Early exercise boundaries for American-style knock-out options

Early Exercise Boundaries for American-Style Knock-Out Options

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Early Exercise Boundaries for American-Style Knock-Out Options ˜ Pedro Vidal Nunes, Joao ˜ Pedro Ruas, Jose´ Carlos Dias Joao PII: DOI: Reference:

S0377-2217(20)30124-7 https://doi.org/10.1016/j.ejor.2020.02.006 EOR 16326

To appear in:

European Journal of Operational Research

Received date: Accepted date:

5 April 2019 4 February 2020

˜ Pedro Vidal Nunes, Joao ˜ Pedro Ruas, Jose´ Carlos Dias, Early ExerPlease cite this article as: Joao cise Boundaries for American-Style Knock-Out Options, European Journal of Operational Research (2020), doi: https://doi.org/10.1016/j.ejor.2020.02.006

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Highlights • Novel representation for the early exercise boundary of double knock-out options. • Valuation reduced to the same complexity as for pricing single barrier contracts. • Direct application to the valuation of expansion and abandonment options. • A new put-call duality is also provided for American-style double knock-out options. • FST, COS and SHP methods enhanced for the valuation of these knock-out options.

1

Early Exercise Boundaries for American-Style Knock-Out Options Jo˜ ao Pedro Vidal Nunesa,b,∗, Jo˜ ao Pedro Ruasa,b,c , Jos´e Carlos Diasa,b a

Instituto Universit´ ario de Lisboa (ISCTE-IUL), Edif´ıcio II, Av. Prof. An´ıbal Bettencourt, 1600-189 Lisboa, Portugal. b Business Research Unit (BRU-IUL), Lisboa, Portugal. c Sociedade Gestora dos Fundos de Pens˜ oes do Banco de Portugal, Av. da Rep´ ublica, no 57, 7o , 1050-189 Lisboa, Portugal.

Abstract This paper proposes a novel representation for the early exercise boundary of American-style double knock-out options in terms of the simpler optimal stopping boundary of a nested single barrier contract. Such representation only requires the existence, continuity and monotonicity (in time) of the nested single barrier exercise boundary, and these requirements are proved for the whole class of single-factor exponential-L´evy processes. To illustrate the practical relevance of our results, a new put-call duality relation is obtained, a real options application is provided and the Fourier space time-stepping method, the COS approximation, and the static hedging portfolio approach are all adapted to the valuation of American-style double knock-out options. Keywords: Finance, American-style knock-out options, Real options, Put-call duality, L´evy processes

1. Introduction As its first and main contribution to the existent literature on option pricing, this paper shows that the early exercise boundary of an American-style double knock-out option can be written in terms of one of its barrier levels and as a function of the optimal stopping boundary of the nested single barrier American-style option associated to the other barrier level. The rational behind our main result is similar to the one used, for instance, by Broadie and Detemple (1995, Theorem 1) or Gao et al. (2000, Theorem 6) to relate the early exercise boundaries of American-style standard and ∗

Corresponding author. Tel.: +351 217650526. Email addresses: [email protected] (Jo˜ ao Pedro Vidal Nunes), [email protected] (Jo˜ ao Pedro Ruas), [email protected] (Jos´e Carlos Dias) Preprint submitted to the European Journal of Operational Research

February 11, 2020

single barrier option contracts. Therefore, we are able to reduce the valuation of American-style double knock-out options to the same complexity level as the one faced when pricing simpler single barrier contracts. Our first result is only based on a few mild assumptions: the existence of a risk-neutral measure (i.e. on no-arbitrage, in the Harrison and Pliska (1981) sense); and the existence, continuity and monotonicity of a unique single barrier early exercise boundary, assumptions that are shown to be valid for the whole class of single-factor exponential-L´evy processes—yielding our second contribution to the literature. Such general framework accounts for the existence of jumps (and heavy tails) in the empirical distribution of returns, explains different implied volatility smile patterns, and is consistent with an incomplete market setup. Hence, the pricing of (simpler European-style) barrier options, under exponential-L´evy processes, has been extensively covered in the operations research literature over the last two decades—see, for instance, Feng and Linetsky (2008), Fusai et al. (2016), Phelan et al. (2018a) and Phelan et al. (2018b). Under the simpler geometric Brownian motion (hereafter, GBM) assumption of Black and Scholes (1973) and Merton (1973), Gao et al. (2000) and AitSahlia et al. (2003) provide analytic approximations for valuing single barrier American-style knock-out options—based on the integral representation method initially offered by Kim (1990), Jacka (1991), Carr et al. (1992), and Jamshidian (1992) in the context of standard American-style options—, AitSahlia et al. (2003) and Chang et al. (2007) extend the quadratic approximation method originally proposed by MacMillan (1986) and Barone-Adesi and Whaley (1987), and Detemple et al. (2019) price American knock-out (and knock-in) step options. Bearing in mind that only a few numerical approaches have been proposed so far for pricing American-style double barrier options—as, for instance, the lattice method of Ritchken (1995) or the partial differential equation scheme of Boyle and Tian (1998)—and that most of them are confined to the GBM assumption, our results should be useful for other option pricing models that also generate viable pricing solutions for single barrier American-style options. And even though these knock-out options are only traded in the OTC market, the pricing tools proposed in this paper are also relevant for other purposes besides option pricing. For instance, Das and Kim (2015, Equation (10)) express the debt discount as a function of a European-style double knock-out put for a firm exposed to (exogenous) ratchet and swap down debt covenants; however, such double knock-out put would become an American-style option if we further allow 3

for endogenous bankruptcy. The previous two results also yield, as our third contribution to the literature, a new put-call duality relation for American-style double knock-out options. A put-call duality relation holds whenever the price of a put option can be recovered from the price of a call option (or the other way around) through a suitable modification of the technical features of the option contracts. In the case of standard American-style options, and following Bjerksund and Stensland (1993), the put-call duality was first demonstrated by McDonald and Schroder (1998), under the usual GBM assumption. Based on the change of numeraire technique proposed in Geman et al. (1995), Schroder (1999, Proposition 1) shows that the put-call duality holds for a much larger class of models, including multifactor and jump-diffusion models. Fajardo and Mordecki (2006) are able to completely characterize the probability distribution of the corresponding dual process, for both European and American-style standard options, under exponential-L´evy option pricing models.1 The extension of the put-call duality for American-style single knock-out options was achieved by Gao et al. (2000), under the GBM assumption, and by Detemple (2001), under a much more general setup (including non-Markovian diffusion models with stochastic coefficients). Framing the duality analysis under the class of exponential-L´evy processes and using the representation offered by Fajardo and Mordecki (2006, Lemma 1) for the characteristic exponent (under the equivalent martingale measure that takes as numeraire the cum-dividend asset price process), we are also able to connect the prices of American-style double knock-out call and put options by linking their different barrier levels and early exercise boundaries. This new duality result is useful, for example, to overcome non-convergence issues that might arise when pricing one option contract (e.g. a call with its unbounded payoff) by valuing its counterpart instead (e.g. a put option). To illustrate the potential of our findings, and as our fourth contribution, both the Fourier space time-stepping (hereafter, FST) method of Jackson et al. (2008) and the Fourier-cosine (hereafter, COS) approximation of Fang and Oosterlee (2009) are adapted to the valuation of American-style double knock-out options under exponential-L´evy processes. In the first case, the adaptation is very simple: it is only necessary to impose the knock-out conditions to the option continuation value (in the real space). Under the COS method, the density function of the asset log-return is 1

