Research

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Refereed journal articles



  • Pilz, K. & Schlögl, E. 2013, 'A hybrid commodity and interest rate market model', Quantitative Finance, vol. 13, no. 4, pp. 543-560.
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    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.
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    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.
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    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.
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    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.
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    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.
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    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.
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    The defaultable forward rate is modelled as a jump diffusion process within the Schönbucher [26,27] general Heath, Jarrow and Morton [20] 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.
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    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.
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    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.
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    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.
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    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 Žfitted to initial term structures given by a Žfinite number of data points, interpolating endogenously. Closed form and near closed form solutions for a large class of Žfixed 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 Žfitted 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.
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    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.
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    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.
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    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.
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  • 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.
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    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.


Working papers



  • Chang, Y. & E. Schlögl 2014, 'A Consistent Framework for Modelling Basis Spreads in Tenor Swaps'
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    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'
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    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'
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    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.