Abstracts

• Dr Eric Benhamou, Pricing Partners

Smart Expansion and Fast Calibration for Jump Diffusion


Using Malliavin calculus techniques, we derive an analytical formula for the price of European options, for any model including local volatility and Poisson jump process. We show that the accuracy of the formula depends on the smoothness of the payoff function. Our approach relies on an asymptotic expansion related to small diffusion and small jump frequency/size. Our formula has excellent accuracy (the error on implied Black-Scholes volatilities for call option is smaller than 2 bp for various strikes and maturities). Additionally, model calibration becomes very fast.

This is joint work with E. Gobet and M. Miri.

[paper]

• Philipp Beyer, Universty of Konstanz and Postbank

Model Risk in Pricing and Hedging Exotic Equity Derivatives


We consider the following models for equity derivatives:

• Heston Stochastic Volatility Model
• Merton Jump Diffusion
• Variance Gamma (VG)
• VG with a Gamma Ornstein-Uhlenbeck clock (VG-OU)
• VG wiht a CIR clock (VG-CIR)
• Normal Inverse Gaussian (NIG)
• Normal Inverse Gaussian with a Gamma Ornstein-Uhlenbeck clock (NIG-OU)
• Normal Inverse Gaussian with a CIR clock (NIG-CIR)

First, we consider the calibration to quoted option prices within these models. We use the calibrated models to compute the prices of some exotic options including cliquet options and compare the prices generated by each model. We proceed by computing the forward characteristic functions for the models and study the forward implied volatility surface for each model. The different shapes of the forward volatility surfaces resulting from applying different models are studied. Furthermore, we show how the intial model parameters affect the shape of the forward volatility surface within a single model. Our conclusion is that if pricing and risk manage exotic options is considered one has to keep track of the forward volatility surface and make sure not only which model to use but adjust the parameters within a chosen model reasonably.

This is joint work with Jörg Kienitz (Deutsche Postbank AG).


• Dr Peter Carr, Bloomberg L.P.

Leverage Effect, Volatility Feedback, and Self-Exciting Market Disruptions:Disentangling the Multi-dimensional Variations in S\&P 500 Index Options

The level of an equity index and the volatility of an equity index interact through several distinct channels. First, holding business risk fixed, an increase in the level of financial leverage raises the level of the equity volatility. Second, regardless of the level of financial leverage, a positive shock to business risk increases the cost of capital and reduces the valuation of future cash flows, generating an instantaneous negative correlation between asset returns and asset volatility. Finally, the market experiences both small continuous movements and large market disruptions. The large and negative market disruptions often generate self-exciting behaviors. The occurrence of one disruption induces more disruptions to follow, thus raising market volatility. We propose an equity index dynamics that capture all three channels of interactions through the separate modeling of the asset return dynamics and the financial leverage variation. We analyze how the different sources of variations impact the index options behaviors differently across a wide range of strikes, maturities, and calendar days.

This is joint work with Liuren Wu.





• Dr Kai Detlefsen, Commerzbank Financial Engineering

Approximations of the Forward Smile

We explore the prices of forward starting options via a Taylor approximation. The resulting representation explains the main risks in ratchets, ie identifies important greeks. Using approximations for implied volatility and skew, we look at qualitative features of forward implied volatilities and skews in the Heston model.



• Prof Raquel M. Gaspar, ISEG, Technical University Lisbon

Convexity Adjustments - A Unified Framework

This article aims to clarify the notion of convexity in fixed income markets. The main challenge is to provide a unified framework for all the different “convexity adjustments” that exist out there. We explain the basic and appealing idea behind the use of convexity adjustments and focus on the situations we believe are of particular importance to practitioners: yield convex- ity adjustments, forward versus futures convexity adjustments, timing and quanto convexity adjustments.

We claim that the appropriate way to look into any of these adjustments is as a side effect of a measure change, as proposed by Plesser (2003). When using the appropriate setup, there may be no immediate urge to do Taylor approximations or fall into too unrealistic assumptions. By using one unified framework, we hope to clarify some issues and help the reader realize that some of the assumptions that are sometimes imposed may be unnecessary.

For fixed income markets, convexity has emerged as an intriguing and challenging notion. Tak- ing this effect into account correctly could provide financial institutions with a competitive advantage. The idea underlying the notion of a convexity adjustment is quite intuitive and can be easily explained in the following terms. Many fixed income products are non-standard with respect to aspects such as the timing, the currency or the rate of payment. This leads to complex pricing formulas, many of which are hard to obtain in closed-form. Examples of such products include in-arreas or in-advance products, quanto products, CMS products, or equity swaps, among others. Despite their non-standard features, these products are quite similar to plain vanilla ones whose price can either be directly obtained from the market or at least computed in closed-form. Their complexity can be understood as introducing some sort of bias into the pricing of plain vanilla instruments. That is, we may decide to use the price of plain products and adjust it somehow to account for the complexity of non-standard products. This adjustment is what is known as convexity adjustment.