The option literature has used the terms duality and symmetry interchangeably. In this paper, the put-call

symmetry label is reserved to describe the less common feature of symmetric volatility smile curves, in the sense of Carr and Chesney (1996) or Schroder (1999, Example 3).

4

approximated through its Fourier-cosine series expansion, and we just have to adjust the Fouriercosine series coefficients to price American-style knock-out options. Finally, and as our last contribution, the static hedging portfolio (hereafter, SHP) approach of Chung et al. (2013) is extended—for the first time, to the authors knowledge—from single to double barrier options, and our first result is applied to the valuation of expansion and abandonment (real) options as American-style double knock-out calls and puts. It is argued that there are often physical and economic boundaries to the value of the underlying project (arising from, for example, limited production capacity or limited capital at risk), and, hence, the usual approach of treating real options as plain-vanilla contracts (with unbounded positive payoffs) is likely to overstate their values. Alternatively, and to avoid such upward bias, our first result allows the underlying real asset price process to be stopped at the upper or lower limits of its admissible domain.

2. Modelling assumptions and contractual features This section lists the mild assumptions required by the novel representation of the early exercise boundary for American-style double barrier knock-out options, i.e. options that are canceled (or exercised) if the underlying asset price crosses either a lower (L) or an upper (U ) barrier level. The contractual features of the knock-out option contracts considered in the next sections are also specified. 2.1. Main assumptions Most of the modelling assumptions adopted will be borrowed from Schroder (1999, Section 1). It will be assumed throughout that the financial market is frictionless, and that trading takes place continuously on the time-interval T := [0, T ], for some fixed time T (> 0). Uncertainty is

represented by a complete probability space (Ω, F, Q), where the information accruing to all the agents in the economy is described by the complete filtration F = {Ft : t ∈ T } that satisfies the usual conditions of right-continuity (i.e. Ft = ∩l>t Fl ) and completeness (i.e. F0 contains all Q-null sets). Assumption 1. There exists a (not necessarily unique) risk-neutral measure Q such that the relative reinvested price of every asset, with respect to the “money-market account” numeraire, is a Q-martingale. 5

Moreover, Assumption 2. The risk-free interest rate r is a positive constant, and For all option contracts to be considered, the underlying asset price will be denoted by S, and can be taken as the price of a stock, a stock index, a currency or a commodity. Furthermore, Assumption 3. The underlying asset price process (St )t∈T is modelled as a nonnegative singlefactor stochastic process generating the complete and right-continuous filtration F, and Assumption 4. The underlying asset pays a continuously compounded (and constant) dividend yield q (≥ 0). The single-factor assumption is required because it will be easier to define the early exercise boundary of the American-style option as a function of time alone. Hence, Assumption 3 has a narrower scope than the framework adopted by Schroder (1999, Page 1159), which accommodates multifactor option pricing models; nevertheless, it is still possible to cope with single-factor diffusion models as well as with the whole class of (single-factor) exponential-L´evy models. Since this paper deals with the valuation of American-style options, we further adopt (and adapt to the context of knock-out options) Schroder (1999, Assumption 2). We recall that the holder of a one-touch knock-out American-style option owns a plain-vanilla option as long as no barrier is breached during the option’s lifetime; otherwise, the option contract is canceled (i.e. it is knocked out), and a cash rebate may also be paid (at the knock-out date or at the maturity date of the contract). However, and to contain the paper length, next definitions assume zero rebates; moreover, following Gao et al. (2000, Footnote 15), and for the knock-out and exercise events to be well defined, the option contracts must be specified in such a way that when the asset price first crosses a barrier, the option holder has the option to either exercise or let the option be knocked out. For this purpose, and representing by Stt0 ,x the time-t realization of the process (St )t∈T when such process is initialized at St0 = x (for some t0 ∈ [0, t]), n o t0 ,x τL;U := inf t > t0 : Stt0 ,x ∈ / ]L, U [

(1)

will denote its first passage time through the (constant and strictly positive) barrier levels L or U , with L < x < U and inf ∅ = ∞. Note that the double barrier hitting time (1) can also be used for 6

single barrier contracts: it is only necessary to take L = 0 (for up-and-out options) or U → +∞ (for down-and-out options). t0 ,x Therefore, and for t0 < τL;U , the time-t0 price of an American-style knock-out option allowing +  t0 ,x the holder to exercise and receive, at any F-stopping time τ ∈ [t0 , T ], the payoff φK − φS t0 ,x , τ ∧τL;U

where K ∈ ]L, U [ is the strike price, while φ = 1 (resp., φ = −1) for a put (resp., call) option, is given by

"

Vt0 = ess sup EQ e τ ∈F[t0 ,T ]

# + t0 ,x φK − φS t0 ,x Ft0 , τ ∧τL;U

  t0 ,x −r τ ∧τL;U −t0

(2)

where FA denotes the set of all F-stopping times taking values in A ⊆ R, and α+ ≡ α ∨ 0 is the positive part of α ∈ R. The Snell envelope (2) is obtained, for instance, by Karatzas (1988,

t0 ,x t0 ,x Theorem 5.4) in a complete market setting (and for τL;U → +∞), adapted (also for τL;U → +∞)

by Pham (1997, Equation (1.5)) to a jump-diffusion framework, and defined (for L = 0, T → +∞ and φ = 1) by Karatzas and Wang (2000, Equation (2.7)) under the GBM assumption. Our main results—in Propositions 1 and 2—will be further based on the existence, uniqueness, continuity and monotonicity of an early exercise boundary t 7−→ E (t) that separates two com plement sets: the continuation region C := (S, t) ∈ [0, ∞[ × [0, T ] : Vt > (φK − φSt )+ and the  exercise (or stopping) region E := (S, t) ∈ [0, ∞[ × [0, T ] : Vt = (φK − φSt )+ of the Americanstyle knock-out option contract. These are mild assumptions that are satisfied by most of the option pricing models already proposed in the literature. Under a single-factor diffusion model, Detemple and Tian (2002, Proposition 1) prove the existence of the exercise boundary for plainvanilla options, which is only a function of time (and a continuous function, as long as the risk-free interest rate is deterministic), while Pham (1997, Theorems 3.2 and 4.1) extends the previous result to a jump-diffusion setup. Therefore, and similarly to Karatzas and Wang (2000, Equation (2.15)), it is assumed that Assumption 5. There exists a unique early exercise boundary t 7−→ E (t) such that equation (2) can be rewritten as "

