### Browsing by Author "Ouwehand, Peter"

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- ItemOpen AccessAccelerated Adjoint Algorithmic Differentiation with Applications in Finance(2017) De Beer, Jarred; Ouwehand, Peter; Kuttel, Michelle MaryAdjoint Differentiation's (AD) ability to calculate Greeks efficiently and to machine precision while scaling in constant time to the number of input variables is attractive for calibration and hedging where frequent calculations are required. Algorithmic adjoint differentiation tools automatically generates derivative code and provide interesting challenges in both Computer Science and Mathematics. In this dissertation we focus on a manual implementation with particular emphasis on parallel processing using Graphics Processing Units (GPUs) to accelerate run times. Adjoint differentiation is applied to a Call on Max rainbow option with 3 underlying assets in a Monte Carlo environment. Assets are driven by the Heston stochastic volatility model and implemented using the Milstein discretisation scheme with truncation. The price is calculated along with Deltas and Vegas for each asset, at a total of 6 sensitivities. The application achieves favourable levels of parallelism on all three dimensions implemented by the GPU: Instruction Level Parallelism (ILP), Thread level parallelism (TLP), and Single Instruction Multiple Data (SIMD). We estimate the forward pass of the Milstein discretisation contains an ILP of 3.57 which is between the average range of 2-4. Monte Carlo simulations are embarrassingly parallel and are capable of achieving a high level of concurrency. However, in this context a single kernel running at low occupancy can perform better with a combination of Shared memory, vectorized data structures and a high register count per thread. Run time on the Intel Xeon CPU with 501 760 paths and 360 time steps takes 48.801 seconds. The GT950 Maxwell GPU completed in 0.115 seconds, achieving an 422⇥ speedup and a throughput of 13 million paths per second. The K40 is capable of achieving better performance.
- ItemOpen AccessAlternatives to the Black-Scholes model(2001) Durrell, Fernando; Ouwehand, PeterIn this paper, I consider alternative models to the one posited by Black and Scholes. I consider discontinuous security price movements, non-constant volatility, and models very different from the Black-Scholes model. I found that most of the model prices for the close to at-the-money options are very different from the market prices. In general, the models did poorly in producing similar prices as the market.
- ItemOpen AccessAnalysis of convertible bonds(2004) Thompson, Kevin; Ouwehand, PeterIncludes bibliographical references ( leaves 90-92).
- ItemOpen AccessApplication of Effective Markovian Projection to SABR and Heston Models(2023) Bagraim, Jacques; Ouwehand, Peter; Mc Walter ThomasModel flexibility is often at odds with tractable pricing, and models with tractable pricing often lack flexibility. This poses an issue when calibrating a model to market data where tractability and flexibility are both required. We investigate an approach that allows one model to be projected onto another, potentially allowing for a flexible model to be represented by a tractable one. Here, Effective Markovian Projection is used to obtain equivalent Heston model parameters from a range of SABR models with different skew parameters using two distinct point-matching algorithms. The implied parameters are used to price European claims under a variety of schemes in order to outline the efficacy in this context. We see that this technique is accurate when the underlying probability densities of both models match closely, i.e., when the SABR skew parameter approaches unity, as is seen by comparing prices of claims using Classic Markovian Projection where the underlying SABR processes share the same density. PDE and perturbation SABR prices match closely while Heston characteristic function prices become unstable at lower skew parameters and far in-the-money and out-the-money values of the strike. Lastly, a potential improvement to this application involving error-correction terms is proposed for further application.
- ItemOpen AccessApplications of Gaussian Process Regression to the Pricing and Hedging of Exotic Derivatives(2021) Muchabaiwa, Tinotenda Munashe; Ouwehand, PeterTraditional 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.
- ItemOpen AccessAsymptotics of the Rough Heston Model(2021) Hayes, Joshua J; Ouwehand, PeterThe 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).
