# Professor Erik Schlogl

**Director**, Quantitative Finance Research Centre**Core Member**, Quantitative Finance Research Centre

DipVw (Bonn), PhD (Bonn)

Member, Bachelier Finance Society

**Email:** Erik.Schlogl@uts.edu.au

**Phone:** +61 2 9514 7785

**Fax:**

**Room:** CB08.07.15 (map)

**Mailing address: **

**Links:**
http://www.schlogl.com

## Biography

Erik Schlögl currently is Professor and Director of the Quantitative Finance Research Centre at the University of Technology, Sydney (UTS), Australia. Erik received his doctorate in Economics from the University of Bonn, Germany, for work on term structure models and the pricing of fixed income derivatives and has gained broad-based experience in computational financial engineering. He has consulted for financial institutions and software developers in Europe, Australia and in the US, and served as an expert witness in cases before the Federal Court of Australia. His research interests cover a broad area of quantitative finance, in particular model calibration, interest rate term structure modelling, credit risk and the integration of multiple sources of risk. His research articles have been published in a number of international journals, including Finance & Stochastics, Quantitative Finance, Risk and the Journal of Economic Dynamics and Control. He is also the chairman of the organising committee of the Sydney Financial Mathematics Workshop (SFMW) and one of the co-organisers of the annual conference Quantitative Methods in Finance (QMF). In addition to UTS, he held positions at the University of New South Wales, Australia, and the University of Bonn, Germany.

## Research

**Research interests**

Derivative securities pricing, term structure of interest rates, quantitative finance techniques, credit risk modelling, computational finance.

**Research supervision: ** Yes

**Postgraduate research degree students supervised:**

Samson Assefa

King Ming Chan

In Hwan (David) Chung

Du Ke

Troy Morgan

Tao Peng

Liang Zhao

## Projects

### Selected Peer-Assessed Projects

A New Theoretical Approach to Pension Fund Economics, Asset Management and Insurance - DP130104074

Pricing and Hedging Extreme Maturity Contracts

A New Integrated Approach to Managing Risk in Financial Markets

An Integrated Approach to Credit Risk Management and the Valuation of Credit Derivatives

## Publications

### Books

Schlogl, E. 2014, *Quantitative finance: An object-oriented approach in C++*, 1st, Taylor and Francis, Florida, USA.

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Quantitative Finance: An Object-Oriented Approach in C++ provides readers with a foundation in the key methods and models of quantitative finance. Keeping the material as self-contained as possible, the author introduces computational finance with a focus on practical implementation in C++. Through an approach based on C++ classes and templates, the text highlights the basic principles common to various methods and models while the algorithmic implementation guides readers to a more thorough, hands-on understanding. By moving beyond a purely theoretical treatment to the actual implementation of the models using C++, readers greatly enhance their career opportunities in the field.

### Journal articles

Pilz, K. & Schlogl, 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-interpretationï½ 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.

Schlogl, 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

### Conferences

Pilz, K. & Schlogl, E. 2012, 'Calibration of multi-currency LIBOR market model: An orthonormal procrustes problem'.

Pilz, K. & Schlogl, E. 2012, 'Calibration of multi-currency LIBOR market model: An orthonormal procrustes problem'.

Schlogl, E. & Chang, Y. 2012, 'Carry trade and liquidity risk: Evidence from forward and cross-currency swap markets'.

### Journal articles

Nielsen, J.A., Sandmann, K. & Schlogl, 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 investors 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.

### Chapters

Chung, I., Dun, T. & Schlogl, 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.

### Other

Nielsen, J., Sandmann, K. & Schlogl, E. 2010, 'Equity-Linked Pension Schemes with Guarantees', *QFRC*.

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This paper demonstrates an efficient numerical (semi-analytical) method for valuing return guarantees embedded in equity-linked pension schemes and analyses the qualitative properties of such contracts. We study and compare the pricing of three types of pension schemes with guarantees. These are long term investment plans, in which the investor typically puts in periodic payments of cash over a long time. The possibility of early contract termination is modelled by a mortality distribution. These schemes are closely related to equity-linked life insurance contracts, and the results presented here are also applicable to the latter.

Nielsen, J.A., Sandman, K. & Schlogl, E. 2010, 'Equity-linked pension schemes with guarantees', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 270 Abstract: 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 participation surplus. 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 contracts pecifications 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 a 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.

### Conferences

Schlogl, E. 2010, 'CalibrationÂ ofÂ LIBORÂ marketÂ modelsÂ incorporatingÂ multipleÂ sourcesÂ ofÂ riskÂ'.

Schlogl, E., Nielsen, J. & Sandman, K. 2010, 'Equity-linked pension schemes with guarantees'.

### Chapters

Schlogl, L. & Schlogl, E. 2010, 'Duffie-Singleton Model' in al, R.C.E. (ed), *Encyclopedia of Quantitative Finance*, Wiley, US, pp. 499-501.

