## Contributed Session Thu.3.H 3027

#### Thursday, 15:15 - 16:45 h, Room: H 3027

**Cluster 7: Finance & economics** [...]

### Applications in finance

**Chair: Janos Mayer**

**Thursday, 15:15 - 15:40 h, Room: H 3027, Talk 1**

**Jonas Ekblom**

Optimal hedging of foreign exchange risk in uncertain cash flows using stochastic programming

**Coauthor: Jörgen Blomvall**

**Abstract:**

We build a stochastic programming framework for optimal hedging of foreign exchange risk in uncertain cash flows. By incorporating term premia we are able to estimate the cost of hedging, and we determine the optimal hedge by minimizing a convex combination of risk (measured as CVaR) and cost. The importance of expected returns for the optimal hedge is verified through numerical results. In this framework, trades are made at market prices and transaction costs are included. The framework offers great flexibility regarding distributional assumptions of the underlying risk factors and the types of financial instruments which can be included.

**Thursday, 15:45 - 16:10 h, Room: H 3027, Talk 2**

**Mathias Barkhagen**

An optimization based method for arbitrage-free estimation of the implied risk neutral density surface

**Coauthor: Jörgen Blomvall**

**Abstract:**

Accurate pricing of OTC derivatives which is consistent with noisy market prices presents a major challenge. The pricing accuracy will crucially depend on using arbitrage-free inputs to the pricing engine. To this end we develop a general optimization based framework for estimation of the option implied risk neutral density (RND) surface, while satisfying no-arbitrage constraints. The developed framework is a generalization of existing models such as the Heston model. Thus, the method considers all types of realistic surfaces and is hence not constrained to a certain function class. Instead the RND is discretized making it possible to use standard solvers for the problem. The approach leads to an optimization model where it is possible to formulate the constraints as linear constraints. The linear constraints and the form of the objective function leads to an inherent problem structure which may be utilized to speed up calculations. We show that our method produce smooth local volatility surfaces that can be used for pricing and hedging of OTC derivatives. Statistical tests demonstrate that our method gives better results than the Heston model in terms of yielding stable RNDs.

**Thursday, 16:15 - 16:40 h, Room: H 3027, Talk 3**

**Janos Mayer**

Portfolio optimization with objective functions from cumulative prospect theory

**Coauthor: Thorsten Hens**

**Abstract:**

We consider portfolio optimization problems with several assets, involving objective functions from the cumulative prospect theory (CPT) of Tversky and Kahneman (1992). These are numerically difficult optimization problems since the objective function to be maximized is neither concave

nor smooth. We have implemented an adaptive simplex grid method for the solution of this type of problems and report on the results of a numerical study. Levy and Levy (2004) proved that under the assumption of normally distributed returns the CPT efficient set is a subset of the

mean-variance (MV) frontier. In fact the authors state that there is no need for separate solution algorithms for CPT-optimization, since those problems are readily solvable by maximizing the CPT objective function along the MV frontier. We compare this suggestion with the direct CPT-optimization, for a real life data-set and for several investors and find that the two approaches lead to substantially different portfolios. This difference increases dramatically if we add a call option to our data-set

and it diminishes almost completely for a data-set obtained by sampling from the corresponding normal distribution.