Vt0 = EQ e

  t0 ,x −r T ∧τe ∧τL;U −t0

# + φK − φS t0 ,x t0 ,x Ft0 , T ∧τe ∧τL;U

(3)

where τe is the first passage time of the underlying asset price process through the early ext0 ,x ercise boundary, and it is assumed hereafter that φSt0 > φE (t0 ) and τL;U > t0 . Therefore,

7

C = {(S, t) ∈ [0, ∞[ × [0, T ] : φSt > φE (t)} and E = {(S, t) ∈ [0, ∞[ × [0, T ] : φSt ≤ φE (t)}, with E (t) = inf {S ≥ 0 : Vt > G1 (S)}, for φ = 1, or E (t) = sup {S ≥ 0 : Vt > G−1 (S)} for φ = −1,

and where Gφ (S) := (φK − φS)+ is the option payoff function.

The monotonicity (in time) of the early exercise boundary—proved by Gao et al. (2000, Theorem 2) for the GBM case—is now assumed for the “out-of-the-money” single knock-out options. Assumption 6. The early exercise boundary t 7−→ E (t) is nondecreasing (resp., nonincreasing) on T , for φ = 1 and L = 0, i.e. for the up-and-out put (resp., for φ = −1 and as U → +∞, i.e. for the down-and-out call). Finally, it will be also necessary to assume that the early exercise boundary—of the nested single knock-out option associated to the “out-of-the-money” barrier level—is a continuous function of time. Assumption 7. For up-and-out puts (i.e. when φ = 1 and L = 0) as well as for down-and-out calls (i.e. when φ = −1 and as U → +∞), the early exercise boundary t 7−→ E (t) is a continuous function of calendar time. 2.2. One-touch knock-out options The aim of the present paper is essentially to reduce the exact valuation of American-style double barrier knock-out options to the same complexity level as the one faced with the corresponding solutions for single barrier contracts. For this purpose, three types of American-style knock-out options—with a unit face value—are first defined in the following lines. Starting with single barrier up-and-out contracts that are knocked-out or exercised if the underlying asset price crosses the up barrier (from below), Definition 1. The time-t0 price of a zero rebate American-style up-and-out put (if φ = 1) or call (if φ = −1) on the asset price S, with strike K, upper barrier level U , and maturity at time T uo(φ)

(≥ t0 ) is given by equation (3) for τe = τe

t0 ,x and τL;U = τU , where

t0 ,x τU := inf {t > t0 : St ≥ U } = τ0;U

is the first passage time through the up barrier level, and n o τeuo(φ) := inf t > t0 : φSt ≤ φE uo(φ) (t) 8

(4)

(5)

is the optimal stopping time at the exercise boundary t 7−→ E uo(φ) (t) of the American-style upand-out option, with φSt0 > φE uo(φ) (t0 ). Therefore, equation (3) implies that the time-t0 option value is equal to Vtuo 0



St0 , K, U, T, E

uo(φ)





; φ := EQ e

  uo(φ) ∧τU −t0 −r T ∧τe

Gφ



  ST ∧τ uo(φ) ∧τ Ft0 . e U

(6)

For American-style down-and-out options, the reasoning is similar: the option contract is knocked-out or exercised if the down barrier is crossed (from above) by the underlying asset price; otherwise, the option can be exercised at or before the expiry date. Hence, Definition 2. The time-t0 price of a zero rebate American-style down-and-out put (if φ = 1) or call (if φ = −1) on the asset price S, with strike K, lower barrier level L, and maturity at time T do(φ)

(≥ t0 ) is given by equation (3) for τe = τe

t0 ,x = τL , where and τL;U

t0 ,x τL := inf {t > t0 : St ≤ L} = lim τL;U

(7)

U →+∞

is the first passage time through the down barrier level, and n o τedo(φ) := inf t > t0 : φSt ≤ φE do(φ) (t)

(8)

is the optimal stopping time at the exercise boundary t 7−→ E do(φ) (t) of the American-style down-

and-out option, with φSt0 > φE do(φ) (t0 ). Therefore, equation (3) implies that the time-t0 option value is equal to Vtdo 0



St0 , K, L, T, E

do(φ)





; φ := EQ e

  do(φ) −r T ∧τe ∧τL −t0

Gφ



  ST ∧τ do(φ) ∧τ Ft0 . e L

(9)

Finally, an American-style double knock-out option is knocked-out or exercised if any of the two barrier levels is crossed by the underlying asset price; otherwise, it can be also exercised at or before the expiry date. Therefore, Definition 3. The time-t0 price of a zero rebate American-style double knock-out put (if φ = 1) or call (if φ = −1) on the asset price S, with strike K, lower barrier level L (< K), upper barrier level ko(φ)

U (> K), and maturity at time T (≥ t0 ) is given by equation (3) for τe = τe where

n o τeko(φ) := inf t > t0 : φSt ≤ φE ko(φ) (t) 9

t0 ,x and τL;U ≡ τLU ,

(10)

is the optimal stopping time at the exercise boundary t 7−→ E ko(φ) (t) of the American-style double

knock-out option, with φSt0 > φE ko(φ) (t0 ). Therefore, equation (3) implies that the time-t0 option value is equal to Vtko 0



St0 , K, L, U, T, E

ko(φ)





  ko(φ) −r T ∧τe ∧τLU −t0

; φ := EQ e

Gφ



3. Early exercise boundaries for double knock-out options

  Ft . ST ∧τ ko(φ) ∧τ 0 e LU

(11)

The novel results presented in Propositions 1 and 2 should reduce the valuation of Americanstyle double knock-out options to the same complexity level as the one faced when pricing single barrier American-style option contracts, and are applicable to any single-factor pricing model satisfying (the usual) Assumptions 5, 6 and 7. Next proposition recovers the double barrier put exercise boundary E ko(1) from the one associated to the simpler American-style up-and-out put option (and from the “in-the-money” barrier level L). Proposition 1. Under Assumptions 1 to 7, the early exercise boundary t 7−→ E ko(1) (t) of an American-style double knock-out put option on the asset price S, with strike K, lower barrier level L, upper barrier level U , and maturity at time T (≥ t), is equal to E ko(1) (t) = L ∨ E uo(1) (t) ,