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### Journal articles

Bruti Liberati, N., Nikitopoulos Sklibosios, C., Platen, E. & Schlogl, 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.

### Conferences

Bruti Liberati, N., Nikitopoulos Sklibosios, C., Platen, E. & Schlogl, E. 2009, 'Alternative defaultable term structure models'.

### Other

Bruti Liberati, N., Nikitopoulos Sklibosios, C., Platen, E. & Schlogl, E. 2009, 'Alternative Defaultable Term Structure Models', *Quantitative Finance Research Paper Series*.

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.

Pilz, K. & Schlogl, E. 2009, 'A hybrid commodity and interest rate', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 261 Abstract: 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 âre-interpretationâ? 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.

### Conferences

Schlogl, E. 2009, 'Quantitative finance before and after the crisis'.

### Chapters

Schlogl, E. & Schlogl, L. 2009, 'Factor Distributions Implied by Quoted CDO Spreads' in Cont, R. (ed), *Frontiers in Quantitative Finance*, John Wiley and Sons, New Jersey, USA, pp. 217-234.

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### Journal articles

Mahayni, A.B. & Schlogl, 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.

### Conferences

Nikitopoulos Sklibosios, C., Bruti Liberati, N., Platen, E. & Schlogl, E. 2008, 'Real-world pricing for defaultable term structure models'.

Peng, T. & Schlogl, E. 2008, 'Dynamic default correlation models: Binomial lattices, cross entropy and perfect match'.

Peng, T. & Schlogl, E. 2008, 'Dynamic default correlation models: Binomial lattices, cross entropy and perfect match'.

Schlogl, E. 2008, 'Design patterns and objects in monte carlo simulation'.

### Chapters

Schlogl, 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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### Conferences

Schlogl, E. 2008, 'Option Pricing where Assets Follow a Gram-Charlier Density of Arbitrary Order'.

Schlogl, E. 2008, 'Option pricing where assets follow a Gram-Charlier density of arbitrary order'.

Schlogl, E. 2008, 'Option pricing where the underlying assets follow a Gram/Charlier density of arbitrary order'.

Bruti Liberati, N., Nikitopoulos Sklibosios, C., Platen, E. & Schlogl, E. 2007, 'Defaultable term structure models under the benchmark approach'.

Bruti Liberati, N., Nikitopoulos Sklibosios, C., Platen, E. & Schlogl, E. 2007, 'Real-World Pricing for Defaultable Term Structure Models'.

### Journal articles

Chiarella, C., Nikitopoulos Sklibosios, C. & Schlogl, 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. & Schlogl, 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 Schonbucher [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.

### Conferences

Peng, T. & Schlogl, E. 2007, 'Default correlation modelling: Binomial lattices, cross entropy and perfect match'.

Schlogl, E. 2007, 'Option pricing where the underlying assets follow a Gram/Charlier density of arbitrary order'.

### Other

Schlogl, E. & SchlÃ¶gl, L. 2007, 'Factor distributions implied by quoted CDO spreads tranche pricing', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 190 Abstract: 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 âsmilesâ? and âskewsâ?. We explore how this analogy can be taken a step further to extract implied factor distributions from the market quotes for synthetic CDO tranches.

### Conferences

Schlogl, E. 2006, 'Fitting the market: Tractable approximations and calibrating models to multiple volatility smiles', *Quantitative Methods in Finance 2006 Conference*.

Schlogl, E. 2006, 'Generic implementation of control variates in option pricing', *5th National Symposium on Financial Mathematics*.

Schlogl, E. & Schlogl, L. 2006, 'Fitting CDO factor distributions to quoted synthetic tranche spreads', *2006 Symposium on Credit Risk, Extreme Values and Actural Studies*.

### Other

Chiarella, C., Nikitopoulos Sklibosios, C. & Schlogl, E. 2005, 'A control variate method for Monte Carlo simulations of Heath-Jarrow-Morton with jumps (QFRC paper #167)'.

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ISSN 1441-8010 www.business.uts.edu.au/qfrc/research/research_papers/rp167.pdf

### Chapters

Choy, B., Dun, T. & Schlogl, E. 2005, 'Correlating market models' in Dunbar, N. (ed), *Derivatives Trading and Option Pricing*, Risk Books, London, UK, pp. 303-322.

### Conferences

Schlogl, E. 2005, 'Factor distributions and correlations implied by market quotes for synthetic CDO', *4th National Symposium on Financial Mathematics*, -, -.

Schlogl, E. 2005, 'Spoken and implied: factor distributions implied by quotes CDO spreads and the pricing of bespoke tranches', *Quantitative Methods in Finance 2005 Conference*, -, -.

### Other

Chiarella, C., Schlogl, E. & Nikitopoulos Sklibosios, C. 2004, 'A Markovian defaultable term structure model with state dependent volatilities (QFRC paper #135)'.