[executive summary]

• Dr Andreas Grau, Thetaris GmbH

Computer Aided Finance - Another Journey in the Quest for the Holy Grail of Financial Engineering

With a unified theory for pricing any derivative still eluding the financial engineering community, a new approach to product modelling is presented that may help the industry deal with the new trends towards ever-increasing volume and complexity of the products traded. We propose a product description language that is both simple and general. It is suitable for computer processing, enabling tools to automatically derive pricing algorithms. Important features of this language are shown and examples for a wide range of derivatives presented. This is joint work with Stefan Dirnstorfer.

[paper]

• Dr Jörg Kienitz, Postbank

CMS - First, Second and Third Generation Products

The demand for structured interest-rate products has led to a wide range of products based on CMS rates. Such products involve caps and floors on the coupon. Therefore, we need formulae to price such options, risk manage the positions and construct hedges.

We start by describing first generation products. The coupons of these basic CMS products are linked to an n-year swap rate. We review the pricing and hedging and show how the full swaption-smile is incorporated. We proceed by looking at second generation products. Expressing views on a steepening or a flattening of the curve involve coupons linked to an n-year and m-year swap rate. We consider again caps and floors and use an analytic solution taking into account the smile of the single rates as well as the fact that the market prices each strike with different volatility. This is commonly known as the correlation smile. Finally, we extend the solution to the case of third generation products. Such products are used to trade the curvature and involve coupons linked to an l-year, m-year and n-year swap rate. Such products may be used for expressing the shape of the curve from the central banks driven front end to the pension funds driven long end. Again analytic solutions for caps and floors are derived.

Finally, we compare the prices for caps and floors obtained from our analytic solutions to those obtained by using some well known (calibrated) term structure models.

This is joint work with Manuel Wittke (University of Bonn).



• Dr Tilman Huhne, d-fine GmbH

Commodity Derivatives - Modelling and Pricing in Practice


Taking a practicioner's viewpoint we survey modelling and pricing approaches for commodity derivatives currently employed by banks and industrial companies. After giving a brief overview of the underlying commodity markets, we focus on spot and forward curve price dynamics and identify key risk drivers for the asset groups energy, agricultural commodities, and industrial metals. Based on this we review popular modelling approaches that take into account empirical commodity price features such as seasonality, volatility dynamics, convenience yield dynamics and spikes. We comment on validation approaches and selection criteria for derivative pricing models.

[slides]

• Dr Jan Maruhn, UniCredit Markets & Investment Banking

Nonparametric Local Volatility Models and their Calibration

In contrast to parametric local volatility models, which by definition possess a certain degree of smoothness, nonparametric model variants suffer of a high degree of ill-posedness which may result in rugged local volatility surfaces. To obtain a smooth output surface, one either has to sufficiently presmooth the input data or to directly control variations of the surface during the calibration process. In this talk we illustrate new variants for both methods. On the one hand we present a presmoothing approach which is phrased in the implied volatility rather than the usually considered call price space. On the other hand we show that adjoint techniques, which have recently been introduced by Giles and Glasserman in the finance community, are very efficient methods to speed up calibrations with direct control of the ruggedness of the local volatility surface. The presented results are joint work with Christian Boehm, Andre Loerx and Ekkehard Sachs.



• Dr Tamas Mayer, Ernst & Young Zürich

Risk Sharing in Insurance Groups

In this talk we investigate solvency models of insurance groups, consisting of a parent company and its subsidiaries. We develop a model for the Swiss Solvency Test (SST), which takes regulatory requirements and other relevant effects into account. These include the credit risk related to the parent company, the limited liability of the parent company with respect to its subsidiaries, a possible ordination between the subsidiaries and a realistic premium principle for risk transfer. We study the capital and risk transfer structure of small insurance groups, in particular we investigate the impact of the above mentioned effects on the group solvency requirement. This is a joint work with Dr. Andreas Kull (AXA Winterthur), Dr. Philipp Keller (Ernst & Young Zürich) and Helga Portmann (Bundesamt für Gesundheit, BAG Schweiz)



• Prof Anthony Neuberger, University of Warwick

The Covariance between Returns and Implied Volatility


The skew in implied volatility has variously been interpreted as evidence of asymmetric risk preferences, transaction costs, the presence of jumps and the correlation between returns and the volatility of returns. We explore the relation between the skew and contracts on the covariance between returns and implied volatility.



