<abstract><p>In this paper, we propose an adaptive neural network surrogate method to solve the implied volatility of American put options, respectively. For the forward problem, we give the linear complementarity problem of the American put option, which can be transformed into several standard American put option problems by variable substitution and discretization in the temporal direction. Thus, the price of the option can be solved by primal-dual active-set method using numerical transformation and finite element discretization in spatial direction. For the inverse problem, we give the framework of the general Bayesian inverse problem, and adopt the direct Metropolis-Hastings sampling method and adaptive neural network surrogate method, respectively. We perform some simulations of volatility in the forward model with one- and four-dimension to compare the point estimates and posterior density distributions of two sampling methods. The superiority of adaptive surrogate method in solving the implied volatility of time-dependent American options are verified.</p></abstract>

<abstract><p>In this paper, an efficient numerical algorithm is proposed for the valuation of unilateral American better-of options with two underlying assets. The pricing model can be described as a backward parabolic variational inequality with variable coefficients on a two-dimensional unbounded domain. It can be transformed into a one-dimensional bounded free boundary problem by some conventional transformations and the far-field truncation technique. With appropriate boundary conditions on the free boundary, a bounded linear complementary problem corresponding to the option pricing is established. Furthermore, the full discretization scheme is obtained by applying the backward Euler method and the finite element method in temporal and spatial directions, respectively. Based on the symmetric positive definite property of the discretized matrix, the value of the option and the free boundary are obtained simultaneously by the primal-dual active-set method. The error estimation is established by the variational theory. Numerical experiments are carried out to verify the efficiency of our method at the end.</p></abstract>