- 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 AccessCongruences and amalgamation in small lattice varieties(1998) Ouwehand, Peter; Rose, HenryWhen it became apparent that many varieties of algebras do not satisfy the Amalgamation Property, George Grätzer introduced the concept of an amalgamation class of a variety . The bulk of this dissertation is concerned with the amalgamation classes of residually small lattice varieties, with an emphasis on lattice varieties that are ﬁnitely generated. Our main concern is whether the amalgamation classes of such varieties are elementary classes or not. Chapters 0 and 1 provide a more detailed guide and summary of new and known results to be found in this dissertation. Chapter 2 is concerned with a cofinal sub-class of the amalgamation class of a residually small lattice variety, namely the class of absolute retracts, and completely characterizes the absolute retracts of ﬁnitely generated lattice varieties. Chapter 3 explores the strong connection between amalgamation and congruence extension properties in residually small lattice varieties. In Chapter 4, we investigate the closure of the amalgamation class under ﬁnite products. Chapter 5 is concerned with the amalgamation class of the variety generated by the pentagon. We prove that this amalgamation class is not an elementary class, but that, surprisingly, the class of all bounded members of the amalgamation class is a ﬁnitely axiomatizable Horn class. Chapters 6 and 7 introduce two techniques for proving that the amalgamation class of a residually small lattice variety is not an elementary class, and we give many examples. Finally, in Chapter 8, we look at the amalgamation classes of some residually large varieties, namely those generated by a ﬁnite dimensional simple lattice.
- ItemOpen AccessConstrained portfolio selection with Markov and non-Markov processes and insiders(2007) Durrell, Fernando; Ouwehand, Peter; Abraham, HaimWord processed copy. Includes bibliographical references (p. 158-168).
- ItemOpen AccessDeep Calibration of Option Pricing Models(2022) Dadah, Sahil; Ouwehand, PeterThis dissertation investigates the calibration efficiency of short rate models using deep neural networks. The main focus is on the calibration of one-and-two factor Hull-White models to caplets and swaptions data, where the inputs are interest rate derivative prices or implied volatilities, and the outputs are the model parameters. A direct and indirect neural network calibration framework is adopted. The former method involves a direct inversion of the standard option pricing function using neural network. The indirect framework uses two consecutive steps; the first step estimates the option pricing function using a neural network. This is followed by applying the pre-trained model in a calibration procedure to fit the model parameters to a set of market observables. The neural networks are trained using simulated data and an optimum set of hyperparameters is obtained via the Bayesian optimization. The best set of hyperparameters is used to train the networks and tested on out-of-sample actual market yield curves data. It is shown that the direct method has substantial improvements in time with a sacrifice in accuracy (a mean relative error of 2.88%). On the other hand, using the indirect method, it is shown that the calibrated parameters reprice the set of options to a mean relative error of less than 0.1% (similar to numerical calibration), with a significant improvement in speed whose execution is twenty-six times faster compared to the conventional calibration procedures currently used.
- ItemOpen AccessDeep Hedging of basis risk(2022) Adewusi, Olatomiwa Ayooluwa; Ouwehand, PeterBasis risk arises when the writer of a contingent claim cannot trade in the underlying asset and must use a correlated proxy asset to hedge the contingent claim. Suppose the proxy asset is not perfectly correlated to the underlying. In that case, there is a risk that the hedge portfolio does not precisely track the contingent claim, which may lead to significant losses at maturity. There are several existing approaches to hedging and pricing of contingent claims in the presence of basis risk. The existing approaches considered in this dissertation are based on the quadratic and exponential utility functions. This dissertation compares these current approaches to a new policy that parameterises the hedge parameters as a recurrent neural network at each rebalancing date. This new approach is called Deep Hedging, and under this approach, the hedge parameters are determined in a model agnostic way. This is achieved using Long Short-Term Memory networks written in TensorFlow. This allows one to make the hedge parameters at each time point a function of current market data and previous hedging decisions. Deep Hedging is Greek-free and more easily allows for the incorporation of other market frictions, like transaction costs, compared to existing approaches. Lastly, we can find optimal hedging strategies under coherent risk metrics, like expected shortfall, using the Deep Hedging approach and given a price. By fixing the volatility and correlation parameters, Deep Hedging produces results that are comparable to the best existing strategies, in both complete and incomplete market settings, across a variety of moneyness levels.
