Nonlinear valuation under credit risk, collateral margins, funding costs and multiple curves
Following a quick introduction to derivatives markets and the classic theory of valuation, we describe the changes triggered by post 2007 events. We re-discuss the valuation theory assumptions and introduce valuation under counterparty credit risk, collateral posting, initial and variation margins, and funding costs. A number of these aspects had been investigated well before 2007. We explain model dependence induced by credit effects, hybrid features, contagion, payout uncertainty, and nonlinear effects due to replacement closeout at default and possibly asymmetric borrowing and lending rates in the margin interest and in the funding strategy for the hedge of the relevant portfolio. Nonlinearity manifests itself in the valuation equations taking the form of semi-linear PDEs or Backward SDEs. We discuss existence and uniqueness of solutions for these equations. We present an invariance theorem showing that the final valuation equations do not depend on unobservable risk free rates, that become purely instrumental variables. Valuation is thus based only on real market rates and processes. We also present a high level analysis of the consequences of nonlinearities, both from the point of view of methodology and from an operational angle, including deal/entity/aggregation dependent valuation probability measures and the role of banks treasuries. Finally, we hint at how one may connect these developments to interest rate theory under multiple discount curves, thus building a consistent valuation framework encompassing most post-2007 effects.
Counterparty risk and funding: Immersion and beyond
This is joint work with Shiqi Song.
A basic counterparty risk reduced-form approach hinges on an immersion property of the reference (or market) filtration into the full model filtration enlarged by the default times of the counterparties, also involving the continuity of some of the data at default time. This is too restrictive for applications to credit derivatives, which are characterized by strong wrong-way risk, i.e. adverse dependence between the exposure and the credit riskiness of the counterparties, and gap risk, i.e. slippage between the portfolio and its collateral during the so called cure period that separates default from liquidation. We show how a suitable extension of the basic approach can be devised so that it can be applied in dynamic copula models of counterparty risk on credit derivatives. More generally, this method is applicable in any marked default times intensity setup satisfying a suitable integrability condition. The integrability condition expresses that no mass is lost in a related measure change. The changed probability measure is not needed algorithmically. All one needs in practice is an explicit expression for the intensities of the marked default times.
Sensitivity analysis in Lévy fixed income theory
First a brief introduction into the Lévy Libor and the Lévy forward process model is given. Basic properties of these two frameworks are discussed. The main goal is to derive formulas for price sensitivities of standard fixed income derivatives. Two approaches are discussed. The first approach is based on the integration–by–parts formula, which lies at the core of the application of the Malliavin calculus to finance. The second approach consists in using Fourier based methods for pricing derivatives. We illustrate the result by applying the formulas to a caplet price where the underlying model is driven by a time–inhomogeneous Gamma process and alternatively by a Variance Gamma process. A comparison between the two approaches which come from totally different mathematical fields is made.
OIS discounting, interest rate derivatives, and the modeling of stochastic interest rate spreads
Prior to 2007, derivatives practitioners used a zero curve that was bootstrapped from LIBOR swap rates to provide “risk-free” rates when pricing derivatives. In the last few years, when pricing fully collateralized transactions, practitioners have switched to using a zero curve bootstrapped from overnight indexed swap (OIS) rates for discounting. This paper explains the calculations underlying the use of OIS rates and investigates the impact of the switch on the pricing of plain vanilla caps and swap options. It also explores how more complex derivatives providing payoffs dependent on LIBOR, or any other reference rate, can be valued. It presents new results showing that they can be handled by constructing a single tree for the evolution of the OIS rate.
Computation of value adjustments in affine LIBOR models with multiple curves
In the first part of this talk, we present an extension of the LIBOR market model with stochastic basis spreads in the spirit of the affine LIBOR models. This multiple curve model satisfies the main no-arbitrage and market requirements (such as nonnegative LIBOR-OIS spreads) by construction. The use of multidimensional affine processes as driving motions ensures the analytical tractability of the model. We provide pricing formulas for caps, swaptions and basis swaptions and discuss an efficient numerical implementation. In the second part, the discrete tenor LIBOR model is extended to the continuous tenor by introducing the corresponding interpolating function which enables us to derive a representation for the forward rate. Under certain conditions on the original affine LIBOR model an interpolating function can be found such that the extended model fits any initial forward curve. Further, we show that under a change to the spot measure implied by the forward rate, the affine structure is preserved. Finally, we apply our results to consistently compute value adjustments (the so-called XVA) for interest rate derivatives.
