Efficient and Robust Combinatorial Option Pricing Algorithms on the Trinomial Lattice for Polynomial and Barrier Options

Wang, Jr-Yan; Wang, Chuan-Ju; Dai, Tian-Shyr; Chen, Tzu-Chun; Liu, Liang-Chih; Zhou, Lei

doi:https://doi.org/10.1155/2022/5843491

Mathematical Problems in Engineering

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Abstract Introduction Preliminaries Experimental Results Conclusion Appendix Data Availability Conflicts of Interest References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 5843491 | https://doi.org/10.1155/2022/5843491

Efficient and Robust Combinatorial Option Pricing Algorithms on the Trinomial Lattice for Polynomial and Barrier Options

Jr-Yan Wang,¹Chuan-Ju Wang,²Tian-Shyr Dai,³Tzu-Chun Chen,⁴Liang-Chih Liu,⁵and Lei Zhou³

Academic Editor: Kazem Nouri

Received03 Aug 2021

Revised19 Jan 2022

Accepted29 Mar 2022

Published21 May 2022

Abstract

Options can be priced by the lattice model, the results of which converge to the theoretical option value as the lattice’s number of time steps approaches infinity. The time complexity of a common dynamic programming pricing approach on the lattice is slow (at least ), and a large is required to obtain accurate option values. Although -time combinatorial pricing algorithms have been developed for the classical binomial lattice, significantly oscillating convergence behavior makes them impractical. The flexibility of trinomial lattices can be leveraged to reduce the oscillation, but there are as yet no linear-time algorithms on trinomial lattices. We develop -time combinatorial pricing algorithms for polynomial options that cannot be analytically priced. The commonly traded plain vanilla and power options are degenerated cases of polynomial options. Barrier options that cannot be stably priced by the binomial lattice can be stably priced by our -time algorithm on a trinomial lattice. Numerical experiments demonstrate the efficiency and accuracy of our -time trinomial lattice algorithms.

1. Introduction

Options are important financial derivatives whose values depend on other more fundamental financial assets, which are called the underlying assets, for instance, stocks, indexes, and currencies [1]. In this study, for convenience, the underlying asset is assumed to be a stock. To improve the market completeness and efficiency, many complex options have been structured to meet specific financial goals. However, most complex options do not have analytical pricing formulae and must be priced using numerical methods [2]. Exploring efficient and accurate numerical pricing methods is thus important in both theory and practice.

The lattice method is a popular numerical pricing method. It can easily deal with American-style features like early redemption and early exercise that are found in many option contracts. It is also flexible, since only nominal changes are needed to price complex, nonstandard options, which do not have simple closed-form solutions [3] (Indeed, the flexibility of the lattice method makes it widely adopted in recent literature, such as the evaluation of American stock options [6], variable annuity products [7, 8], catastrophe equity puts [9], employee stock options [10], swing options [11], or corporate bonds [12–15]). Technically speaking, a lattice divides the option life into discrete equal-length time steps and simulates stock price movements discretely at each time step. Four-time-step binomial and trinomial lattices are shown in Figure 1. While each node (except the nodes at the last time step) in the binomial lattice has two branches connected to the two following nodes at the next time step, each node in the trinomial lattice has three branches. The most classical binomial lattice is proposed by Cox, Ross, and Rubinstein [4] (CRR model hereafter) (Rubinstein [16] reviews the relation between binomial and trinomial lattices and shows that they are equivalent to explicit finite difference methods.), where a node with stock price can move to or () at the next time step with probabilities and , respectively. Similarly, a node with price in a trinomial lattice can move to , , and (assuming in Figure 1) at the next time step with probabilities , , and , respectively. The branching probabilities and the stock price multiplicative factors (like , , , and in the binomial lattice and , , , , and in the trinomial lattice) are determined to match the first two moments of distribution of the stock price process, so that the pricing results generated by lattice models converge to the theoretical value as [4, 5].

(a)

(b)

Figure 1

Illustration of binomial and trinomial lattice models. Four-time-step binomial (a) and trinomial (b) lattices are shown. The initial stock prices for both lattices are S₀. In the binomial lattice, the upward and downward multiplicative factors for the stock price are U and D, respectively, where UD = 1 based on Cox et al. [4], and the upward and downward branching probabilities are P_U and P_D, respectively. In the trinomial lattice, the upward and downward multiplicative factors for the stock price are u and d (assuming ud = 1), respectively, and the upward, middle, and downward branching probabilities are p_u, p_m, and p_d, respectively.

Pricing options on the lattice model can be achieved by dynamic programming. The value for each node in the lattice is recursively evaluated from the last-time-step back to the root node. Thus, it takes at least time (or quadratic time of ) (For an -time pricing method, the computational time quadruples when doubles) to price options on both the binomial and trinomial lattice models, since there are nodes on an -time-step lattice. The dynamic programming approach can be quite inefficient as the pricing results may converge slowly in , and thus, a large value for is required. Fortunately, Dai et al. [3] and Lyuu [17] proposed combinatorial approaches that can efficiently price a variety of options on the CRR binomial lattice in time (or linear time of ). The computational time of an (O(n))-time pricing method doubles when O(n) doubles. (therefore, an O(n)-time pricing method runs much faster than a O(n² ) one when n is large.)

