Browsing by Author "Rudd, Ralph"

Now showing 1 - 8 of 8
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    Approximating the Heston-Hull-White Model
    (2019) Patel, Riaz; Rudd, Ralph
    The hybrid Heston-Hull-White (HHW) model combines the Heston (1993) stochastic volatility and Hull and White (1990) short rate models. Compared to stochastic volatility models, hybrid models improve upon the pricing and hedging of longdated options and equity-interest rate hybrid claims. When the Heston and HullWhite components are uncorrelated, an exact characteristic function for the HHW model can be derived. In contrast, when the components are correlated, the more useful case for the pricing of hybrid claims, an exact characteristic function cannot be obtained. Grzelak and Oosterlee (2011) developed two approximations for this correlated case, such that the characteristics functions are available. Within this dissertation, the approximations, referred to as the determinist and stochastic approximations, were implemented to price vanilla options. This involved extending the Carr and Madan (1999) method to a stochastic interest rate setting. The approximations were then assessed for accuracy and efficiency. In determining an appropriate benchmark for assessing the accuracy of the approximations, the full truncation Milstein and Quadratic Exponential (QE) schemes, which are popular Monte Carlo discretisation schemes for the Heston model, were extended to the HHW model. These schemes were then compared against the characteristic function for the uncorrelated case, and the QE scheme was found to be more accurate than the Milstein-based scheme. With the differences in performance becoming increasingly noticeable when the Feller (1951) condition was not satisfied and the maturity and volatility of the Hull-White model (⌘) was large. In assessing the accuracy of the approximations against the QE scheme, both approximations were similarly accurate when ⌘ was small. In contrast, when ⌘ was large, the stochastic approximation was more accurate than the deterministic approximation. However, the deterministic approximation was significantly faster than the stochastic approximation and the stochastic approximation displayed signs of potential instability. When ⌘ is small, the deterministic approximation is therefore recommended for use in applications such as calibration. With its shortcomings, the stochastic approximation could not be recommended. However, it did show promising signs of accuracy that warrants further investigation into its efficiency and stability.
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    Optimal tree methods
    (2014) Rudd, Ralph; McWalter, Thomas; Taylor, David
    Although 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.
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    Option pricing with physics-informed neutral networks (PINNS)
    (2024) Zamxaka, Nichume; Rudd, Ralph
    We investigate the application of physics-informed neural networks (PINNs) to option pricing. PINNs are neural networks that are trained to numerically solve partial differential equations (PDEs) by obeying the dynamics induced by the PDE as well as the initial/terminal conditions of the PDE. They are mesh-free to an extent and compute the derivatives of the PDE through backward-propagation. We construct a PINN toy example to solve the Black-Scholes-Merton PDE for a vanilla European option. The numerical solutions from the PINN are compared against the true analytical solution – the Black-Scholes-Merton equation. The problem is also extended by incorporating a local volatility model. Here, we derive the PDE of a vanilla European option under the constant elasticity of variance (CEV) model. We then construct and train a PINN to solve the PDE and compare it to the true analytical solution of a special case of the CEV model, the square-root process.
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    Parameter learning with particle filters
    (2020) Pather, Vegan; Rudd, Ralph; Soane, Andrew
    Common applications of asset-pricing models in practice rely on recalibrating model parameters periodically for effective risk management. Yet, these model parameters are often assumed to be constant over time, thereby countering the notion of readjusting these values. A possible solution to this problem is to recalibrate at times where observed market prices cannot realistically match model prices based on parameter values at those times. This dissertation aims to test the effectiveness of a possible algorithm which can be used in optimally identifying such times. An overview is provided of the recently proposed particle filter with accelerated adaptation which has demonstrated rapid time detection for changes in parameter values and has been applied to regime-shifting and stochastic volatility models. Numerical and graphical evidence of parameter and volatility estimation will be provided under regime-shifting parameters for the Heston (1993) stochastic volatility model. The filter demonstrates rapid adaptation in estimating parameter values and accurate estimation of the volatility process. Furthermore, we provide a discussion for possible extensions towards a metric for optimal recalibration times.
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    Pricing a Bermudan option under the constant elasticity of variance model
    (2017) Rwexana, Kwaku; McWalter, Thomas; Rudd, Ralph
    This 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.
