Research

Quantitative Finance

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Peer-reviewed journal articles

  • Gellert, K. & Schlögl, E. 2021, ‘Parameter Learning and Change Detection Using a Particle Filter with Accelerated Adaptation’, Risks, 9(12).
    View/Download from: Publisher’s site (open access)/Working paper version on SSRN
    Click arrow to view description This paper presents the construction of a particle filter, which incorporates elements inspired by genetic algorithms, in order to achieve accelerated adaptation of the estimated posterior distribution to changes in model parameters. Specifically, the filter is designed for the situation where the subsequent data in online sequential filtering does not match the model posterior filtered based on data up to a current point in time. The examples considered encompass parameter regime shifts and stochastic volatility. The filter adapts to regime shifts extremely rapidly and delivers a clear heuristic for distinguishing between regime shifts and stochastic volatility, even though the model dynamics assumed by the filter exhibit neither of those features.
  • Feng, Y., Rudd, R., Baker, C., Mashalaba, Q., Mavuso, M. & Schlögl, E. 2021, ‘Quantifying the Model Risk Inherent in the Calibration and Recalibration of Option Pricing Models’, Risks, 9(13).
    View/Download from: Publisher’s site (open access)/Working paper version on SSRN
    Click arrow to view description 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.
  • Alfeus, M., Grasselli, M. & Schlögl, E. 2020, ‘A Consistent Stochastic Model of the Term Structure of Interest Rates for Multiple Tenors’, Journal of Economic Dynamics and Control, 114.
    View/Download from: Publisher’s site/Working paper version on SSRN
    Click arrow to view description Starting from the observation that single-currency swap basis spreads contradict classical arbitrage arguments, we construct a framework where this basis arises due to the presence of “roll-over risk.” This risk consists of two components: (1) facing a higher credit spread (e.g. due to a credit downgrade) when rolling over short-term borrowing (2) heightened borrowing costs due to an absence of market liquidity. The model simultaneously fits OIS, interest rate swap and basis swap market quotes. Including CDS market quotes allows the two components of roll-over risk to be explicitly separated. This is highly relevant to the current LIBOR transition, illustrating why alternative benchmarks are fundamentally different from the rates they may be replacing.
  • Alfeus, M. & Schlögl, E. 2019, ‘On Spread Option Pricing Using Two-Dimensional Fourier Transform’, International Journal of Theoretical and Applied Finance, 22(5).
    View/Download from: Publisher’s site/Working paper version on SSRN
    Click arrow to view description

    Spread options are multi-asset options whose payoffs depend on the difference of two underlying financial variables. In most cases, analytically closed form solutions for pricing such payoffs are not available, and the application of numerical pricing methods turns out to be non-trivial. We consider several such non-trivial cases and explore the performance of the highly efficient numerical technique of Hurd and Zhou (2010), comparing this with Monte Carlo simulation and the lower bound approximation formula of Caldana and Fusai (2013). We show that the former is in essence an application of the two–dimensional Parseval Identity.

    As application examples, we price spread options in a model where asset prices are driven by a multivariate normal inverse Gaussian (NIG) process, in a threefactor stochastic volatility model, as well as in examples of models driven by other popular multivariate Lévy processes such as the variance Gamma process, and discuss the price sensitivity with respect to volatility. We also consider examples in the fixed–income market, specifically, on cross–currency interest rate spreads and on LIBOR/OIS spreads. In terms of FFT computation, we have used the FFTW library (see Frigo and Johnson (2010)) and we document appropriate usage of this library to reconcile it with the MATLAB ifft2 counterpart.

