# Abstracts

**Valuation and hedging of OTC contracts with funding costs and collateralization**

*Tomasz Bielecki (Illinois Institute of Technology, USA)*

Thursday June 5, 14:00-14:30 | session 8.3 | Hedging | room EF

The research presented in this work is motivated by recent papers by Brigo at al., Crepey, Burgard and Kjaer, Fujii and Takahashi and Piterbarg. Our goal is to provide a sound theoretical underpinning for some results presented in these papers by developing a unified martingale framework for the non-linear approach to hedging and pricing of OTC financial contracts. The impact that various funding bases and margin covenants exert on the values and hedging strategies for OTC contracts is examined.

**On an optimization problem related to static super-replicating strategies**

*Xinliang Chen (ING Financial Markets, Belgium)*

Thursday June 5, 14:30-15:00 | session 8.3 | Hedging | room EF

In this paper, we investigate an optimization problem related to super-replicating strategies for European-type call options written on a weighted sum of asset prices, for example Asian or basket options. Firstly, we proved that in general, the optimal solution is non-unique. This observation is useful since it allows some flexibility to compose the optimal super-replicating strategies in a real market situation, which often has some constraints in trading. Secondly, a generalized optimization problem with random weights has been studied. Using these results, we derived optimal static super-replicating strategies for different kind of options in a stochastic interest rate setting. Thirdly, the co-existence of the comonotonicity property and the martingale property was studied. We have seen that in the case of a single underlying asset, they can co-exist and additionally a linear relationship of the underlying random variables was found in certain situations. However, there can also exist a contradiction between them. For Asian options e.g., the price of the optimal static super-replicating strategies may not be reached by the Asian option price.

**Evaluation of individual equity values for basket type first passage time structural models**

*Yuji Yamada (University of Tsukuba, Japan)*

Thursday June 5, 15:00-15:30 | session 8.3 | Hedging | room EF

In this paper, we consider problems of hedging barrier-style basket options using individual options based on our previously proposed optimal hedging strategy in Yamada (2010—2013) for European options. The optimal hedging problem is formulated as follows: Find optimal smooth functions of individual options to minimize the mean square error from the payoff of down-and-out basket option. For solving the problem, we shall take advantage of a necessary and sufficient condition expressed in terms of linear equations of conditional expectations involving multivariate and path-dependent underlying asset in barrier options. We provide an efficient computational procedure by performing the Brownian bridge decomposition in the expression of underlying assets, and show that the computations involving conditional expectations of basket type barrier options may be reduced to those of unconditional expectations.

Then, we apply our methodology to the Black and Cox (1976) type first passage time structural model, in which a dafaultable company possesses/runs multiple assets/projects and the default may occur the first time the asset value hits a certain lower threshold before the maturity. We discuss how to evaluate individual equity values using smooth functions of optimal hedging problem, and examine the behavior of terminal values of individual equity values approximated by the optimal smooth functions based on the conditional expectations given default or survival. We provide the notion of unbiased payoff, in which the payoff defined by the sum of individual options is said to be unbiased with respect to the barrier option payoff if their conditional expectations are equal given default or survival. We show that the unbiased payoff is constructed by adding a systematic term related to the default event for adjusting the conditional expectations. Moreover, such an unbiased payoff is shown to achieve a smaller mean square error in the objective function of the optimal hedging problem. Finally, we demonstrate how to evaluate contributions of individual equity values to default or survival using the unbiased payoff via certain normalization for conditional expectations. Numerical experiments illustrate that, in the case where the firm is still survival under default risk, the source of excess profit is mainly brought from a high volatility project (or asset), whereas a negatively larger equity value is obtained with a higher volatility project when the firm is default.

**Model-independent hedging under portfolio constraints**

*Yu-Jui Huang (Dublin City University, Ireland)*

Thursday June 5, 15:30-16:00 | session 8.3 | Hedging | room EF

We consider a finite discrete-time market with one risky asset. An investor can hold static positions in vanilla options written on the risky asset, and also trade dynamically the risky asset under portfolio constraints. Without specifying any particular model for the risky asset, we derive a model-independent duality for superhedging. With the aid of this duality, we obtain a robust version of the fundamental theorem of asset pricing under portfolio constraints. Finally, we establish the model-independent 'facelifting phenomenon': superhedging a payoff under portfolio constraints is equivalent to superhedging a facelifted payoff without constraints.