### Browsing by Author "Backwell, Alex"

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- ItemOpen AccessAccounting for roll-over risk in the pricing of caps and floors(2022) Vidima, Sizwe; Backwell, AlexThe peak of the global financial crisis necessitated practitioners to rethink the single curve approach to pricing interest-rate derivatives. This was as a result of a violation in spot-forward parity relationships thereby prompting markets to realise the presence of a new type of risk and subsequently the need for a multi-curve pricing framework. The roll-over risk framework is one that accounts for liquidity constraints and default risk thereby providing a cogent explanation for the spotforward parity violation that led to the need for multiple curves. The primary objective of this work is to price XIBOR-based caps and floors under a framework which accounts for roll-over risk. This reformulation of interest-rate derivatives is achieved using Fourier Transform methods as well as Monte Carlo simulations for comparison. We found that the results obtained using the two approaches were comparable even though the two methods are different in nature. This agreement in prices is compelling evidence that the computations are correct.
- ItemOpen AccessAn application of short rate modelling involving roll-over risk to caplet pricing(2022) Montgomery, Thomas; Backwell, AlexThe concept of roll-over risk encapsulates the risk that a bank sitting on an interbank panel may be unable to borrow at the interbank overnight reference rate at some point in the future. Roll-over risk is comprised of two separate risks: the risk that the bank may deteriorate in credit quality relative to the 'average' bank sitting on the interbank panel and the risk that the bank may experience worse liquidity than the 'average' panel bank. Roll-over risk has been offered as a possible explanation of basis spreads which have proliferated since the Global Financial Crisis. This dissertation makes use of the established methods in order incorporate rollover risk in the pricing of a caplet based on an underlying reference rate. The caplet pricing function is compared to traditional discretised Monte Carlo techniques. The performance of the function proves more accurate and computationally efficient.
- ItemOpen AccessAn introduction to interest rate jumps at deterministic times(2022) Bastick, Kirk; Backwell, AlexThe observation of jumps in empirical interest-rate data has prompted the inclusion of these jumps in recent term-structure models. This dissertation focusses on explaining the effects of jumps that occur at known times on the pricing of bonds. Filipovic (2009) affirms that the transition from the physical measure to the riskneutral measure is key to the pricing of bonds and other financial instruments. Jumps in the interest rate at known times add a layer of complexity to this measurechange process. A simplified version of the term-structure model proposed by Kim and Wright (2014) is employed to analyse the effect of the jumps on the one-year point on the yield curve. Jumps at deterministic times are found to have a material effect on the one-year yield with an increasing effect as time approaches a deterministic jump date.
- ItemOpen AccessCalibrating Term Structure Models to an Initial Yield Curve(2020) Sylvester, Matthew; Backwell, AlexThe modelling of the short rate offers many advantages, with the models explored in this dissertation all offering closed-form, analytic formulae for bond prices and for options on bonds. Often, a vital primary condition is for a model to be calibrated to the initial term structure and to recover the bond prices observed in the market – that is, to be calibrated to the initial yield curve. Under the two exogenous models explored in this dissertation, the Hull-White and the CIR++, the effect of increasing the volatility parameter of the SDE increases the mean of the short rate. Increasing volatility of an SDE is a common approach to stress testing a model, as such, the consequences of bumping volatility in a calibrated model is a vital concern. The Hull-White model and CIR++ model were calibrated to market data, with the former being able to match the observed cap prices, while the latter failed, displaying an upper bound on cap prices. Investigating this, under CIR++ model, bond option prices are shown to not be straightforward increasing functions of the volatility parameter. In fact, for high volatility, bond option prices display an upper limit before decreasing, thus providing a limit to the level of cap prices too. This dissertation points to the reason residing in the underlying CIR model from which the CIR++ is based on, and the manner in which the model is extended
- ItemOpen AccessCredit default swaps in a roll-over risk framework(2021) Petersen, Nicholas; Backwell, AlexSpreads between swap legs referencing floating cashflows of different tenors have widened significantly since the global financial crisis of 2008. This frequency basis can be explained by the presence of “roll-over risk”. Defining the roll-over risk state variables in an affine form, this dissertation prices a credit default swap using an “affine transform” methodology. This price is then compared to that obtained from a traditional Monte Carlo simulation approach. The former is shown to produce accurate results with greater computational efficiency, providing a useful way to price complex financial instruments when the state variables are defined in an appropriate form.