  • Alfeus, M., Overbeck, L. & Schlögl, E. 2019 ‘Regime Switching Rough Heston Model’, Journal of Futures Markets, 39(5), 535-631.
    View/Download from: Publisher’s site
    Click arrow to view description This model combines two important stylized features of volatility, the rough behavior consistent with a Hurst parameter less than 0.5, and the regime switching property consistent with more long-term economic considerations. It is nevertheless highly tractable in the sense of semianalytic formulae for European options, and permits a partial Monte Carlo method of similar computational speed as the semianalytic formula (at an appropriate number of Monte Carlo simulations). While option prices are relatively insensitive to the choice of Hurst parameter, introducing rough volatility allows for a better fit to the at-the-money skew.
  • Cheng, B., Nikitopoulos, C. S. & Schlögl, E. 2019, ‘Interest rate risk in long-dated commodity options positions: To Hedge or Not to Hedge?', Journal of Futures Markets, 39(1), 109-127.
    View/Download from: Publisher’s site/Working paper version on SSRN
    Click arrow to view description We empirically assess hedging interest rate risk beyond the conventional delta, gamma, and vega hedges in long-dated crude oil options positions. Using factor hedging in a model featuring stochastic interest rates and stochastic volatility, interest rate hedges consistently provide an improvement beyond delta, gamma, and vega hedges. Under high interest rate volatility and/or when a rolling hedge is used, combining interest rate and delta hedging improves performance by up to four percentage points over the common hedges of gamma and/or vega. Thus, contrary to common practice, hedging interest rate risk should have priority over these “second-order” hedges.
  • Cheng, B., Nikitopoulos, C. S. & Schlögl, E. 2018, ‘Pricing of long-dated commodity derivatives: Do stochastic interest rates matter?', Journal of Banking and Finance, 95, 148-166.
    View/Download from: Publisher’s site/Working paper version on SSRN
    Click arrow to view description Does modelling stochastic interest rates, beyond stochastic volatility, improve pricing performance on long-dated commodity derivatives? To answer this question, we consider futures price models for commodity derivatives that allow for stochastic volatility and stochastic interest rates and a correlation structure between the underlying variables. We examine the empirical pricing performance of these models on pricing long-dated crude oil derivatives. Estimating the model parameters from historical crude oil futures prices and option prices, we find that stochastic interest rate models improve pricing performance on long-dated crude oil derivatives, when the interest rate volatility is relatively high. Furthermore, increasing the model dimensionality does not tend to improve the pricing performance on long-dated crude oil option prices, but it matters for long-dated futures prices. We also find empirical evidence for a negative correlation between crude oil futures prices and interest rates that contributes to improving fit to long-dated crude oil option prices.
  • Karlsson, P., Pilz, K. F. & Schlögl, E. 2017, ‘Calibrating a Market Model with Stochastic Volatility to Commodity and Interest Rate Risk’, Quantitative Finance, 17(6), 907-925.
    View/Download from: Publisher’s site/Working paper version on SSRN
    Click arrow to view description Based on the multi-currency LIBOR Market Model, this paper constructs a hybrid commodity interest rate market model with a stochastic local volatility function allowing the model to simultaneously fit the implied volatility surfaces of commodity and interest rate options. Since liquid market prices are only available for options on commodity futures, rather than forwards, a convexity correction formula for the model is derived to account for the difference between forward and futures prices. A procedure for efficiently calibrating the model to interest rate and commodity volatility smiles is constructed. Finally, the model is fitted to an exogenously given correlation structure between forward interest rates and commodity prices (cross-correlation). When calibrating to options on forwards (rather than futures), the fitting of cross-correlation preserves the (separate) calibration in the two markets (interest rate and commodity options), while in the case of futures a (rapidly converging) iterative fitting procedure is presented. The fitting of cross-correlation is reduced to finding an optimal rotation of volatility vectors, which is shown to be an appropriately modified version of the ‘orthonormal Procrustes’ problem in linear algebra. The calibration approach is demonstrated in an application to market data for oil futures.
