DH 1st floor SR

In the Heston stochastic volatility model, the variance process is given by a mean-reverting square-root process. It is known that its transition probability density can be represented by a non-central chi-square density. There are fundamental differences in the behaviour of the variance process depending on the number of degrees of freedom: if the number of degrees of freedom is larger or equal to 2, the zero boundary is unattainable; if it is smaller than 2, the zero boundary is attracting and attainable.

We focus on the attainable zero boundary case and in particular the case when the number of degrees of freedom is smaller than 1, typical in foreign exchange markets. We prove a new representation for the density based on powers of generalized Gaussian random variables. Further we prove that Marsaglia's polar method extends to the generalized Gaussian distribution, providing an exact and efficient method for generalized Gaussian sampling. Thus, we establish a new exact and efficient method for simulating the Cox-Ingersoll-Ross process for an attracting and attainable zero boundary, and thus establish a new simple method for simulating the Heston model.

We demonstrate our method in the computation of option prices for parameter cases that are described in the literature as challenging and practically relevant.

The static two price economy of conic finance is first employed to

define capital, profit, and subsequently return and leverage. Examples

illustrate how profits are negative on claims taking exposure to loss

and positive on claims taking gain exposure. It is argued that though

markets do not have preferences or objectives of their own, competitive

pressures lead markets to become capital minimizers or leverage

maximizers. Yet within a static context one observes that hedging

strategies must then depart from delta hedging and incorporate gamma

adjustments. Finally these ideas are generalized to a dynamic context

where for dynamic conic finance, the bid and ask price sequences are

seen as nonlinear expectation operators associated with the solution of

particular backward stochastic difference equations (BSDE) solved in

discrete time at particular tenors leading to tenor specific or

equivalently liquidity contingent pricing. The drivers of the associated

BSDEs are exhibited in complete detail.

We present theory and numerics of affine processes and several of their applications in finance. The theory is appealing due to methods from probability theory, analysis and geometry. Applications are diverse since affine processes combine analytical tractability with a high flexibility to model stylized facts like heavy tails or stochastic volatility.

Emma Warneford: "Formation of Zonal Jets and the Quasigeostrophic Theory of the Thermodynamic Shallow Water Equations"

Georgios Anastasiades: "Quantile forecasting of wind power using variability indices"

We consider an optimal stopping problem arising in connection with the exercise of an executive stock option by an agent with inside information.

The agent is assumed to have noisy information on the terminal value of the stock, does not trade the stock or outside securities, and maximises the expected discounted payoff over all stopping times with regard to an enlarged filtration which includes the inside information. This leads to a stopping problem governed by a time-inhomogeneous diffusion and a call-type reward. Using stochastic flow ideas we establish properties of the value function (monotonicity, convexity in the log-stock price), conditions under which the option value exhibits time decay, and derive the smooth fit condition for the solution to the free boundary problem governing the maximum expected reward. From this we derive the early exercise decomposition of the value function. The resulting integral equation for the unknown exercise boundary is solved numerically and this shows that the insider may exercise the option before maturity, in situations when an agent without the privileged information may not.

In portfolio management, there are specific strategies for trading between two assets that are cointegrated. These are commonly referred to as pairs-trading or spread-trading strategies. In this paper, we provide a theoretical framework for portfolio choice that justifies the choice of such strategies. For this, we consider a continuous-time error correction model to model the cointegrated price processes and analyze the problem of maximizing the expected utility of terminal wealth, for logarithmic and power utilities. We obtain and justify an extra no-arbitrage condition on the market parameters with which one obtains decomposition results for the optimal pairs-trading portfolio strategies.

Optimisation problems involving objective functions defined on function spaces often have a natural interpretation as a variational problem, leading to a solution approach via calculus of variations. An equally natural alternative approach is to approximate the function space by a finite-dimensional subspace and use standard nonlinear optimisation techniques. The second approach is often easier to use, as the occurrence of absolute value terms and inequality constraints poses no technical problem, while the calculus of variations approach becomes very involved. We argue our case by example of two applications in mathematical finance: the computation of optimal execution rates, and pre-computed trade volume curves for high frequency trading.