The singular points binominal method for pricing American path-dependent options

Pricing Ratchet Equity-Indexed Annuities with Early Surrender Risk in a CIR++ Model

“…the expected value of the discounted cash flow obtained not surrendering the contract, and then we can compare it with the payoff of the early surrender case. We solve the problem of pricing simple Ratchet EIAs with early surrender risk using the bivariate tree introduced in the previous section and the framework of the singular points introduced in [7].…”

“…However, it can be problematic from a computational point of view because of the high number of singular points generated by the procedure. We can use the argument proposed in [7], based on the the convexity of the piecewise linear functions v i,j,k , to reduce drastically such number controlling the error involved in the elimination procedure. Let us remark that deleting some internal points from the sequence of the singular points which characterizes v i,j,k , the new function is still piecewise linear and convex.…”

“…So it is sufficient to choose the maximal level of error h depending on n and such that h(n)n → 0 as n → ∞, for obtaining the convergence of the values obtained by the elimination procedure to the continuous value. For a detailed discussion we refer to [7] (see the argument for obtaining the upper estimates) and [4].…”

“…In this case, the pricing problem leads to a non-recombining tree. In order to treat the problem we use the framework of singular points techniques introduced in Gaudenzi, Lepellere and Zanette [7] (see also Costabile, Gaudenzi, Massabò and Zanette [4]). The proposed algorithm is based on appropriate jump conditions at each node corresponding to each reset date.…”

Pricing Discretely Monitored Asian Options by Maturity Randomization

“…Among analytical approaches, we mention the Laplace transform approach in Geman and Yor (1993), the spectral expansion derived by Linetsky (2004), and the approximation of the average distribution by …tting integer moments in Turnbull and Wakeman (1991), Lévy (1992), Milevsky and Posner (1998) or logarithmic moments as in Fusai and Tagliani (2002). Another approach uses binomial trees, such as Gaudenzi et al (2007). However, a large number of contracts specify discrete time monitoring, and the impact of the continuous-time assumption can be substantial for some path-dependent derivatives, see for instance the literature on lookback and barrier options, Kat (2001).…”