# Abstracts

**Robustness of quadratic hedging strategies to the variation of the model, discrete and continuous time**

*Asma Khedher (Technical University of munich, Germany)*

Tuesday June 3, 11:00-11:30 | session 1.3 | Hedging | room EF

We study the robustness of quadratic hedging strategies to the variation of model. Starting from a jump-diffusion model, we get discrete time models by performing a simple Euler discretization scheme. First, the error caused by the discretization method is analyzed. Second, we describe discrete time backward stochastic differential equations with jumps related to quadratic hedging strategies within the discretized models. Then, we investigate the convergence of the discrete time quadratic hedging strategies to quadratic hedging strategies in the continuous time framework. Moreover, when discretizing we take into account the approximation of small jumps by a continuous martingale.

**Robustness of quadratic hedging strategies in finance via Fourier transforms**

*Catherine Daveloose (Ghent University, Belgium)*

Tuesday June 3, 11:30-12:00 | session 1.3 | Hedging | room EF

Asset price models based on Lévy processes, e.g. exponential Lévy, are well established in literature. Several methods are available to price and hedge related options. One of these methods is based on Fourier transforms. This leads to formulas for the option price and the Greeks in terms of the characteristic function of the driving Lévy process and of the Fourier transform of the payoff function for European vanilla options, see [3]. On the other hand since Lévy models imply incomplete markets, perfect hedging is impossible. However partial hedging can be reached via quadratic hedging strategies. For markets observed in a martingale, resp. semimartingale, setting the quadratic hedging strategies are computed by Fourier transform techniques in [5], resp. [4].

To compute the option price, the Greeks, or the position in the quadratic hedging strategies at any date before time of maturity, simulated prices of the underlying asset are required. However simulation of Lévy processes with infinite activity is hard. The approximation introduced in [1], based on replacing the jumps with absolute size smaller than s of a Lévy process by a scaled Brownian motion, facilitates this simulation issue. One could also look at this approximation from a modelling point of view, since one can choose to consider infinitely small variations coming either from a Brownian motion, or from a Lévy process with infinite activity. For s tending to zero, the approximation clearly converges in distribution to the original Lévy process. Even though convergence of asset prices does not necessarily imply the convergence of option prices, it was proved in [2] that for the considered models the related option prices and the deltas are robust. The question remained whether quadratic hedging is also robust.

We reconsidered the robustness conditions in [2] and proved the convergence of the quadratic hedging strategies. Moreover we computed convergence rates. In other words, it is justified to use the approximation and facilitate simulations with it. Besides, we discuss the integrability conditions appearing in the Fourier based approach more carefully.

[1] Asmussen, S., Rosinski, J.: Approximations of small jump Lévy processes with a view towards simulation. J. Appl. Probab. 38, 482-493 (2001)

[2] Benth, F., Di Nunno, G., Khedher, A.: Robustness of option prices and their deltas in markets modelled by jump-diffusions. Comm. Stoch. Anal. 5(2), 285-307 (2011)

[3] Eberlein, E., Glau, K., Papapantoleon, A.: Analysis of Fourier transform valuation formulas and applications. Appl. Math. Fin. 17(3), 211-240 (2010)

[4] Hubalek, F., Kallsen, J., Krawczyk, L.: Variance-optimal hedging for processes with stationary independent increments. Ann. Appl. Probab. 16, 853-885 (2006)

[5] Tankov, P.: Pricing and hedging in exponential Lévy models: review of recent results. Paris-Princeton Lecture Notes in Mathematical Finance (2010)

**On the Weak Convergence of Risk-Minimizing Option Hedging Strategies**

*Jean-Luc Prigent (University of Cergy-Pontoise, France)*

Tuesday June 3, 12:00-12:30 | session 1.3 | Hedging | room EF

Option hedging is one of the main problem in finance, both in discrete and continuous time. For complete markets, this problem is rather straightforward but most financial markets are incomplete, meaning that the risk-neutral probability is no longer unique and contingent claims are not all attainable. In incomplete markets, a first approach to price and hedge an option H is to use the superhedging method as introduced by Kreps (1981) and further studied by El Karoui and Quenez (1995), Karatzas (1997)...Its cost is given by the supremum of the expected values of the claim under all equivalent martingale measures. The corresponding process is a supermartingale under any equivalent martingale measure and the superhedging strategy is determined by the 'optional decomposition' of such an universal supermartingale (e.g. Kramkov, 1996). However, the cost of superhedging is too high in practice.

An alternative approach is to measure the risk by the quadratic risk criterion (see Föllmer and Schweizer, 1991; Schweizer, 1991). Usually, we obtain explicit and tractable formula. Nevertheless, one drawback of this approach is that we penalize both situations where the terminal wealth is smaller or larger than H. Other approaches are based on quantile and expected shortfall hedging. This latter criterion seems to be more interesting in practice than the superhedging (see Föllmer and Leukert, 1999, 2000).

It has been previously proved that both prices and hedging strategies associated to the locally risk-minimizing criteria are stable under weak convergence (see Prigent,1999; Prigent and Scaillet, 2002; Jacod et al., 2000). In this paper, we analyze and compare the weak convergence (for the Skorokhod topology) of main risk-minimizing option hedging strategies. Both complete and incomplete financial markets are considered. First, using the optional decomposition of supermartingales, we examine the weak convergence of superhedging prices, showing that generally it does not hold. Second, we extend previous results of Prigent (1999) and Prigent and Scaillet (2002) about the stability under convergence of hedging strategies associated to the locally quadratic risk-minimizing criteria. Finally, we examine the convergence of quantile and expected shorfall minimizing hedging strategies. It is proved that the stability under convergence is generally satisfied in the complete case. Nevertheless, for the incomplete case, usually there is no longer stability under weak convergence. We detail also several particular examples, leading for instance in continuous-time to Lévy processes dynamics and stochastic volatility models.

**Static Hedging and Early Exercise Boundaries for American-style Barrier Options**

*João Pedro Nunes (ISCTE-IUL, Portugal)*

Tuesday June 3, 12:30-13:00 | session 1.3 | Hedging | room EF

This paper provides three contributions to the literature on the pricing and hedging of American-style barrier options. First and most importantly, a novel representation is proposed for the early exercise boundary of American-style double knock-out options in terms of the simpler optimal stopping boundary of a nested single barrier American-style contract. Therefore, we are able to extend the static hedge portfolio (SHP) approach to the valuation of American-style double knock-out options by simply adding an extra value-matching condition on the barrier not attached to the nested single barrier contract.

Secondly, we also extend the SHP approach to the pricing of American-style double knock-in options. Based on the knowledge of the early exercise boundary for the corresponding American-style plain-vanilla contract, we simply match the value of the hedging portfolio along both lower and upper barriers.

Finally, all the previous results are tested through an extensive numerical analysis run not only under the constant elasticity of variance (CEV) model but also under the more general jump to default CEV (JDCEV) framework. Hence, this paper offers efficient and robust pricing and hedging solutions for American-style single and double barrier options on defaultable equity.