  • Pilz, K. & Schlögl, E. 2013, ‘A hybrid commodity and interest rate market model’, Quantitative Finance, vol. 13, no. 4, pp. 543-560.
    View/Download from: Publisher’s site/Working paper version on SSRN
    Click arrow to view description A joint model of commodity price and interest rate risk is constructed analogously to the multi-currency LIBOR Market Model (LMM). Going beyond a simple ‘re-interpretation’ of the multi-currency LMM, issues arising in the application of the model to actual commodity market data are specifically addressed. Firstly, liquid market prices are only available for options on commodity futures, rather than forwards, thus the difference between forward and futures prices must be explicitly taken into account in the calibration. Secondly, we construct a procedure to achieve a consistent fit of the model to market data for interest options, commodity options and historically estimated correlations between interest rates and commodity prices. We illustrate the model by an application to real market data and derive pricing formulas for commodity spread options.
  • Schlögl, E. 2013, ‘Option pricing where the underlying assets follow a Gram/Charlier density of arbitrary order’, Journal of Economic Dynamics and Control, vol. 37, no. 3, pp. 611-632. View/Download from: Publisher’s site/Working paper version on SSRN
    Click arrow to view description If a probability distribution is sufficiently close to a normal distribution, its density can be approximated by a Gram/Charlier Series A expansion. In option pricing, this has been used to fit risk-neutral asset price distributions to the implied volatility smile, ensuring an arbitrage-free interpolation of implied volatilities across exercise prices. However, the existing literature is restricted to truncating the series expansion after the fourth moment. This paper presents an option pricing formula in terms of the full (untruncated) series and discusses a fitting algorithm, which ensures that a series truncated at a moment of arbitrary order represents a valid probability density. While it is well known that valid densities resulting from truncated Gram/Charlier Series A expansions do not always have sufficient flexibility to fit all market-observed option prices perfectly, this paper demonstrates that option pricing in a model based on these densities is as tractable as the (far less flexible) original model of Black and Scholes (1973), allowing non-trivial higher moments such as skewness, excess kurtosis and so on to be incorporated into the pricing of exotic options: Generalising the Gram/Charlier Series A approach to the multiperiod, multivariate case, a model calibrated to standard option prices is developed, in which a large class of exotic payoffs can be priced in closed form. Furthermore, this approach, when applied to a foreign exchange option market involving several currencies, can be used to ensure that the volatility smiles for options on the cross exchange rate are constructed in a consistent, arbitrage-free manner.
  • Nielsen, J.A., Sandmann, K. & Schlögl, E. 2011, ‘Equity-linked pension schemes with guarantees’, Insurance: Mathematics and Economics, vol. 49, no. 3, pp. 547-564. View/Download from: Publisher’s site/Working paper version on SSRN
    Click arrow to view description This paper analyses the relationship between the level of a return guarantee in an equity-linked pension scheme and the proportion of an investor’s contribution needed to finance this guarantee. Three types of schemes are considered: investment guarantee, contribution guarantee and surplus participation. The evaluation of each scheme involves pricing an Asian option, for which relatively tight upper and lower bounds can be calculated in a numerically efficient manner. We find a negative (and for two contract specifications also concave) relationship between the participation in the surplus return of the investment strategy and the guarantee level in terms of a minimum rate of return. Furthermore, the introduction of the possibility of early termination of the contract (e.g. due to the death of the investor) has no qualitative and very little quantitative impact on this relationship.
  • Bruti Liberati, N., Nikitopoulos Sklibosios, C., Platen, E. & Schlögl, E. 2009, ‘Alternative defaultable term structure models’, Asia-Pacific Financial Markets, vol. 16, no. 1, pp. 1-31. View/Download from: UTS OPUS/Working paper version on SSRN