(12)

for all t ∈ T , where t 7−→ E uo(1) (t) denotes the optimal stopping boundary of the American-style up-and-out put option defined in equation (6), and L < K < U . Proof. Similarly to Gao et al. (2000, Theorems 5 and 6), the analysis is divided into three mutually exclusive cases. Case A First, assume that E uo(1) (t) ≥ L, for all t ∈ T . Since the value of the double knock-out cannot exceed the value of the single knock-out, the value of the double knock-out put is maximized by taking E ko(1) (t) = E uo(1) (t) = L ∨ E uo(1) (t), for all t ∈ T , because that makes the value of the two knock-out options equal. Alternatively, equation (12) can also be proved by contradiction, showing that arbitrage is possible if Proposition 1 does not hold. For this purpose, and in contradiction to equation (12), suppose that E ko(1) (t) > E uo(1) (t). To show that such scenario cannot prevail in an arbitrage-free 10

market, a self-financing portfolio Π is formed by going long ω units of the double barrier option, while financing this position by shorting one unit of the nested up-and-out put. Hence, ω must be such that the time-t value of the portfolio is zero, i.e. ω :     Πt = ω × Vtko St , K, L, U, T, E ko(1) ; 1 − Vtuo St , K, U, T, E uo(1) ; 1 = 0,

uo(1)

for t < τe

(13)

∧τU ∧τL ∧T . Note that the solution of equation (13) must be such that ω > 1 because

the scenario E ko(1) (t) > E uo(1) (t) implies that the double knock-out put would be cheaper than the up-and-out put. This portfolio is maintained until the short position, i.e. the one unit of the up-and-out put, uo(1)

is exercised, is knocked-out, or expires; that is until time τΠ := τe

∧ τU ∧ T . Consequently,

there are three possible final payouts for the self-financing portfolio Π. First, if τΠ = τU , both put options are knocked-out and the terminal payoff of the portfolio equals ΠτU = ω × 0 − 0 = 0.

Second, if τΠ = T , then ΠT = ω × (K − ST )+ − (K − ST )+ > 0 since ω > 1, and an arbitrage uo(1)

opportunity exists for ST < K. Finally, if τΠ = τe , then both options are exercised, Πτ uo(1) = e     ω × K − Sτ uo(1) − K − Sτ uo(1) > 0 because ω > 1, and again a riskless profit is available. e

e

In summary, if E ko(1) (t) > E uo(1) (t), arbitrage is possible, and since E ko(1) (t) < E uo(1) (t)

cannot happen also—as it would mean that the double knock-out put would be more expensive than the up-and-out put—then it is necessary that E ko(1) (t) = E uo(1) (t) = L ∨ E uo(1) (t), for all t ∈ T and whenever E uo(1) (t) ≥ L; this now yields ω = 1 as the solution of equation (13) and,

therefore, ensures that ΠτΠ = 0. Case B The second scenario assumes that E uo(1) (t) < L, for all t ∈ T . Since the knock-out

event triggers the early exercise of the double barrier option, the condition E ko(1) (t) ≥ L must be

observed for all t ∈ T ; however, it is still necessary to find whether E ko(1) (t) > L or E ko(1) (t) = L. Using Gao et al. (2000, Equation (B.1)), and since E uo(1) (T ) < L < K < U , then   rK rK uo(1) ∧K = , E (T ) = U ∧ q q

(14)

and, therefore, this Case B can only happen if r < q. To show that the scenario E ko(1) (t) > L cannot prevail, the following portfolio is formed at any time t where both (single and double) barrier puts are still alive (i.e. where St ∈ ]L, U [), and comprises: a long position on the double barrier put; a long position on the underlying asset; and a loan for K monetary units (accruing 11

interest at the constant interest rate r). Such portfolio is held until time τΠ := τU ∧ τL ∧ T , and yields a terminal payoff of +

ΠτΠ = (K − SτΠ ) +

Z

τΠ

t

(qSv − rK) er(τΠ −v) dv > 0

that is strictly positive because equation (14) implies that Sv > L >

rK q ,

for all v ∈ [t, τΠ ].

Therefore, the absence of arbitrage opportunities imposes that the initial (time-t) portfolio value
must be negative, i.e. that −Vtko St , K, L, U, T, E ko(1) ; 1 −St +K < 0, which means that (St , t) ∈ C for all St > L. Consequently, the scenario E ko(1) (t) > L is not arbitrage-free, and, hence, it follows that E ko(1) (t) = L = L ∨ E uo(1) (t) for all t ∈ T .

  Case C Finally, it is necessary to consider the possibility of L ∈ E uo(1) (0) , E uo(1) (T ) . Since

Assumption 7 implies that E uo(1) (t) is a continuous function of t, then there exists a unique  t∗ := t ∈ T : E uo(1) (t) = L . For any time t ≥ t∗ , the same situation as in Case A arises (i.e. E uo(1) (t) ≥ L), and hence it follows that E ko(1) (t) = E uo(1) (t) = L ∨ E uo(1) (t). For t < t∗ , and since equation (14) implies that L <

rK q ,

the portfolio from Case B might not

yield an arbitrage opportunity any longer. Instead, and using the same argument as Broadie and Detemple (1995, Figure 6), Assumptions 6 and 7 guarantee that there always exists another single barrier up-and-out put on the same asset, with the same strike and upper barrier level, but with uo(1)

a shorter maturity T0 ∈ ]t, T [, whose (shrank) early exercise boundary t 7−→ E0

(t) is such that

uo(1)

(t) > L = L ∨ E uo(1) (t) . (15)   uo(1) The second inequality in equation (15) implies that E0 (t) , t is also an admissible exercise
uo(1) policy for the double knock-out put, and, therefore, Vtko St , K, L, U, T, E ko(1) ; 1 ≥ K − E0 (t). St > E0

Furthermore, the first inequality in equation (15) yields   uo(1) (t) Vtko St , K, L, U, T, E ko(1) ; 1 ≥ K − E0 > K − St ,

and, hence, (St , t) ∈ C for all St > L, i.e. E ko(1) (t) = L = L ∨ E uo(1) (t) for t < t∗ . Remark 1. Equation (12) mimics the relation between the early exercise boundaries of down-andout and standard American-style puts that is offered by Gao et al. (2000, Theorem 6) under the GBM assumption. However, our Proposition 1 relies on much lighter assumptions concerning the underlying asset price process. 12