However, option pricing results generated by the lattice may oscillate wildly with [18, 19]. For instance, when applying the CRR binomial lattice to price barrier options, the significant oscillation renders it impractical, as shown in Figure 2. To suppress this oscillation, the lattice structure should be adjusted to fit the option’s specifications. Since the trinomial lattice is more flexible than the binomial lattice, the structure of the trinomial lattice can be adjusted according to the option contacts to suppress the oscillation [2, 5, 19] (In the literature, various methods are used to mitigate the oscillating convergent behavior of the CRR binomial model; for example, Tian [22] and Klassen [23] for pricing vanilla options and Bolye and Lau [24] for pricing barrier options. Note that the modifications in Tian [22] and Klassen [23] lead to a nonhorizontal binomial tree, i.e., the asset price layers are no longer parallelly horizontal. However, the wide applications of combinatorial option pricing approaches depend crucially on the existence of horizontal layers, especially when they are used to price barrier-dependent options. Consequently, this study develops combinatorial option pricing algorithms based on trinomial lattices consisting of a set of parallelly horizontal layers. Bolye and Lau [24], meanwhile, proposed a method to locate a layer of nodes as close as possible to the barrier. Nevertheless, Ritchken [19] demonstrated that the trinomial lattice adopted in this study outperforms Bolye and Lau’s [24] method in terms of a faster convergence rate. In addition, the exponential error convergence rate of Fourier transform methods has led to their wide adoption in recent derivative pricing literature, such as discretely monitored barrier options [25, 26], continuously monitored barrier options [27], α-quantile and perpetual early exercise options [28], and Asian options [29]. Although lattice methods have slow error convergence rates as in Talponen and Turunen [30], their flexibility has led to their wide adoption in recent literature, as mentioned in footnote 1.). Although the trinomial lattice can alleviate the price oscillation problem, no efficient combinatorial algorithms for the trinomial lattice have been proposed and the inefficient dynamic programming approach, which takes at least time, must be applied. To address this problem, we develop combinatorial pricing algorithms on the trinomial lattice, such that this novel option pricing framework inherits both the efficiency of the combinatorial algorithm and the accuracy of the trinomial lattice.

Figure 2

Comparison of convergence behavior between binomial and proposed trinomial lattice models for pricing barrier options. The x-axis and y-axis denote the number of time steps n and the option values, respectively. The dotted and dashed lines denote the pricing results generated by the binomial and our trinomial lattices, respectively. The solid line stands for the theoretical value based on the analytical pricing formula [20, 21]. Clearly, the trinomial lattice model alleviates the price oscillation problem with respect to the number of time steps, n.

Developing a combinatorial algorithm for the trinomial lattice is much more difficult than that for the binomial one because the additional middle branch in the trinomial lattice complicates combinatorial formulae. Consider, for example, the number of price paths that start at the root and then finally reach the terminal stock price at the fourth-time-step, as shown in Figure 1. The number of price paths based on the binomial lattice (Figure 1(a)) is because all price paths in this case should consist of exactly two upward and two downward movements. In contrast, the number of price paths based on the trinomial lattice (Figure 1(b)) is , where each term represents a different composition of upward, middle, and downward movements for the price paths reaching the terminal node . For instance, the second term indicates that the price paths consist of two middle movements, one upward movement, and one downward movement. This complicated nature might be why no linear-time algorithm has been proposed for the trinomial lattice before this paper.

We address this first by noting that the option pricing formulae for a wide range of options on the trinomial lattice can be decomposed into certain regular summation forms. Then, we prove in two theorems that each term in the summation form can be represented by a simple recursive formula of its preceding term; thus, the summation form can be evaluated in time. We take polynomial options as the first example to develop an -time combinatorial algorithm for the trinomial lattice because polynomial options are general and able to accommodate widely traded power and vanilla call or put options as special cases, but polynomial options do not have closed-form solutions unless their payoffs satisfy certain restricted criteria [31, 32] (Kim et al. [33] and Zhang et al. [34] price power options as special cases of polynomial options). Other reasons motivating us to consider polynomial options are their flexible applications in practice. For instance, since the unusual payoff functions of exotic path-independent options (or even option portfolios) can be approximated by a polynomial function, the pricing method for the polynomial option can be employed to approximate the values of those exotic options that might not have closed-form solutions. In addition, if a portfolio is exposed to a risk factor in a nonlinear manner, the polynomial option can provide a more appropriate payoff to hedge this portfolio; such a hedge strategy is usually much cheaper than hedging with a combination of more fundamental derivatives. As a result, developing efficient and accurate algorithms for pricing polynomial options is an important and valuable issue. Furthermore, to demonstrate the versatility of the theorems proposed in this study, we develop the reflection principle on the trinomial lattice and then extend the combinatorial pricing algorithms on the binomial lattice in the study by Dai et al. [3] to evaluate barrier options on the trinomial lattice. The numerical results verify that our pricing algorithms run in time and show that our algorithms are superior in terms of accuracy and efficiency.

The structure of this study is as follows. We introduce fundamental knowledge about option pricing and the combinatorial approach in Section 2 and propose two novel combinatorial theorems for option pricing in Section 3. In Section 4, we apply these theorems to develop linear-time algorithms for polynomial, power, plain vanilla, and barrier options. We present the experimental results in Section 5 and conclude in Section 6.