- ItemOpen AccessHedging performance of interest-rate models(2016) Ziervogel, Graham; Backwell, AlexThis dissertation is a hedging back-study which assesses the effectiveness of interest- rate modelling and the hedging of interest-rate derivatives. Caps that trade in the Johannesburg swap market are hedged using two short-rate models, namely the Hull and White (1990) one-factor model and the subsequent Hull and White (1994) two-factor extension. This is achieved by using the equivalent Gaussian additive-factor models (G1++ and G2++) outlined by Brigo and Mercurio (2007). The hedges are constructed using different combinations of theoretical zero-coupon bonds. A flexible factor hedging method is proposed by the author and the bucket hedging technique detailed by Driessen, Klaasen and Melenberg (2003) is tested. The results obtained support the claims made by Gupta and Subrahmanyam (2005), Fan, Gupta and Ritchken (2007) and others in the literature that multi-factor models outperform one-factor models in hedging interest-rate derivatives. It is also shown that the choice of hedge instruments can significantly influence hedge performance. Notably, a larger set of hedge instruments and the use of hedge instruments with the same maturity as the derivative improve hedging accuracy. However, no evidence to support the finding of Driessen et al. (2003) that a larger set of hedge instruments can remove the need for a multi-factor model is found.
- ItemOpen AccessImplementation of Bivariate Unspanned Stochastic Volatility Models(2018) Cullinan, Cian; Backwell, AlexUnspanned stochastic volatility term structure models have gained popularity in the literature. This dissertation focuses on the challenges of implementing the simplest case – bivariate unspanned stochastic volatility models, where there is one state variable controlling the term structure, and one scaling the volatility. Specifically, we consider the Log-Affine Double Quadratic (1,1) model of Backwell (2017). In the class of affine term structure models, state variables are virtually always spanned and can therefore be inferred from bond yields. When fitting unspanned models, it is necessary to include option data, which adds further challenges. Because there are no analytical solutions in the LADQ (1,1) model, we show how options can be priced using an Alternating Direction Implicit finite difference scheme. We then implement an Unscented Kalman filter — a non-linear extension of the Kalman filter, which is a popular method for inferring state variable values — to recover the latent state variables from market observable data
- ItemOpen AccessImplementing short-rate models with jumps at deterministic times(2022) Shibduth, Darvesh Yogandar; Backwell, AlexMacroeconomic announcements have a direct impact on short-term interest rates during a financial year. However, this is not directly reflected in the continuous-time interest rate models. In this paper, we work with short-rate models which include the possibility of jumps at deterministic times. An application of the finite-difference method enables the pricing of bonds and bond options in these short-rate models with different types of jump distributions. A closed-form solution for bond prices, when the jumps are normally distributed, is available in the literature, but not for other jump distributions. The Monte Carlo method is used to compare the finite-difference calculations for these cases. An illustration of varying important model parameters is provided in which we observe that an increase in option prices could result from an increase in the jump variances and/or volatility parameters.
- ItemOpen AccessInterest-Rate Option Pricing Accounting For Jumps At Deterministic Times(2021) Allman, Timothy; Backwell, AlexThe short rate is central in the context of interest-rate markets as well as broader finance. As such, accurate modelling of this rate is of particular importance in the pricing of interest-rate options, especially during times of high volatility where increased demand is seen for simpler and lower risk investments. Recent interest has moved away from models of a pure continuous nature towards models that can account for discontinuities in the short rate. These are more representative of real world movements where the short rate is seen to jump due to current and scheduled market information. This dissertation examines this phenomenon in the context of a Vasicek short rate model and accounts for random-sized jumps at deterministic times following ideas similar to those introduced by Kim and Wright (2014). Finite difference methods are used successfully to find PDE solutions via backwards diffusion of the option value equation to its initial state. This procedure is implemented computationally and compared to Monte Carlo benchmark methods in order to assess its accuracy. In both non-jump and jump settings the method constructed was able to accurately price the call option specified and proved to be a viable means for pricing interest-rate options when stochastically-sized discontinuities are present at known times between inception and expiry. Furthermore the method showed that the stochastic discontinues in the short rate most notably affect the option price in the region around and just out of the money.
