A Cubature Method for Solving Stochastic Equations : A Modern Monte-Carlo Approach with Applications to Financial Market

• 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.

Ämnesord

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

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Mathematics/Applied Mathematics

matematik/tillämpad matematik

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