On multi-curve models for the term structure
This is joint work with Zorana Grbac. We present possible extensions, to a post-crisis multi-curve setup, of the classical short rate and forward rate models for the term structure of interest rates. We discuss pricing of linear and optional derivatives and, for linear derivatives, we exhibit an adjustment factor allowing one to pass directly from the one-curve to the multi-curve prices.
First loss structures for hedge fund investments
The 2/20 fee structure of the hedge fund world is being challenged by new trends. In this talk we will review some innovative fee structures and evaluate their valuation characteristics from the investor and the fund perspectives.
Conic finance explained and applied
We give an introduction to conic finance. Conic Finance is a brand new quantitative finance theory and is incorporating in a fundamental way bid and ask pricing. We provide the basics and its connection with the concept of acceptability and coherent risk measures. Distorted expectations are employed to actually calculate bid and ask prices. We elaborate on various applications of the theory like conic hedging, conic portfolio theory and show how conic finance can be used for systemic risk measurement, liquidity measurement and how the counter-intuitive effects of booking profits due to your own credit deterioration (also referred to as Debt Valuation Adjustment or DVA) are mitigated under the conic bid and ask pricing theory.
Panel Discussion: Counterparty Credit Risk
Ralf Werner (Ancorman)
Ralf Werner is currently professor for business mathematics at Augsburg University. Prior to his previous position as professor for mathematical modelling at the University of Applied Science Munich he was heading the global Risk Methods & Valuation department at a large European real estate bank where he was in charge of risk methodology, financial engineering and economic capital modelling. Ralf also held various positions at an international insurance group and with an asset manager as consultant, financial engineer and prop trader.
- Damiano Brigo
- Christian Fries
- John Hull
- Daniel Sommer
CVA Risk Charge and P&L Volatility Trade-Off: Proposal of a Unified Steering
As a consequence of the recent financial crisis, Basel III introduced a new capital charge, the CVA risk charge to cover the risk of future CVA fluctuations (CVA volatility). Although Basel III allows for hedging the CVA risk charge, mismatches between the regulatory (Basel III) and accounting (IFRS) rules lead to the fact that hedging the CVA risk charge is challenging. The reason is that the hedge instruments reducing the CVA risk charge cause additional Profit and Loss (P&L) volatility. In the present article, we propose a solution to the problem of determining the optimal hedge amount in the sense that a maximal CVA charge reduction will be achieved on the one hand and the additional P&L volatility will be minimized on the other hand.
Developments in credit and derivative risks: Challenges & lessons learned
From the point of view of practical issues in credit portfolio management and financial derivatives recent developments as well as challenges in credit and derivative risks will be discussed, relevant examples will be provided and explained and consequences as well as take-aways will be elaborated.
Extracting the implied correlation from quanto derivatives
This is joint work with Laura Balotta and Grégory Rayée.
In this paper we apply the multivariate construction for Lévy processes introduced by Ballotta and Bonfiglioli (2014) in a multidimensional FX market. We show that the proposed construction is consistent in terms of symmetries with respect to inversion and triangulation, and provides an insight into the quanto adjustment showing that this is affected by higher order cumulants of the pure jump part of the systematic risk factor. Using the Esscher transform, we relate Quanto options to vanilla call and put options, which allows for a fast calibration method to the vanilla and the Quanto market. A joint calibration to the CME USD denominated Quanto futures on the Nikkei 225 index and both the Nikkei 225 and USDJPY market implied volatilities allows to extract an implied correlation between the log-returns of the index and the FX rate. We illustrate this method for the case of Variance Gamma processes.
A general HJM framework for multiple yield curve modeling
This is joint work with Christa Cuchiero and Alessandro Gnoatto.
We propose a general framework for modeling multiple yield curves which have emerged since the last financial crisis. In a general semimartingale setting, we provide an HJM approach to model the term structure of multiplicative spreads between (normalized) FRA rates and simply compounded OIS risk-free forward rates. We derive an HJM drift and consistency condition ensuring absence of arbitrage and we show how to construct models such that multiplicative spreads are greater than one and ordered with respect to the tenor's length.