    Click arrow to view description The objective of this paper is to consider defaultable term structure models in a general setting beyond standard risk-neutral models. Using as numeraire the growth optimal portfolio, defaultable interest rate derivatives are priced under the real-world probability measure. Therefore, the existence of an equivalent risk-neutral probability measure is not required. In particular, the real-world dynamics of the instantaneous defaultable forward rates under a jump-diffusion extension of a HJM type framework are derived. Thus, by establishing a modelling framework fully under the real-world probability measure, the challenge of reconciling real-world and risk-neutral probabilities of default is deliberately avoided, which provides significant extra modelling freedom. In addition, for certain volatility specifications, finite dimensional Markovian defaultable term structure models are derived. The paper also demonstrates an alternative defaultable term structure model. It provides tractable expressions for the prices of defaultable derivatives under the assumption of independence between the discounted growth optimal portfolio and the default-adjusted short rate. These expressions are then used in a more general model as control variates for Monte Carlo simulations of credit derivatives.
  • Mahayni, A.B. & Schlögl, E. 2008, ‘The Risk Management of Minimum Return Guarantees’, BuR - Business Research, vol. 1, no. 1, pp. 55-76.
    View/Download from: Published paper via SSRN
    Click arrow to view description Contracts paying a guaranteed minimum rate of return and a fraction of a positive excess rate, which is specified relative to a benchmark portfolio, are closely related to unit-linked life-insurance products and can be considered as alternatives to direct investment in the underlying benchmark. They contain an embedded power option, and the key issue is the tractable and realistic hedging of this option, in order to rigorously justify valuation by arbitrage arguments and prevent the guarantees from becoming uncontrollable liabilities to the issuer. We show how to determine the contract parameters conservatively and implement robust risk-management strategies.
  • Chiarella, C., Nikitopoulos Sklibosios, C. & Schlögl, E. 2007, ‘A Control Variate Method for Monte Carlo Simulations of Heath-Jarrow-Morton Models with Jumps’, Applied Mathematical Finance, vol. 14, no. 5, pp. 365-399.
    View/Download from: Publisher’s site/Working paper version on SSRN
    Click arrow to view description This paper examines the pricing of interest rate derivatives when the interest rate dynamics experience infrequent jump shocks modelled as a Poisson process. The pricing framework adapted was developed by Chiarella and Nikitopoulos to provide an extension of the Heath, Jarrow and Morton model to jump-diffusions and achieves Markovian structures under certain volatility specifications. Fourier Transform solutions for the price of a bond option under deterministic volatility specifications are derived and a control variate numerical method is developed under a more general state dependent volatility structure, a case in which closed form solutions are generally not possible. In doing so, a novel perspective is provided on control variate methods by going outside a given complex model to a simpler more tractable setting to provide the control variates.
  • Chiarella, C., Nikitopoulos Sklibosios, C. & Schlögl, E. 2007, ‘A Markovian Defaultable Term Structure Model with State Dependent Volatilities’, International Journal of Theoretical and Applied Finance, vol. 10, no. 1, pp. 155-202.
    View/Download from: Publisher’s site/Working paper version on SSRN
    Click arrow to view description The defaultable forward rate is modelled as a jump diffusion process within the Schönbucher general Heath, Jarrow and Morton framework where jumps in the defaultable term structure fd(t, T) cause jumps and defaults to the defaultable bond prices Pd(t, T). Within this framework, we investigate an appropriate forward rate volatility structure that results in Markovian defaultable spot rate dynamics. In particular, we consider state dependent Wiener volatility functions and time dependent Poisson volatility functions. The corresponding term structures of interest rates are expressed as finite dimensional affine realizations in terms of benchmark defaultable forward rates In addition, we extend this model to incorporate stochastic spreads by allowing jump intensities to follow a square-root diffusion process. In that case the dynamics become non-Markovian and to restore path independence we propose either an approximate Markovian scheme or, alternatively, constant Poisson volatility functions. We also conduct some numerical simulations to gauge the effect of the stochastic intensity and the distributional implications of various volatility specifications.
  • Choy, B., Dun, T. & Schlögl, E. 2004, ‘Correlating market models’, Risk, vol. September, pp. 124-129.
    View/Download from: Working paper version on SSRN
