How much do negative probabilities matter in option pricing? A case of a lattice-based approach for stochastic volatility models

In this paper, we focus on two-factor lattices for general diffusion processes with state-dependent volatilities. Although it is common knowledge that branching probabilities must be between zero and one in a lattice, few methods can guarantee lattice feasibility, referring to the property that all branching probabilities at all nodes in all stages of a lattice are legitimate. Some practitioners have argued that negative probabilities are not necessarily 'bad' and may be further exploited. A theoretical framework of lattice feasibility is developed in this paper, which is used to investigate how negative probabilities may impact option pricing in a lattice approach. It is shown in this paper that lattice feasibility can be achieved by adjusting a lattice's configuration (e.g., grid sizes and jump patterns). Using this framework as a benchmark, we find that the values of out-of-the-money options are most affected by negative probabilities, followed by in-the-money options and at-the-money options. Since legitimate branching probabilities may not be unique, we use an optimization approach to find branching probabilities that are not only legitimate but also can best fit the probability distribution of the underlying variables. Extensive numerical tests show that this optimized lattice model is robust for financial option valuations.