• Dr Bereshad Nonas, Commerzbank Financial Engineering

The SABR Model in Approximation


We look at the properties of the stochastic volatility CEV model first introduced by P. Hagan et al. in 2002 that by now is being widely used in the financial community as a way of describing implied volatility surfaces. The success of the model is based on using dynamics that are close to market behaviour as well as on an analytic approximation that allows fast calculation of European option values. Unfortunately this approximation is known to fail under certain input parameter combinations. We put the original approximation in context with more recent literature and present a new workaround that builds on the current solutions but provides more stable results for a wider range of parameters.



• Prof Gilles Pagès, Université Paris 6

Quantization of Probability Distributions and its Applications to Mathematical Finance


Quantization consists in finding the best approximation to a random variable in the mean sense using only a finite number of elements. If we equip those elements with weights corresponding to the underlying distribution, we arrive at a deterministic cubature formula, which is optimal for the class of Lipschitz-continuous functionals. Constructing a cubature formula in this way, we are able to numerically compute expectations with respect to finite and infinite dimensional random variables very efficiently. This will be demonstrated in the case of European and American option pricing for finite dimensional quantization and the pricing of exotic options for infinite dimensional quantization. Finally, we present a hybrid MC-Quantization approach, which combines the benefits from deterministic as well as randomized cubature.

This is a joint presentation with Benedikt Wilbertz.



• Prof Wim Schoutens, K.U.Leuven

Implied Levy Volatility

We introduce implied Levy volatility and study its use and behavior. The concept of implied volatility in the Black-Scholes model is one of the key points to its success and its wide spread use. The implied volatility is very intuitive to use. However, the Black-Scholes model is not really founded by empirical historical data; stock returns tend to be more skewed and have fatter tails than the normal distribution can provide. Hence blind trust in a single implied volatility number and all the numbers derived from that, like deltas and other hedge parameters could be dangerous. Here we try to develop a similar concept but now under a Levy framework and therefore based on empirical more founded distributions. More precisely, we introduce Levy implied time and space volatility and make a study about the shape of implied Levy volatilities. Further, we analyze its performance in delta hedging strategies for a battery of Levy settings. Typically in the Levy setting there are some additional degrees of freedom, i.e. parameters that can be set freely. We look for the historical optimal settings on the basis of a delta hedge study of short term ATM vanilla on the SP500. We show that under such parameter settings the model perform systematically better. We illustrate this by looking to the daily hedge error distribution and by noting that for the Levy models under investigation its empirical mean is closer to zero and its the empirical variance is smaller than under the Black-Scholes setting.

[slides]

• Prof Steven E. Shreve, Carnegie Mellon University

A Mixture Model for Exotic Options

The Black-Scholes option pricing formula is based on the assumption that the underlying asset price has a log-normal distribution under a so-called risk-neutral (or martingale) probability measure. However, this assumption leads to option prices that do not agree with those observed in the market. A solution to this dilemma adopted in practice is to assume that the distribution of the underlying asset price is a mixture of log-normal distributions.
In this talk, we discuss how to construct a dynamic model that is consistent with this practice, and then extend the construction of the model to price exotic options. The essence of the extended construction is to create a Markov process so that the joint distribution of the process and its maximum-to-date agrees with the joint distribution of a given Ito process and its maximum-to-date at each fixed time. This is joint work with Gerard Brunick.

• Prof George Skiadopoulos, University of Piraeus

Asset Allocation with Option-Implied Distributions: A Forward-Looking Approach

We address the empirical implementation of the static asset allocation problem by employing forward-looking information from market option prices. To this end, constant maturity one-month S&P 500 implied distributions are extracted and subsequently transformed to the corresponding risk-adjusted ones. The optimal portfolio strategy is obtained for the cases where direct maximisation of the expected utility and its truncated Taylor series expansion is performed separately. We find that the use of the risk-adjusted implied distributions makes the investor significantly better o¤ compared with the case where she uses the historical distribution of returns to calculate her optimal strategy. The results hold under a number of evaluation metrics and utility functions and carry through even when transaction costs are taken into account. This is joint work with Alexandros Kostakisy and Nikolaos Panigirtzoglou.

Check the working papers on http://web.xrh.unipi.gr/faculty/gskiadopoulos/

[paper]

• Dr Radu Tunaru, Cass Business School

Deterministic Approximation Algorithms for European Options Pricing

European options pricing can be recast as an expectation calculation. I show how to generate closed form approximation algorithms for pricing any European contingent claim. Our results can be applied for any models where the distribution of the underlying is known at maturity. The formulas are proved with results from probability theory, mainly based on weak convergence of probability measures and central limit theorems. The algorithms proposed here are based on the binomial distribution, negative binomial distribution and the Poison generalized binomial distribution for single-asset options and on multinomial distribution for the multi-asset options. Moreover, the formulas derived in this fashion are deterministic and can handle complex payoffs. Since no simulation is involved some of the common pitfalls associated with Monte Carlo techniques are avoided. Furthermore, computational costs can be saved when calculating derivatives prices contingent on the same underlying. The algorithms are related to grid sets that are derived from weak convergence results. We prove that those sets are dense in the space of outcomes of the random quantity defining the model under which the no-arbitrage pricing is realised. For example, under a Black-Scholes model, the grid is dense in the set of real numbers representing the possible states of the Wiener process. We apply our algorithms to spread options and compare our method with other methods in the literature. The numerical results indicate a very good precision. In addition, we show how the same formulas can be applied to calculate other integrals in finance related to a stochastic frontier model.



