Andrew P. Carverhill

Flows of stochastic dynamical systems: ergodic theory

We present a version of the Multiplicative Ergodic (Oseledec) Theorem for the flow of a nonlinear stochastic system definedon a smooth compact manifold. This theorem establishes the existence of a Lyapunov spectrum for the flow, which characterises the asymptotic behaviour of the derivative flow. Then we establish the existence of stable manifolds for the flow (on which trajectories cluster) associated with the Lyapunov spectrum. This work is a generalisation of that of Ruelle who deals with ordinary dynamical systems. Finally we give an example of a stochastic system for which the flow is calculated explicitly, and which illustrates the behaviour predicted by the abstract results.

Lyapunov exponents for a stochastic analogue of the geodesic flow

New invariants for a Riemannian manifold are defined as Lya- punov exponents of a stochastic analogue of the geodesic flow. A lower bound is given reminiscent of corresponding results for the geodesic flow, and an upper bound is given for surfaces of positive curvature. For surfaces of con- stant negative curvature a direct method via the Doob rt-transform is used to determine the full Lyapunov structure relating the stable manifolds to the horocycles.

Furstenberg's theorem for nonlinear stochastic systems

SummaryWe extend Furstenbergs theorem to the case of an i.i.d. random composition of incompressible diffeomorphisms of a compact manifold M. The original theorem applies to linear maps {Xi}i∈N on ℝm with determinant 1, and says that the highest Lyapunov exponent

Pricing and Integration of the CDX Tranches in the Financial Market

\beta \equiv \mathop {\lim }\limits_{n \to \infty } \frac{1}{n}||X_n \circ ... \circ X_1 ||

A nonrandom Lyapunov spectrum for nonlinear stochastic dynamical systems

is strictly positive unless there is a probability measure on the projective (m-1)-space which is a.s. invariant under the action of Xi. Our extension refers to a probability measure on the projective bundle over M.We show that when our diffeomorphism is the flow of a stochastic differential equation, the criterion for β>0 is ensured by a Lie algebra condition on the induced system on the principal bundle over M.

Global solutions of the Navier-Stokes equation with strong viscosity

This paper solves for a firms optimal cash holding policy within a continuous time, contingent claims framework that has been extended to incorporate most of the significant contracting frictions that have been identified in the corporate finance literature. Under the optimal policy the firm targets a level of cash holding that is a non-monotonic function of business conditions and an increasing function of the amount of long-term debt outstanding. By allowing firms to either issue equity or to borrow short-term, we show how share issue and dividends on the one hand and cash accumulation and bank borrowing on the other are all mutually interlinked. We calibrate the model and show that it matches closely a wide range of empirical benchmarks including cash holdings, leverage, equity volatility, yield spreads, default probabilities and recovery rates. Furthermore, we show the predicted dynamics of cash and leverage are in line with the empirical literature. Despite the presence of significant contracting frictions we show that the model exhibits a near irrelevance of long-term capital structure property. Furthermore, the optimal policy exhibits a state-dependent hierarchy among financing alternatives that is consistent with recent explorations of pecking order theory. We calculate the agency costs generated by the conflict of interest between shareholders and creditors regarding the firms liquidity policy and show that bond covenants that establish an earnings restriction on dividend payments may be value increasing.

Indonesia's stock market: evolving role, growing efficiency

This paper first designs an efficient procedure to value Credit Default Swap Index tranches using an intensity-based model. The tranche spreads are effectively explained by a three-factor version of this model, both before and during the financial crisis of 2008. We then construct tradable tranche portfolios to track the intensity factors and compare the pricing of the tranches with equities and their derivatives. Our results show that the senior tranche spreads do not offer returns in excess of the common risk compensations in the equity and derivatives markets, while the junior tranche is not spanned by these standard factors.

A note on the efficiency of the binomial option pricing model

One theoretical implication of cointegration, according to Granger (1986), is that asset prices in an efficient market cannot be cointegrated. Using price data on US Treasury STRIPS with maturities from 2/15/1997 to 8/15/2015, it is found that a set of three STRIPS series is often cointegrated. In addition, by setting up a costless hedge portfolio from three STRIPS with three different maturities, it is found that the hedge portfolio is often stationary and thus arbitrage opportunities are likely to occur. That is, because the hedge portfolio is costless and stationary, cash in can be done when the value of the hedge portfolio is either positive or negative. However, when taking liquidity, tax effects, and transaction costs into consideration, these arbitrage profits would be unlikely. Hence, it is concluded that the US Treasury STRIPS market is efficient.