- ItemOpen AccessEnlargement of Filtration, Backward Stochastic Differential Equations and Optimal Stopping Problems(2022) Soane, Andrew; Ouwehand, PeterThis thesis focuses on the application of the enlargement of filtration to backward stochastic differential equations (BSDEs) and optimal stopping problems. In particular, the thesis develops the theory of the progressive enlargement of filtration with multiple random times and their associated marks. Several extensions of the classical progressive enlargement of filtration are derived, including a semimartingale decomposition theorem and a martingale representation theorem. The extensions then allow for the study of BSDEs and optimal stopping problems in an enlarged filtration. BSDEs are a very useful tool in stochastic optimal control and mathematical finance, the usefulness in the latter being that the solutions provide simultaneous calculation of derivative prices and their corresponding hedging strategies. Enlargement of filtration has a very intuitive application to BSDEs in a financial context, it models the effect that additional information has on the valuation of derivatives and their hedging strategies. This thesis develops certain classical results on BSDEs in the context of enlargement of filtration. The thesis then progresses to studying the effect of additional information on the value process of an optimal stopping problem. This again has an intuitive application to finance, as the effect of valuing American contingent claims in the presence of additional information. A very useful decomposition of the Snell envelope is derived. The thesis is rounded out with several applications of certain key results to topical fields in mathematical finance such as utility optimisation, risk metrics and Snell envelopes.
- ItemOpen AccessEquity options and stochastic interest rates : error in Black-Scholes prices and hedges for European- and American-style equity options when short rates are Ornstein-Uhlenbeck(2006) Acott, David M; Ouwehand, PeterThis dissertation considers the errors when using Black-Scholes prices and hedges for European equity options (Black&Scholes (1973), Merton (1973)) and American equity options (Karatzas (1988)) in an economy with stochastic interest rates. In particular, we consider an economy with Vasicek (1977) type interest rates.
- 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 AccessEstimation of Shadow-Rate Term Structure Models Near the Zero-Lower Bound(2019) Esmail, Shabbirhussein; Ouwehand, PeterThough it is customary to use standard Gaussian term structure models for term structure modelling, this becomes theoretically implausible in cases when nominal interest rates are near zero: Gaussian models can have arbitrarily large negative rates, whereas arbitrage considerations dictate that rates should remain positive (or very slightly negative at most). Black (1995) suggests that interest rates include an optionality which restricts them to non-negative values. This introduces a non-linearity at the zero-lower bound that makes these so-called shadow-rate models a computational challenge. This dissertation analyses the shadow-rate approximations suggested by Krippner (2013) and Priebsch (2013) for the Vasicek and ˇ arbitrage-free Nelson-Siegel (AFNS) models. We also investigate and compare the accuracy of the iterated extended Kalman filter (IEKF) with that of the unscented Kalman filter (UKF). We find that Krippner’s approach approximates interest rates within reasonable bounds for both the 1-factor Vasicek and AFNS models. Prieb- ˇ sch’s first-cumulant method is more accurate than Krippner’s method for a 1-factor Vasicek model, while Priebsch’s second-cumulant method is deemed impractical ˇ because of the computational time it takes. In a multi-factor AFNS model, only Krippner’s framework is feasible. Moreover, the IEKF outperforms the UKF in terms of filtering with no significant difference in run-time.
- ItemOpen AccessFactorization properties of universal algebras(2010) Wiggins, Harry; Ouwehand, Peter; Künzi, Hans-Peter A; Gilmour, Christopher Robert AndersonThis dissertation deals with algebraic structures that can be written as the product of directly indecomposable algebras in a unique way up to isomorphism, known as the Unique Factorization Property. Here we undertake the task of collecting all the major results discovered by a few mathematicians (A. Tarski, B. J´onsson, R. Mckenzie, C. Chang, G. Birkhoff, L. Lovasz, etc.) over the past century. Another goal of this thesis was to highlight important and to introduce fresh techniques. The scope of most of them is still unknown and hopefully they can be utilised further to yield new results to revive this beautiful branch of mathematics.
- ItemOpen AccessFourier pricing of two-asset options: a comparison of methods(2018) Roberts, Jessica Ellen; Ouwehand, Peter; Huang, Chun-SungFourier methods form an integral part in the universe of option pricing due to their speed, accuracy and diversity of use. Two types of methods that are extensively used are fast Fourier transform (FFT) methods and the Fourier-cosine series expansion (COS) method. Since its introduction the COS method has been seen to be more efficient in terms of rate of convergence than its FFT counterparts when pricing vanilla options; however limited comparison has been performed for more exotic options and under varying model assumptions. This paper will expand on this research by considering the efficiency of the two methods when applied to spread and worst-of rainbow options under two different models - namely the Black-Scholes model and the Variance Gamma model. In order to conduct this comparison, this paper considers each option under each model and determines the number of terms until the price estimate converges to a certain level of accuracy. Furthermore, it tests the robustness of the pricing methodologies to changes in certain discretionary parameters. It is found that although under the Black-Scholes model the COS method converges in fewer terms than the FFT method for both spread options (32 versus 128 terms) and the rainbow options (64 versus 512 terms), this is not the case under the more complex Variance Gamma model where the terms to convergence of both methods are similar. Both the methodologies are generally robust against changes in the discretionary variables; however, a notable issue appears under the implementation of the FFT methodology to worst-of rainbow options where the choice of the truncated integration region becomes highly influential on the ability of the method to price accurately. In sum, this paper finds that the improved speed of the COS method against the FFT method diminishes with a more complex model - although the extent of this can only be determined by testing for increasingly complex characteristic functions. Overall the COS method can be seen to be preferable from a practical point of view due to its higher level of robustness.
