This chapter deals with an extension to the developed novel cubature methods of degrees 5 on Wiener space. It examines cubature formulae that are exact for all multiple Stratonovich integrals up to dimension equal to the degree. Cubature method reduces solving a stochastic differential equation to solving a finite set of ordinary differential equations. The chapter aims to compare the numerical solutions with the Black's and Black–Scholes models' analytical solutions. It examines the convergence of the sequences of constructed trinomial model to a geometric Brownian motion. The chapter also examines the conditions that make the probability measure in our trinomial model a martingale measure, i.e. risk-neutral probability measure. The constructed model has practical usage in pricing American options and American-style derivatives. The chapter emphasizes that the constructed trinomial tree has practical usage and applications in pricing path-dependent and American-style options.

Cubature is an effective way to calculate integrals in a finite dimensional space. Extending the idea of cubature to the infinite-dimensional Wiener space would have practical usages in pricing financial instruments. In this paper, we calculate and use cubature formulae of degree 5 and 7 on Wiener space to price European options in the classical Black–Scholes model. This problem has a closed form solution and thus we will compare the obtained numerical results with the above solution. In this procedure, we study some characteristics of the obtained cubature formulae and discuss some of their applications to pricing American options.

In our previous studies, we developed novel cubature methods of degree 5 on the Wiener space in the sense that the cubature formula is exact for all multiple Stratonovich integrals up to dimension equal to the degree. In this paper, we apply the above methods to the modeling of fixed-income markets via affine models. Then, we apply the obtained results to price interest rate derivatives in the Hull-White one-factor model.

Before the financial crisis started in 2007, the forward rate agreement contracts could be perfectly replicated by overnight indexed swap zero coupon bonds. After the crisis, the simply compounded risk-free overnight indexed swap forward rate became less than the forward rate agreement rate. Using an approach proposed by Cuchiero, Klein, and Teichmann, we construct an arbitrage-free market model, where the forward spread curves for a given finite tenor structure are described as a mild solution to a boundary value problem for a system of infinite-dimensional stochastic differential equations. The constructed financial market is large: it contains infinitely many overnight indexed swap zero coupon bonds and forward rate agreement contracts with all possible maturities. We also investigate the necessary assumptions and conditions which guarantee existence, uniqueness and non-negativity of solutions to the obtained boundary value problem. 

The multi-curve extension of the Heath–Jarrow–Morton framework is a popular method for pricing interest rate derivatives and overnight indexed swaps in the post-crisis financial market. That is, the set of forward curves is represented as a solution to an initial boundary value problem for an infinite-dimensional stochastic differential equation. In this paper, we review the post-crisis market proxies for interest rate models. Then, we consider a simple model that belongs to the above framework. This model is driven by a single Wiener process, and we discretize the space of trajectories of its driver by cubature method on Wiener space. After that, we discuss possible methods for numerical solution of the resulting deterministic boundary value problem in the finite-dimensional case. Finally, we compare the obtained numerical solutions of cubature method with the classical Monte Carlo simulation.

After the financial crisis of 2007, significant spreads between interbank rates associated to different maturities have emerged. To model them, we apply the Heath--Jarrow--Morton framework. The price of a financial instrument can then be approximated using cubature formulae on Wiener space in the infinite-dimensional setting. We present a short introduction to the area and illustrate the methods by examples.