• Carlos Veiga, Frankfurt School of Finance & Management

Closed Formula for Options with Discrete Dividends and its Derivatives

We present a closed pricing formula for European options under the Black-Scholes model and formulas for its partial derivatives. The formulas are developed making use of Taylor series expansions and by expressing the spatial derivatives as expectations under special measures, as in Carr, together with an unusual change of measure technique that relies on the replacement of the initial condition. The closed formulas are attained for the case where no dividend payment policy is considered. Despite its little practical relevance, a digital dividend policy case is also considered which yields approximation formulas. The results are readily extensible to time dependent volatility models but no so for local-vol type models. For completeness, we reproduce the numerical results in Vellekoop and Nieuwenhuis using the formulas here obtained. The closed formulas presented here allow a fast calculation of prices or implied volatilities when compared with other valuation procedures that rely on numerical methods.



• Dr Thomas Weber, Weber und Partner

Speeding Up - Parallelisation Of Derivative Pricing Models

Latest developments with low cost GPUs makes parallel methods available for the masses and accelerates the execution of a derivative pricing model by factors of 20-100x. But to benefit from the new hardware opportunity new numerical techniques have to be developed and/or adapted. These techniques will be discussed and results will be shown for Monte Carlo simulation, PDE solutions, and calibration.



• Manuel Wittke, University of Bonn

CMS - First, Second and Third Generation Products

The demand for structured interest-rate products has led to a wide range of products based on CMS rates. Such products involve caps and floors on the coupon. Therefore, we need formulae to price such options, risk manage the positions and construct hedges.

We start by describing first generation products. The coupons of these basic CMS products are linked to an n-year swap rate. We review the pricing and hedging and show how the full swaption-smile is incorporated. We proceed by looking at second generation products. Expressing views on a steepening or a flattening of the curve involve coupons linked to an n-year and m-year swap rate. We consider again caps and floors and use an analytic solution taking into account the smile of the single rates as well as the fact that the market prices each strike with different volatility. This is commonly known as the correlation smile. Finally, we extend the solution to the case of third generation products. Such products are used to trade the curvature and involve coupons linked to an l-year, m-year and n-year swap rate. Such products may be used for expressing the shape of the curve from the central banks driven front end to the pension funds driven long end. Again analytic solutions for caps and floors are derived.

Finally, we compare the prices for caps and floors obtained from our analytic solutions to those obtained by using some well known (calibrated) term structure models.

This is joint work with Jörg Kienitz (Deutsche Postbank AG).



• Dr Benedikt Wilbertz, Université Paris 6

Quantization of Probability Distributions and its Applications to Mathematical Finance


Quantization consists in finding the best approximation to a random variable in the mean sense using only a finite number of elements. If we equip those elements with weights corresponding to the underlying distribution, we arrive at a deterministic cubature formula, which is optimal for the class of Lipschitz-continuous functionals. Constructing a cubature formula in this way, we are able to numerically compute expectations with respect to finite and infinite dimensional random variables very efficiently. This will be demonstrated in the case of European and American option pricing for finite dimensional quantization and the pricing of exotic options for infinite dimensional quantization. Finally, we present a hybrid MC-Quantization approach, which combines the benefits from deterministic as well as randomized cubature.

This is a joint presentation with Gilles Pagès.



• Prof Uwe Wystup, Frankfurt School of Finance & Management

Riester-Rente - a Comparative Study

When saving for retirement the market for Riester-savings plans is currently booming in Germany. Many suppliers try to get their share in the market. Besides insurance companies banks have also entered the Riester market. In this paper we compare the performance of four representative investment concepts used to guarantee the minimum payoff of a Riester plan, which is the sum of the investors payments. We take a look at DWS Riesterrente Premium, AXA TwinStar Rente Invest, Nürnberger Funds-linked Doppel-Invest and Allianz Riesterrente with Funds and Guaranty. We simulate the final capital available over an investment horizon of 35 years. The simulation model is a displaced double-exponential jump diffusion. We consider optimistic, pessimistic and mixed market scenarios and two types of investors. As a result we learn that one of the main contributors to the success of the investment plan is the contract and management fees of the supplier. Among the considered investment strategies the CPPI-approach performed by DWS and the variable annuity approach of AXA outperform classic insurance plans.

This is joint work with Andreas Weber (MathFinance AG).

[paper] [slides]

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