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    Quantifying the Model Risk Inherent in the Calibration and Recalibration of Option Pricing Models
    (2021-01-04) Feng, Yu; Rudd, Ralph; Baker, Christopher; Mashalaba, Qaphela; Mavuso, Melusi; Schlögl, Erik
    We focus on two particular aspects of model risk: the inability of a chosen model to fit observed market prices at a given point in time (calibration error) and the model risk due to the recalibration of model parameters (in contradiction to the model assumptions). In this context, we use relative entropy as a pre-metric in order to quantify these two sources of model risk in a common framework, and consider the trade-offs between them when choosing a model and the frequency with which to recalibrate to the market. We illustrate this approach by applying it to the seminal Black/Scholes model and its extension to stochastic volatility, while using option data for Apple (AAPL) and Google (GOOG). We find that recalibrating a model more frequently simply shifts model risk from one type to another, without any substantial reduction of aggregate model risk. Furthermore, moving to a more complicated stochastic model is seen to be counterproductive if one requires a high degree of robustness, for example, as quantified by a 99% quantile of aggregate model risk.
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    Recursive marginal quantization: extensions and applications in finance
    (2018) Rudd, Ralph; Kienitz, Jorg; Platen, Eckhard
    Quantization techniques have been used in many challenging finance applications, including pricing claims with path dependence and early exercise features, stochastic optimal control, filtering problems and the efficient calibration of large derivative books. Recursive marginal quantization of an Euler scheme has recently been proposed as an efficient numerical method for evaluating functionals of solutions of stochastic differential equations. This algorithm is generalized and it is shown that it is possible to perform recursive marginal quantization for two higher-order schemes: the Milstein scheme and a simplified weak-order 2.0 scheme. Furthermore, the recursive marginal quantization algorithm is extended by showing how absorption and reflection at the zero boundary may be incorporated. Numerical evidence is provided of the improved weak-order convergence and computational efficiency for the geometric Brownian motion and constant elasticity of variance models by pricing European, Bermudan and barrier options. The current theoretical error bound is extended to apply to the proposed higher-order methods. When applied to two-factor models, recursive marginal quantization becomes computationally inefficient as the optimization problem usually requires stochastic methods, for example, the randomized Lloyd’s algorithm or Competitive Learning Vector Quantization. To address this, a new algorithm is proposed that allows recursive marginal quantization to be applied to two-factor stochastic volatility models while retaining the efficiency of the original Newton-Raphson gradientdescent technique. The proposed method is illustrated for European options on the Heston and Stein-Stein models and for various exotic options on the popular SABR model. Finally, the recursive marginal quantization algorithm, and improvements, are applied outside the traditional risk-neutral pricing framework by pricing long-dated contracts using the benchmark approach. The growth-optimal portfolio, the central object of the benchmark approach, is modelled using the time-dependent constant elasticity of variance model. Analytic European option prices are derived that generalize the current formulae in the literature. The time-dependent constant elasticity of variance model is then combined with a 3/2 stochastic short rate model to price zerocoupon bonds and zero-coupon bond options, thereby showing the departure from risk-neutral pricing.
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    Testing of an Arbitrage-free Volatility Surface
    (2022) Tarr, Grant; Rudd, Ralph
    The Ensemble Carr-Pelts surface, which is a weighted mixture of standard CarrPelts surfaces, is an arbitrage-free parameterization of an implied volatility surface proposed by Antonov, Konikov and Spector (2019). This dissertation aims to investigate the additional benefits provided by using the Ensemble Carr-Pelts surface as opposed to the standard Carr-Pelts surface. We also show its validity in comparison to stochastic volatility inspired Gatheral (2004) surface, which is widely used by practitioners. The approach adopted was done in three stages, with each stage calibrating to an increasingly complicated surface. Surfaces considered were a flat volatility surface, a surface changing with strike only, and a surface changing with both strike and maturity. Testing revealed that as complexity increased for the implied volatility surface, the Ensemble Carr-Pelts calibrated better than CarrPelts. When compared to the widely accepted stochastic volatility inspired surface; considering no-arbitrage was not enforced, the Ensemble Carr-Pelts performed adequately. However, the Ensemble Carr-Pelts takes significantly longer to calibrate.