Abstract
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.
Highlights
The lattice approach is a popular one for valuing derivative securities, as it is normally simple to implement and has an intuitive appeal
To investigate the impact of negative probabilities in option valuations, we focus on using a two-factor lattice to represent general diffusion processes such as the Heston stochastic volatility (SV) model (Heston 1993)
We focus on two-factor lattices for general diffusion processes where volatilities can be state-dependent, including stochastic volatility models
Summary
The lattice (or tree) approach is a popular one for valuing derivative securities, as it is normally simple to implement and has an intuitive appeal. The authors further showed that the popular two-factor interest rate tree proposed by H&W (Hull and White 1994) for valuing interest rate derivatives can only guarantee lattice feasibility when the correlation is no greater than 0.2 This means that negative probabilities may occur far more often than we know in using lattices to value derivatives in real practice. To investigate the impact of negative probabilities in option valuations, we focus on using a two-factor lattice to represent general diffusion processes such as the Heston stochastic volatility (SV) model (Heston 1993). Since the MC method generates stochastic paths, not lattices, to model the evolution of the underlying uncertainties, there is no issue of lattice feasibility or negative probability. All proofs of propositions and theorems are given in the Appendix A of this paper
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have