Large losses - probability minimizing approach

The probability minimizing problem of large losses of portfolio in discrete and continuous time models is studied. This gives a generalization of quantile hedging presented in [3].Comment: 13 page

On the shortfall risk control -- a refinement of the quantile hedging method

The issue of constructing a risk minimizing hedge under an additional almost-surely type constraint on the shortfall profile is examined. Several classical risk minimizing problems are adapted to the new setting and solved. In particular, the bankruptcy threat of optimal strategies appearing in the classical risk minimizing setting is ruled out. The existence and concrete forms of optimal strategies in a general semimartingale market model with the use of conditional statistical tests are proven. The well known quantile hedging method as well as the classical Neyman-Pearson lemma are generalized. Optimal hedging strategies with shortfall constraints in the Black-Scholes and exponential Poisson model are explicitly determined.Comment: 24 pages in Statistics & Risk Modeling. ISSN (Online) 2196-7040, ISSN (Print) 2193-140

Monotonicity of the collateralized debt obligations term structure model

The problem of existence of arbitrage free and monotone CDO term structure models is studied. Conditions for positivity and monotonicity of the corresponding Heath-Jarrow-Morton-Musiela equation for the $x$-forward rates with the use of the Milian type result are formulated. Two state spaces are taken into account - of square integrable functions and a Sobolev space. For the first the regularity results concerning pointwise monotonicity are proven. Arbitrage free and monotone models are characterized in terms of the volatility of the model and characteristics of the driving L\'evy process.Comment: 28 page

Incompleteness of the bond market with L\'evy noise under the physical measure

The problem of completeness of the forward rate based bond market model driven by a L\'evy process under the physical measure is examined. The incompleteness of market in the case when the L\'evy measure has a density function is shown. The required elements of the theory of stochastic integration over the compensated jump measure under a martingale measure is presented and the corresponding integral representation of local martingales is proven.Comment: 25 page

Approximations for solutions of L\'evy-type stochastic differential equations

The problem of the construction of strong approximations with a given order of convergence for jump-diffusion equations is studied. General approximation schemes are constructed for L\'evy type stochastic differential equation. In particular, the paper generalizes the results of Platen Kloeden and Gardo\n. The Euler and the Milstein schemes are shown for finite and infinite L\'evy measure.Comment: 33 page

Quantile hedging on markets with proportional transaction costs

In the paper a problem of risk measures on a discrete-time market model with transaction costs is studied. Strategy effectiveness and shortfall risk is introduced. This paper is a generalization of quantile hedging presented in [4].Comment: 15 page

Quantile hedging for basket derivatives

The problem of quantile hedging for basket derivatives in the Black-Scholes model with correlation is considered. Explicit formulas for the probability maximizing function and the cost reduction function are derived. Applicability of the results for the widely traded derivatives as digital, quantos, outperformance and spread options is shown.Comment: 30 page

Integral representations of risk functions for basket derivatives

The risk minimizing problem $\mathbf{E}[l((H-X_T^{x,\pi})^{+})]\overset{\pi}{\longrightarrow}\min$ in the multidimensional Black-Scholes framework is studied. Specific formulas for the minimal risk function and the cost reduction function for basket derivatives are shown. Explicit integral representations for the risk functions for $l(x)=x$ and $l(x)=x^p$, with $p>1$ for digital, quantos, outperformance and spread options are derived.Comment: 25 page

Heath-Jarrow-Morton-Musiela equation with L\'evy perturbation

The paper studies the Heath-Jarrow-Morton-Musiela equation of the bond market. The equation is analyzed in weighted spaces of functions defined on $[0,+\infty)$. Sufficient conditions for local and global existence are obtained . For equation with the linear diffusion term the conditions for global existence are close to the necessary ones.Comment: 42 page

Heath-Jarrow-Morton-Musiela equation with linear volatility

The paper is concerned with the problem of existence of solutions for the Heath-Jarrow-Morton equation with linear volatility. Necessary conditions and sufficient conditions for the existence of weak solutions and strong solutions are provided. It is shown that the key role is played by the logarithmic growth conditions of the Laplace exponent