2. Preliminaries

2.1. Basic Framework

2.1.1. Stock Price Process

Suppose an option initiates at time 0 and matures at time . Let denote the stock price at time , where . In addition, this study adopts the most common assumption for the process on option pricing, which states that under the risk-neutral measure, follows a log-normal diffusion process with drift (this paper takes a non-dividend-paying stock for example. It is straightforward to extend our model to consider a stock paying a continuously compounding dividend yield) and volatility , where is the risk-free rate.

2.1.2. Kamrad and Ritchken Lattice (KRL) Model

Kamrad and Ritchken [5] proposed a trinomial lattice model, as shown in Figure 1(b). Denote as the length of each time step. The KRL model parameterizes upward and downward multiplication factors and aswhere is the stretch parameter that can be adjusted to fit the option’s specification to alleviate the oscillation problem. To ensure that the trinomial lattice matches the drift () and volatility () of the stock price process, Kamrad and Ritchken set the upward, middle, and downward branching probabilities , , and as

According to Ritchken [19], the oscillation problem can be significantly reduced by causing the lattice to coincide with a certain price level . Following this idea, we define as

Note that when (there must be a stock price layer coinciding with ) or is not available (there is no specific level in the option contract to fit), we follow Kamrad and Ritchken [5] in setting as 1.224745, such that all branching probabilities converge to 1/3 when , which streamlines convergence for pricing options.

2.2. Polynomial Option

One major contribution of this study is the pricing of polynomial options—which do not have closed-form solutions unless polynomial payoffs satisfy some restricted criteria [31, 32]—with the trinomial lattice model in linear time. A general payoff form is considered aswhere denotes the stock price on the maturity date , and can be any polynomial function defined as , where ’s can be any real numbers and ’s are the nonidentical real numbers. Typically, the curve may intersect the -axis several times as shown in Figure 3. Consequently, there may be several separate regions that can generate positive payoffs, such as regions i, ii, and iii in Figure 3.

2.3. Power Option

A power option grants the holder a right to buy or sell the power of the stock price for the exercise price at the maturity date . On the other hand, the holder can junk the call option (or the put option) if the power of the stock price is lower (or higher) than the exercise price at time . The payoff of a power option can be defined aswhere can be any real number and equals 1 for call options and −1 for put options.

2.4. Vanilla Option

A vanilla option, a special case of the power option by setting , is a standard option without unusual features. The payoff for a vanilla option at time iswhere equals 1 for call options and −1 for put options. Since the payoff function of the polynomial option can accommodate the payoffs of the power and vanilla options as special cases, it is straightforward to employ our combinatorial pricing algorithm for the polynomial option to evaluate these two options. Moreover, to alleviate the oscillation problem when pricing power and vanilla options, the KRL lattice requires a price level that coincides with (i.e., in equation (3)) [23].

2.5. Barrier Option

A barrier option is an option whose payoff depends on whether the stock price path ever hits a certain price level called a barrier. Our study focuses on “up-and-in” barrier options; the extension to other types of barrier options is straightforward. Furthermore, we consider the case of for convenience. Thus, an “up-and-in” barrier option can only be exercised at maturity when the stock price path has risen to hit the barrier during the option life. Define . The payoff function for a barrier option at time iswhere equals 1 for call options and −1 for put options.

To alleviate the oscillation problem, Ritchken [19] suggests that a price level of the lattice should coincide with the barrier (i.e., in equation (3)).

2.6. Option Pricing Based on the KRL Model

The theoretical value of an option equals the discounted expected payoff of the option under the risk-neutral measure:

The above formula can be evaluated via dynamic programming by computing the option value of each node on the lattice model from time step back to time step 0. Take the polynomial option, for example. The value of each node at maturity can be evaluated by equation (4). In Figure 4, the examined node, say D, has three descendant nodes, say A, B, and C. Define as the option value for any node . The value of nodes other than the nodes at maturity can be computed by following backward induction formula.

The pricing result is the value of the root of the lattice computed recursively by the aforementioned formula. This naive pricing method takes at least time as there are nodes in the -time-step KRL model. When pricing barrier options, one must consider not only the stock price but also other path-dependent variables; thus, the dynamic programming approach is more complex, and the order of computational time complexities could be higher than two [4, 35].

Instead of evaluating the option value of each node, the combinatorial approach improves efficiency by grouping stock price paths with the same payoff and calculating their contribution to the option value. Define G as the set of all possible stock paths from time step 0 to time step . If we define as a subset of G that contains price paths with middle movements, upward movements, and downward movements, then the set of subsets {} forms a partition of G. Thus, the theoretical value of a polynomial option can be expressed aswhere denotes the payoff corresponding to path at maturity anddenotes the number of paths in the subset . The second equality is based on the fact that the probability of each path in is and the payoff of each path in is the same as . The following sections will show how to evaluate equation (9) in linear time of . Moreover, similar techniques plus the reflection principle can also be applied to develop time pricing algorithms for barrier options.

2.7. Reflection Principle on the KRL Trinomial Lattice Model

The reflection principle efficiently counts the number of paths that hit a specific price level before reaching a certain node at maturity in the KRL model. This principle is essential when pricing barrier options using combinatorial methods. In contrast to applying the reflection principle on the binomial lattice to price barrier options in the study by Dai et al. [3] and Lyuu [17], this study extends their work to the trinomial lattice. The grid in Figure 5 represents the structure of the KRL model, where x and y-axes denote the time step and the stock price level, respectively. Each node can move to node (upward movement), node (middle movement), or node (downward movement) at the next time step. The reflection principle can help us count the number of paths from node to node that hit the barrier . Without loss of generality, we assume that , ≥ 0. Consider one such path, , that hits barrier at node for the first time. We can reflect the path with respect to barrier to obtain (the dashed path). Each path from node to node maps to a unique path from node to node and vice versa. Thus, the number of paths from node to node equals the number of paths from node to node . As a result, the desired number of paths moving from node to node and hitting barrier equals the number of paths from node to node . This is the celebrated reflection principle.