Remark 2. Even though equation (12) is proved in a single-factor setup, Appendix E of the supplementary file to this paper shows that Proposition 1 is still valid under a multifactor framework with stochastic interest rates and stochastic volatility. Proposition 2 writes the double knock-out call early exercise boundary E ko(−1) in terms of the simpler one associated to an American-style down-and-out call option (and as a function of the “in-the-money” barrier level U ). Proposition 2. Under Assumptions 1 to 7, the early exercise boundary t 7−→ E ko(−1) (t) of an American-style double knock-out call option on the asset price S, with strike K, lower barrier level L, upper barrier level U , and maturity at time T (≥ t), is equal to E ko(−1) (t) = U ∧ E do(−1) (t) ,

(16)

for all t ∈ T , where t 7−→ E do(−1) (t) denotes the exercise boundary of the American-style downand-out call option defined in equation (9), and L < K < U . Proof. The proof of Proposition 2 is similar to the one presented for Proposition 1, and is available upon request. Remark 3. Equation (16) mimics the relation between the exercise boundaries of capped and plainvanilla American-style calls given by Broadie and Detemple (1995, Theorem 1), under the GBM assumption. Again, our Proposition 2 relies on much lighter assumptions. Equipped with Propositions 1 and 2, it is now possible to price American-style double barrier knock-out options as long as there is a viable valuation method for American-style standard options as well as the knowledge of the optimal exercise boundaries for American-style up-and-out puts or down-and-out calls. As shown in the next proposition, it is only necessary to adjust the exercise boundaries through the relations offered by equations (12) and (16). Proposition 3. Under Assumptions 1 to 7, the time-t0 value of the American-style double knockout put defined in equation (11) is equal to     ko(1) uo uo(1) Vtko S , K, L, U, T, E ; 1 = V S , K, U, T, L ∨ E ; 1 , t t t0 0 0 0

whereas for the associated double knock-out call     ko(−1) do do(−1) Vtko S , K, L, U, T, E ; −1 = V S , K, L, T, U ∧ E ; −1 . t0 t0 t0 0 13

(17)

(18)

Proof. Please see Appendix A. 4. American-style knock-out options under exponential-L´ evy processes Propositions 1 and 2 heavily rely on Assumptions 5, 6 and 7, and a general framework where such assumptions should be easily verified is the set of log-asset price processes with stationary independent increments. Such general framework accommodates the existence of jumps and heavy tails in the empirical distribution of returns, explains different implied volatility smile and skew patterns observed in different option markets—through the degree of symmetry amongst jump sizes—, and is consistent with incomplete markets (and non-redundant option contracts). 4.1. Additional assumption Assumption 3 is now restricted to the whole class of exponential-L´evy processes: Assumption 8. Hereafter, the time-t underlying asset price is defined as St = St0 eYt ,

(19)

where (Yt )t≥t0 is a L´evy process (with Yt0 = 0) whose law is determined, under the martingale measure Q, by the characteristic exponent Ψ : C0 → C such that, for all θ ∈ C0 :=      w ∈ C : EQ eRe(w)Yt Ft0 < ∞ , EQ eθYt Ft0 = e(t−t0 )Ψ(θ) , with Z   1 2 2 eθz − 1 − θz11{|z|≤1} ν (dz) , Ψ (θ) = aθ + σ θ + 2 R

where a and σ(≥ 0) are real constants, and ν is a positive Radon measure on R\ {0} satisfying
R 2 ν (dz) < ∞. R 1∧z Therefore, and to ensure that the process e−(r−q)(t−t0 ) St Z

|z|≥1

ez ν (dz) < ∞




t≥t0

is a Q-martingale, the restrictions

and Ψ (1) = r − q must be imposed, i.e. the following no-arbitrage condition arises: Z
1 2 a=r−q− σ − ez − 1 − z11{|z|≤1} ν (dz) . 2 R

(20)

(21)

Note that Assumptions 2, 4 and 8 still nest, as particular cases, most of the single-factor and time-homogeneous diffusion models (such as the celebrated GBM), (finite activity) jump-diffusion 14

models, and (infinite activity) pure jump processes already proposed in the literature. Moreover, it is easy to prove the existence, uniqueness, monotonicity and continuity of the optimal exercise boundary under the class of exponential-L´evy processes adopted—please see Appendix D of the supplementary file to this paper. 4.2. Put-call duality for double knock-out options To further illustrate the relevance of Propositions 1 and 2, and based on Proposition 3 and Fajardo and Mordecki (2006, Lemma 1), this subsection offers a novel put-call duality relation for American-style double knock-out options and under any exponential-L´evy process. We borrow the put-call duality label from Fajardo and Mordecki (2006) to distinguish the underlying more general concept from the put-call symmetry in the sense of Carr and Chesney (1996) or Carr and Lee (2009). The put-call duality holds whenever the price of a put option can be recovered from the price of a call option (and vice versa) through a suitable change in its arguments (e.g. the spot price and the strike). Such relations are important for practitioners since numerical instabilities eventually found when pricing one option contract (e.g. a call) can be often overcome by valuing its counterpart instead (e.g. a put option). For instance, Zhang and Oosterlee (2012) note that the COS approximation is accurate for put options with early-exercise features—and as long as the transition density function of the underlying asset price is smooth enough—but convergence is not guaranteed for call options due to its unbounded payoff. Hence, Zhang and Oosterlee (2012) adapt the COS approximation for Bermudan-style calls by applying put-call parity and duality relations to the Fourier cosine coefficients of European-style options. Alternatively, our results allow the duality relation to be applied directly to Bermudan-style knock-out calls. Following Schroder (1999) or Detemple (2001), our put-call duality relations will arise through ¯ that takes as a change from the risk-neutral measure Q to the equivalent martingale measure Q
numeraire of the economy the cum-dividend asset price process eq(t−t0 ) St t≥t0 . For this purpose,

the following fundamental result is borrowed from Fajardo and Mordecki (2006, Lemma 1) that specializes Schroder (1999, Corollary 1) to the context of L´evy processes. Lemma 1. Let St = St0 eYt , where (Yt )t≥t0 is a L´evy process whose law is determined by the characteristic exponent Ψ : C0 → C under the risk-neutral measure Q. For any stopping time 15

τ ∈ F [t0 , T ] and constant K ∈ R+ ,

i i h h
+ EQ e−r(τ −t0 ) (Sτ − K)+ Ft0 = EQ¯ e−q(τ −t0 ) St0 − S¯τ Ft0 ,

where the dual asset price process S¯t

and Y¯t ≡ −Yt




t≥t0




t≥t0

(22)

is defined as

KSt0 ¯ S¯t := = KeYt , St

(23)

is another L´evy process whose law is determined by the characteristic exponent

¯ such that, for any z, 1 − z ∈ C0 , ¯ : C0 → C under the equivalent martingale measure Q, Ψ ¯ (z) = Ψ (1 − z) − Ψ (1) . Ψ

