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by DAVIS BUNDI NTWIGA Numerical methods form an important part of the pricing of financial derivatives and especially in cases where there is no closed form analytical formula. We begin our work with an introduction of the mathematical tools needed in the pricing of financial derivatives. Then, we discuss the assumption of the log-normal returns on stock prices and the stochastic differential equations. These lay the foundation for the derivation of the Black Scholes differential equation, and various Black Scholes formulas are thus obtained. Then, the model is modified to cater for dividend paying stock and for the pricing of options on futures. Multi-period binomial model is very flexible even for the valuation of options that do not have a closed form analytical formula. We consider the pricing of vanilla options both on non dividend and dividend paying stocks. Then show that the model converges to the Black-Scholes value as we increase the number of steps. We discuss the Finite difference methods quite extensively with a focus on the Implicit and Crank-Nicolson methods, and apply these numerical techniques to the pricing of vanilla options. Finally, we compare the convergence of the binomial model, the Implicit and Crank Nicolson methods to the analytical Black Scholes price of the option
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Notes by George Chalamandaris This chapter introduces the reader to definitions and key properties of stochastic processes that are important in finance. The discussion starts from the description of Brownian motion that describes the idea of a continuous random walk and proceeds to Ito processes that incorporate both trend and volatility. The emphasis of the exposition is the applicability of stochastic processes in financial modeling. The paper demonstrates that ordinary calculus cannot tackle the problems that arise in continuous time financial economics because of the presence of randomness. We offer a brief presentation of the main concepts of stochastic calculus by reviewing the Ito integral and the Ito formula. Finally, the Binomial tree model is presented as an intuitive way to approximate a stochastic process in discrete time
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Arne Karsten Strauss Jump-diusion models can under certain assumptions be expressed as partial integrodi erential equations (PIDE). Such a PIDE typically involves a convection term and a nonlocal integral like for the here considered models of Merton and Kou. We transform the PIDE to eliminate the convection term, discretize it implicitly using nite dierences and the second order backward dierence formula (BDF2) on a uniform grid. The arising dense linear system is solved by an iterative method, either a splitting technique or a circulant preconditioned conjugate gradient method. Exploiting the Fast Fourier Transform (FFT) yields the solution in only O(n log n) operations and just some vectors need to be stored. Second order accuracy is obtained on the whole computational domain for Merton's model whereas for Kou's model rst order is obtained on the whole computational domain and second order locally around the strike price. The solution for the PIDE with convection term can oscillate in a neighborhood of the strike price depending on the choice of parameters, whereas the solution obtained from the transformed problem is stabilized.
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M.A.H. DEMPSTER J.P. HUTTON Abstract: We investigate numerical valuation of cross-currency interest rate-based derivatives under Babbs' extended Vasicek-style model by numerical solution of the associated partial differential equation (PDE), in particular, we consider the terminable differential (diff) swap. Firstly we precisely formulate, in terms of their cash ows, various types of single and cross-currency swaps and swaptions. We describe Babbs' model for the domestic and foreign term structures and the exchange rate, its formulation in terms of three correlated driftless Gaussian processes and the associated three state variable parabolic PDE. We then formulate nine difference approximations to the PDE, and discuss explicit and implicit methods. With this discrete approximation to the valuation problem in a period, we proceed to value the terminable diff swap and other deals numerically by backwards recursion through the payment dates, and investigate the solutions found graphically. We conclude that it is certainly practical, on a fast workstation, to solve for the value function of a wide range of cross-currency derivative securities by solution of explicit nite difference approximations of the PDE.
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Artur Sepp Mail: artursepp@hotmail.com, Web: www.hot.ee/seppar 30 April 2002 Abstract We implement the finite-difference (FD) solver and the Hull-White (HW) tree for numerical treatment of the pricing problem under the Hull-White interest rate model. We find that the FD solver is superior to the HW tree.
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Ph.D. Thesis Maya Briani A 150 page thesis including following topics- Numerical approximations: the Integro-Differential case Implicit-Explicit Schemes Discuss this paper
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