Spread modeling in a general multiple-curve HJM framework
In the context of the general HJM framework of Cuchiero et al. (2014), we present concrete examples of multiple-curve interest rate models. We specify the driving semimartingale as an affine process, which provides a flexible Markovian structure allowing for tractable valuation formulas for most interest rate derivatives. We also show how the framework of Cuchiero et al. (2014) allows to unify and extend several recent approaches to multiple yield curve modeling.
Efficient Estimation of Sensitivities for Counterparty Credit Risk with the Finite Difference Monte-Carlo Method
Kees de Graaf
According to Basel III, financial institutions have to charge a Credit Valuation Adjustment (CVA) to account for a possible counterparty default. Calculating this measure and its sensitivities is one of the big challenges in risk management. Here we introduce an efficient method for the estimation of CVA and its sensitivities for a portfolio of financial derivatives. We use the Finite Difference Monte-Carlo (FDMC) method to measure exposure profiles and consider the computationally challenging case of FX barrier options in the context of the Black-Scholes as well as the Heston Stochastic Volatility model for a wide range of parameters. Our results show that FDMC is an accurate method compared to the semi-analytic COS method and has as an advantage that it can compute multiple options on one grid, which paves the way for real portfolio level risk analysis.
Capital Optimisation through an innovative CVA Hedge
Michael Hünseler and Dirk Schubert
One of the lessons of the financial crisis as of late was the inherent credit risk attached to the value of derivatives. Since not all derivatives can be cleared by central counterparties, a significant amount of OTC derivatives will be subject to increased regulatory capital charges. Since these charges cover both current and future unexpected losses, the capital costs for derivatives transactions can become substantial if not prohibitive. At the same time, capital optimisation through CDS hedging of counterparty risks will result in a hedge position beyond the economic risk (“overhedging”) required to meet Basel II/III rules. In addition, IFRS accounting rules again differ from Basel, creating a mismatch when hedging CVA. Even worse, CVA hedging using CDS may introduce significant profit and loss volatility while satisfying the conditions for capital relief. An innovative approach to hedging CVA aims to solve the issue of not corresponding with both Basel III and accounting requirements.
Implied volatilities from strict local martingales
Several authors have proposed to model price bubbles in stock markets by specifying a strict local martingale for the risk-neutral stock price process. Such models are consistent with absence of arbitrage (in the NFLVR sense) while allowing fundamental prices to diverge from actual prices and thus modeling investors exuberance during the appearance of a bubble. We show that the strict local martingale property as well as the "distance to a true martingale" can be detected from the asymptotic behavior of implied option volatilities for large strikes, thus providing a model-free asymptotic test for the strict local martingale property of the underlying. This talk is based on joint work with Antoine Jacquier.
The impact of a scrip dividend on an equity forward
We consider an equity forward contract on a stock which pays a dividend during the forward’s lifetime. Furthermore, the stock owner is assumed to have the right to opt for either cash or scrip dividend. In the latter case, the stock owner receives the dividend in the form of additional shares and the number of shares to be received depends on the average stock price in a certain time period. The decision between scrip or cash must be made by the stock owner at some time point during the averaging period. Within a Black-Scholes-type setup we derive a closed formula for the fair strike price of such an equity forward contract in dependence on the stock volatility parameter. It is demonstrated how the optionality for the stock owner can have a non-negligible value which lowers the forward equity strike.
Multiple curve dynamics impact in credit and funding adjustments
We present a consistent dynamical approach to justify the market practice of extrapolating different term structures from different instruments. In particular, we include into term structure modelling the impact of credit, collateral and funding risks by extending the risk-neutral pricing framework to incorporate margining procedures and treasury operational models. Numerical examples with single-curve and multiple-curve models are presented to price interest-rate swaps and tenor basis swaps and to highlight the impact of a dynamical model for the tenor basis in valuation adjustments due to counterparty credit risk and funding effects. Both bilateral and CCP-clearead trades are considered.