(24)

Proof. Please see Fajardo and Mordecki (2006, Lemma 1). Based on Lemma 1, Fajardo and Mordecki (2006) have already obtained put-call duality relations for European and American-style standard options under exponential-L´evy processes. Next corollary extends the literature by offering a put-call duality relation for double knock-out American-style options that follows easily from Proposition 3 and Fajardo and Mordecki (2006, Lemma 1). For this purpose, each option price will be written as an explicit function also of the risk-free interest rate r, of the dividend yield q, and of the characteristic exponent associated to the underlying exponential-L´evy process. Corollary 1. Under Assumptions 1, 2, 4 and 8, the time-t0 value of the American-style double knock-out call option defined in equation (11) is equal to   ko(−1) Vtko S , K, L, U, T, r, q, E ; −1; Ψ t0 0   St0 K KSt0 KSt0 ko ¯ , = Vt0 K, St0 , , , T, q, r, ko(−1) ; 1; Ψ U L E

(25)

¯ are defined in Lemma 1. where the characteristic exponents Ψ and Ψ Proof. Please see Appendix B. Remark 4. For single-barrier options, taking the limit of equation (25) as U → +∞, and using Proposition 2, then Vtdo 0



St0 , K, L, T, r, q, E

do(−1)

   KSt0 St0 K uo ¯ ; −1; Ψ = Vt0 K, St0 , , T, q, r, do(−1) ; 1; Ψ , L E 16

(26)

as already shown, for instance, by Gao et al. (2000, Theorem 3) or Detemple (2006, Proposition 50) under the GBM assumption. Likewise, and following the same steps as in the proof of Corollary 1, it is also possible to show that Vtuo 0



St0 , K, U, T, r, q, E

uo(−1)

   KSt0 St0 K do ¯ ; −1; Ψ = Vt0 K, St0 , , T, q, r, uo(−1) ; 1; Ψ , U E

(27)

as already derived by Detemple (2001, Corollary 8) in a diffusion setting.

5. Numerical results This section illustrates the usefulness of the novel representations proposed in Propositions 1 and 2. The numerical analysis is framed under the jump-diffusion model of Kou (2002) and under the Variance Gamma (hereafter, VG) pure jump model of Madan et al. (1998). This way both finite and infinite activity L´evy processes are considered. 5.1. Results under the Kou (2002) model Table 1 prices American-style up-and-out and double knock-out put options under the Kou (2002) model. Under this L´evy process, the Q-measured characteristic exponent equals   σ2 σ2 Ψ (z; r, q, σ, λ, α1 , α2 , p) = r − q − − λ (g (1) − 1) z + z 2 + λ (g (z) − 1) , 2 2 for z ∈ C0 , where λ =

R

Rν

(dz) is the intensity of the Poisson jump counter, g (z) :=

pα1 α1 −z

(28)

2 + (1−p)α α2 +z

is the moment generating function of the jump size, p (≥ 0) is the (risk-neutral) probability of an upward jump, and α1 > 1 (resp., α2 > 0) is the inverse of the mean upward (resp., downward) jump size. Note that the arguments of the characteristic exponent on the left-hand side of equation (28) have been augmented to highlight its dependency on the option pricing model parameters. The option parameters are borrowed from Kou and Wang (2004, Table 1), yielding a total of 24 single and double barrier contracts: St0 = 100, r = 5%, σ = 20%, T − t0 = 0.25 years, p = 0.6, q ∈ {0%, 7%}, λ ∈ {3, 7}, and α1 , α2 ∈ {25, 50}. The up barrier is always set at U = 115, but the strike price and the lower barrier level vary amongst the three different panels (A, B and C) contained in Table 1 that correspond to the three cases shown in the proof of Proposition 1: K = 110 and L = 80 for Panel A; K = 90 and L = 85 for Panel B; and K = 100 and L = 90 for Panel C. 17

Table 1: American-style knock-out put options under the jump-diffusion model of Kou (2002).

#

q

λ

α1

1 2 3 4 5 6 7 8

0 0 0 0 0 0 0 0

3 3 3 3 7 7 7 7

25 25 50 50 25 25 50 50

9 10 11 12 13 14 15 16

0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07

3 3 3 3 7 7 7 7

25 25 50 50 25 25 50 50

17 0 3 18 0 3 19 0 3 20 0 3 21 0 7 22 0 7 23 0 7 24 0 7 CPU (seconds)

25 25 50 50 25 25 50 50

Double knock-out puts FST COS COS1 COS2 Panel A (K = 110, L = 80 and U = 115) 25 10.391 10.391 10.391 10.391 10.391 50 10.358 10.358 10.358 10.358 10.358 25 10.286 10.286 10.286 10.286 10.286 50 10.255 10.255 10.255 10.255 10.255 25 10.649 10.649 10.649 10.649 10.649 50 10.568 10.568 10.568 10.568 10.568 25 10.392 10.392 10.392 10.392 10.392 50 10.315 10.315 10.315 10.315 10.315 Panel B (K = 90, L = 85 and U = 115) 25 1.016 1.016 0.915 0.915 0.915 50 0.920 0.920 0.830 0.831 0.831 25 0.934 0.934 0.852 0.853 0.853 50 0.837 0.837 0.766 0.766 0.766 25 1.337 1.337 1.167 1.168 1.168 50 1.126 1.126 0.987 0.987 0.987 25 1.141 1.141 1.027 1.028 1.028 50 0.924 0.924 0.836 0.837 0.837 Panel C (K = 100, L = 90 and U = 115) 25 3.850 3.850 3.827 3.827 3.827 50 3.741 3.741 3.722 3.722 3.722 25 3.693 3.693 3.681 3.681 3.681 50 3.580 3.580 3.570 3.570 3.570 25 4.316 4.316 4.253 4.254 4.254 50 4.089 4.089 4.038 4.039 4.039 25 3.970 3.970 3.946 3.946 3.946 50 3.719 3.719 3.702 3.703 3.703 3,264 15,040 3,253 23,094 15,040 α2

Up and out puts FST COS

This table reports prices of American-style knock-out put options under the Kou (2002) model, with St0 = 100, r = 5%, σ = 20%, T − t0 = 0.25 years, and p = 0.6. Columns 1 to 5 show the contract number, the dividend yield, the jump intensity, and the inverse of the means from both exponential random variables, respectively. The 24 option contracts are divided into three panels representing the tree cases outlined in the proof of Proposition 1. The sixth and eighth columns implement the Fourier space time-stepping (FST) algorithm of Jackson et al. (2008) for American-style up-and-out and double knock-out puts, respectively, using 216 space and time steps, and a maximum log asset return of 7.5. The seventh column values American-style up-and-out put options through the Fourier-cosine (COS) approximation of Fang and Oosterlee (2009), using the Fourier-cosine series coefficients given by equations (C.4) and (C.5), 215 time steps and Fourier series terms, and a range of integration defined through the first four cumulants of the log-rate of return. American-style double knock-out puts are also priced in the last two columns through the COS approximation: using the Fourier-cosine series coefficients given by equations (C.1) and (C.3), in the penultimate column (“COS1”); and by changing, in the last column (“COS2”) and according to Proposition 1, the early exercise boundary obtained—in the seventh column—for the up-and-out puts. The last line contains the running times (in seconds) for the whole set of contracts.

