Optimal Numeraires for Risk Measures

Can the usage of a risky numeraire with a greater than risk free expected return reduce the capital requirements in a solvency test? I will show that this is not the case. In fact, under a reasonable technical condition, there exists no optimal numeraire which yields smaller capital requirements than any other numeraire.

Polynomial Diffusions and Applications in Finance

This paper provides the mathematical foundation for polynomial diffusions. They play an important role in a growing range of applications in finance, including financial market models for interest rates, credit risk, stochastic volatility, commodities and electricity. Uniqueness of polynomial diffusions is established via moment determinacy in combination with pathwise uniqueness. Existence boils down to a stochastic invariance problem that we solve for semialgebraic state spaces. Examples include the unit ball, the product of the unit cube and nonnegative orthant, and the unit simplex.Comment: This article is forthcoming in Finance and Stochastic

Regularity of finite-dimensional realizations for Evolution Equations

We show that a continuous local semiflow of $C^k$-maps on a finite-dimensional $C^k$-manifold M can be embedded into a local $C^k$-flow on M under some weak (necessary) assumptions. This result is applied to an open problem in [fil/tei:01]. We prove that finite-dimensional realizations for interest rate models are highly regular objects, namely given by submanifolds M of $D(A^{\infty})$, where A is the generator of a strongly continuous semigroup

Option Pricing with Orthogonal Polynomial Expansions

We derive analytic series representations for European option prices in polynomial stochastic volatility models. This includes the Jacobi, Heston, Stein-Stein, and Hull-White models, for which we provide numerical case studies. We find that our polynomial option price series expansion performs as efficiently and accurately as the Fourier transform based method in the nested affine cases. We also derive and numerically validate series representations for option Greeks. We depict an extension of our approach to exotic options whose payoffs depend on a finite number of prices.Comment: forthcoming in Mathematical Finance, 38 pages, 3 tables, 7 figure

On the Group Level Swiss Solvency Test

In this paper we elaborate on Swiss Solvency Test (SST) consistent group diversification effects via optimizing the web of capital and risk transfer (CRT) instruments between the legal entities. A group level SST principle states that subsidiaries can be sold by the parent company at their economic value minus some minimum capital requirement. In a numerical example we examine the dependence of the optimal CRT on this minimum capital requirement. Our findings raise the question of how to actually implement this group level SST principle and how to define the respective level of minimum capital requirements, in particular.convex optimization; group diversification; minimum capital requirement; Swiss solvency test

Modeling Credit Risk by Affine Processes

In this paper, the treasury rates and the credit migrations are jointly modeled by multi-dimensional affine processes. In order to capture the entire information, including credit migrations and default events, we construct non-conservative regular affine processes to model credit migrations and characterize the default by the death of the processes. In particular, two specific cases: purely jump affine models and affine diffusion models with potentials, are discussed. This affine approach not only produces the explicit formulas for the prices of corporate bonds and other credit derivatives, but also directly incorporates the credit rating information as a parameter into the pricing formulas. Moreover, our affine models allow to consider the joint credit migrations within an analytically tractable framework in order to capture the correlations of credit movements between firms. Finally, the empirical testing results of a simple affine model are presented to support the effectiveness of our models.Credit Risk Models, Credit Migrations, Affine Processes