Suppose the number of upward and downward movements is and , respectively, and the sum of upward and downward movements is (given the number of middle movements to be ). Thereby,

The first equation is from the definition of , , and , and the second equation ensures that the number of downward movements exceeds the number of upward movements by , which is the distance in y-axis between node and node . By solving the aforementioned system of equations for and , we have

Thus, the number of paths that hit barrier before arriving node from and that have middle movements iswhere if is not a nonnegative integer. Thus, the total number of paths hitting barrier before arriving from is

3. Linear-Time Algorithms for Evaluating Summations in Option Pricing Formulae

Trinomial lattice option pricing formulae such as equation (9) can be expressed in terms of a linear combination of summations , where follows certain forms (introduced later) and is a nonnegative integer smaller than , the number of time steps in the lattice model. These summations can be evaluated in linear time (i.e., in time) if the first term can be evaluated in time and each following term (i.e., for ) can be iteratively evaluated in time. The aforementioned linear-computation property is the main idea to construct our linear-time option pricing algorithms. Define the set to contain all the sequences (e.g., ) that have this linear-computation property. The following proofs show that the series expressed in certain forms (this will be used to develop pricing algorithms later) belong to .

Lemma 1. Define as , where can be evaluated in time, is a real number, and is a nonnegative running integer variable. Then, the sequence belongs to the set .

Proof. Suppose is the first term of the sequence , where is a nonnegative integer smaller than . Then, can be evaluated in time since , , , , and can all be evaluated in time. In addition, each following term can be iteratively computed in time by the recurrence formula for . Consequently, we can obtain , and thus, can be evaluated in time.

Lemma 2. Define, where and are the nonnegative running integer variables, and are the real-number constants, and is a nonnegative integer-valued function. Then, the following formula holds:

Proof. First, and can be expressed asThe sum of the second term in equation (16) and the first term in equation (17) can be simplified aswhere the third equality is due to the combinatorial addition identity. Thus,As a result, Lemma 2 can be obtained by rewriting the above formula.
The following theorems and corollaries show that the summation in certain formats can be evaluated in time, which is useful for deriving linear-time combinatorial option pricing algorithms.

Theorem 1. If the sequence , then the sequence defined in Lemma 2 also belongs to under the premise that , that is, can be evaluated in time given that is a nonnegative integer smaller than .

Proof. To show that , we first prove that the first term of the sequence, , can be evaluated in time. This can be achieved by showing that the sequence , the terms in the summation, belongs to . Since can be evaluated in time and suggests that each can be evaluated in time given is known, we can infer that . Thus, can be evaluated in time since both and can be evaluated in time.
The next task is to express in terms of , such that can be iteratively computed in time given the value of . Rewrite asBy replacing with Lemma 2, can be expressed in terms of asDefine functions and asThen, we have , since and each can be parallelly evaluated in constant time during the calculation of given . Following the same logic, because given , can be evaluated in constant time in parallel when evaluating given .
Since can be expressed in terms of as can be computed in time through equation (23) by evaluating and in parallel.
In the following corollaries, we consider the different integer-valued function and show the resulting sequence .

Corollary 1. can be computed in time when .

Proof. Similarly, can be expressed as equation (20) except that :By replacing with the formula in Lemma 2, can be further expressed asDefine to be a function as follows:Clearly, since and can be evaluated in constant time given in parallel with the calculation of given . Since can be expressed in terms of as can be computed in time via equation (27) by parallelly evaluating and .

Corollary 2. can be computed in time when 0 or 1 (Note that the value is determined by the nonnegative integral running variable . It is not a random variable).

Proof. Since , , , and the first term of and , can be evaluated in time as mentioned in Theorem 1, each following term (note that .) can be recursively evaluated from the precedent term in time by equation (21) or (23) if 1 or 0, respectively. Thus, can be computed in time.

Theorem 2. Let belonging to , , and be the nonnegative running integer variables, and be the real-number constants, and both and be the nonnegative integer-valued functions with the properties or 1 and or 1. Then, a more general series with the definition belongs to , given that is a nonnegative integer smaller than

Proof. We can rewrite asWe next define and , since both and can be evaluated in time by specifying the upper bounds of the running variable to be and in Theorem 1 (including Corollary 1 and 2), respectively. As a consequence, can be evaluated in time and .

3.1. Combinatorial Option Pricing Algorithms

We will first develop a linear-time pricing algorithm for polynomial options (with the payoff in equation (4)) by calculating the discounted expected payoffs with the aforementioned theorems and corollaries. Since power and vanilla options are special cases of polynomial options (their payoffs are shown in equations (5) and (6), respectively), our pricing algorithm for polynomial options can be easily extended to price power options or vanilla options. In addition, the reflection principle on the KRL trinomial lattice is applied to price barrier options. Our algorithms focus on call options; the extension to put options is straightforward.