To benchmark the accuracy of our results, not one, but two different valuation approaches are adopted and adapted to price knock-out options: the FST method of Jackson et al. (2008) 18

and the COS approximation proposed by Fang and Oosterlee (2009). Both pricing methods are implemented by running Matlab (R2019a) programs on an Intel Core i7-8700K 3.7 GHz processor. The main (and clever) idea of the FST method is to take the Fourier transform of the partial integro-differential equation satisfied by the option price, yielding a simpler linear system of ordinary differential equations, and thus avoiding the need to deal with its non-local term. Nevertheless, and as described by Jackson et al. (2008, Figure 1), for American-style or barrier options, it will be necessary to switch between the real and the Fourier space: the early exercise and barrier conditions are imposed in the real space but the time-stepping is performed in the Fourier space. For this purpose, the time to maturity of the option contract, the asset price domain, and the frequency domain are partitioned into a finite mesh of points. To adapt the FST method for the valuation of the American-style knock-out options under analysis, and using the Jackson et al. (2008) notation, it is only necessary to replace Jackson et al. (2008, Equations (14) or (25)) by the following boundary constraint in the time-stepping algorithm:   
v m−1 = max HL,U FFT−1 FFT [v m ] · eΨ∆t , v M ,

(29)

where v m and v M are N -dimensional vectors of option values at time m∆t (with m = 0, . . . , M ) and at the maturity date, respectively, ∆t is the dimension (in years) of each time step, Ψ is the characteristic exponent of the underlying L´evy process, FFT [v] and FFT−1 [v] are, respectively, the discrete direct and inverse Fourier transforms of v, · denotes the inner product in RN , HL,U (v) := v · 11

ln



L St 0



≤ln



S St 0



≤ln



U St 0

 ,

S is the N -dimensional vector of admissible asset

prices, and the maximum and indicator functions are taken componentwise.2 Based on the characteristic exponent (28), the sixth and eighth columns of Table 1 implement

the FST algorithm for American-style up-and-out (with L = 0) and double knock-out puts, respectively. And, even though Jackson et al. (2008) claim a convergence rate of 2 in space and 1 in time, we use an equal (and very large) number of space and time steps, as accuracy (and not CPU time) is our only concern: N = M = 216 (with a maximum log asset return of 7.5). The COS approximation is proposed by Fang and Oosterlee (2008) for the valuation of Europeanstyle options, and later extended by Fang and Oosterlee (2009) for the pricing of early-exercisable 2

Equation

(29)

is

the

only

modification

we

make

to

the

Matlab

code

available

at

https://sourceforge.net/projects/fst-framework/files/ and provided by Vladimir Surkov; anyway, all Matlab codes used in this paper can be found at http://home.iscte-iul.pt/˜jpvn/weblinks/CodesADKO.zip.

19

or discretely-monitored barrier options. This Fourier-cosine approximation replaces the density function of the log-ratio between the asset price and the strike price by its Fourier-cosine series expansion, and truncates both the integration range (over the density function) and the number of terms in the series expansion. Like the FST method, the COS approximation is a very general pricing approach—applicable to the whole class of exponential-L´evy asset pricing models under analysis—that only requires the knowledge of the underlying asset price characteristic function for the efficient computation of the Fourier series coefficients. To adapt the COS approximation to the valuation of American-style knock-out options, and as shown in Appendix C, it is only necessary to change the formulae presented by Fang and Oosterlee (2009, Equations (16) and (17)) for the computation of the Fourier-cosine series coefficients of the option price. Hence, we use the Matlab code from Kienitz and Wetterau (2012)—distributed via the MathWorks file-exchange—with only two changes: the new formulae for the Fourier-cosine series coefficients presented in Appendix C; and the use of the “newtsol” procedure of Kelley (2003) to determine the early exercise points—instead of the function xstar.m. Based on equations (C.4) and (C.5), the seventh column of Table 1 values American-style up-and-out put options through the COS approximation. Using exactly the same algorithm but modifying the early exercise boundary obtained for the up-and-out puts according to Proposition 1, the last column of Table 1 (labeled as “COS2”) prices the corresponding double knock-out puts. Finally, and to check the accuracy of the results obtained, the penultimate column of Table 1 (labeled as “COS1”) reprices the same double knock-out puts through equations (C.1) and (C.3), yielding exactly the same results. Note that, instead of using the 4-point Richardson extrapolation scheme proposed by Fang and Oosterlee (2009, Equation (76)), our COS approximations are implemented with a large number of time steps (M = 215 ) and of Fourier series terms (N = 215 ), as our main concern is the accuracy of the results obtained and not the efficiency of the valuation methods adopted; in other words, we approximate American-style options through Bermudan options with a large enough set of exercise dates. The range of integration is defined through the first four cumulants of the log-rate of return, which are obtained from Fang and Oosterlee (2009, Table 9).

20

21

0 0 0 0 0 0 0 0 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0.07 0 0 0 0 0 0 0 0

r

q 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05

σ 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2 0.2

λ 3.029 3.051 2.991 3.013 7.067 7.120 6.978 7.031 3.029 3.051 2.991 3.013 7.067 7.120 6.978 7.031 3.029 3.051 2.991 3.013 7.067 7.120 6.978 7.031

α1 26 51 26 51 26 51 26 51 26 51 26 51 26 51 26 51 26 51 26 51 26 51 26 51

α2 24 24 49 49 24 24 49 49 24 24 49 49 24 24 49 49 24 24 49 49 24 24 49 49

p 0.381 0.386 0.386 0.390 0.381 0.386 0.386 0.390 0.381 0.386 0.386 0.390 0.381 0.386 0.386 0.390 0.381 0.386 0.386 0.390 0.381 0.386 0.386 0.390

L 95.652 95.652 95.652 95.652 95.652 95.652 95.652 95.652 78.261 78.261 78.261 78.261 78.261 78.261 78.261 78.261 86.957 86.957 86.957 86.957 86.957 86.957 86.957 86.957

