abstracts

• Alexander Antonov, Numerix

Skew and smile calibration using Markovian projection

We briefly remind a subject of Markovian projection from the Gyongy theorem to Piterbarg’s research. Then we address a universal approach to the Markovian projection to two main processes in the Mathematical finance: a displaced diffusion and a Heston process. The first process is able to capture option skew effects and the second one can reproduce both skew and smile.

We develop a systematic view on the Markovian projection of these two processes, and work out a set of computationally efficient formulas valid for a large class of non-Markovian underlying processes.

We illustrate the theory with multiple examples including calculation of FX options in cross-currency models, swaption pricing in LIBOR Market Models and a spread of two correlated Heston models.

Theory and applications of the Markovian projection to a displaced diffusion can be found in Antonov and Misirpashaev, "Markovian Projection onto a Displaced Diffusion: Generic Formulas with Applications" (October 9, 2006).

Available at SSRN: http://ssrn.com/abstract=937860

[paper] [slides]

• Jörg Behrens, Ernst & Young Switzerland

Economic Capital Models under Solvency II

Solvency II places various challenges to the insurance industry. Despite of similarities with Basel II and experience with internal insurance models, various modeling and implementation challenges remain. We select some technical aspects of Solvency II and discuss options for implementation within an internal solvency model.

[slides]

• Denis Belomestny, Weierstrass Institute Berlin

True upper bounds for Bermudan products via non-nested Monte Carlo

We present a generic non-nested Monte Carlo procedure for computing true upper bounds for Bermudan products, given an approximation of the Snell envelope. The pleonastic true stresses that, by construction, the estimator is biased above the Snell envelope. The key idea is a regression estimator for the Doob martingale part of the approximative Snell envelope, which preserves the martingale property. The so constructed martingale may be employed for computing dual upper bounds without nested simulation. In general, this martingale can also be used as a control variate for simulation of conditional expectations. In this context, we develop a variance reduced version of the nested primal-dual estimator (Anderson & Broadie (2004)) and nested consumption based (Belomestny & Milstein (2006)) methods. Numerical experiments indicate the efficiency of the non-nested Monte Carlo algorithm and the variance reduced nested one. This is joint work with Christian Bender and John G.M. Schoenmakers.

[paper] [slides]

• Oliver Caps, Dresdner Bank

On the Valuation of Power-Reverse Duals and Equity-Rates Hybrids

For the valuation of cross-asset products like the popular power-reverse dual variants or equity-rates hybrids it is important to take into account pricing effects of asset smiles, stochastic rates or interest rate smiles appropriately.

In this talk we discuss requirements for a flexible hybrid model and describe a stochastic volatility (Heston) model for asset processes (like FX or equity) with stochastic rates (Markovian HJM dynamics like the Hull-White dynamics) satisfying these needs and allowing for an efficient calibration. We describe how several numerical problems can be managed when implementing the model, show calibration results, and give an impression of the price impact for hybrid products.

[slides] [handouts]

• Sergio Dutra, Commerzbank

Realistic Interest-rate Smile in a Cross-Currency Markov Functional Model

This general multicurrency extension preserves the simple driftless dynamics of the hidden Markovian state variables that is one of the main reasons behind the practical usefulness of the original single currency Markov functional model. This model describes each interest rate market in a completely non-parametric way leading to realistic modelling of interest-rate smile, whilst still allowing practical calibration. It models the FX rate in a semiparametric way allowing a simple and transparent interpretation for the correlations between the driving Brownian factors in terms of correlations between actual financial quantities observable in the market. As an example, the model is applied to multi-callable quanto swaps.

[handouts]

• Ernst Eberlein, University of Freiburg

A Cross-Currency Lévy Market Model

The Lévy Libor or market model which was introduced in Eberlein and Özkan (2005) is extended to a multi-currency setting. As an application we derive closed form pricing formulas for crosscurrency derivatives. Foreign caps and floors and cross-currency swaps as well as quanto caplets are studied in detail. Numerically efficient pricing algorithms based on bilateral Laplace transforms are derived. A calibration example is given for a two-currency setting (EUR, USD). This is joint work with Nataliya Koval.

[slides]

• Gabriele Guehring, d-fine

Stochastic processes for implied volatilities

Pricing models specifying a stochastic process for the implied volatility are investigated. No-arbitrage arguments lead to an expression specifying the drift of this stochastic implied volatility process. The drift restriction formula connects implied volatility, spot volatility and the volatility of the implied volatility process. It is possible to show that this formula is also satisfied for every explicitly specified stochastic spot volatility process. The proof of this result gives an explicit formula for the volatility of the implied volatility in terms of the volatility of the spot volatility. Consequences in case the implied volatility or the spot volatility are uncorrelated with the asset process are derived. We show that a spot volatility process which is not correlated with the process for the asset price, does not imply that the process for the implied volatility is also not correlated with the asset price. Nevertheless, we obtain a symmetric smile function in both cases.

