Title A lattice algorithm for pricing moving average barrier options

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A lattice algorithm for pricing moving average barrier options

Dai, M; Li, P; Zhang, JE

Journal Of Economic Dynamics And Control, 2010, v. 34 n. 3, p. 542-554

2010

http://hdl.handle.net/10722/85602 Journal of Economic Dynamics and Control. Copyright © Elsevier BV.; This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Moving Average Barrier Options

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A Lattice Algorithm for Pricing Moving Average Barrier Options1 Min Daia, 2 , Peifan Lia , Jin E. Zhangb a

b

Department of Mathematics, National University of Singapore, Singapore School of Economics and Finance, The University of Hong Kong, Pokfulam Road, Hong Kong

First Version: January 2008 Final Version: October 2009 Forthcoming in Journal of Economic Dynamics and Control

Abstract This paper presents a lattice algorithm for pricing both European- and American-style moving average barrier options (MABOs). We develop a finite-dimensional partial differential equation (PDE) model for discretely monitored MABOs and solve it numerically by using a forward shooting grid method. The modeling PDE for continuously monitored MABOs has infinite dimensions and cannot be solved directly by any existing numerical method. We find their approximate values indirectly by using an extrapolation technique with the prices of discretely monitored MABOs. Numerical experiments show that our algorithm is very efficient. JEL classification: C63; G13 Keywords: Barrier option; Moving average; Lattice algorithm; Forward shooting grid method; Extrapolation

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We are especially grateful to the three anonymous referees whose helpful comments substantially improved the paper. We also acknowledge helpful comments from Carl Chiarella (editor). Min Dai (MD) is also an affiliated member of Risk Management Institute (RMI), National University of Singapore (NUS). MD has been partially supported by the Singapore MOE AcRF grant (No. R-146-000-096-112) and the NUS RMI grant (No. R-146-000-117/124-720/646). Part of the work was done when MD visited Hong Kong University of Science and Technology and City University of Hong Kong in 2002. Jin E. Zhang (JEZ) thanks Yimin Zhang for special help. JEZ has been partially supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. HKU 7549/09H). 2 Corresponding author. Tel: (65) 6516 2754; fax: (65) 6779 5452. E-mail addresses: [email protected] (M. Dai), [email protected] (P.-F. Li), [email protected] (J.E. Zhang).

Moving Average Barrier Options

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Introduction

Options on the moving average price3 (MAP) are widely used in corporate finance. For example, moving average call options are often used to design a poison pill, a strategy in business that increases the likelihood of negative results over positive ones against a party that attempts any kind of takeover. The moving average calls, issued to existing shareholders, would be triggered by the event of a hostile takeover. The French investment bank Compagnie Financi´ere Indosuez and the French construction company Bouygues have, for example, successfully issued such options/warrants to protect themselves against potentially unfriendly investors, see e.g., Bouaziz, Briys and Crouhy (1994). Options on the MAP are currently traded as part of structured products in the overthe-counter financial markets. They are particularly welcome by investors who believe in technical analysis4 . Those investors use the MAP as a measure of market inertia who predict the future trends of a stock with information from its historical MAPs. Therefore, it is very natural to include the MAP as an element in designing financial products to attract such investors. For instance, an investor might make a trading rule that he will buy a stock whenever its 50-day MAP rises above a target level and sell it whenever the MAP drops below this level. A financial institution could issue to the investor an up-and-in call and a down-and-in put triggered by the event that the MAP has hit the target level, which serves as a barrier. Like the normal average in regular Asian options, using the MAP could also effectively alleviate the impact of short-term price fluctuation and protect investors from price manipulation. 3

The moving average price is computed based on the current price and on those in the most recent periods over a fixed length, denoted as D in equation (1) in this paper. 4 The usefulness of technical analysis has been well justified in the finance literature. For example, Lo, Mamaysky and Wang (2000) find that technical analysis has added value to the investment process based on their novel approach comparing the distribution conditional on technical patterns, such as headand-shoulders and double-bottoms, with the unconditional distribution. Zhu and Zhou (2009) analyze the widely employed moving average trading rule from an asset allocation perspective. They show that technical analysis adds value no matter if stock returns are predictable.

