Alet Roux

An Improved Theta-method for Systems of Ordinary Differential Equations

The / -method of order 1 or 2 (if / =1/2) is often used for the numerical solution of systems of ordinary differential equations. In the particular case of linear constant coefficient stiff systems the constraint 1/2 h / h 1, which excludes the explicit forward Euler method, is essential for the method to be A -stable. Moreover, unless / =1/2, this method is not elementary stable in the sense that its fixed-points do not display the linear stability properties of the fixed-points of the involved differential equation. We design a non-standard version of the / -method of the same order. We prove a result on the elementary stability of the new method, irrespective of the value of the parameter / ] [0,1]. Some absolute elementary stability properties pertinent to stiffness are discussed.

The Fundamental Theorem of Asset Pricing Under Proportional Transaction Costs

We extend the fundamental theorem of asset pricing to a model where the risky stock is subject to proportional transaction costs in the form of bid-ask spreads and the bank account has different interest rates for borrowing and lending. We show that such a model is free of arbitrage if and only if one can embed in it a friction-free model that is itself free of arbitrage, in the sense that there exists an artificial friction-free price for the stock between its bid and ask prices and an artificial interest rate between the borrowing and lending interest rates such that, if one discounts this stock price by this interest rate, then the resulting process is a martingale under some non-degenerate probability measure. Restricting ourselves to the simple case of a finite number of time steps and a finite number of possible outcomes for the stock price, the proof follows by combining classical arguments based on finite-dimensional separation theorems with duality results from linear optimisation.

Derivative Pricing in Discrete Time

Derivative Pricing and Hedging.- A Simple Market Model.- Single-Period Models.- Multi-Period Models: No-Arbitrage Pricing.- Multi-Period Models: Risk-Neutral Pricing.- The Cox-Ross-Rubinstein model.- American Options.- Advanced Topics.

Linear Vector Optimization and European Option Pricing Under Proportional Transaction Costs

A method for pricing and superhedging European options under proportional transaction costs based on linear vector optimisation and geometric duality developed by Lohne & Rudloff (2014) is compared to a special case of the algorithms for American type derivatives due to Roux & Zastawniak (2014). An equivalence between these two approaches is established by means of a general result linking the support function of the upper image of a linear vector optimisation problem with the lower image of the dual linear optimisation problem.

American options with gradual exercise under proportional transaction costs

American options in a multi-asset market model with proportional transaction costs are studied in the case when the holder of an option is able to exercise it gradually at a so-called mixed (randomised) stopping time. The introduction of gradual exercise leads to tighter bounds on the option price when compared to the case studied in the existing literature, where the standard assumption is that the option can only be exercised instantly at an ordinary stopping time. Algorithmic constructions for the bid and ask prices and the associated superhedging strategies and optimal mixed stoping times for an American option with gradual exercise are developed and implemented, and dual representations are established.

Pricing And Hedging Game Options In Currency Models With Proportional Transaction Costs

The pricing, hedging, optimal exercise and optimal cancellation of game or Israeli options are considered in a multi-currency model with proportional transaction costs. Efficient constructions for optimal hedging, cancellation and exercise strategies are presented, together with numerical examples, as well as probabilistic dual representations for the bid and ask price of a game option.

A counter-example to an option pricing formula under transaction costs

In the paper by Melnikov and Petrachenko (Finance Stoch. 9: 141–149, 2005), a procedure is put forward for pricing and replicating an arbitrary European contingent claim in the binomial model with bid-ask spreads. We present a counter-example to show that the option pricing formula stated in that paper can in fact lead to arbitrage. This is related to the fact that under transaction costs a superreplicating strategy may be less expensive to set up than a strictly replicating one.

Parallel binomial valuation of american options with proportional transaction costs

We present a multi-threaded parallel algorithm that computes the ask and bid prices of American options with the asset transaction costs being taken into consideration. The parallel algorithm is based on the recombining binomial tree model, and is designed for modern shared-memory multi-core processors. Although parallel pricing algorithms for American options have been well studied, the cases with transaction costs have not been addressed. The parallel algorithm was implemented via POSIX Threads, and was tested. The results demonstrated that the approach was efficient and light-weighted. Reasonable speedups were gained on problems of small sizes.

Game options with gradual exercise and cancellation under proportional transaction costs

ABSTRACT Game (Israeli) options in a multi-asset market model with proportional transaction costs are studied in the case when the buyer is allowed to exercise the option and the seller has the right to cancel the option gradually at a mixed (or randomized) stopping time, rather than instantly at an ordinary stopping time. Allowing gradual exercise and cancellation leads to increased flexibility in hedging, and hence tighter bounds on the option price as compared to the case of instantaneous exercise and cancellation. Algorithmic constructions for the bid and ask prices, and the associated superhedging strategies and optimal mixed stopping times for both exercise and cancellation are developed and illustrated. Probabilistic dual representations for bid and ask prices are also established.

The Cox-Ross-Rubinstein Model

This chapter examines the well-known Cox-Ross-Rubinstein model, which is a multi-period binary model with one stock and one bond in which all nodes behave in the same way. It is shown how this allows substantial simplification in pricing and replicating the large class of path-dependent derivatives. The chapter also gives a detailed demonstration of how option prices in suitably scaled Cox-Ross-Rubinstein models converge to option prices in the Black-Scholes continuous-time model.