[slides]

• Susanne Griebsch, Frankfurt School of Finance & Management

Closed-Form Exotic Option Pricing in the Heston Model

In our study we focus on closed-form option pricing under stochastic volatility models, particularly with regard to the Heston model. In Heston's model closed-form formulas exist only for a few options. Most of these closed-form solutions are constructed from characteristic functions for the calculation of the product's expected payoff values.

In our work we follow this approach and derive multivariate characteristic functions dependent on at least two spot values for different points in time. The derived characteristic functions are then applied to closed-form option pricing of Fader options and discretely monitored Barrier options.

The derived formulas are evaluated with different numerical methods and compared to Monte Carlo values with regard to accuracy and computational times.

A paper will appear in the near future.



• Reinhard Hirsch, d-fine

Modelling and Forecasting of Prices and Forward Curves for Energy and Commodities

Energy and Commodities Derivatives are an old but also very actual topic with a strong growing market and an exiting development during the last years. All kinds of Investors from pension funds to private persons are today exploring at least the exchange traded commodities. Enormous profits combined with huge losses of hedge funds in these markets punctuate the need of detailed understanding and modelling the pice processes and risk measurement for commodities.

We give an brief introduction to the characteristics of energy and commodity markets and show an analysis of the special elements of risk measurement and modelling of price processes and the forward curve dynamics of crude oil, refined products and base metals. The forecasting power of these models is analysed for crude oil and compared to multivariate models for the forecasting of price levels and the shape of the forward curve, i.e. backwardation / contango strength of the forward curve.

[slides]

• Martin Keller-Ressel, Vienna University of Technology

Yield Curve Shapes and the Asymptotic Short Rate Distribution in Affine One-Factor Models

We consider a model for interest rates where the short rate is given by a time-homogenous, one-dimensional affine process in the sense of Duffie, Filipovic and Schachermayer. We show that in such a model yield curves can only be normal, inverse or humped (i.e. endowed with a single local maximum). Each case can be characterized by simple conditions on the present short rate. We give conditions under which the short rate process will converge to a limit distribution and describe the limit distribution in terms of its cumulant generating function. We apply our results to the Vasicek model, the CIR model, a CIR model with added jumps and a model of Ornstein-Uhlenbeck type.

[paper] [slides]

• Jörg Kienitz, Postbank

Monte Carlo Simulation Software and Application to CPPI

Monte Carlo Simulation has become a key technology in the financial sector. It can be applied in a variety of settings. To cope a wide range of applications an efficient, robust and generic Monte Carlo Engine is necessary.

We consider a CPPI approach to a basket of IR linked funds as an example. CPPI is an abbrevation for "Constant Proportion Portfolio Insurance". It is a portfolio management technique aimed on the one hand side at maximizing returns for the investor and on the other hand side protecting the principal. It has been applied in the equity and hedge fund business since the early 80's. It now has been expanded to other asset classes like commodities, interest rates or credit. Our example will focus on the simulation of the basket and on selecting the "optimal" starting configuration.

The key to the simulation of portfolios in different market settings is a robust and efficient implementation of the Monte Carlo simulation engine. We will describe a general framework in C++ which can be applied to this setting and easily extends to other problems, e.g. derivatives pricing and other portfolio management settings. The framework consist of a bunch of loosely coupled C++ classes and the application of design patterns.

Daniel J. Duffy, Jörg Kienitz, Monte Carlo Methods in Quantitative Finance Generic and Efficient MC Solver in C++, Wilmott Magazine, 2005

Daniel J. Duffy, Jörg Kienitz, Software Frameworks in Quantitative Finance, Part I Fundamental Pricinples and Applications to Monte Carlo Methods, Wilmott Magazine, 2007

Daniel J. Duffy, Jörg Kienitz, Efficient and Robust Monte Carlo Methods in Financial Engineering, Wiley and Sons, forthcoming

[slides]

• Roger Lord, Rabobank International

A Fast and Accurate FFT-based Method for Pricing Early-exercise Options under Lévy Processes

A fast and accurate method for pricing early exercise and certain exotic options in computational finance is presented in this paper. The method is based on a quadrature technique and relies heavily on Fourier transformations. The main idea is to reformulate the well-known risk-neutral valuation formula by recognising that it is a convolution. The resulting convolution is dealt with numerically by using the Fast Fourier Transform (FFT). This novel pricing method, which we dub the Convolution method, CONV method for short, is applicable to a wide variety of payoffs and only requires the knowledge of the characteristic function of the model. As such the method is applicable within exponentially Lévy models, including the exponentially affine jump-diffusion models. For an M-times exercisable Bermudan option, the overall complexity is O(MN log(N)) with N grid points used to discretise the price of the underlying asset. It is shown how to price American options efficiently by applying Richardson extrapolation to the prices of Bermudan options.

