Modified implicit method embedded in a two-dimensional space for pricing brazilian interest rate derivatives

Resumo

 In the Brazilian fixed income market there is a standardized derivative product known as IDI (Interfinancial Deposits Index) Option, commonly used by corporations to manage risk. It is a financial Option of Asian type and, as such, the payoff depends of the path followed by this index. Among other works, [3] derived a closed form solution to price this contract based on a Martingale approach, assuming that the updating of the IDI is continuous. Based on the BM&FBovespa definition we propose a discrete daily monitoring of the IDI, via a PDE approach and relaxing the continuity restriction. We use the Vasicek [2] model, namely dr  a(b r)dtσdW , to represent the random movements of the instantaneous short term interest rate. This Ornstein-Uhlenbeck stochastic process pull the short rate to a level b at a rate a together with a normally distributed random term σdW , where dW is the standard Brownian Motion. We assume that the contract value is a function of the time t, the short rate r and the IDI y only, and denote it by V (r, y, t). So, we set up a portfolio containing one unit of the Option we want to price, and  units of another option of similar type and different maturity we use to hedge the position. Applying Ito’s Lemma we have: […]