### Journal articles

Choy, B., Dun, T. & Schlogl, E. 2004, 'Correlating market models', *Asia Risk*, vol. October, pp. 53-59.

Choy, B., Dun, T. & Schlogl, E. 2004, 'Response to comments by Stephen Blyth and Maciej Sawicki', *Risk*, vol. 17, no. 11, p. 118.

### Conferences

Frishling, V. & Schlogl, E. 2004, 'Credit derivatives pricing models: Overview and firm value models', *Sydney financial mathematics workshop*, -, -.

Kazakov, V., Schlogl, E. & Schlogl, L. 2004, 'Gram-charlier expansions, edgeworth expansions and multivariate distributions implied by options prices.', *3rd National symposium on financial mathematics*, -, -.

Schlogl, E. 2004, 'Advance credit risk measurement and modelling techniques for effective portfolio credit risk management', *Risk magazine workshop*, -, -.

Schlogl, E. 2004, 'Default correlation modelling', *Sydney financial mathematics workshop*, -, -.

Schlogl, E. 2004, 'Factor distribution and correlations implied by market quotes for synthetic CDO tranches', *-*, -, -.

Schlogl, E. 2004, 'Modelling default correlation for portfolio credit risk', *Derivatives and risk management*, -, -.

Schlogl, E. 2004, 'Understanding the key issues and concerns in modelling portfolio credit risk', *Credit risk forum 2004*, -, -.

Chiarella, C., Nikitopoulos Sklibosios, C. & Schlogl, E. 2003, 'A Markovian defaultable term structure model with state dependent volatilities', *Quantitative Methods in Finance 2003 Conference*, --, --.

Chiarella, C., Schlogl, E. & Nikitopoulos Sklibosios, C. 2003, 'A Markovian defaultable term structure model with state dependent volatilities', *CREDIT 2003 Conference on Dependence Modelling for Credit Portfolios*, --, --.

### Other

Choy, S., Dun, T. & Schlogl, E. 2003, 'Correlating market models (QFRC paper #105)'.

Mahayni, A.B. & Schlogl, E. 2003, 'The risk management of minimum return guarantees (QFRC paper #102)'.

### Conferences

Mahayni, A.B. & Schlogl, E. 2003, 'The risk management of power options embedded in life-insurance products', *20th AFFI International Conference*, The Association Francaise de Finance, Lyon, France, pp. 1-27.

Schlogl, E. 2003, 'The risk management of minimum return guarantees', *Quantitative Methods in Finance 2003 Conference*, --, --.

Schlogl, E. & Kazakov, V. 2003, 'Implied volatilities, implied correlations, implied distributions: Information contained in options prices', *2nd National Symposium on Financial Markets*, --, --.

Schlogl, E. & Mahayni, A.B. 2003, 'The risk management of minimum return guarantees: Retirement provision in scary markets', *11th Australian Colloquium of Superannuation Researchers*, --, --.

### Journal articles

Schlogl, E. 2002, 'A multicurrency extension of the lognormal interest rate market models', *Finance and Stochastics*, vol. 6, no. 2, pp. 173-196.

### Conferences

Schlogl, E. 2002, 'Arbitrage-free interpolation in models of market observable interest rates', *2nd World Congress of the Bachelier Finance Society*.

### Chapters

Schlogl, E. 2002, 'Arbitrage-free interpolation in models of market observable interest rates' in Sandmann, K. & Schonbucher, P.J. (eds), *Advances in Finance and Stochastics: essays in honour of Dieter Sondermann*, Springer-Verlag Berlin Heidelberg, Berlin, Germany, pp. 197-218.

### Reports

Schlogl, E. 2002, *Extracting the joint volatility structure of foreign exchange & interest rates from option prices*.

### Conferences

Schlogl, E. 2002, 'Integration of interest rate and currency risk across markets: model calibration, derivatives pricing & risk management', *RISK 2002 Conference*.

Schlogl, E. 2002, 'Joint calibration of volatilities & correlations in interest rate and FX markets', *Quantitative Methods in Finance 2002 Conference*.

### Journal articles

Dun, T., Barton, G.W. & Schlogl, 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 prot 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.

### Other

Schlogl, E. 2001, 'Arbitrage-free interpolation in models of market observable interest rates', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 71 Abstract: 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 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.

Dunne, T. & Schlogl, E. 2000, 'Simulated swaption delta-hedging in the lognormal forward Libor model', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 40 Abstract: Alternative approaches to hedging swaptions are explored and tested by simulation. Hedging methods implied by the Balck 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 profit 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 financial practitioners - it would be the preferred method in practice.

Schlogl, E. & Schlogl, L. 1999, 'A square-root interest rate model fitting discrete initial term structure data', *Research Paper Series, Quantitative Finance Research Centre, University of Technology, Sydney*.

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Research Paper Number: 24 Abstract: This paper presents the one- and the 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. This 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.