An Irregular Grid Approach for Pricing High-Dimensional American Options

We propose and test a new method for pricing American options in a high-dimensional setting.The method is centred around the approximation of the associated complementarity problem on an irregular grid.We approximate the partial differential operator on this grid by appealing to the SDE representation of the underlying process and computing the root of the transition probability matrix of an approximating Markov chain.Experimental results in five dimensions are presented for four different payoff functions.option pricing;inequality;markov chains

Reduced basis methods for pricing options with the Black-Scholes and Heston model

In this paper, we present a reduced basis method for pricing European and American options based on the Black-Scholes and Heston model. To tackle each model numerically, we formulate the problem in terms of a time dependent variational equality or inequality. We apply a suitable reduced basis approach for both types of options. The characteristic ingredients used in the method are a combined POD-Greedy and Angle-Greedy procedure for the construction of the primal and dual reduced spaces. Analytically, we prove the reproduction property of the reduced scheme and derive a posteriori error estimators. Numerical examples are provided, illustrating the approximation quality and convergence of our approach for the different option pricing models. Also, we investigate the reliability and effectivity of the error estimators.Comment: 25 pages, 27 figure

Systemic Risk and Default Clustering for Large Financial Systems

As it is known in the finance risk and macroeconomics literature, risk-sharing in large portfolios may increase the probability of creation of default clusters and of systemic risk. We review recent developments on mathematical and computational tools for the quantification of such phenomena. Limiting analysis such as law of large numbers and central limit theorems allow to approximate the distribution in large systems and study quantities such as the loss distribution in large portfolios. Large deviations analysis allow us to study the tail of the loss distribution and to identify pathways to default clustering. Sensitivity analysis allows to understand the most likely ways in which different effects, such as contagion and systematic risks, combine to lead to large default rates. Such results could give useful insights into how to optimally safeguard against such events.Comment: in Large Deviations and Asymptotic Methods in Finance, (Editors: P. Friz, J. Gatheral, A. Gulisashvili, A. Jacqier, J. Teichmann) , Springer Proceedings in Mathematics and Statistics, Vol. 110 2015

Pricing and Hedging Asian Basket Options with Quasi-Monte Carlo Simulations

In this article we consider the problem of pricing and hedging high-dimensional Asian basket options by Quasi-Monte Carlo simulation. We assume a Black-Scholes market with time-dependent volatilities and show how to compute the deltas by the aid of the Malliavin Calculus, extending the procedure employed by Montero and Kohatsu-Higa (2003). Efficient path-generation algorithms, such as Linear Transformation and Principal Component Analysis, exhibit a high computational cost in a market with time-dependent volatilities. We present a new and fast Cholesky algorithm for block matrices that makes the Linear Transformation even more convenient. Moreover, we propose a new-path generation technique based on a Kronecker Product Approximation. This construction returns the same accuracy of the Linear Transformation used for the computation of the deltas and the prices in the case of correlated asset returns while requiring a lower computational time. All these techniques can be easily employed for stochastic volatility models based on the mixture of multi-dimensional dynamics introduced by Brigo et al. (2004).Comment: 16 page

An option-theoretic valuation model for residential mortgages with stochastic conditions and discount factors

Standard mathematical mortgage valuation models consist of three components: the future promised payments, the financial option to default, and the financial option to prepay. In this thesis we propose and analyze new concepts introduced into the standard models. The new concepts include discount factors, coherent boundary conditions, and stochastic terms. In this framework, the value of a mortgage satisfies a Black-Scholes type stochastic PDE. The approximate solution to our model involves a numerical method based on the Wiener-Ito chaos expansion, which breaks the stochastic PDE into a sequence of deterministic PDEs. These PDEs involve a free boundary, are discretized by finite differences, and solved through the PSOR method. Finally, extensions to MBS valuation are discussed. This work represents a timely study of mortgage valuation in the wake of the recent MBS/financial crisis. This thesis is broadly organized as follows: In chapter 1, we briefly introduce some concepts that are part of the foundations of the standard mortgage models. In chapter 2, we review the standard mortgage valuation PDE models. In chapter 3, we discuss the discount factors, the coherent boundary conditions, and the stochastic terms. In chapter 4 we give a quick overview of the Wiener-Ito chaos expansion. In chapter 5 we analyze the simulation of our model and present some numerical results. Finally, in chapter 6 we make some remarks regarding the valuation of MBS

Pricing High-Dimensional American Options Using Local Consistency Conditions

We investigate a new method for pricing high-dimensional American options. The method is of finite difference type but is also related to Monte Carlo techniques in that it involves a representative sampling of the underlying variables.An approximating Markov chain is built using this sampling and linear programming is used to satisfy local consistency conditions at each point related to the infinitesimal generator or transition density.The algorithm for constructing the matrix can be parallelised easily; moreover once it has been obtained it can be reused to generate quick solutions for a large class of related problems.We provide pricing results for geometric average options in up to ten dimensions, and compare these with accurate benchmarks.option pricing;inequality;markov chains

Pricing American Interest Rate Options in a Heath-Jarrow-Morton Framework Using Method of Lines

We consider the pricing of American bond options in a Heath-Jarrow-Morton framework in which the forward rate volatility is a function of time to maturity and the instantaneous spot rate of interest. We have shown in Chiarella and El-Hassan (1996) that the resulting pricing partial differential operators are two dimensional in the spatial variables. In this paper we investigate an efficientnumerical method to solve there partial differential equations for American option prices and the corresponding free exercise surface. We consider in particular the method of lines which other investigators (eg Carr and Faguet (1994) and Van der Hoek and Meyer (1997)) have found to be efficient for American option pricing when there is one spatial variable. In extending this method for the two dimensional case, we solve the pricing equation by discretising the time variable and one state varialbe and using the spot rate of interest as a continuous variable. We compare our method with the lattice method of Li, Ritchken and Sankarasubramanian (1995).

Liquidity risks on power exchanges

Financial derivatives are important hedging tool for asset’s manager. Electricity is by its very nature the most volatile commodity, which creates big incentive to share the risk among the market participants through financial contracts. But, even if volume of derivatives contracts traded on Power Exchanges has been growing since the beginning of the restructuring of the sector, electricity markets continue to be considerably less liquid than other commodities. This paper tries to quantify the effect of this insufficient liquidity on power exchange, by introducing a pricing equilibrium model for power derivatives where agents can not hedge up to their desired level. Mathematically, the problem is a two stage stochastic Generalized Nash Equilibrium and its solution is not unique. Computing a large panel of solutions, we show how the risk premium and player’s profit are affected by the illiquidity.illiquidity, electricity, power exchange, artitrage, generalized Nash Equilibrium, equilibrium based model, coherent risk valuation