pricing method, the backward finite difference method, the forward finite difference method as well as the Markov-chain Monte-Carlo method based on the random grids approach. Similarly, consistent prices are obtained under the stochastic local volatility model for the backward finite difference method, the forward finite difference method and the Markov-chain Monte-Carlo method based on the random grids approach.
- ItemOpen AccessPricing swaptions on amortising swaps(2018) Masutha, Ndinae Nico; McWalter, ThomasIn this dissertation, two efficient approaches for pricing European options on amortising swaps are explored. The first approach is to decompose the pricing of a European amortising swaption into a series of discount bond options, with an assumption that the interest rate follows a one-factor affine model. The second approach is using a one-dimensional numerical integral technique to approximate the price of European amortising swaption, with an assumption that the interest rate follows an additive two-factor affine model. The efficacy of the two methods was tested by making a comparison with the prices generated using Monte Carlo methods. Two methods were used to accelerate the convergence rate of the Monte Carlo model, a variance reduction method, namely the control variates technique and a method of using deterministic low-discrepancy sequences (also called quasi-Monte Carlo methods).