Efficient trinomial trees for local‐volatility models in pricing double‐barrier options

• Author: Yuh‐Dauh Lyuu,U Hou Lok
• Date: 01 April 2020
• Published date: 01 April 2020
• DOI: http://doi.org/10.1002/fut.22080

J Futures Markets. 2020;40:556–574.wileyonlinelibrary.com/journal/fut556

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© 2019 Wiley Periodicals, Inc.

Received: 4 June 2019

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Accepted: 22 November 2019

DOI: 10.1002/fut.22080

RESEARCH ARTICLE

Efficient trinomial trees for local‐volatility models

in pricing double‐barrier options

U Hou Lok

1

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Yuh‐Dauh Lyuu

2

1

College of Business, National Taipei

University of Business, Taipei, Taiwan

2

Department of Finance and Department

of Computer Science & Information

Engineering, National Taiwan University,

Taipei, Taiwan

Correspondence

U Hou Lok, College of Business, National

Taipei University of Business, No. 321,

Sec. 1, Jinan Rd, Taipei 10051, Taiwan.

Email: miguellok@ntub.edu.tw

Funding information

Ministry of Science and Technology,

Taiwan, Grant/Award Numbers:

106‐2221‐E‐002‐049‐MY3,

108‐2410‐H‐141‐019‐

Abstract

A local‐volatility (LV) model captures the volatility smile while retaining the

preference freedom of the Black–Scholes model. Past attempts to construct a

smile‐consistent tree for the LV surface do not guarantee validity. This paper

presents an efficient and valid smile‐consistent tree for the LV model. The only

assumption is that the LV surface be upper‐and lower‐bounded. With this tree,

double‐barrier options can be priced with fast convergence even in the presence

of volatility smile. This is confirmed numerically. An implied tree is also

presented. It recovers the LV surface reasonably well.

KEYWORDS

double‐barrier option, implied tree, local‐volatility model, stochastic volatility, trinomial tree

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INTRODUCTION

The classic model of Black and Scholes (BS; 1973) prices options by assuming constant volatility for the stock price.

Closed‐form pricing formulas are then available for vanilla European options. However, the constant‐volatility

assumption is not supported by empirical data (Rubinstein, 1994). Instead, options with different times to maturity or

strike prices have different volatilities when inverted by the BS formula. These volatilities are called implied volatilities,

which form the implied volatility surface. This surface typically exhibits complex dependency on both the maturity and

strike price of an option, a phenomenon known as the volatility smile or simply the smile. Models that can reproduce

the smile are desirable because exotic options can then be priced and hedged consistently against the benchmark

options. The local‐volatility (LV) model is one such model.

The LV model makes the local volatility (also called the instantaneous volatility or the deterministic volatility)

depend on the stock price and time (Derman & Kani, 1994; Dupire, 1994; Rubinstein, 1994). This assumption makes the

LV model preference‐free, like the BS model. As a result, the market is complete and options can be valued by the no‐

arbitrage argument without the need to estimate the market price of risk, which is difficult (Fengler, 2005; Rebonato,

2004). Indeed, the LV model is the only smile‐consistent model that is complete (Bennett, 2014). An option pricing

model like the LV model can be used in two ways (Hull, 2012; Lyuu, 2002). (a) A stochastic process for the underlying

asset is assumed and the options are priced thereof. (b) Option prices are used to infer the stochastic process that

reproduces the smile as closely as possible (a step called calibration), after which other options can be priced

consistently. Empirical studies of LV models can be found in Crépey (2004), Dumas, Fleming, and Whaley (1998), Lim

and Zhi (2002), Linaras and Skiadopoulos (2005). Applications of the LV model can be found in Derman, Miller, and

Park (2016).

Options without closed‐form pricing formulas require numerical methods, such as trees. A calibrated tree converges

to the underlying continuous‐time model as the number of time steps increases (Duffie, 2001). Trees have been widely

used to price options because (a) trees can handle the early‐exercise feature of American options and (b) trees are

reasonably efficient if the options are not strongly path‐dependent. The best‐known tree is the Cox–Ross–Rubinstein

(CRR) tree of Cox, Ross, and Rubinstein (1979). Many alternative trees have also been proposed (Chance, 2008).

The continuous local volatilities form the LV surface. A tree that fits the LV surface is called an LV tree. Common LV

trees are either binomial or trinomial. Their up, middle (if applicable), and down moves as well as the associated

transition probabilities are such that the local volatilities on the tree match the LV surface’s. A tree is valid if (a) its

transition probabilities lie between 0 and 1, and (b) the stock prices are all positive.