4. Linear-Time Algorithm for Polynomial Options on the KRL Model

For convenience, we first introduce useful notation for developing our combinatorial pricing algorithm. On the KRL trinomial lattice, , , and denote the initial stock price, the upward and the downward multiplication factors, respectively. In addition, each price path is described with middle movements, upward movements, and downward movements; thus the corresponding stock price at maturity is , for . As a consequence, there are a total of attainable stock prices at maturity listed from the highest to the lowest as follows: . Thus, we number the stock price layer from to according to the exponent of of the attainable stock prices at maturity. Finally, we identify the stock price layer and that makes a largest interval bracketed by a positive-payoff region through the following formulae:

Note that neither nor can be less than or greater than since the highest and the lowest stock prices attainable at maturity are and , respectively. Therefore, to ensure that equation (29) is well-defined under all scenarios, two extreme cases are considered in our model: if , is set to , and if , is set to .

Next, we classify each positive-payoff region of a polynomial option into one of the following three cases: , , and . For each case, we can develop a combinatorial algorithm to evaluate the option value contributed by the positive-payoff region in linear time of . Thus, the total value of a polynomial option can also be evaluated in linear time of by accumulating the option values contributed by all positive-payoff regions.

Case 1.
We first derive the constraints for the price paths that end up in the positive-payoff region. Due to the conditions , the corresponding price layers and can be derived based on equation (29), and should satisfy . Since the stock price at maturity can be expressed as , and must be within the range , we derive the first constraint for and asIn addition, since the number of downward movements should be no less than 0 and no larger than the sum of the numbers of upward and downward movements, , we derive the second constraint for and as . Finally, a stock price path should have at least downward movements to ensure that ; thus the sum of the numbers of upward and downward movements, , should be no less than and no larger than the number of time steps, , i.e., , which is the third constraint. The option value contributed from the positive-payoff region can be expressed as the sum of the values contributed by the stock price paths that satisfy the above three constraints aswhereand denotes the payoff of the polynomial option given that the maturity stock price is .

Case 2.
According to the condition and the definitions in equation (29), we can obtain the stock price layers and corresponding to and , respectively, and they should satisfy . Similar to Case 1, we can derive three constraints for and as , , and . As a consequence, the option value contributed from the region can again be expressed as the sum of the values contributed by the paths that satisfy the above three constraints as

Case 3.
Since , the corresponding stock price layers and derived from equation (29) should satisfy . Next, we simply decompose the positive-payoff region into two parts: one includes the stock price layers from to and the other includes the stock price layers from 0 to . The first part can be evaluated as Case 1 by setting as and as , and the second part can be evaluated as Case 2 by setting as and as 0. Finally, the option value contributed from region can be evaluated as the sum of the values contributed by the aforementioned two parts.
To show the efficiency of our algorithm, we next prove that Case 1 and Case 2 can be evaluated in linear time of . Note that this implies that Case 3 can be evaluated in linear time since Case 3 can be expressed as a combination of Case 1 and Case 2. The value of a polynomial option is simply the sum of the values contributed by all positive-payoff regions and thus can be evaluated in linear time of . Our proofs take advantage of Theorems 1 and 2 developed in Section 3.
The linear-time pricing algorithm for each positive-payoff region can be proved by showing that equations (31) and (33) can both be evaluated in linear time of . We first show that the second summation term in equation (33) can be evaluated in linear time according to Theorem 1. The polynomial payoff of this term can be decomposed into the sum of its component terms aswherefor , andNote that each of the () summation terms in equation (34) can be evaluated in time. Specifically, can be evaluated in time according to Theorem 1 by setting , (According to Lemma 1, can be obtained by setting and ) and . In addition, also can be evaluated in linear time by setting , (According to Lemma 1, can be obtained by setting and .) and in Theorem 1. As a result, the second summation term in equation (33) can be evaluated in linear time of .
For both of the first summation terms in equations (31) and (33), since the lower bounds for the index are integer-valued functions of another running variable rather than constants, Theorem 2 is used to prove that they can be evaluated in linear time of . We set and , for , and then mimic the method to evaluate equation (34) by specifying the appropriate terms for , , and to evaluate each component term in the polynomial payoff in linear time with Theorem 2. Similarly, for the second summation terms in equation (31), we set , , and , , and to appropriate terms and next apply Theorem 2 to evaluate each component term in the polynomial payoff in linear time of .
As a result, each of the four summation terms in equation (31) of Case 1 and equation (33) of Case 2 can be evaluated in linear time of . Therefore, Case 3 can be evaluated in linear time of as well. Since the option value of any polynomial option contributed from each positive-payoff region can be computed as either Case 1, 2, or 3 in linear time of , the total value of any polynomial option can be obtained in time with our combinatorial pricing algorithm. Thus our approach is more efficient than other lattice models, which price polynomial options in at least time.
Note that the above algorithm is quite general and able to price not only polynomial options but also power and vanilla call or put options in time. To the best of our knowledge, this paper is also the first in the literature to obtain -time combinatorial pricing algorithms for polynomial, power, and vanilla options on the trinomial lattice model.