    Click arrow to view description While swaption prices theoretically contain information on interest rate correlation, Bruce Choy, Tim Dun and Erik Schlögl argue that, for any practical purpose, this information cannot be extracted. Care must therefore be taken when pricing correlation-sensitive instruments in a model calibrated to caps and swaptions. The good news is that Bermudan swaptions do not fall into this category: their prices do not react substantially even when radically different correlation structures are fed into the model.
  • Schlögl, E. 2002, ‘A multicurrency extension of the lognormal interest rate market models’, Finance and Stochastics, vol. 6, no. 2, pp. 173-196.
    View/Download from: Publisher’s site/Working paper version on SSRN
    Click arrow to view description The Market Models of the term structure of interest rates, in which forward LIBOR or forward swap rates are modelled to be lognormal under the forward probability measure of the corresponding maturity, are extended to a multicurrency setting. If lognormal dynamics are assumed for forward LIBOR or forward swap rates in two currencies, the forward exchange rate linking the two currencies can only be chosen to be lognormal for one maturity, with the dynamics for all other maturities given by no–arbitrage relationships. Alternatively, one could choose forward interest rates in only one currency, say the domestic, to be lognormal and postulate lognormal dynamics for all forward exchange rates, with the dynamics of foreign interest rates determined by no-arbitrage relationships.
  • Dun, T., Barton, G.W. & Schlögl, E. 2001, ‘Simulated Swaption Delta-Hedging in the Lognormal Forward Libor Model’, International Journal of Theoretical & Applied Finance, vol. 4, no. 4, pp. 677-709.
    View/Download from: Publisher’s site/Working paper version on SSRN
    Click arrow to view description Alternative approaches to hedging swaptions are explored and tested by simulation. Hedging methods implied by the Black swaption formula are compared with a lognormal forward LIBOR model approach encompassing all the relevant forward rates. The simulation is undertaken within the LIBOR model framework for a range of swaptions and volatility structures. Despite incompatibilities with the model assumptions, the Black method performs equally well as the LIBOR method, yielding very similar distributions for the hedging pro t and loss even at high rehedging frequencies. This result demonstrates the robustness of the Black hedging technique and implies that being simpler and generally better understood by nancial practitioners it would be the preferred method in practice.
  • Schlögl, E. and L. Schlögl 2000, ‘A Square-Root Interest Rate Model Fitting Discrete Initial Term Structure Data’, Applied Mathematical Finance vol. 7, no. 3, pp. 183-209.
    View/Download from: Publisher’s site/Working paper version on SSRN
    Click arrow to view description This paper presents one-factor and multifactor versions of a term structure model in which the factor dynamics are given by Cox/Ingersoll/Ross (CIR) type ‘square root’ diffusions with piecewise constant parameters. The model is fitted to initial term structures given by a finite number of data points, interpolating endogenously. Closed form and near closed form solutions for a large class of fixed income derivatives are derived in terms of a compound noncentral chi-square distribution. An implementation of the model is discussed where the initial term structure of volatility is fitted via cap prices.
  • Schlögl, E. and D. Sommer 1998, ‘Factor Models and the Shape of the Term Structure, The Journal of Financial Engineering vol. 7, no. 1, March 1998, pp. 79-88.
    View/Download from: Working paper version on SSRN
    Click arrow to view description The present paper analyses a broad range of one- and multifactor models of the term structure of interest rates. We assess the influence of the number of factors, mean reversion, and the factor probability distributions on the term structure shapes the models generate, and use spread options as an aggregate measure of the relative importance assigned to rising and falling forward rate curves by the models considered. We derive valuation formulas for these contingent claims in the multifactor Gaussian and CIR- models. Our main result is that the specification of mean reversion and the number of factors are both much more important for the relative movements of interest rates than the distributional characteristics of the factors. To the extent that interest rate risk depends on the movements of different parts of the term structure relative to one another rather than on shifts of its absolute level, the distributional assumption on the factor dynamics is found to be essentially irrelevant.
  • Sandmann, K. and E. Schlögl 1996, ‘Zustandspreise und die Modellierung des Zinsänderungsrisikos (State Prices and the Modelling of Interest Rate Risk)', Zeitschrift für Betriebswirtschaft vol. 66, no. 7, July 1996, pp. 813-836.