A few LV trees have been studied. Amin (1993) presents trees for LV surfaces that depend only on time. The LV

surface of the constant elasticity of variance (CEV) model, on the other hand, depends only on the stock price in the

power form (Cox, 1975). Its trees are given by Lu and Hsu (2005) and Nelson and Ramaswamy (1990). Kamp (2009)

proposes trees for LV surfaces which depend on both stock price and time, but they are not guaranteed to be valid.

Guthrie’s (2011) tree maintains constant up and down moves as well as constant transition probabilities by varying the

durations of the time steps. However, the tree may not match the LV surface. Lok and Lyuu’s (2017) provably valid

binomial tree matches only separable LV surfaces.

An implied tree aims to recover the unknown LV surface from option prices (equivalently, the implied volatility

surface). The option prices are supposed to be generated by some LV model. This tree is thus the result of model

calibration, an inverse problem, which is in general ill‐posed (Atkinson, 1989). Once the implied tree is in place, it can

be used in pricing. To avoid ambiguities, an implied tree will not be called an (implied) LV tree in this paper.

Many implied trees have been proposed. Rubinstein (1994) suggests a binomial implied tree from options with the

same maturity. The probabilities are determined by nonlinear optimization. But this tree can only deal with options

with the same maturity, and it may not match the implied volatilities at maturity. Jackwerth (1997) relaxes Rubinstein’s

path‐independence assumption. This greatly increases the degree of freedom and allows the tree to tackle the whole

implied volatility surface. Still, the tree may not match the implied volatility surface.

Derman and Kani’s (1994) alternative implied‐tree methodology is widely adopted. Option prices with different strike

prices and times to maturity are used to determine both the geometry and the transition probabilities of the binomial implied

tree. But their binomial tree (DK henceforth) contains invalid transition probabilities and may not fit the implied volatility

surfaceexactly.Toaddressthisissue,stockpricesthatviolatetheno‐arbitrage principle are replaced via ad hoc procedures.

But the resulting tree may still fail to fit the surface exactly. Derman, Kani, and Chriss (1996) propose a trinomial implied

tree which has more degrees of freedom. However, negative probabilities remain an issue. Barle and Cakici (1999) improve

the DK tree with a better node‐placement strategy to reduce, but not eliminate, the occurrences of invalid probabilities.

Exotic options are used in hedging, speculation, or, much less often, model calibration (Ayache, Henrotte, Nassar, &

Wang, 2004). With the LV tree or the implied tree in place, such options can be priced consistently with the vanilla

options. Barrier options are among the most popular exotic options (Bennett, 2014; Rubinstein & Reiner, 1991). Their

payoff depends on whether the stock price ever touches certain price levels called barriers. As an example, the knock‐

out double income (KODI) option issued by Polaris Securities in Taiwan features two barriers (Tan, Chiu, & Tseng,

2005). Merton (1973) derives analytical formulas for single‐barrier options under the BS model. Funahashi and Kijima

(2016) provide approximation formulas for single‐barrier options under LV models whose volatility depends on stock

price but not time. There are no approximation formulas for double‐barrier options under such models. The analytical

formulas for double‐barrier options in Geman and Yor (1996), Kunitomo and Ikeda (1992), Luo (2001), and Sidenius

(1998) work for the BS model only. The approximation formulas of Boyarchenko and Levendorskiǐ(2009) and

Levendorskiǐ(2017) price single‐barrier options for Lévy‐type models, which nest the BS model but not the LV model.

Because analytical formulas for double‐barrier options under LV models are not available, numerical methods become

necessary. Andersen and Brotherton‐Ratcliffe (1997), Boyle and Thangaraj (2000), and Hull and Suo (2002) price vanilla

and single‐barrier options under LV models with the implicit finite‐difference method. Their method may not match

the implied volatility surface as regularization is imposed. Moreover, their method is complicated compared with trees.

Pricing barrier options accurately with trees is nontrivial even in the constant‐volatility case. For instance, the prices

calculated by the CRR tree oscillate significantly because this tree lacks the flexibility to align the tree levels with the

barriers (Boyle & Lau, 1994; Dai, Liu, & Lyuu, 2008; Tavella & Randall, 2000). Various trees have been proposed to

address this problem. Ritchken (1995) proposes a trinomial tree to align the tree levels with barriers. Figlewski and

Gao’s (1999) adaptive mesh model is another alternative, but it is complex and does not align the tree level with the

barrier. The bino‐trinomial tree of Dai and Lyuu (2010) is very efficient and can align the tree levels with two barriers

when volatilities are constant. Their central ideas will be used in this paper. None of the above trees address

non‐constant volatilities.

LOK AND LYUU

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