Abstracts

We introduce a continuous model for the limit order book density with infinitesimal tick size, where the evolution of buy and sell side is described by a semilinear second-order SPDE. The mid price process defines a free boundary separating buy and sell side. Price changes are assumed to be determined by the bid-ask imbalance. Following empirical observations by Lipton, Pesavento and Sotiropoulos (2013) we allow this dependency to be nonlinear. The resulting limit order book model can be considered as a generalization of the linear stochastic Stefan problem introduced by Kim, Sowers and Zheng (2012).
In order to show existence of a solution we transform the problem into a stochastic evolution equation, where the boundary interaction leads to an additional non-Lipschitz drift. There is no chance to control this term in any reasonable Banach space, however, smoothing properties of the heat semigroup allow to make use of the mild formulation of the equation. Despite of the non-standard setting for the stochastic evolution equation, we show existence of a unique maximal mild solution of the general model; extending results of Kim, Sowers and Zheng. We show that this solution is continuous, and, up to a stopping time, solves the equation even in the analytically strong sense. Additional assumptions on the boundary interaction then yield non-explosion and global existence. Finally, we obtain that a Nagumo-type condition is sufficient for positivity of the order volume, a natural property any limit order book model should satisfy.

The purpose of the paper consists in developing a novel closed form approximation scheme for pricing European basket and spread options when the joint characteristic function of the corresponding log-prices is known in closed form. The idea consists in expanding the risk neutral density of the terminal underlying value in a Gauss-Hermite series around the normal density. Recently, Necula, Drimus and Farkas (2013, A General Closed Form Option Pricing Formula, SSRN id 2210359) have pointed out that the Gauss-Hermite expansion is more appropriate for heavy tailed log-return distributions than the Gram-Charlier expansion and, at the same time, it allows for a closed form European option pricing formula. In this paper, we obtain closed form pricing formulas for basket and spread options that depend on the Gauss-Hermite expansion coefficients of the risk neutral distribution of the terminal underlying value. We take advantage of the fact that the terminal underlying value in the case of these options is a linear combination of asset prices and provided that the joint characteristic function of the corresponding log-prices is known in closed form, we devise an efficient method for obtaining these Gauss-Hermite expansion coefficients. Our approach has several advantages over existing approximations. First, it is valid both for basket and spread options since no restriction is imposed that the terminal underlying value should be positive. Second, it can be employed beyond the simple case when log-prices are normally distributed since many heavy-tailed distributions have closed form characteristic functions. Finally, our approximation method is feasible for basket/spread options with more than two assets and hence it is appropriate for pricing energy spreads between three commodities such as clean spark spreads. We also conduct a study of the performance of the new approximation method.
Moreover, the option pricing formulas obtained in this paper can be interpreted as generalized Bachelier option pricing formulas, since the risk neutral density of the terminal underlying value (and not that of log-returns) is expanded around the normal density. As a curiosity, the method also allows to obtain a generalized Bachelier formula for pricing European options in the context of the Black-Scholes model, by expanding the log-normal density of the terminal price in a Gauss-Hermite series around the normal density.

J. Gáll, Gy. Pap and M. V. Zuijlen (2004) described a special interest rate model which driven by a geometric spatial AR sheet ([1]) and introduce a new type of Heath-Jarrow-Morton forward interest rate model. In this model we give the no-arbitrage criteria and we estimate parameters of the model (for example volatility) on special samples by maximum likelihood estimation. Finally we observe the asymptotic behaviour of the maximum likelihood estimator in each cases.

Basket options are contingent claims on a group of assets such as equities, commodities, currencies and other vanilla derivatives. From a modelling point of view, the framework to price and hedge these options ought to be multidimensional since baskets of 15 to 30 assets are frequently traded. Many pricing methods that seem to work well for single assets cannot be easily expanded to a multidimensional set-up, mainly due to computational difficulties.
In the paper, we present a general computational solution to the problem of multidimensional models which lack closed-form formulae or models that require burdensome numerical procedures, avoiding most of the strong assumptions made by recent techniques. We generalize the approach in [S. Borovkova, F. Permana and H.V. Weide. A closed form approach to valuing and hedging basket and spread options. JOD (2007) 14, 8-24] by employing the Hermite polynomial expansion that replaces the risk-neutral density implied by the model and matches exactly its first m moments. Our approach elaborates on some variants of the method in [A. Leccadito, P. Toscano and R. Tunaru. Hermite binomial trees: a novel technique for derivative pricing. IJTAF (2012) 15,1-36] to deal with baskets that may take on negative values. In particular, the moment matching is carried on three different return quantities and consequently three different methods are presented. The methods can even be applied to price and hedge multi-asset derivatives in situations when some assets follow one diffusion model and other assets follow different ones, with the only assumption being able to calculate the moments of the basket in closed form. We use, as exemplification, the shifted log-normal process with jumps in pricing basket options.
Furthermore, we propose a test to evaluate the performances of methods for pricing and hedging European-style contingent claims. Through this test, we show that no more than four moments of the underlying assets processes are needed in pricing and hedging options. Moreover, the test shows that one of the proposed methods returns the best price in 84\% of cases and has a pricing error smaller than 5\% in 95\% of the cases. On the hedging side, our methods tend to slightly sub-hedge but their average error is much smaller when compared to other methods. Consequently, the methods are shown to provide superior results not only with respect to pricing but also for hedging.

Polynomial preserving processes are defined as time-homogeneous Markov jump-diffusions whose generator leaves the space of polynomials of any fixed degree invariant. The moments of their transition distributions are polynomials in the initial state. The coefficients defining this relationship are given as solutions of a system of nested linear ordinary differential equations. Polynomial processes include affine processes, whose transition functions admit an exponential-affine characteristic function. These processes are attractive for financial modeling because of their tractability and robustness. In this work we study approximations of polynomial preserving processes with finite-state Markov processes via a moment-matching methodology. This approximation aims to exploit the defining property of polynomial preserving processes in order to reduce the complexity of the implementation of such models. More precisely, we study sufficient conditions for the existence of finite-state Markov processes that match the moments of a given polynomial preserving process. We first construct discrete time finite-state Markov processes that match moments of arbitrary order. This discrete time construction relies on the existence of long-run moments for the polynomial process and cubature methods over these moments. In the second part we give a characterization theorem for the existence of a continuous time finite-state Markov process that matches the moments of a given polynomial preserving process. This theorem illustrates the complexity of the problem in continuous time by combining algebraic and geometric considerations. We show the impossibility of constructing in general such a process for polynomial preserving diffusions, for high order moments and for sufficiently many points in the state space. We provide however a positive result by showing that the construction is possible when one considers finite-state Markov chains on lifted versions of the state space.

Contingent convertibles (CoCos) are a new loss-absorbing bond class which receive high interest from financial regulators. When developing models to price these Cocos, it is important to see which factors drive the CoCo's value. Possible factors are the price of the stock on which the Coco is based, the interest rate and the credit default swap spread, but other factors can also be included. A way to look at this sensitivity is by fitting a regression model with multiple factors as covariates based on historical data. We perform a full regression analysis to see which factors are relevant when determining the CoCo price and should therefore be included in our pricing model. Hereby, it is also necessary to study the interaction between included factors. When developing models for different CoCo types, we can determine which types are sensitive to changes in certain factors (such as the volatility) which leads to a differentiation between CoCo types based on these market parameters. To compensate for possible outliers in the data, we also focus on robust regression techniques. The use of these techniques also allows us to identify deviating trading periods.