International Journal of Theoretical and Applied Finance

We consider the problem of pricing step double barrier options with binomial lattice methods. We introduce an algorithm, based on interpolation techniques, that is robust and efficient, that treats the "near barrier" problem for double barrier options and permits the valuation of step double barrier options with American features. We provide a complete convergence analysis of the proposed lattice algorithm in the European case.

Volume weighted average price (VWAP) options are a popular security type in many countries, but despite their popularity very few pricing models have been developed so far for VWAP options. This can be explained by the fact that the VWAP pricing problem is set in an incomplete market since there is no underlying with which to hedge the volume risk, and hence there is no uniquely defined price. Any price, which is obtained will include a market price of volume risk which must be determined from the corresponding volume statistics. Our analysis strongly supports the hypothesis that the empirical volume statistics of ASX equities can be described reasonably well by fitted gamma distributions. Based on this observation we suggest a simple gamma process-based model that allows for the exact analytic pricing of VWAP options in a rather straightforward way.

We present an accurate and easy-to-compute approximation of zero-coupon bonds and Arrow–Debreu (AD) prices for the Black–Karasinski (BK) model of interest rates or default intensities. Through this procedure, dubbed exponent expansion, AD prices are obtained as a power series in time to maturity. This provides remarkably accurate results — for time horizons up to several years — even when truncated to the first few terms. For larger time horizons the exponent expansion can be combined with a fast numerical convolution to obtain extremely accurate results.

This paper proposes a new estimation algorithm for the uni-variate Cox–Ingersoll–Ross (CIR) model in the state-space framework. The selection criterion among parameters is the likelihood but some parameters may have the same value; thus the initialization of the optimization routine is important especially if deterministic solvers are used. The algorithm aims at combining likelihood and two additional criteria based on the Kullback–Leibler divergence in order to find initial values in a grid search. The likelihood is then optimized in a restricted parameter set. A numerical experiment consists of generating data given a parameter set varying the length of the time series and the observation noise and then estimating the parameters with the algorithm. The results are discussed showing different performance levels for each parameter.

Risk capital allocation is based on the assumption that the risk of a homogeneous portfolio is scaled up and down with the portfolio size. In this article we show that this assumption is true for large portfolios, but has to be revised for small ones. On basis of numerical examples we calculate the minimum portfolio size that is necessary to limit the error of gradient risk capital allocation and the resulting error in a portfolio optimization algorithm or pricing strategy. We show the dependency of this minimum portfolio size on different parameters like the probability of default and on the credit risk model that is used.

A shared belief in the financial industry is that markets are driven by two types of regimes: bull markets, characterized by high returns and low volatility, and bear markets, characterized by low returns coupled with high volatility. Modeling the dynamics of different asset classes (stocks, bonds, commodities and currencies) with a Markov switching (MS) model and using a density-based test, we reject the hypothesis that two-regimes are enough to capture asset return evolutions for many of the investigated assets. Once the accuracy of our test methodology has been assessed through Monte Carlo experiments, our empirical results point out that between two and five regimes are required to capture the features of each asset's distribution. Moreover, we show that only a part of the underlying number of regimes is explained by the distributional characteristics of the returns such as kurtosis. A thorough out-of-sample analysis provides additional evidence that there are more than just bulls and bears in financial markets. Finally, we highlight that taking into account the real number of regimes allows both improved portfolio returns and density forecasts.

In this paper, we determine the lowest cost strategy for a given payoff in Lévy markets where the pricing is based on the Esscher martingale measure. In particular, we consider Lévy models where prices are driven by a normal inverse Gaussian (NIG)- or a variance Gamma (VG)-process. Explicit solutions for cost-efficient strategies are derived for a variety of vanilla options, spreads, and forwards. Applications to real financial market data show that the cost savings associated with these strategies can be quite substantial. The empirical findings are supplemented by a result that relates the magnitude of these savings to the strength of the market trend. Moreover, we consider the problem of hedging efficient claims, derive explicit formulas for the deltas of efficient calls and puts and apply the results to German stock market data. Using the time-varying payoff profile of efficient options, we further develop alternative delta hedging strategies for vanilla calls and puts. We find that the latter can provide a more accurate way of replicating the final payoff compared to their classical counterparts.