Moving Average Barrier Options

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There are a few different ways to use MAP in designing exotic options. The simplest one would be the moving average barrier option (MABO), the up-and-in call and down-and-in put discussed above. The MABO has a regular European option payoff, but is triggered by the event that the MAP hits a prescribed barrier level. The MAP can also be used as an underlying asset in designing options with American, lookback and reset features. A European option on MAP reduces to a known product of a forward starting Asian option or an Asian tail option. In this paper, we focus on the pricing of MABOs and leave the pricing of other MAP-related options for future research. The study on pricing regular barrier options has a long history. Merton (1973) first studied the price of a down-and-out call option. Conze and Viswanathan (1991) derived the pricing formulas of the up-and-out call and down-and-out put by applying the joint distribution of historical extremum and terminal stock price. Rubinstein and Reiner (1991) provided formulas for many other barrier options. Gao, Huang and Subrahmanyam (2000) proposed an alternative approach for pricing and hedging American barrier options using the decomposition technique. Dai and Kwok (2004) derived analytical formulas for knockin American options. These papers dealt with continuously monitored barrier options. However, in practice, many barrier options traded on markets are discretely monitored. Broadie, Glasserman and Kou (1997) proposed a continuity correction for the discretely monitored barrier option and justified the correction both theoretically and numerically. Kou (2003) extended the results by covering more cases and giving a simpler proof. To resolve the pricing issue of discretely monitored barrier options fully, one has to use a numerical approach. Zvan, Vetzal and Forsyth (2000) presented an implicit method for solving PDE models of contingent claims prices with general algebraic constraints on the solution. Examples of constraints include barriers and early-exercise features. Recently, the literature has evolved into pricing barrier options in a more complicated economy. Fusai and Recchioni (2007) provided an analysis of a quadrature method combined with

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an interpolation procedure for pricing discrete barrier options in GBM, CEV and variance gamma models. Bernard, Le Courtois and Quittard-Pinon (2008) developed a general valuation approach to price barrier options in a stochastic interest-rate environment. Feng and Linetsky (2008) proposed an extrapolation approach for the pricing of barrier options in jump-diffusion models. Pricing arithmetic Asian options has become a subfield of its own in the discipline of computational finance. The problem is difficult because there is no known closed-form expression for the distribution of the arithmetic average of the lognormal process. For continuously sampled Asian options, Ingersoll (1987) first derived a partial differential equation with 2+1 dimensions.5 Roger and Shi’s (1995) transformation uncovered the symmetry of the problem so that they could reduce the problem by one dimension, but it was still nontrivial to solve the resulting 1+1 dimensional PDE numerically (see Zvan, Forsyth and Vetzal (1998) for flux limiting techniques). Zhang (2001) observed that one difficulty in solving Roger and Shi’s PDE came from the singularity embedded in the initial condition. He used a novel approach to remove the singularity in the manner of the perturbation method and obtained highly accurate numerical values of continuous average rate options. The problem indeed has a quasi-analytical solution in a few different forms. Geman and Yor’s (1993) analytical expression required calculating inverse Laplace transformations of a confluent hypergeometric function, which was shown by Geman and Eydeland (1995) to be a challenging task. Dufresne’s (2000) Laguerre series was mathematically appealing, but it had some problems in the low-volatility case. Linetsky’s (2004) elegant spectral expansions of Whittaker functions were alternative representations of Dufresne’s Laguerre series. Both Dufresne’s and Linetsky’s methods were relatively easy to implement in a symbolic system like Maple or Mathematica, but none of them provided a particularly effi5

By 2+1 dimensions, we mean two asset price (stock and cumulative sum) dimensions and one time dimension.