Book chapters

  • Chung, I., Dun, T. & Schlögl, E. 2010, ‘Lognormal forward market model (LFM) volatility function approximation’ in Chiarella, C; Novikov, A (eds), Contemporary Quantitative Finance: Essays in Honour of Eckhard Platen, Springer, Germany, pp. 369-405.
    Download from: Publisher’s site
    Click arrow to view description In the lognormal forward Market model (LFM) framework, the specification for time-deterministic instantaneous volatility functions for state variable forward rates is required. In reality, only a discrete number of forward rates is observable in the market. For this reason, traders routinely construct time-deterministic volatility functions for these forward rates based on the tenor structure given by these rates. In any practical implementation, however, it is of considerable importance that volatility functions can also be evaluated for forward rates not matching the implied tenor structure. Following the deterministic arbitrage-free interpolation scheme introduced by Schlögl in (Advances in Finance and Stochastics: Essays in Honour of Dieter Sondermann. Springer, Berlin 2002) in the LFM, this paper, firstly, derives an approximate analytical formula for the volatility function of a forward rate not matching the original tenor structure. Secondly, the result is extended to a swap rate volatility function under the lognormal forward rate assumption.
  • Schlögl, E. & Schlögl, L. 2009, ‘Factor Distributions Implied by Quoted CDO Spreads’ in Cont, R (eds), Frontiers in Quantitative Finance, John Wiley and Sons, New Jersey, USA, pp. 217-234.
    Download from: UTS OPUS
    Click arrow to view description The rapid pace of innovation in the market for credit risk has given rise to a liquid market in synthetic collateralised debt obligation (CDO) tranches on standardised portfolios. To the extent that tranche spreads depend on default dependence between different obligors in the reference portfolio, quoted spreads can be seen as aggregating the market views on this dependence. In a manner reminiscent of the volatility smiles found in liquid option markets, practitioners speak of implied correlation “smiles” and “skews” . We explore how this analogy can be taken a step further to extract implied factor distributions from the market quotes for synthetic CDO tranches.
  • Schlögl, E. 2008, ‘Markov Models for CDOs’ in Meissner, G. (ed), The Definitive Guide to CDOs: Market, Application, Valuation and Hedging, Risk Books, Cambridge, UK, pp. 319-340.
  • Schlögl, E. 2002, ‘Arbitrage-free interpolation in models of market observable interest rates’ in Sandmann K; Schönbucher PJ (eds), Advances in Finance and Stochastics: Essays in honour of Dieter Sondermann, Springer-Verlag Berlin Heidelberg, Berlin, Germany, pp. 197-218.
    Download from: Publisher’s site/Working paper version on arXiv
    Click arrow to view description Models which postulate lognormal dynamics for interest rates which are compounded according to market conventions, such as forward LIBOR or forward swap rates, can be constructed initially in a discrete tenor framework. Interpolating interest rates between maturities in the discrete tenor structure is equivalent to extending the model to continuous tenor. The present paper sets forth an alternative way of performing this extension; one which preserves the Markovian properties of the discrete tenor models and guarantees the positivity of all interpolated rates.

Conference proceedings

  • MacNamara, S., Schlögl, E. & Botev, Z. 2021, ‘Estimation When Both Covariance and Precision Matrices Are Sparse’; in: Kim, S., B. Feng, K. Smith, S. Masoud, Z. Zheng, C. Szabo, & M. Loper (Eds.): Proceedings of the 2021 Winter Simulation Conference.
    View/Download from: Working paper version on arXiv
    Click arrow to view description We offer a method to estimate a covariance matrix in the special case that \textit{both} the covariance matrix and the precision matrix are sparse — a constraint we call double sparsity. The estimation method is maximum likelihood, subject to the double sparsity constraint. In our method, only a particular class of sparsity pattern is allowed: both the matrix and its inverse must be subordinate to the same chordal graph. Compared to a naive enforcement of double sparsity, our chordal graph approach exploits a special algebraic local inverse formula. This local inverse property makes computations that would usually involve an inverse (of either precision matrix or covariance matrix) much faster. In the context of estimation of covariance matrices, our proposal appears to be the first to find such special pairs of covariance and precision matrices.

Working papers

  • Gellert, K. & Schlögl, E. 2021, ‘Short Rate Dynamics: A Fed Funds and SOFR perspective’.
    View/Download from: Working paper version on SSRN
    Click arrow to view description

    The Secured Overnight Funding Rate (SOFR) is becoming the main Risk–Free Rate benchmark in US dollars, thus interest rate term structure models need to be updated to reflect the key features exhibited by the dynamics of SOFR and the forward rates implied by SOFR futures. Historically, interest rate term structure modelling has been based on rates of substantially longer time to maturity than overnight, but with SOFR the overnight rate now is the primary market observable. This means that the empirical idiosyncrasies of the overnight rate cannot be ignored when constructing interest rate models in a SOFR–based world.

    As a rate reflecting transactions in the Treasury overnight repurchase market, the dynamics of SOFR are closely linked to the dynamics Effective Federal Funds Rate (EFFR), which is the interest rate most directly impacted by US monetary policy target rate decisions. Therefore, these rates feature jumps at known times (Federal Open Market Committee meeting dates), and market expectations of these jumps are reflected in prices for futures written on these rates. On the other hand, forward rates implied by Fed Funds and SOFR futures continue to evolve diffusively. The model presented in this paper reflects the key empirical features of SOFR dynamics and is calibrated to futures prices. In particular, the model reconciles diffusive forward rate dynamics with piecewise constant paths of the target short rate.

