We develop a stochastic programming framework for hedging currency and interest rate risk, with market traded currency forward contracts and interest rate swaps, in an environment with uncertain cash flows. The framework captures the skewness and kurtosis in exchange rates, transaction costs, the systematic risks in interest rates, and most importantly, the term premia which determine the expected cost of different hedging instruments. Given three commonly used objective functions: variance, expected shortfall, and mean log profits, we study properties of the optimal hedge. We find that the choice of objective function can have a substantial effect on the resulting hedge in terms of the portfolio composition, the resulting risk and the hedging cost. Further, we find that unless the objective is indifferent to hedging costs, term premia in the different markets, along with transaction costs, are fundamental determinants of the optimal hedge. Our results also show that to reduce risk properly and to keep hedging costs low, a rich-enough universe of hedging instruments is critical. Through out-of-sample testing we validate the findings of the in-sample analysis, and importantly, we show that the model is robust enough to be used on real market data. The proposed framework offers great flexibility regarding the distributional assumptions of the underlying risk factors and the types of hedging instruments which can be included in the optimization model.

Hedging of an option book in an incomplete market with transaction costs is an important problem in finance that many banks have to solve on a daily basis. In this paper, we develop a stochastic programming (SP) model for the hedging problem in a realistic setting, where all transactions take place at observed bid and ask prices. The SP model relies on a realistic modeling of the important risk factors for the application, the price of the underlying security and the volatility surface. The volatility surface is unobservable and must be estimated from a cross section of observed option quotes that contain noise and possibly arbitrage. In order to produce arbitrage-free volatility surfaces of high quality as input to the SP model, a novel non-parametric estimation method is used. The dimension of the volatility surface is infinite and in order to be able solve the problem numerically, we use discretization and principal component analysis to reduce the dimensions of the problem. Testing the model out-of-sample for options on the Swedish OMXS30 index, we show that the SP model is able to produce a hedge that has both a lower realized risk and cost compared with dynamic delta and delta-vega hedging strategies.

In this paper we develop a methodology for simultaneous recovery of the real-world probability density and liquidity premia from observed S&P500 index option prices. Assuming the existence of a numeraire portfolio for the US equity market, fair prices of derivatives under the benchmark approach can be obtained directly under the real-world measure. Under this modeling framework there exists a direct link between observed call option prices on the index and the real-world density for the underlying index. We use a novel method for estimation of option implied volatility surfaces of high quality which enables the subsequent analysis. We show that the real-world density that we recover is consistent with the observed realized dynamics of the underlying index. This admits the identication of liquidity premia embedded in option price data. We identify and estimate two separate liquidity premia embedded in S&P500 index options that are consistent with previous findings in the literature.

The vast size of real world stochastic programming instances requires sampling to make them practically solvable. In this paper we extend the understanding of how sampling affects the solution quality of multistage stochastic programming problems. We present a new heuristic for determining good feasible solutions for a multistage decision problem. For power and log-utility functions we address the question of how tree structures, number of stages, number of outcomes and number of assets affect the solution quality. We also present a new method for evaluating the quality of first stage decisions.

In this paper we present a nonlinear dynamic programming algorithm for the computation of forward rates within the maximum smoothness framework. The algorithm implements the forward rate positivity constraint for a one-parametric family of smoothness measures and it handles price spreads in the constraining data set. We investigate the outcome of the algorithm using the Swedish Bond market showing examples where the absence of the positive constraint leads to negative interest rates. Furthermore we investigate the predictive accuracy of the algorithm as we move along the family of smoothness measures. Among other things we observe that the inclusion of spreads not only improves the smoothness of forward curves but also significantly reduces the predictive error.

In [Euro. J. Operat. Res. 143 (2002) 452, Opt. Meth. Software 17 (2002) 383] a Riccati-based primal interior point method for multistage stochastic programmes was developed. This algorithm has several interesting features. It can solve problems with a nonlinear node-separable convex objective, local linear constraints and global linear constraints. This paper demonstrates that the algorithm can be efficiently parallelized. The solution procedure in the algorithm allows for a simple but efficient method to distribute the computations. The parallel algorithm has been implemented on a low-budget parallel computer, where we experience almost perfect linear speedup and very good scalability of the algorithm. © 2003 Elsevier Science B.V. All rights reserved.

We build an investment model based on Stochastic Programming. In the model we buy at the ask price and sell at the bid price. We apply the model to a case where we can invest in a Swedish stock index, call options on the index and the risk-free asset. By reoptimizing the portfolio on a daily basis over a ten-year period, it is shown that options can be used to create a portfolio that outperforms the index. With ex post analysis, it is furthermore shown that we can create a portfolio that dominates the index in terms of mean and variance, i.e. at given level of risk we could have achieved a higher return using options.