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cient way to overcome the low-volatility problem. Because the pricing issue cannot be fully settled by the analytical methods, researchers have tried to develop a few approximation methods. Keman and Vorst’s (1990) Monte Carlo simulation with variance reduction and Turnbull and Wakeman’s (1991) and Levy’s (1992) lognormal approximations were among the earliest attempts at pricing average rate options. Curran’s (1994) approximate formula was derived by calculating the expectation conditional on the geometrical mean. Milevsky and Posner’s (1998) reciprocal gamma and Posner and Milevsky’s (1998) shifted-lognormal and shifted-arcsinh-normal approximations were based on the idea of moment-matching. Built on Roger and Shi’s (1995) bounds, Thompson (2002) found some new bounds on the value of fixed-strike and floating-strike Asian options. Ju’s (2002) accurate approximate formula obtained via Taylor expansion worked for both discrete and continuous Asian options. Zhang (2003) provided higher-order terms of Zhang’s (2001) leading approximation with a perturbation method. For discretely sampled Asian options, Veˇceˇr (2001) observed that the Asian option is a special case of the option on a traded account. He obtained a 1+1 dimensional PDE with piecewise continuous coefficients, which can be regarded as a generalization of Roger and Shi’s PDE. Benhamou and Duguet’s (2003) small dimension PDE for the homogeneous case was equivalent to Veˇceˇr’s PDE. Nielsen and Sandmann (2003) developed and compared bounds on the pricing formulas. Recently, Boyle and Potapchik (2008) provided an excellent comprehensive survey of current methods for pricing Asian options and computing their sensitivities to the key input parameters. With extensive numerical exercises, they concluded that Ju’s (2002) Taylor expansion and Zhang’s (2001) semi-analytical method are two of the most efficient methods to calculate the prices and Greeks of arithmetic Asian options. All of these papers studied options on average prices with a fixed starting date, instead of those on the MAP considered in this paper. Pricing continuously monitored MABOs is notoriously difficult due to the curse of dimensionality. Only Heritage (2002) studied the pricing of European MABOs. He derived

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an approximate pricing formula for the options by working with the expectations. Unfortunately, we will see later that his formula only applies to a certain range of parameter values. Other related literature includes Kao and Lyuu (2003) and Broadie and Cao (2008). Kao and Lyuu (2003) developed a numerical algorithm based on the binomial tree model to price the moving average lookback option and the moving average reset option, which were listed on the Taipei Stock Exchange in 1999. However, their algorithm is potentially very time consuming because the resulting number of computations grows exponentially with the increasing number of time steps. Broadie and Cao (2008) introduced new variance reduction techniques and computational improvements to the Monte Carlo methods for pricing American-style moving window Asian options. In this paper, we derive the PDE model and implement the forward-shooting grid method (FSGM) to price discretely monitored MABOs. However, we emphasize that it is not straightforward to apply the FSGM to price continuously monitored MABOs. The reason is that the FSGM essentially relies on a PDE model, but no PDE models are available for continuously monitored MABOs. To overcome the difficulty, we make use of an extrapolation technique together with the prices of discretely monitored MABOs to provide an approximation to that of the continuously monitored counterpart. Numerical results are presented to demonstrate the efficiency of our method. In addition, we show that the approximation formula proposed by Heritage (2002) for European-style MABOs works within a certain range of parameter values but loses power when the window horizon enlarges or the barrier level approaches the spot price. We also find that a simple extension of the least squares Monte Carlo approach proposed by Longstaff and Schwartz (2001) yields lower biased prices for American-style MABOs. This paper makes three contributions to the literature. First, this paper is one of few studies that consider the pricing of MABOs by bringing together the two strands of literature on pricing barrier and Asian options. For discretely monitored MABOs, it is possible

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to solve the problem numerically because the modeling PDE has finite dimensions. For continuously monitored MABOs, the modeling PDE has infinite dimensions and cannot be solved directly by any existing numerical method. Pricing continuously monitored MABOs, especially American-style MABOs, is an extremely difficult task. Second, we provide a practical way to approximate the solution to this classically difficult computational problem. Our FSGM method for discretely monitored MABOs and our extrapolation method for continuously monitored MABOs are shown to be very efficient. Third, our numerical results can be used by financial institutions and investors as references to determine the premia of options embedded in a moving average trading rule discussed above. Our numerical values can be also used by researchers as a benchmark to test the accuracy of approximate formulas developed based on Heritage (2002) with some other analytical methods. Finding a more accurate analytical approximation for the price of MABOs is an interesting and challenging problem for future research. The rest of this paper is organized as follows. Section 2 presents PDE formulations for discretely monitored MABOs and explains why PDE models are not available for continuously monitored MABOs. Numerical algorithms are introduced in Section 3. We conduct an extensive numerical study to investigate the performance of our algorithms in Section 4. Section 5 concludes with a short summary.

2

PDE models for MABOs

As usual, we assume that the risk-neutral process of the underlying asset price, St , is governed by dSt = rdt + σdWt , St where r and σ represent the riskless rate and the volatility, respectively, and Wt is a standard one-dimensional Brownian motion.