Available at SSRN: http://ssrn.com/abstract=966046

[paper] [slides]

• Christian Menn, Sal. Oppenheim

Spot and Derivative Pricing in the EEX Power Market

Using spot and futures price data from the German EEX Power market, we test the adequacy of various models for electricity spot prices. The models are com- pared along two different dimensions: (1) We assess their ability to explain the major data characteristics and (2) the forecasting accuracy for expected future spot prices is analyzed. We find that the regime switching models clearly outperform its competitors in almost all respects. Furthermore, for short and medium-term periods our results underpin the frequently stated hypothesis that electricity futures quotes are consistently greater than the expected future spot, a situation which is denoted as contango.

Joint work with Michael Bierbrauer, Svetlozar T. Rachev and Stefan Trück

[paper] [slides]

• Attilio Meucci, Lehman Brothers

Beyond Black-Litterman: Views on Non-Normal Markets - the Copula-Opinion Pooling Approach

The pathbreaking technique by Black and Litterman allows portfolio managers to smoothly blend their subjective views on the market with a prior market distribution. Nevertheless, the BL approach suffers from two drawbacks. In the first place, in BL both the market prior and the manager's views are normally distributed. For most markets the normal assumption is too strong: fat tails, skewness and high dependence among extreme events characterize the joint distribution of market risk factors in many contexts. Secondly, in BL managers express views on the parameters that determine the market distribution. In reality, except in normal markets, it is arguably more natural to express views directly on the possible realizations of the market. We rely on the opinion pooling, rather than Bayesian, theory to expand on BL in the above directions. We use opinion pooling criteria to determine the marginal distribution of each view separately, whereas the joint co-dependence, i.e. the copula, among the views is directly inherited from the prior market structure. Finally, a suitable change of coordinates allows us to translate the joint distribution of the views into a joint posterior distribution for the market. First we introduce the theory behind the copula-opinion pooling approach; then we detail an algorithm to implement the COP approach under virtually any distributional assumption on the market and the views; we conclude with an application to the management of a fixed-income portfolio.

[slides] [BeyondBlackLitterman] [BeyondBlackLittermanInPractice]

• Natalie Packham, Frankfurt School of Finance & Management

Dependence in Credit Risk and Credit Derivatives Pricing: a Technique for Multivariate Stratified Sampling

When valuing a claim with Monte Carlo simulation, the variance of the estimator is a key figure for assesing the quality of the simulation. We present a variance reduction technique for sampling from the joint probability distribution of several random variables whose estimator is unbiased and consistent. Furthermore, the strategy does not require specific knowledge about the problem at hand, but can be applied in a very general sense. A practical application is the valuation of claims that are sensitive to the dependence structure of the underlying securities, such as first-to-default baskets and CDO tranches.

[slides]

• Jianwei Zhu, Sal. Oppenheim

An Extended Libor Market Model with Nested Stochastic Volatility

In this paper we extend standard Libor Market Model (LMM) with nested stochastic volatilities. The stochastic volatility of each Libor follows a mean-reverting process as in Schoebel and Zhu (1999) or in Heston (1993) under the individual forward measure of each Libor. Other than the existing stochastic volatility models, every volatility in the extended LMM is correlated with its Libor individually, and the parameters of stochastic volatility are also different over all Libors, however, are nested by some deterministic functions. With a nesting function, the same type of parameter such as mean level in all volatility processes share a certain term structure. In this model set-up, we can still derive the stochastic processes for Libors and volatilities under an arbitrary forward measure. In line with the stochastic volatility models for equity options, we obtain a closed-form solution via Fourier transform for caplets and floorlets. Finally, we use factor representation to express Libors and swap rates by some independent factors, namely principle components. The approximated analytical pricing formula for swaption can then be derived by using the characteristic functions that are just a product of the characteristic function of each factor. The numerical implementation of the nested stochastic volatility model is efficient and identical to the existing stochastic volatility models.

Available at SSRN: http://ssrn.com/abstract=955352

[paper] [slides]

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