- ItemOpen AccessLevel Dependence in Volatility in Linear-Rational Term Structure Models(2019) Ramnarayan, Kalind; Backwell, AlexThe degree of level dependence in interest rate volatility is analysed in the linearrational term structure model. The linear-rational square-root (LRSQ) model, where level dependence is set a priori, is compared to a specification where the factor process follows CEV-type dynamics which allows a more flexible degree of level dependence. Parameters are estimated using an unscented Kalman filter in conjunction with quasi-maximum likelihood. An extended specification for the state price density process is required to ensure reliable parameter estimates. The empirical analysis indicates that the LRSQ model generally overestimates level dependence. Although the CEV specification captures the degree of level dependence in volatility more accurately, it has a trade-off with analytical tractability. The optimal specification, therefore, depends on the type of model implementation and general economic conditions.
- ItemOpen AccessLinear-Rational Term Structure Models With Flexible Level-Dependent Volatility(2018) Schwellnus, Adrian; Backwell, AlexThe Linear-Rational Framework for the modelling of interest rates is a framework which allows for the addition of spanned and unspanned factors, while maintaining a lower bound on rates and tractable valuation of interest rate derivatives, particularly swaptions. The advantages of having all these properties are significant. This dissertation presents the Linear-Rational Framework, and specializes the factor process to a class of diffusion models which allows for the degree of state dependence of volatility to be estimated. This dissertation then finds that the estimated state dependent volatility structure is significantly different to that of typical models, where it is set it a priori. The effect the added degree of freedom has on the model implied swaption skew is then analysed.
- ItemOpen AccessPricing with Bivariate Unspanned Stochastic Volatility Models(2019) Wort, Joshua; Backwell, AlexUnspanned stochastic volatility (USV) models have gained popularity in the literature. USV models contain at least one source of volatility-related risk that cannot be hedged with bonds, referred to as the unspanned volatility factor(s). Bivariate USV models are the simplest case, comprising of one state variable controlling the term structure and the other controlling unspanned volatility. This dissertation focuses on pricing with two particular bivariate USV models: the Log-Affine Double Quadratic (1,1) – or LADQ(1,1) – model of Backwell (2017) and the LinearRational Square Root (1,1) – or LRSQ(1,1) – model of Filipovic´ et al. (2017). For the LADQ(1,1) model, we fully outline how an Alternating Directional Implicit finite difference scheme can be used to price options and implement the scheme to price caplets. For the LRSQ(1,1) model, we illustrate a semi-analytical Fourierbased method originally designed by Filipovic´ et al. (2017) for pricing swaptions, but adjust it to price caplets. Using the above numerical methods, we calibrate each (1,1) model to both the British-pound yield curve and caps market. Although we cannot achieve a close fit to the implied volatility surface, we find that the parameters in the LADQ(1,1) model have direct control over the qualitative features of the volatility skew, unlike the parameters within the LRSQ(1,1) model.
- ItemRestrictedRecovery theorem: expounded and applied(2014) Backwell, Alex; Taylor, DavidThis dissertation is concerned with Ross' (2011) Recovery Theorem. It is generally held that a forward-looking probability distribution is unobtainable from derivative prices, because the market's risk-preferences are conceptually inextricable from the implied real-world distribution. Ross' result recovers this distribution without making the strong preference assumptions assumed necessary under the conventional paradigm. This dissertation aims to give the reader a thorough understanding of Ross Recovery, both from a theoretical and practical point of view. This starts with a formal delineation of the model and proof of the central result, motivated by the informal nature of Ross' working paper. This dissertation relaxes one of Ross' assumptions and arrives at the equivalent conclusion. This is followed by a critique of the model and assumptions. An a priori discussion only goes so far, but potentially problematic assumptions are identified, chief amongst which being time additive preferences of a representative agent. Attention is then turned to practical application of the theorem. The author identifies a number of obstacles to applying the result { some of which are somewhat atypical and have not been directly addressed in the literature { and suggests potential solutions. A salient obstacle is calibrating a state price matrix. This leads to an implementation of Ross Recovery on the FTSE/JSE Top40. The suggested approach is found to be workable, though certainly not the final word on the matter. A testing framework for the model is discussed and the dissertation is concluded with a consideration of the findings and the theorem's applicability.