Post-crisis interest rates: XIBOR mechanics and basis spreads
In this talk we present a structural model for interbank money market rates (XIBOR rates) that is able to endogenously generate the basis spreads that characterize post-crisis fixed income markets: XIBOR-OIS spreads, tenor basis spreads, and the forward basis. In contrast to existing reduced-form multi-curve models, which impose basis spreads exogenously, our approach is based on a micro-model for interbank cash transactions and the relevant credit and liquidity risk factors. Our framework thus provides a consistent arbitrage-free explanation for the emergence of basis spreads and, in particular, a theoretical underpinning for reduced-form multi-curve models. We derive closed-form Black-type pricing formulae for caplets and swaptions in a lognormal version of our model. Moreover we demonstrate the fit of our model to pre- and post-crisis interest rate market data.
Asymptotic methods for model validation
Since no single model can be used to price and manage the risk of all options in a bank's portfolio, different exotic options are typically priced and hedged with different models. In this context, model validation boils down to identifying the risk factors and/or market imperfections which are important for a given derivative product and must be accounted for by the model. Usually, this is done by repricing the same product within different models, but for this, each model must be implemented and calibrated to market data: a highly non-trivial task, which is itself subject to model risk. We propose a new paradigm for model validation and selection based on indifference pricing and asymptotic analysis. Instead of repricing the exotic in each new model, we suggest to evaluate the importance of a risk factor using the spread between the seller's and the buyer's indifference price in a model involving this risk factor. This spread is computed asymptotically, assuming that the new model is a small perturbation of a reference model. For example, a stochastic volatility model with small volatility of volatility can be seen as a perturbation of the Black-Scholes model. This leads to an explicit approximate formula for the spread, which can be evaluated within the reference model and is to a certain extent independent of the specific model for the risk factor. In the talk, we will present the main ideas of the approach, and illustrate them using the examples of volatility risk and jump risk.
Calibration of the Nelson-Siegel and the Svensson model
This is joint work with Dirk Banholzer and Jörg Fliege.
The Nelson-Siegel and the Svensson model are two of the most widely used models for the term structure of interest rates. Even though both models are quite simple and intuitive, their calibration to available market data turns out to be a numerically very challenging task, various difficulties have been reported. In this presentation we propose an enhanced technique for calibrating these models to market rates. It is based on the observation that the related optimisation problem can be formulated as a separable nonlinear least-squares problem in which the linear parameters can be implicitly eliminated. The resulting problem is a reduced optimisation problem in the nonlinear parameters only, which can effectively be solved by specific global optimisers. By applying the enhanced technique to yield curve data from the Deutsche Bundesbank, we perform a numerical analysis in which we first point out the potential ill-conditioning of the problem. We then provide calibration results and show that the suggested technique is efficient and improves traditional calibration methods considerably. We close by emphasizing the consequences of some numerical findings on econometric analysis of interest rates.
Please note that we can no longer accept proposals for contributed talks.
On the joint dynamics between the spot and the implied volatility surface
Sofiene El Aoud
This is joint work with Frederic Abergel.
In this paper, we revisit the “Smile Dynamics” problem. In [Bergomi (2009)], Bergomi built a class of linear stochastic volatility models in which he specified the joint dynamics between the underlying and its instantaneous forward variances. The author introduced a quantity, which he called the Skew Stickiness ratio, in order to relate two quantities of interest: the first quantity is the correlation between the increments of the at-the-money implied volatility of maturity T and the log-returns of the underlying, while the second quantity is the implied skew of the same maturity T. In our work, we continue the study of the Skew stickiness ratio both from theoretical and empirical point of view. First, we provide a method to estimate the SSR (skew stickiness ratio) from option prices, this measure is called the implied SSR as it is conducted under the risk-neutral pricing measure Q. Next to that, we recall how to measure the realized SSR under the real-world probability measure P and we point out empirically that there is a discrepancy between the implied SSR and the realized SSR. The empirical study shows also that the implied SSR, in the limit of short maturities, can take a value superior to 2 which is in discordance with the results obtained in linear stochastic volatility models. For this reason, we show that the positive quantity (SSR_Implied - 2) is coherent with the presence of jumps in a stochastic volatility model.
Counterparty credit risk measurement: dependence effects, mitigating clauses and gap risk
This is joint work with Gianluca Fusai and Daniele Marazzina.