  • Backwell, A., Macrina, A. Schlögl, E. & Skovmand, D. 2019, ‘Term Rates, Multicurve Term Structures and Overnight Rate Benchmarks: a Roll–Over Risk Approach’.
    View/Download from: Working paper version on SSRN
    Click arrow to view description Modelling the risk that a financial institution may not be able to roll over its debt at the market reference rate, the so–called “roll–over risk”, we construct a model framework for the dynamics of reference term rates (e.g., LIBOR) and their spread vis–à–vis benchmarks based on overnight reference rates, e.g., rates implied by overnight index swaps (OIS). In this framework, different interest rate term structures are endogenously generated for each tenor, that is, a different term structure for each choice of the length of the interest rate accrual period, be it overnight (e.g., OIS), three–month LIBOR, six–month LIBOR, etc. A concrete model instance in this framework can be calibrated simultaneously to available market instruments at a particular point in time, but more importantly, we explicitly obtain dynamics of term rates such as LIBOR. Thus models in our framework are amenable to econometric estimation. For a model class based on affine dynamics, we conduct an empirical analysis on EUR data for OIS, interest–rate swaps, basis swaps and credit default swaps. Our model achieves a better fit to time series data than other models proposed in prior literature. We find that credit risk typically contributes only about 30% of the IBOR/OIS spread, with the balance of the spread due to the funding liquidity component of roll–over risk. Looking ahead, we show that, even if credit risk is entirely mitigated by repo transactions, the presence of roll–over risk confounds attempts to obtain term rates from overnight rate benchmarks. As various jurisdictions transition away from panel–based term rate benchmarks towards transaction–based overnight ones (such as SOFR in the United States), the framework presented in this paper thus provides important insights into some of the consequences of this transition.
  • Kang, B., Nikitopoulos, C. S., Schlögl, E. & Taruvinga, B. 2019 ‘The Impact of Jumps on American Option Pricing: The S&P 100 Options Case’.
    View/Download from: Working paper version on SSRN
    Click arrow to view description This paper analyzes the importance of asset and volatility jumps in American option pricing models. Using the Heston (1993) stochastic volatility model with asset and volatility jumps and the Hull and White (1987) short rate model, American options are numerically evaluated by the Method of Lines. The calibration of these models to S&P 100 American options data reveals that jumps, especially asset jumps, play an important role in improving the models’ ability to fit market data. Further, asset and volatility jumps tend to lift the free boundary, an effect that augments during volatile market conditions, while the additional volatility jumps marginally drift down the free boundary. As markets turn more volatile and exhibit jumps, American option holders become more prudent with their exercise decisions, especially as maturity of the options approaches.
  • Feng, Y. & Schlögl, E. 2018, ‘Model Risk Measurement under Wasserstein Distance’.
    View/Download from: Working paper version on SSRN
    Click arrow to view description The paper proposes a new approach to model risk measurement based on the Wasserstein distance between two probability measures. It formulates the theoretical motivation resulting from the interpretation of fictitious adversary of robust risk management. The proposed approach accounts for all alternative models and incorporates the economic reality of the fictitious adversary. It provides practically feasible results that overcome the restriction and the integrability issue imposed by the nominal model. The Wasserstein approach suits for all types of model risk problems, ranging from the single-asset hedging risk problem to the multi-asset allocation problem. The robust capital allocation line, accounting for the correlation risk, is not achievable with other non-parametric approaches.
  • Chiarella, C., Nikitopoulos, C. S., Schlögl, E. & Yang, H. 2016, ‘Pricing American Options under Regime Switching Using Method of Lines’.
    View/Download from: Working paper version on SSRN
    Click arrow to view description This paper considers the American option pricing problem under regime-switching by using the method-of-lines (MOL) scheme. American option prices in each regime involve prices in all other regimes. We treat the prices from other regimes implicitly, thus guaranteeing consistency. Iterative procedures are required but very few iterative steps are needed in practice. Numerical tests demonstrate the robustness, accuracy and efficiency of the proposed numerical scheme. We compare our results with Buffington and Elliott (2002)‘s analytical approximation under two regimes. Our MOL scheme provides improved results especially for out-of-the money options, possibly because they use a separation of variable approach to the PDEs which cannot hold around the early exercise region. We also compare our results with those of Khaliq and Liu (2009) and suggest that their implicit scheme can be improved.