- ItemOpen AccessRobustness of bond portfolio optimisation(2016) Pillay, Divanisha; Backwell, Alex; Ouwehand, PeterKorn and Koziol (2006) apply the Markowitz (1952) mean-variance framework to bond portfolio selection by proposing the use of term structure models to estimate the time-varying moments of bond returns. Duffee (2002) introduces a distinction between completely affine and essentially affine term structure models. A completely affine model uses a market price of risk specification that is proportional to the volatility of the risk factors. However, this assumption of proportionality of the market price of risk contradicts the observed behaviour of bond returns. In response, Duffee (2002) introduces a more flexible essentially affine market price of risk specification by breaking the strict proportionality of the completely affine specification. Essentially affine models better represent the empirical features of bond returns whilst preserving the tractability of completely affine models. However, Duffee and Stanton (2012) find that the increased flexibility of the essentially affine model comes at the expense of real-world parameter estimation. Given these parameter estimation issues, this dissertation investigates whether the difficulty in estimating an essentially affine specification is outweighed by the empirical preferability, and whether, all these issues considered, the Markowitz (1952) approach to bond portfolio optimisation is robust. The results indicate that the superior capability of an essentially affine model to forecast expected returns outweighs real-world parameter estimation issues; and that the estimation and mean-variance optimisation procedures are worthwhile.
- ItemOpen AccessThe Credit Risk in Stock-Based Loans(2018) Korula, Febin; McWalter, Thomas; Backwell, AlexStock-based loans are an increasingly popular form of loan that are collateralised using stocks. Since these loans are often non-recourse loans, the lenders are subject to the risk that the collateral is worth less than the loan, and the borrower defaults. This dissertation will consider the credit risk faced by lenders when issuing these loans. To achieve this, this dissertation will propose different models to quantify this risk using various credit measures. A sensitivity analysis to key model parameters is then conducted. Some brief comments about capital requirements will also be made.
- ItemOpen AccessThe Effects of Dilutions and Payout Policy on Equity- and Stock-linked Call Options on a Firm with Leverage(2022) Brill, Nicola; Backwell, AlexCapital-structure models are useful tools for pricing claims on equity. They provide insight into the effects of changing capital-debt structures on the value of options. Backwell et al. (2022) developed a capital structure framework which, in addition to a typical structural model, specifically considers the number of outstanding shares. This allows for the differentiation between options on total firm equity and options on share price. The extension by Backwell et al. (2022) also allows for the effects of dilutions and buyback policy on stock-linked options to be explored. Using the framework developed by Backwell et al. (2022) with asset value dynamics presented by Leland (1994), the capital-debt structure of a firm is modelled. Finitedifference methods utilising a generalised version of the Black-Scholes equation are then used to value and compare call options on total equity and call options on share price. Under the presented model, dilutions have little to no effect on stocklinked call option value in firms with low levels of leverage. However, dilutions clearly decrease the value of call options in firms with higher levels of leverage. Share buybacks significantly improve the value of stock-linked call options, particularly in lower leveraged firms where there is more available cash flow. This indicates that while shareholders are indifferent between cash dividends and share buybacks in a perfect market, holders of options on share price are not indifferent. In fact, option holders prefer payout policies that favour buybacks over dividends. Finally, the leverage effect is demonstrated by calculating implied volatilities under various levels of firm leverage.
- ItemOpen AccessThe Lifted Heston Stochastic Volatility Model(2020) Broodryk, Ryan; Backwell, Alex; Soane, AndrewCan we capture the explosive nature of volatility skew observed in the market, without resorting to non-Markovian models? We show that, in terms of skew, the Heston model cannot match the market at both long and short maturities simultaneously. We introduce Abi Jaber (2019)'s Lifted Heston model and explain how to price options with it using both the cosine method and standard Monte-Carlo techniques. This allows us to back out implied volatilities and compute skew for both models, confirming that the Lifted Heston nests the standard Heston model. We then produce and analyze the skew for Lifted Heston models with a varying number N of mean reverting terms, and give an empirical study into the time complexity of increasing N. We observe a weak increase in convergence speed in the cosine method for increased N, and comment on the number of factors to implement for practical use.