The aim of the paper is to provide a valuation framework for counterparty credit risk based on a structural default model which incorporates jumps and dependence between the assets of interest. In this framework default is caused by the firm value falling below a prespecified threshold following unforeseeable shocks, which deteriorate its liquidity and ability to meet its liabilities. The presence of dependence between names captures wrong-way risk and right-way risk effects. The structural model traces back to Merton (1974), who considered only the possibility of default occurring at the maturity of the contract; first passage time models starting from the seminal contribution of Black and Cox (1976) extend the original framework to incorporate default events at any time during the lifetime of the contract. However, as the driving risk process used is the Brownian motion, all these models suffers of vanishing credit spreads over the short period - a feature not observed in reality. As a consequence, the CVA would be underestimated for short term deals as well as the so-called gap risk, i.e. the unpredictable loss due to a jump event in the market. Improvements aimed at resolving this issue include for example random default barriers, time dependent volatilities, and jumps. In this contribution, we adopt Lévy processes and capture dependence via a linear combination of two independent Lévy processes representing respectively the systematic risk factor and the idiosyncratic shock. We then apply this framework to the valuation of CVA and DVA related to equity contracts such as forwards and swaps. The main focus is on the impact of correlation between entities on the value of CVA and DVA, with particular attention to wrong-way risk and right-way risk, the inclusion of mitigating clauses such as netting and collateral, and finally the impact of gap risk. Particular attention is also devoted to model calibration to market data, and development of adequate numerical methods for the complexity of the model considered.
A Note on CVA and Wrong Way Risk
Gaetano La Bua
This is joint work with Roberto Baviera and Paolo Pellicioli.
Hull White approach to Wrong Way Risk in the computation of the Credit Value Adjustment is considered the most straightforward generalization of the standard Basel approach. The model is financially intuitive and it can be implemented by a slight modification of existing algorithms for the calculation of standard CVA. However, path dependency in the key quantities has non elementary consequences in the calibration of some model parameters. In this note we discuss in detail the implications of this path dependency and propose a simple and fast approach for computing them via a recursion formula. We show calibration methodology on real market data and CVA computations in two relevant cases: a FX forward and an interest rate swap.
A note on dual-curve construction: Mr. Crab’s bootstrap
This is joint work with Roberto Baviera.
Observe crabs in the sand of our beaches: they move forward, backward and then forward again. Before the crisis, the standard bootstrap of interest rate curves was a ‘Forward’-looking iterative algorithm where only information from previous knots was used to find discounts at subsequent dates. In this note we describe a new bootstrapping technique that involves various ‘Backward’ steps, which are reminiscent of a crab’s steps: this new methodology coherently considers now standard dual-curve framework. Two other major results emerge from the bootstrap methodology described: (i) discounts are independent from the chosen interpolation rule for all practical purposes; and (ii) convexity adjustments to Short-Term Interest Rate futures can be dealt with using a methodology in line with market practice.
Liquidity risk in fund management
Liquidty risk in fund management has become an important topic over the last years and the employment of an appropriate liquidity risk management has become mandatory by European guidelines ( and ). In this talk we will describe two possible approaches that are able to cope with these regulatory requirements:
- Static approach: We compute the POT-quantile of the excess distribution of fund redemption data which can be approximated by the GPD (compare ). This risk measure is called Liquidity-at-Risk (compare ) and is tailored for the application to the fund management sector in  including a thorough backtesting analysis. Furthermore we give an automated procedure for the determination of the threshold parameter of a GPD.
- Dynamic approach: In , we define an autoregressive process with GPD tails as generalization of the ARP process as given in  which is able to cover the time structure of fund redemption data. We prove the stationarity of the ARGP process, determine its model parameters and apply it to real data. As an additional liquidity measure/analysis tool we introduce interarrival times within this framework.
 Desmettre, S., Deege, M. (2014). Liquidity at Risk for Mutual Funds, available at SRRN.
 Desmettre, S., de Kock, J., Seifried, F.T.: Generalized Pareto Processes and Liquidity, Working Paper.
 Embrechts, P., Klüppelberg, C., Mikosh, T. (1997). Modeling Extremal Events for Insurance and Finance. Springer.
 The European Commission (2010). Commission Directive 2010/43/EU. Official Journal of the European Union.
 The European Parliament and the Council of the European Union (2011). Directive 2011/61/EU of the Parliament and of the Council. Official Journal of the European Union.