- ItemOpen AccessGaussian process regression approach to pricing multi-asset American options(2022) Mokone, Christoffel Maboe; Ouwehand, PeterThis dissertation explores the problem of pricing American options in high dimensions using machine learning. In particular, the Gaussian Process Regression Monte Carlo (GPR-MC) algorithm developed by Goudenege et al (2019). is explored, and ` its performance, i.e., its accuracy and efficiency, is benchmarked against the Least Squares Regression Method (LSM) developed by Carriere (1996) and popularised by Longstaff and Schwartz (2001). In this dissertation, American options are approximated by Bermudan options due to limited computing power. To test the performance of GPR-MC, an American geometric mean basket put option, an American arithmetic mean basket put option and an American maximum call option are priced under the multi-asset Black-Scholes and Heston models, using both GPRMC and LSM. The algorithms are run a 100 times to obtain mean option values, 95% confidence intervals about the means, and average computational times. Numerical results show that the efficiency of GPR-MC is independent of the number of underlying assets, in contrast to the LSM method which is not. At 10 underlying assets, GPR-MC is shown to be more efficient than LSM. Moreover, GPR-MC is reasonably accurate, producing relative errors that are within reasonable bounds.
- ItemOpen AccessGaussian Process Regression for Option Pricing and Hedging(2022) Patel, Ishani; Ouwehand, PeterRecent literature in the field of quantitative finance has employed machine learning methods to speed up typical numerical calculations including derivative pricing, fitting Greek profiles, constructing volatility surfaces and modelling counterparty credit risk, to name a few. This dissertation aims to investigate the accuracy and efficiency of Gaussian process regression (GPR) compared to traditional quantitative pricing algorithms. The GPR algorithm is applied to pricing a down-and-out barrier call option. Notably, Crepey and Dixon ´ (2019) propose an alternative method for computing the Gaussian process Greeks by directly differentiating the GPR option pricing model. Based on their approach, the GPR algorithm is further extended to compute the delta and vega of the option. Numerical experiments display that option pricing accuracy scores are within a tolerable range and demonstrate increased speed of considerable magnitudes with speed-up factors in the 1 000s. Computing the Greeks convey favourable computational properties; however, the GPR model struggles to obtain accurate predictions for the delta and vega. The trade-off between accuracy and speed is further investigated, where the inclusion of additional GPR input parameters hinder performance metrics whilst a larger training data set improves model accuracy.
- ItemOpen AccessGram-Charlier expansions and option pricing(2022) Knipe, Joshua; Ouwehand, Peter; Mc Walter, ThomasGram-Charlier expansions provide a tractable way of fitting risk-neutral distributions to asset prices. This allows the model to capture skewness, excess kurtosis and higher moments in observed asset returns. Schlogl (2013) proposes a calibration method to ensure the fitted densities are valid and arbitrage free. This method is implemented with standard foreign exchange options and gives an exact fit when enough moments are included in the calibration process. GramCharlier expansions also result in analytic solutions for many exotic option prices through an extremely general framework. This relies on representing an option as a portfolio of the M-binaries defined by Skipper and Buchen (2003). Geometric Asian options are priced using this approach and compared to the corresponding Black-Scholes prices. Numerical examples highlight the effect skewness and excess kurtosis can have on these option prices, particularly for options that are out-the-money. Gram-Charlier distributions are also combined with Monte Carlo simulations to estimate option prices for calls and geometric Asian options. The results show convergence to the analytical solutions for all cases. Additionally, Gram-Charlier estimates for arithmetic Asian options are calculated and compared to Black-Scholes estimates.

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