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Home  >  Transasctions of NAMP VOL2

29. A Stochastic Algorithm and Multiple Scale For Solution to PDE With Financial Application by Bright O. Osu and Okechukwu U. Solomon Transactions of NAMP Vol 2, (Nov., 2016), pp 313 – 324
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This paper presents an application of multiple scale and stochastic approximation method to discretize generic financial PDE. The multiple scale method was adopted in calculating the periodic solutions resulted from a Hopf bifurcation of a discretized generic PDE to monitor and stabilize the oscillatory movement of the market price of stock. Thereafter a stochastic algorithm was formulated to price an American option under the Black-Scholes model through a drifted financial derivative system. With finer discretization, positive periodic solution, space nodes and time nodes, we demonstrate that the drifted financial derivative system can be efficiently and easily solved with high accuracy, by using a stochastic approximation method which proves to be faster in pricing an American option. Multiple scale method is used to obtain the periodic solution which in turn is used to identify the movements of two parameters responsible for oscillatory behavior of the system to be risk-free in conjunction with the stock price. It is discovered that as the stock price decreases, the periodic solution fluctuates according to risk-free rate. An illustrative example is given in concrete setting.  An illustrative example is given in a concrete setting.

Keywords: Financial PDE; Stochastic algorithm; Multiple scale; Drifted system; Option pricing; Spatial discretization. MSC: 65C05, 65D30, 98B28.