 Yeh, H.C., Barry, B.C., Robertson, C.A. (1998). Pareto Processes. Journal of Applied Probability 25, 291-301
 Zeranski, S. (2006). Liquidity at Risk - Quantifizierung extremer Zahlungsstromrisiken. Risikomanager 11(1), 4–9.
A two-Factor cointegrated commodity price model with an application to spread option pricing
This is joint work with Elise Gourier, Robert Huitema and Ciprian Necula.
We introduce a flexible continuous-time model of cointegrated commodity prices, which allows capturing between one and n-1 cointegration relations between the n components of a system. Closed-form expressions for futures and European option prices are derived. The model is estimated using futures prices of ten commodities. We find compelling evidence of multiple cointegration relationships. We calculate the prices of options written on various spreads such as the spark spread and the crack spread. We show that cointegration creates an upward sloping term structure of correlation, which lowers the volatility of spreads and consequently the price of options on them.
Approximated pricing of swaptions in general interest rate models
Anna Maria Gambaro
This is joint work with Ruggero Caldana and Gianluca Fusai.
We propose new lower and upper bounds on the prices of European-style swaptions for a wide class of interest rate models. These methods are applicable whenever the joint characteristic function of the state variables is known in closed form or can be obtained numerically via some efficient procedure. This is the case of affine and quadratic interest rate models. Our lower bound involves the computation of one dimensional Fourier transform independently from the swap length. In addition, we can control the error of our method by providing a new upper bound on swaption price applicable to all affine models. Finally the bounds can be used as a control variable to reduce confidence interval of the Monte Carlo technique. We test our bounds on different affine models, also allowing for jumps, and on a 2-factors quadratic Gaussian model. The bound are found to be accurate and computationally efficient.
Interbank convexity adjustments
This is joint work with José Cruz.
Convexity adjustments are used by practitioners to value non standard products using information on plain vanilla products. The real world interbank market is not populated risk less banks. This became particularly obvious after the financial crisis of 2007-2009. Nonetheless, most theoretical interest rate models, assume the risk in the interbank lending market is negligible, using interest rate sensitive products to build zero-coupon bonds curves. Here we take a different approach and consider the Libor rate L(t,T) is no longer a good approximation to the truly default-free interest rate. Thus, the value of contracts having as underlying the Libor rate, should be adjusted to correct for the true risk existent in the Libor rate. We call that adjustment an interbank convexity adjustment. In this paper we explicitly compute the interbank convexity adjustment of FRAs (Forward Rate Agreements), combining the classical affine term structure (ATS) framework with shot-noise process that are able to capture the counter-party risk of interbank contracts.
Options embedded in life insurance policies: Risk management and optimal asset allocation
This is joint work with An Chen.
In a typical participating life insurance contract, the insurance company is entitled to a share of the return surplus as compensation for the return guarantee granted to policyholders. This call-option-like stake gives the insurance company an incentive to increase the riskiness of its investments at the expense of the policyholders. This conflict of interests can partially be solved by regulation deterring the insurance company from taking excessive risk. In a utility-based framework where default is modeled continuously by a structural approach, we show that a flexible design of regulatory supervision can be beneficial for both the policyholder and the insurance company.
Basket option pricing and implied correlation in a Lévy copula model
This is joint work with Wim Schoutens.
We introduce the Lévy copula model and consider the problem of finding accurate approximations for the price of a basket option. The basket is a weighted sum of dependent stock prices and its distribution function is unknown or too complex to work with. Therefore, we replace the random variable describing the basket price at maturity by a random variable with a more simple structure. Moreover, the Carr-Madan formula can be used to determine approximate basket option prices in a fast and efficient way. In a second part of the paper we show how implied volatility and implied correlation can be defined in the Lévy copula model. In this model, each stock price is described by a volatility parameter and the marginal parameters can be calibrated separately from the correlation parameters using single name option prices. However, the available market prices for basket options together with our newly designed basket option pricing formula enables us to determine implied Lévy correlation estimates. We observe that implied correlation depends on the strike and the so-called implied Lévy correlation smile is flatter than its Gaussian counterpart. The standard technique to price non-traded basket options (or other multi-asset derivatives), is by interpolating on the implied correlation curve. In the Gaussian copula model, this can sometimes lead to non-meaningful correlation values. We show that the Lévy version of the implied correlation solves (at least to some extent) this problem.