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Without loss of generality, we consider an up-and-out moving average barrier call option with maturity T and strike price X. Let H = S0 eb be the predetermined barrier level and D = T /M be the rolling time window for averaging, where M is a positive integer. At time t, the arithmetic MAP, J(t), is defined as 1 J (t) = D

Z

t

Sτ dτ .

(1)

t−D

Let T be the set of monitoring time instants at which whether or not the MAP exceeds the barrier level is monitored. If J(t) ≥ H for some t ∈ T , the option is called a “knock-out” and expires worthless; otherwise, the option has the same payoff as a vanilla call option. The terminal payoff of the option can thus be written as (ST − X)+ 1{J(t) 0, J1 > 0, t ∈ (tk−1 , tk ), k = 1, 2, · · · , M. At maturity, we have the terminal condition V1 (S, J1 , T ) = (S − X)+ 1{J1 0, J1 > 0.

(5)

It remains to prescribe a matching condition at tk . Apparently, V1 (S, J1 , t− k) = 0

if J1 ≥ H.

(6)

On the other hand, when J1 < H, by continuity we have − + + V1 (Stk , J1 (t− k ), tk ) = V1 (Stk , J1 (tk ), tk ),

which, combined with J1 (t+ k ) = limt→t+ k

1 t−tk

Rt tk

Sτ dτ = Stk , yields

+ V1 (S, J1 , t− k ) = V1 (S, S, tk )

if J1 < H.

(7)

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Combination of (6) and (7) gives the matching condition at tk : + V1 (S, J1 , t− k ) = V1 (S, S, tk )1{J1 0, J1 > 0, k = 1, 2, · · · , M − 1. Equations (4), (5) and (8) form a complete pricing model for European up-and-out call MABOs. For American style options, due to the early exercise feature, (4) is replaced by a variational inequality ½

∂V1 S − J1 ∂V1 1 2 2 ∂ 2 V1 ∂V1 min − − − σ S − rS + rV1 , V1 − (S − X)+ 2 ∂t t − tk−1 ∂J1 2 ∂S ∂S

¾ =0

for S > 0, J1 > 0, t ∈ (tk−1 , tk ) , k = 1, 2, · · · , M , and equations (5)-(8) remain unchanged. Francesco, Pascucci and Polidoro (2008) proved the existence of a strong solution to this type of problem. They also showed that the strong solution is a viscosity solution.

2.2

Discrete monitoring with frequency F = 2

Now we consider the case of F = 2. In addition to J1 (t) given in (2), it is necessary to introduce another path-dependent variable J2 (t) =

Z

1

¡ t − tk−1 −

D 2

µ

t

¢ tk−1 − D 2

Sτ dτ , t ∈

D D tk−1 − , tk − 2 2

¶ ,

for each k = 1, 2, · · · , M . The option knocks out provided that J1 (tk ) > H or J2 (tk − D2 ) > H for some k = 1, 2, · · · , M. It is worth pointing out that when t > T − D2 , there is only the last monitoring time, tk = T , remaining and the option value is nothing but V1 (S, J1 , t) as given in the case of F = 1. As a result, we only need to take into consideration the valuation before T −

D 2

and denote the option value by V2 = V2 (S, J1 , J2 , t), t < T −

D . 2

Apparently, we have the terminal condition µ V2

D S, J1 , J2 , T − 2

¶

µ = V1

D S, J1 , T − 2

¶ 1{J2 0, J1 > 0, J2 > 0, t ∈ tk−1 , tk − D2 , k = 1, 2, ..., M ; and S − J1 ∂V2 ∂V2 S − J2 ∂V2 ¡ ¢ + + + ∂t t − tk−1 ∂J1 t − tk − D2 ∂J2 ¡ ¢ for S > 0, J1 > 0, J2 > 0, t ∈ tk − D2 , tk , k = 1,

∂V2 1 2 2 ∂ 2 V2 σ S + rS − rV2 = 0 2 2 ∂S ∂S

(10)

(11)

2, · · · , M − 1.

Similarly, we need matching conditions at t = t1 − D2 , t1 , t2 − D2 , · · · , tM −1 . Using the same argument as in the case of F = 1, we have + V2 (S, J1 , J2 , t− k ) = V2 (S, S, J2 , tk )1{J1