4.1. Linear-Time Algorithm for Barrier Options on the KRL Model

Pricing a barrier option is much more complex than pricing a polynomial option since the barrier option is a path-dependent option and can be exercised only if the stock price has hit during the life of the option. To mitigate the oscillation problem when pricing barrier options on the trinomial lattice model, the parameter in equation (3) is adjusted by setting , such that the barrier can coincide with a layer of the lattice, as shown in Figure 6. Before introducing our combinatorial algorithm, we first define to be the number of net upward movements needed to reach from and also define where can represent the lowest stock price layer which is above or coinciding with the strike price . In the following paragraphs, our linear-time algorithm for barrier options discusses two different cases regarding the relation between the strike price and the barrier .

Case 4.
In this case, a positive payoff (i.e., ) at maturity guarantees the hit on barrier during the option life, so the barrier option degenerates into a vanilla option. The reason behind this phenomenon is explained as follows.where the first equality is the combination of equations (7) and (9). Since is true by definition and holds in Case 4, implies . Thus, the second equality can be reduced to the third one. As a consequence, the barrier option in the case of can be priced by the algorithm for the vanilla option.

Case 5.
In this case, the option value can be further decomposed into two parts: the value contributed by the terminal nodes above (inclusive) and the value contributed by the terminal nodes between (exclusive) and (The value contributed by the nodes below is not taken into account since the option payoff at these nodes is 0.).
First, consider a terminal node, say in Figure 6, above or equal to the barrier . Note that all price paths reaching node must also hit the barrier . Thus, the payoff for this kind of node is the same as the payoff for the vanilla call, i.e., . Furthermore, since in this case, the payoff for nodes above can be simplified to be . In addition, for those paths reaching a node above (inclusive), we have , which implieswhere the third inequality is due to the fact that is an integer and the last equality is according to the definition in equation (31). In addition, since the number of downward movements is a nonnegative integer, the number of the sum of the upward and downward movements should satisfyThereby, the option value contributed by these terminal nodes iswhere the lower bound of the first summation is from equation (39) and the upper bound of the second summation is from equation (38).
Next, we employ Corollary 1 to develop an -time algorithm for evaluating equation (40), which can be rewritten aswhereThe summation term, , can be evaluated in time through Corollary 2 by setting and where can be proved to belong to through Lemma 1 by setting and . Similarly, the summation term, , can be evaluated in time through Corollary 2 by setting , and . Note that can be proved using Lemma 1 by setting and .
Second, consider terminal nodes that are below the barrier (exclusive) but above or equal to , like node in Figure 6. For those terminal nodes, they should satisfy . Therefore, a pair of upper and lower bounds for can be derived asIn addition, for a price path that first reaches the barrier (which takes upward movements) and then falls below the barrier at maturity (which takes at least 1 downward movement), the sum of the number of downward movements and upward movements, , should satisfyMoreover, in order to hit during the option life, the number of upward movements, , must be larger than , which suggests another constraint for :Finally, the number of paths, which starts from the root node, hits the barrier during the option life, and reaches the terminal node below the barrier that can be computed by the reflection principle mentioned in equation (13). Note that in Figure 6, the root of the lattice is step lower than (i.e., in Figure 5) and the vertical distance between the terminal node and the root is ; thus, the y-coordinate for node B is , which implies in Figure 5. The number of the paths that satisfy the aforementioned constraints can be obtained by substituting and into equation (13) to get . Thus, the option value contributed by the terminal nodes that are below the barrier but above or equal to iswhere the lower bound of the first summation is from equation (44) and the lower and upper bounds of the second summation are from equations (43 and 45), respectively.
To evaluate equation (46) in time, the first step is to substitute for , and then, we can rewrite equation (46) by employing as the running variable for the inner summation. Becausewhere , equation (46) can be rewritten aswhereThe first term in equation (48), , can be evaluated in time through Theorem 2 by setting , , and . Note that can be proved via Lemma 1 by rewriting and setting and .
Similarly, the second term can be evaluated in time through Theorem 2 by setting , , , , and . Again, can be proved by Lemma 1 by rewriting and setting and . Thereby, equations (48) and (46) can be evaluated in time.
The price of the barrier option of Case 5 is the sum of equations (40) and (46). We hereby propose an -time algorithm to price barrier options on the KRL model.

5. Experimental Results

In this section, we compare our linear-time combinatorial algorithms with the standard backward induction procedure (which is indeed the dynamic programming approach mentioned above) for pricing polynomial, vanilla, and barrier options on the KRL trinomial lattice. The results demonstrate the superiority of our combinatorial model in terms of efficiency and accuracy.

Since there is no simple closed-form pricing formula for polynomial options, it is necessary to price them numerically. However, pricing polynomial options via dynamic programming suffer from slow convergence, and thus, a large number of is required. Figure 7 shows the comparison of the running time between our -time trinomial combinatorial approach and the -time dynamic programming approach. The x and y-axes denote the logarithmic values of and running time, respectively. The slopes of linear regression formulae denote the estimations for the order of time complexities. Note that the combinatorial approach (slope) is much more efficient than the dynamic programming approach (slope). Moreover, we compare the convergence rates of absolute pricing errors for binomial and trinomial lattices, as shown in Figure 8. Our trinomial lattice generates smaller absolute pricing errors and shows a faster convergence rate than the CRR binomial lattice for pricing polynomial options. To further confirm the stable and fast convergence of our combinatorial algorithm, Figure 9 shows comprehensive sensitivity analyses for the polynomial option analyzed in Figure 7. The absolute pricing errors for the combinatorial approaches based on both the binomial and trinomial lattices are much smaller than the dynamic programming approach on the trinomial lattice and the explicit finite difference method (The implementation issues about applying the finite difference and Fourier transform methods to price polynomial options are discussed in Appendix.) for different settings of the risk-free rates r, stock price volatilities , times to maturity T, exercise prices X, and payoff functions. In addition, the combinatorial approach based on the trinomial lattice performs better than that on the binomial one in terms of less oscillations, as shown in Figure 10.