This is joint work with Olena Burkovska, Kathrin Glau, Maximilian Gaß, Maximilian Mair, Wim Schoutens and Barbara Wohlmuth.
We investigate a market practitioner’s approach to calibration called De–Americanization. Instead of calibrating market models to American option prices directly, these are de-americanized first by calibrating a simple tree model and then using the calibrated tree model to generate respective European option prices. More sophisticated models can then be calibrated to these pseudo-European option prices. The effect of this approximation is tested for a variety of models including the Heston and Lévy driven models. Being able to calibrate these to American option prices directly, we conduct comparisons to the De-Americanization approach. Both, scenarios where the practitioner’s approximation makes sense as well as cases where the routine generates unsatisfactory results are described.
A general closed form option pricing formula
This is joint work with Gabriel Drimus and Walter Farkas.
A new method to retrieve the risk-neutral probability measure from observed option prices is developed and a closed-form pricing formula for European options is obtained by employing a modified Gram-Charlier series expansion, known as the Gauss-Hermite expansion. This new option pricing formula is also an alternative to the inverse Fourier transform methodology and can be employed in general models with probability distribution function or characteristic function known in closed form. We calibrate the model to both simulated and market option prices and find that the resulting implied volatility curve provides a good approximation for a wide range of strikes.
A few remarks on the pricing of contingent convertibles
Due to the equity approach of De Spiegeleer and Schoutens (cf. ) there exists a well-known closed form formula for a risk neutral price of a contingent convertible (CoCo) under the Black-Scholes setting with a given constant volatility. The payoff of a CoCo (at maturity) in particular contains payoffs of exotic down-and-in barrier options, describing (in the case of a trigger event) the conversion from a corporate bond to a certain amount of stocks of the financial institution which issued the CoCo. The embedded down-and-in options start to exist the first time the stock price of the company hits a lower barrier. By using the first passage time approach (“running infimum") and the implicit - debatable - assumption that conversion takes place at maturity, yet not at the trigger time, we explicitly derive the closed-form-formula of De Spiegeleer and Schoutens (cf.  and ). Therefore, regarding a possible generalisation of this model, we sketch ap- proaches how to price American down-and-in barrier options including the use of stochastic time change techniques (cf. ), possibly allowing us to consider a pricing of a CoCo for a large class of semimartingales with jumps.
 W. Schoutens and J. De Spiegeleer. Pricing Contingent Convertibles: A Derivatives Approach. Journal of Derivatives 20, No. 2, pp. 27-36 (2012).
 N. El Karoui and P-A. Meyer. Les changements de temps en theorie generale des processus. Seminaire de probabilites (Strasbourg), tome 11, p. 65-78 (1977).
 M. Musiela and M. Rutkowski Martingale Methods in Financial Modelling - 2nd ed. Springer (2005).
 K. Shang. Understanding Contingent Capital. Casualty Actuarial Society Working Paper (2013).
Lambda value at risk: a new backtestable alternative to VaR
This is joint work with Asmerilda Hitaj.
A new risk measure, the lambda value at risk (Lambda-VaR), has been recently proposed from a theoretical point of view as an immediate generalization of the value at risk (VaR). The Lambda-VaR appears to be attractive for its potential ability to solve several problems of the VaR. This paper presents the first empirical application of the Lambda-VaR to equity markets. Our benchmark approach reveals that the Lambda-VaR is able to capture extreme downward scenarios and react to market fluctuations significantly faster than the VaR and expected shortfall. In addition, we show that the Lambda-VaR satisfies the elicitability property under general conditions. This property guarantees proper backtesting and statistically meaningful comparisons among Lambda-VaR and VaR. Hence, we perform a backtesting exercise by adapting the VaR hypothesis-testing framework. The results display a higher level of accuracy for our Lambda-VaR calibrations. Our comparative analysis also shows that Lambda-VaR performs better than VaR.
