Coherent Price Systems and Uncertainty-Neutral Valuation

We consider fundamental questions of arbitrage pricing arising when the uncertainty model is given by a set of possible mutually singular probability measures. With a single probability model, essential equivalence between the absence of arbitrage and the existence of an equivalent martingale measure is a folk theorem, see Harrison and Kreps (1979). We establish a microeconomic foundation of sublinear price systems and present an extension result. In this context we introduce a prior dependent notion of marketed spaces and viable price systems. We associate this extension with a canonically altered concept of equivalent symmetric martingale measure sets, in a dynamic trading framework under absence of prior depending arbitrage. We prove the existence of such sets when volatility uncertainty is modeled by a stochastic differential equation, driven by Peng's G-Brownian motions

Option Pricing with Transaction Costs Using a Markov Chain Approximation

An e cient algorithm is developed to price European options in the pres- ence of proportional transaction costs, using the optimal portfolio frame- work of Davis (1997). A fair option price is determined by requiring that an in nitesimal diversion of funds into the purchase or sale of options has a neutral e ect on achievable utility. This results in a general option pricing formula, in which option prices are computed from the solution of the investor's basic portfolio selection problem, without the need to solve a more complex optimisation problem involving the insertion of the op- tion payo into the terminal value function. Option prices are computed numerically using a Markov chain approximation to the continuous time singular stochastic optimal control problem, for the case of exponential utility. Comparisons with approximately replicating strategies are made. The method results in a uniquely speci ed option price for every initial holding of stock, and the price lies within bounds which are tight even as transaction costs become large. A general de nition of an option hedg- ing strategy for a utility maximising investor is developed. This involves calculating the perturbation to the optimal portfolio strategy when an option trade is executed

Option pricing with transaction costs using a Markov chain approximation

An efficient algorithm is developed to price European options in the presence of proportional transaction costs, using the optimal portfolio framework of Davis (in: Dempster, M.A.H., Pliska, S.R. (Eds.), Mathematics of Derivative Securities. Cambridge University Press, Cambridge, UK). A fair option price is determined by requiring that an infinitesimal diversion of funds into the purchase or sale of options has a neutral effect on achievable utility. This results in a general option pricing formula, in which option prices are computed from the solution of the investor's basic portfolio selection problem, without the need to solve a more complex optimisation problem involving the insertion of the option payoff into the terminal value function. Option prices are computed numerically using a Markov chain approximation to the continuous time singular stochastic optimal control problem, for the case of exponential utility. Comparisons with approximately replicating strategies are made. The method results in a uniquely specified option price for every initial holding of stock, and the price lies within bounds which are tight even as transaction costs become large. A general definition of an option hedging strategy for a utility maximising investor is developed. This involves calculating the perturbation to the optimal portfolio strategy when an option trade is executed

Valuation of cash flows under random rates of interest: A linear algebraic approach

This paper reformulates the classical problem of cash flow valuation under stochastic discount factors into a system of linear equations with random perturbations. Using convergence results, a sequence of uniform approximations is developed. The new formulation leads to a general framework for deriving approximate statistics of cash flows for a broad class of models of stochastic interest rate process. We show applications of the proposed method by pricing default-free and defaultable cash flows. The methodology developed in this paper is applicable to a variety of uncertain cash flow analysis problems

The valuation of clean spread options: linking electricity, emissions and fuels

The purpose of the paper is to present a new pricing method for clean spread options, and to illustrate its main features on a set of numerical examples produced by a dedicated computer code. The novelty of the approach is embedded in the use of a structural model as opposed to reduced-form models which fail to capture properly the fundamental dependencies between the economic factors entering the production process

Valuing American Put Options Using Chebyshev Polynomial Approximation

This paper suggests a simple valuation method based on Chebyshev approximation at Chebyshev nodes to value American put options. It is similar to the approach taken in Sullivan (2000), where the option`s continuation region function is estimated by using a Chebyshev polynomial. However, in contrast to Sullivan (2000), the functional is fitted by using Chebyshev nodes. The suggested method is flexible, easy to program and efficient, and can be extended to price other types of derivative instruments. It is also applicable in other fields, providing efficient solutions to complex systems of partial differential equations. The paper also describes an alternative method based on dynamic programming and backward induction to approximate the option value in each time period

Efficient option pricing with transaction costs

A fast numerical algorithm is developed to price European options with proportional transaction costs using the utility-maximization framework of Davis (1997). This approach allows option prices to be computed by solving the investor’s basic portfolio selection problem without insertion of the option payoff into the terminal value function. The properties of the value function can then be used to drastically reduce the number of operations needed to locate the boundaries of the no-transaction region, which leads to very efficient option valuation. The optimization problem is solved numerically for the case of exponential utility, and comparisons with approximately replicating strategies reveal tight bounds for option prices even as transaction costs become large. The computational technique involves a discrete-time Markov chain approximation to a continuous-time singular stochastic optimal control problem. A general definition of an option hedging strategy in this framework is developed. This involves calculating the perturbation to the optimal portfolio strategy when an option trade is executed

Migration, credit markets, moral hazard, interlinkage.

A fast numerical algorithm is developed to price European options with proportional transaction costs using the utility maximization framework of Davis (1997). This approach allows option prices to be computed by solving the investor's basic portfolio selection problem, without the inser- tion of the option payo into the terminal value function. The properties of the value function can then be used to drastically reduce the number of operations needed to locate the boundaries of the no transaction region, which leads to very e cient option valuation. The optimization problem is solved numerically for the case of exponential utility, and comparisons with approximately replicating strategies reveal tight bounds for option prices even as transaction costs become large. The computational tech- nique involves a discrete time Markov chain approximation to a continuous time singular stochastic optimal control problem. A general de nition of an option hedging strategy in this framework is developed. This involves calculating the perturbation to the optimal portfolio strategy when an option trade is execute