The Bino-Trinomial Tree: A Simple Model for Efficient and Accurate Option Pricing

Abstract

Most derivatives do not have simple valuation formulas and must be priced by numerical methods such as tree models. Although the option prices computed by a tree model converge to the theoretical value as the number of time steps increases, the distribution error and the nonlinearity error may make the prices converge slowly or even oscillate significantly. This paper introduces a novel tree model, the bino-trinomial tree (BTT), that can price a wide range of derivatives efficiently and accurately. The BTT reduces the nonlinearity error sharply by adapting its structure to suit the derivative's specification; consequently, the pricing results converge smoothly and quickly. Moreover, the pricing of some European-style options on the BTT can be made extremely efficient by combinatorial tools, which are not available to most other tree models.Therefore, the BTT can efficiently reduce the distribution error by picking a large number of time steps. This paper uses a variety of options to demonstrate the effectiveness 1 of the BTT. Extensive numerical experiments show the superiority of the BTT to many other popular numerical models.

The Binomial Interpolated Lattice Method for Step Double Barrier Options

“…In the following, we will use the same binomial parameters ∆T , u, d and p of Dai-Lyuu [5], computed as described in the previous section. Moreover, we modify the number of steps considering a new number of time steps N := M +2 in order to perform a suitable interpolation in time.…”

“…In order to test the efficiency of the Binomial Interpolated Lattice (BIL) approach we first consider the numerical experiments proposed in Day-Lyuu (DL) [5] for pricing knock-out doublebarrier call options and then we propose other comparisons with different parameters. The volatility of the stock price is σ = 0.25, the interest rate is r = 0.1, the time to maturity is T = 1, the strike price is K = 100 and the two barriers are L = 90 and H = 140.…”

“…As described in Section 1, Dai and Lyuu build the grid in the log-space until time t 2 and then they construct a 1-step trinomial tree in the remaining amount of time ∆T using a moment matching procedure. Specifically they select at time t 2 three nodes that are closer to the mean of the logarithmic process at time ∆T and they define the three branching probabilities such that the first two moments of the logarithmic stock price process are matched (see Dai-Lyuu [5] for details). As in the proof of Proposition 1 we call Y i,k = log S i,k and u…”

“…Numerical results show that this method is not able to treat the "near barrier" problem, occurring when the initial asset price is very close to one of the barriers. To overcome this problem we introduce a method (called Binomial Interpolated Lattice) that generates the binomial tree points using the Dai-Lyuu [5] binomial mesh but forgets the trinomial part using just simple interpolations. Moreover, the extension of this method to multi-step double barrier options (including American features) is straightforward and it allows to obtain accurate estimates of the prices in a very short time.…”

“…In order to deal with double barrier options Dai-Lyuu [5] introduce the bino-trinomial tree that is constructed so that both barriers exactly hit two lines of the tree nodes. Numerical results show that this method is not able to treat the "near barrier" problem, occurring when the initial asset price is very close to one of the barriers.…”