  • Chang, Y. & E. Schlögl 2014, ‘A Consistent Framework for Modelling Basis Spreads in Tenor Swaps’
    View/Download from: Working paper version on SSRN
    Click arrow to view description The phenomenon of the frequency basis (i.e. a spread applied to one leg of a swap to exchange one floating interest rate for another of a different tenor in the same currency) contradicts textbook no-arbitrage conditions and has become an important feature of interest rate markets since the beginning of the Global Financial Crisis (GFC) in 2008. Empirically, the basis spread cannot be explained by transaction costs alone, and therefore must be due to a new perception by the market of risks involved in the execution of textbook “arbitrage” strategies. This has led practitioners to adopt a pragmatic “multi-curve” approach to interest rate modelling, which leads to a proliferation of term structures, one for each tenor. We take a more fundamental approach and explicitly model liquidity risk as the driver of basis spreads, reducing the dimensionality of the market for the frequency basis from observed spread term structures for every frequency pair down to term structures of two factors characterising liquidity risk. To this end, we use an intensity model to describe the arrival time of (possibly stochastic) liquidity shocks with a Cox Process. The model parameters are calibrated to quoted market data on basis spreads, and the improving stability of the calibration suggests that the basis swap market has matured since the turmoil of the GFC.
  • Chang, Y. & E. Schlögl 2012, ‘Carry Trade and Liquidity Risk: Evidence from forward and Cross-Currency Swap Markets’
    View/Download from: Working paper version on SSRN
    Click arrow to view description This study empirically examines the effect of foreign exchange (FX) market liquidity risk and volatility on the excess returns of currency carry trades. In contrast to the existent literature, we construct an alternative proxy of liquidity risk - violations of no arbitrage bounds in the forward and currency swap markets. We also use volatility smile data to capture FX-market specific volatility. The sample data cover periods both before and after the Global Financial Crisis (GFC). Both proxies are significant in explaining the abnormal returns of carry trades, particularly after the GFC. Our findings provide substantial evidence that uncovered interest parity (UIP) puzzle can be resolved after controlling for liquidity risk and market volatility.
  • Pilz, K. & Schlögl, E. 2011, ‘Calibration of the Multi-Currency LIBOR Market Model’
    View/Download from: Working paper version on SSRN
    Click arrow to view description This paper presents a method for calibrating a multicurrency lognormal LIBOR Market Model to market data of at-the-money caps, swaptions and FX options. By exploiting the fact that multivariate normal distributions are invariant under orthonormal transformations, the calibration problem is decomposed into manageable stages, while maintaining the ability to achieve realistic correlation structures between all modelled market variables.
  • Mahayni, A. B., Schlögl, E. & Schlögl, L. 1999, ‘Robustness of Gaussian Hedges and the Hedging of Fixed Income Derivatives’.
    View/Download from: Working paper version on SSRN
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    The effect of model and parameter misspecification on the effectiveness of Gaussian hedging strategies for derivative financial instruments is analyzed, showing that Gaussian hedges in the “natural” hedging instruments are particularly robust. This is true for all models the imply Black/Scholes-type formulas for option prices and hedging strategies. In this paper we focus on the hedging of fixed income derivatives and show how to apply these results both within the framework of Gaussian term structure models as well as the increasingly popular market models where the prices for caplets and swaptions are given by the corresponding Black formulas.

    By explicitly considering the behaviour of the hedging strategy under misspecification we also derive the result by El Karoui, Jeanblanc-Picque and Shreve that a superhedge is obtained in the Black/Scholes model if the misspecified volatility dominates the true volatility. Furthermore, we show that the robustness and superhedging result do not hold if the natural hedging instruments are unavailable. In this case, we study criteria for the optimal choice from the instruments available.