Figure 7

Time complexity for pricing polynomial options. This figure is a log-log plot of the running time versus the number of time steps. The solid and dashed lines denote the combinatorial and the dynamic programming approaches, respectively. The linear regression formulae are listed next to the lines. The initial stock price is 5, the exercise price is 5, the risk-free rate is 10% per annum, the volatility of the stock price is 25%, and the time to maturity (T) is one year. The payoff of the examined polynomial option is . The results demonstrate that our combinatorial approach successfully reduces the time complexity for pricing polynomial options on the trinomial lattice to time, comparing to the -time complexity of the dynamic programming approach.

Figure 8

Convergence of absolute pricing errors for pricing polynomial options. The black dots indicate the pricing errors generated by our trinomial lattice, and the gray triangles represent the pricing errors of the CRR binomial lattice. The benchmark option price, 10.8856, is generated by the CRR binomial lattice with a very large n = 10,000. All parameters are identical to those in Figure 7. The trinomial lattice adopted in this study converges faster than the CRR binomial lattice does for pricing polynomial options.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 9

Sensitivity analyses for polynomial options. The x and y-axes represent the running time and the absolute pricing error, respectively. The black and light gray thin curves denote the pricing errors generated by the combinatorial algorithms based on the KRL trinomial and CRR binomial lattice, respectively. The dark gray and light gray thick curves indicate the pricing errors generated by the dynamic programming approach on the CRR binomial lattice and explicit finite difference method. All parameters in (a)–(h) are identical to those in Figure 7 except the parameters specified in the legend. Specifically, the risk-free rates (r) in (a) and (b) are set to 5% and 15%; the volatilities of the stock price in (c) and (d) is 20% and 30%; the times to maturity (T) in (e) and (f) are 0.5 and 1.5 years; the exercise prices (X) in (g) and (h) are 2.5 and 7.5, respectively. The benchmark option value for each parameter setting is listed on the top right corner of each subfigure. (a) r = 5%. (b) r = 15%. (c) σ = 20%. (d) σ = 30%. (e) T = 0.5. (f) T = 1.5. (g) X = 2.5. (h) X = 7.5.

Figure 10

Convergence oscillations for pricing polynomial options. The x-axis and y-axis represent the running time and the absolute pricing error, respectively. The black and light-gray curves denote the pricing errors generated by the combinatorial algorithms based on the KRL trinomial and CRR binomial lattice, respectively. The dark-gray thick curves indicate the pricing errors generated by the dynamic programming approach on the KRL trinomial lattice. The payoff function for the polynomial option is specified as , where the initial stock price S0, the exercise price X, the risk-free rate r, the time to maturity T, and the volatility of the stock price σ are set to 50, 50, 10%, 0.5 years, and 40%, respectively. The benchmark option value for each parameter setting is listed on the top right corner of the figure.

A vanilla option is a special case of a polynomial option, and its value can be analytically evaluated by the Black–Scholes formula [36]. With this property, we can analyze the pricing errors generated by our combinatorial algorithm. Figure 11 shows the comparison of the absolute pricing errors and running times between the dynamic programming approach and the combinatorial approach for pricing vanilla call options. The x and y-axes denote the running time and absolute pricing errors, respectively. The results show that the absolute pricing errors of the combinatorial approach decrease more smoothly and rapidly than the dynamic programming approach.

Figure 11

Comparison of absolute pricing error versus running time for pricing vanilla call options. For the examined vanilla call option, the initial stock price is 90, the exercise price is 100, the risk-free rate is 10% per annum, the volatility of the stock price is 25%, and the time to maturity is 1 year. The benchmark option price is obtained based on Black and Scholes [36]. The black line indicates the absolute pricing errors of our combinatorial approach, and the gray line represents the absolute pricing errors of the dynamic programming approach. The right-hand side diagram is a more detailed illustration for the running time shorter than 0.003 seconds. Compared with the dynamic programming approach, our combinatorial approach shows more rapidly decreasing errors and smoother convergence behavior.

Figure 12 shows the relation between the running time and the number of time steps for our combinatorial and dynamic programming approaches on pricing up-and-in barrier call options. The x and y-axes denote the values after taking the logarithm of the number of time steps and for the corresponding running time, respectively. The regression formula for the dynamic programming approach is , whereas the regression formula for the combinatorial approach is . The slope of the dynamic programming approach is twice as high as that of the combinatorial one. These results again demonstrate that we successfully reduce the time complexity of pricing barrier options on the trinomial lattice from by the dynamic programming approach to by this novel trinomial combinatorial approach. Figure 13 shows the comparison of the absolute pricing errors versus running time for pricing up-and-in barrier call options between the binomial combinatorial approach and the trinomial combinatorial and dynamic programming approaches based on the KRL trinomial lattice. The binomial combinatorial approach is the linear-time pricing algorithm developed in Dai et al. [3] based on the CRR binomial lattice. Even equipped with this efficient linear-time binomial combinatorial approach, the oscillation problem is clearly more pronounced than the other two trinomial lattice-based models. Moreover, the trinomial combinatorial approach converges faster and more smoothly than the trinomial lattice dynamic programming approach and the binomial combinatorial approach in terms of running time.

