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Before the financial crisis started in 2007, there were no significant spreads between the forward rate curves constructed either using the market quotes of overnight indexed swaps or those of forward rate agreements. After the crisis, we observe such spreads in the form of forward spread curves.

In a popular approach pioneered by Heath, Jarrow, and Morton, the above curves satisfy a system of infinite-dimensional stochastic integral equations. In fact, the solution is a random field, or a random function of two real variables. By fixing the value of the second variable, one obtains a finite set of random forward spread curves, one for each maturity. Varying the above value generates the curves “in motion”.

A standard approach to solve such a system is to replace it by a “discrete” version in the following order: first introduce discrete space, then discrete time, and finally, a discrete set of solutions. A modern approach starts by introducing a discrete space of solutions called a “cubature formulae on Wiener space”. An advantage of the modern approach is that the obtained system of equations becomes deterministic rather than stochastic and may be easily solved by standard finite-difference or finite-element methods.

The thesis contains the followings new important results. The market model under consideration is large, that is, it includes infinitely many financial instruments. We reviewed existing approaches for finding conditions of no arbitrage on such a market with only one forward spread curve. First, we extended one of the approaches to the case of multiple curves and proved sufficient conditions for absence of arbitrage on such a large market. Second, we found conditions under which the solution to our system of equations is unique and non-negative. Third, using the theory of free Lie algebra, we found new cubature formulae on Wiener space and extensively tested them using the celebrated Black–Scholes equation as an input. Forth, using the results of cubature formula of degree 5, we evaluated the forward and short rates in the Heath–Jarrow–Morton and Hull–White (one-factor) models. Finally, using the same results, we constructed a new trinomial tree model for Black–Scholes–Merton and Black models.

In future research, we plan to apply the obtained formulae to solve some systems of infinite-dimensional stochastic equations describing mathematical models of spread curves. Further, we plan to use the obtained formulae to deal with backward stochastic differential equations.

This contribution deals with an extension to our developed novel cubature methods of degrees 5 on Wiener space. In our previous studies, we studied cubature formulae that are exact for all multiple Stratonovich integrals up to dimension equal to the degree. In fact, cubature method reduces solving a stochastic differential equation to solving a finite set of ordinary differential equations. Now, we apply the above methods to construct trinomial models and to price different financial derivatives. We will compare our numerical solutions with the Black’s and Black-Scholes models’ analytical solutions. The constructed model has practical usage in pricing American options and American-style derivatives.

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.

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.

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.

The multi-curve extension of 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 a 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 and contrast the obtained numerical solutions of cubature method and classical Monte Carlo simulation.

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.