Model-free discretisation-invariant swaps and S&P 500 higher-moment risk premia
We derive a general multivariate theory for realised characteristics of "model-free discretisation-invariant swaps", so-called because the standard no-arbitrage assumption of martingale forward prices is sufficient to derive fair-value swap rates for such characteristics which have no jump or discretisation errors. This theory underpins specific examples for swaps based on higher moments of a single log return distribution where exact replication is possible via option-implied `fundamental contracts' like the log contact. The common factors determining the S&P 500 risk premia associated with these higher-moment characteristics are investigated empirically at the daily, weekly and monthly frequencies.
The worst-case dependence structure maximizing the bilateral CVA
When calculating the bilateral CVA incorporating wrong way risk, one has to assume some dependence structure between three quantities, the default times of the bank and the counterparty, and the portfolio value at default time. Common approaches are, for example, assuming independence between all quantities or at least between two of them. Other established procedures specify tractable bivariate dependencies between those random variables. Fixing the dependence structure results in an enormous model risk yielding to a huge interval of possible CVA values. In this work, we present a model-free approach to detect the dependence structures leading to the maximal (resp. minimal) possible BCVA.
The minimal entropy martingale measure in a market of traded financial and actuarial risks
This is joint work with Jan Dhaene, Pierre Devolder and Michel Vellekoop.
In arbitrage-free but incomplete markets, the equivalent martingale measure chosen by the market for pricing traded assets is not uniquely determined. A possible approach when it comes to choosing a particular pricing measure is to consider the one that is `closest' to the physical probability measure, where closeness is measured in terms of relative entropy. In this paper, we determine the minimal entropy martingale measure in a market where securities with payoffs depending on financial as well as actuarial risks are traded. In case only purely financial and purely actuarial securities are traded, we prove that financial and actuarial risks are independent under the physical measure if and only these risks are independent under the entropy measure. Moreover, in such a market the entropy measure of the combined financial-actuarial world is the product measure of the entropy measures of the financial and the actuarial subworld, respectively.
On the Heston model with stochastic correlation
The degree of relationship between e.g. financial products, financial instituations must be considered for pricing and hedging. Usually, for the financial products modelled with the specification of a system of stochastic different equations (SDEs), the relationship is represented with the correlated Brownian motions (BMs). For example, the BM of asset price and the BM of stochastic volatility in the Heston model correlates with a deterministic constant. However, market observation indicates that financial quantities are correlated in a strongly nonlinear way, correlation behave even stochastically and unpredictably. This article extends the Heston model by imposing stochastic correlations given by Orstein-Uhlenbeck (OU) or Jacobi processes. By approximating non-affine term we find the characteristic function (CF) in a closed-form which can be used for pricing purposes. Our numerical results and experiment on calibration to market data approves that incorporating stochastic correlations improves the performance of the Heston model.
Minimizing the index tracking error of actively managed fixed income ETFs
This is joint work with Kujtim Avdiu.
We analyse different optimization techniques for minimizing the tracking error of an actively managed fixed income ETF and show that by making use of these techniques an approximation of a Bond index performance by a Bond ETF is possible. The asset allocation targets at minimizing costs, therefore determining the minimum amount of underlyings, which are necessary in order to track the Bond index while keeping the tracking error at a minimum. By using the calibrated CIR model we then simulate the short-rate in order to create a bond index consisting of 10 European T-bills with different maturities. We then investigate which optimization technique creates for which asset allocation the smallest tracking error.
Inside the emerging markets risky spreads and credit default swap - Sovereign bonds basis
The paper considers a no-arbitrage setting for pricing and relative value analysis of risky sovereign bonds. The typical case of an emerging market country that has bonds outstanding both in foreign hard currency (Eurobonds) and local soft currency (treasuries) is inspected. The resulting two yield curves give rise to a credit and currency spread that need further elaboration. We discuss their proper measurement and also derive and analyze the necessary no-arbitrage conditions that must hold. With that characterization we turn attention to the broader issue of the bond price discovery process and relative value diagnostics in that multicurve framework. There the CDS-Bond basis takes a central place. For EM countries it is special both in conceptual background and empirical performance. The paper further focuses on analyzing these peculiarities. If the proper measurement of the basis in the standard case of only hard currency debt being issued is still problematic, the situation is much more complicated in a multicurve setting when a further contingent claim on the sovereign risk in the face of local currency debt curve appears. We investigate the issue and provide relevant theoretical and empirical input.