Figure 12

Time complexity comparison of pricing up-and-in barrier call options. This figure represents the log-log plot of the running time versus the number of time steps. The solid and dashed lines denote the results for the combinatorial approach and the dynamic programming approach, respectively. The linear regression formulae are listed next to the lines. The initial stock price is 90, the exercise price is 85, the higher barrier is 95, the risk-free rate is 10% per annum, the volatility of the stock price is 25%, and the time to maturity is 1 year. The results demonstrate that our combinatorial approach successfully reduces the time complexity for pricing barrier options on the trinomial lattice to time, whereas the time complexity of the dynamic programming approach is .

Figure 13

Comparison of absolute pricing error versus running time for pricing up-and-in barrier call options. The black and gray solid lines indicate the absolute errors of our combinatorial approach and the dynamic programming approach based on the KRL trinomial lattice. The dotted line represents the absolute errors of the linear-time binomial lattice combinatorial approach proposed in the study by Dai et al. [3]. The benchmark option price is derived based on Rich [20] and Rubinstein and Reiner [21]. All parameters are the same as those in Figure 12. The right-hand side diagram is a more detailed illustration for the running time shorter than 0.005 seconds. The oscillation problem of the binomial combinatorial approach is more pronounced than those of the other two trinomial lattice-based models. Moreover, our trinomial combinatorial approach shows most rapidly decreasing errors and smoothest convergence behavior among the three models.

Note that the analyses in Figure 2 demonstrate that the pricing results of the binomial combinatorial approach are more inclined to oscillate and less accurate than those of the trinomial combinatorial approach in terms of both the number of time steps and running time. These comparisons between binomial and trinomial combinatorial approaches support the necessity and contribution of proposing combinatorial option pricing approaches based on the trinomial lattice in this study.

6. Conclusion

Although the trinomial lattice framework for option pricing has been shown to provide better convergence behavior than the binomial lattice framework, only the dynamic programming approach, whose complexity is at least time, has been developed under the trinomial lattice framework to price options. This study proposes a novel combinatorial approach to reduce the time complexity for pricing options on the trinomial lattice of Kamrad and Ritchken [5]. The notion is to decompose the complex option payoff functions into certain regular summation forms, which can be evaluated in time by simple recursive formulae. We successfully develop -time pricing algorithms for a wide range of options which require at least time when priced by the original dynamic programming approach.

Appendix

A. Issues about Pricing Polynomial Options with the Finite Difference and Fourier Transform Methods

Note that for vanilla options, since they are deeply in- and out-of-the-money at the two ends of the examined stock price range, well-performing boundary conditions can be determined based on this property for the finite difference method (abbreviated as the FD method). However, the property may not hold for complex polynomial payoff functions because the in- and out-of-the-money regions may distribute alternately. Therefore, in the FD methods, it is a challenging task to set the boundary conditions for pricing polynomial options. Recall that the payoff of a European call option is , where and denote the price of the underlying asset at maturity and the strike price, respectively. Thus, the option on the boundary with very high underlying asset value at time t is very likely to be in-the-money at maturity; therefore, the option value on the boundary can be trivially solved as by dropping the “+” operator. However, taking advantage of this in-the-money property may lead to pricing errors for complex polynomial option payoff functions. Feasible settings for the boundary position and corresponding option values are also nontrivial. In Figure 9, we set the upper bound to a high stock price level and examine a considerable number of stock prices to ensure the convergence of option values. Perhaps, this is why the FD method converges slightly slower than the dynamic programming method shown in Figure 9.

On the other hand, pricing polynomial options with the Fourier transform method (abbreviated as the FT method) could be an interesting but challenging future work due to complex payoff functions and in-the-money regions. This may explain why there are no FT-based pricing methods for polynomial options to our knowledge. We sketch the FT derivations proposed in Carr and Madan [38] for pricing a European-style vanilla call to illustrate the challenge. They price a European call option with payoff . In this study, they argue that the vanilla call option price can be expressed as an inverse Fourier transformwhere and denotes the Fourier transform of the call price multiplied by , defined aswhere and denote the density function of . One can change the integration order of equation (A.2) to yieldand then analytically evaluate the integration for later substitution into equation (A.1) to solve the option price. Now, we replace the payoff function of a vanilla option withthe payoff function of a polynomial option. Note that is analogous to the strike price X; thus, directly applying the above approach requires a positive to ensure that () is defined. However, a nonpositive does not cause any problem in the lattice-based approach. In addition, the ranges defined by the lower and upper limits of the inner integrals like and in equations (A.2) and (A.3) reflect the in-the-money region of the vanilla option. Multiple in-the-money regions of polynomial options could make the change of the integration order and analytical integration of equation (A.3) intractable. This is because, on the s-x plane, the contours of in-the-money regions are disconnected and nonpiecewise linear. In addition, on pages 587–588 of the study by Kwok et al. [37], we find that the singularity of the integration function for pricing vanilla options arises from the kink in the payoff function. Carr and Madan [38] solved this by multiplying the payoff with an exponential decay function . However, more singularities must be circumvented for pricing polynomial options due to the multiple roots of equation (A.4). These reasons all show why developing a fast convergent FT pricing method is a challenging task.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2022 Jr-Yan Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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