U 137.5 137.5 137.5 137.5 137.5 137.5 137.5 137.5 105.882 105.882 105.882 105.882 105.882 105.882 105.882 105.882 111.111 111.111 111.111 111.111 111.111 111.111 111.111 111.111

Down-and-out calls FST COS 10.391 10.391 10.358 10.358 10.286 10.286 10.255 10.255 10.649 10.649 10.568 10.568 10.392 10.392 10.315 10.315 1.016 1.016 0.920 0.920 0.934 0.934 0.837 0.837 1.337 1.337 1.126 1.126 1.141 1.141 0.924 0.924 3.850 3.850 3.741 3.741 3.693 3.693 3.580 3.580 4.316 4.316 4.089 4.089 3.970 3.970 3.719 3.719 3,285 16,104

Double knock-out calls FST COS 10.391 10.391 10.358 10.358 10.286 10.286 10.255 10.255 10.649 10.649 10.568 10.568 10.392 10.392 10.315 10.315 0.915 0.915 0.830 0.831 0.852 0.853 0.766 0.766 1.167 1.168 0.987 0.987 1.027 1.028 0.836 0.837 3.827 3.827 3.722 3.722 3.681 3.681 3.570 3.570 4.253 4.254 4.038 4.039 3.946 3.946 3.702 3.703 3,248 16,104

This table reports prices of American-style down-and-out and double knock-out call options under the Kou (2002) model. The first twelve columns show, respectively, the contract number, the spot price, the strike price, the interest rate, the dividend yield, the instantaneous volatility, the intensity of the Poisson jump counter, the inverse of the mean upward and downward jump sizes, the risk-neutral probability of an upward jump, and both the lower and upper barrier levels. All the parameter constellations adopted correspond to the “dual market” associated to the “L´evy market” described in Table 1, and are obtained from Table 1 using equations (25) and (30). Columns thirteenth and fifteenth price down-and-out and double knock-out calls through the Fourier space time-stepping (FST) algorithm of Jackson et al. (2008), using 216 space and time steps, and a maximum log asset return of 7.5. The Fourier-cosine (COS) approximation of Fang and Oosterlee (2009) is implemented in the fourteenth column—for down-and-out calls—and in the last column—for double knock-out calls—using 215 time steps and Fourier series terms, a range of integration defined through the first four cumulants of the log-rate of return, and the Fourier-cosine series coefficients given by equations (C.6) and (C.7) or (C.1) and (C.3), respectively. The last line contains the running times (in seconds) for the whole set of contracts.

# St0 K 1 110 100 2 110 100 3 110 100 4 110 100 5 110 100 6 110 100 7 110 100 8 110 100 9 90 100 10 90 100 11 90 100 12 90 100 13 90 100 14 90 100 15 90 100 16 90 100 17 100 100 18 100 100 19 100 100 20 100 100 21 100 100 22 100 100 23 100 100 24 100 100 CPU (seconds)

Table 2: American-style knock-out call options under the jump-diffusion model of Kou (2002).

As expected, the prices of single and double barrier knock-out options are exactly the same in Panel A of Table 1. The last five columns are exactly equal (in Panel A) because the lower barrier level (L = 80) is always below the early exercise boundary of the up-and-out put. For the other panels (B and C), the price of the double knock-out is strictly lower than the price of the corresponding single knock-out option as the early exercise boundary of the two barrier options is no longer the same. Note also that both valuation methods employed (FST and COS schemes) are extremely accurate as the maximum absolute difference between them, for all the contracts in the three panels, is equal to only 0.1 cents of a dollar. In Table 2, the up-and-out and the double knock-out puts of Table 1 are repriced through the put-call duality relations established in equation (26) and Corollary 1, respectively. Hence, columns 2 to 5 and 11 to 12 follow directly from the parameter constellations used in Table 1 and from equation (25). Columns 6 to 10 are defined through Lemma 1, since equation (24) implies that, under the Kou (2002) jump-diffusion model,   λγ (1 − p) α (α − 1) 2 1 ¯ (z; r, q, σ, λ, α1 , α2 , p) = Ψ z; q, r, σ, Ψ , α2 + 1, α1 − 1, , (α1 − 1) (α2 + 1) γ (30) where γ := (1 − p) α2 (α1 − 1) + pα1 (α2 + 1). In summary, the first twelve columns of Table 2 describe the “dual market” associated to the “L´evy market” described in Table 1. Consequently, and as expected, the values of the down-and-out calls—in columns thirteenth and fourteenth of Table 2—are exactly equal to the prices of the up-and-out puts from Table 1, and the prices of the double knock-out calls—in the last two columns of Table 2—coincide with the values of the double knock-out puts from Table 1. Again, all the contracts are valued using both the FST method (with N = M = 216 ) and the COS approximation—with N = M = 215 and based on equations (C.6) and (C.7), for down-and-out calls, or on equations (C.1) and (C.3), for double knock-out calls. 5.2. Results under the VG model The numerical analysis is closed with an example of a jump model with infinite activity: the VG model of Madan et al. (1998) that extends the symmetric VG process of Madan and Seneta (1990), and is obtained by evaluating a Brownian motion (with constant drift θ ∈ R and constant

volatility σ ∈ R+ ) at a random time given by a gamma process with unit mean rate and variance

rate ω ∈ R+ . Under the VG model, the Q-measured characteristic exponent (of the log returns on 22

the underlying asset) equals   1  1  ω  2ω z − ln 1 − θωz − σ 2 z 2 , Ψ (z; r, q, ω, θ, σ) = r − q + ln 1 − θω − σ ω 2 ω 2

(31)

for z ∈ C0 —see, for instance, Madan et al. (1998, Equation (7))—and, again, we have augmented the arguments’ list of the characteristic exponent on the left-hand side of equation (31) to include all the option pricing model parameters. Table 3 prices American-style double knock-out call and put options under the VG model, and is divided into two panels. Panel A considers 12 calls and puts under the following parameter configurations, which are based on the setup described in Jackson et al. (2008, Table 10): St0 ∈ {95, 100, 105}, K = 100, r = 5.49%, q = 1.1%, (T − t0 ) ∈ {0.25, 0.5, 0.75, 1}, L = 80, U = 120, σ = 19.071%, θ = −0.28113, and ω = 49.083%. Panel B contains 12 additional put and call options under the corresponding dual (or auxiliary) market associated to the L´evy market specified in Panel A. For this purpose, the first ten columns in Panel B of Table 3 were obtained from the put-call duality relation offered by Lemma 1 and Corollary 1, as equation (24) implies that, under the VG model, θ + σ2 σ ¯ (z; r, q, ω, θ, σ) = Ψ z; q, r, ω, − Ψ ,p 1 − θω − σ 2 ω2 1 − θω − σ 2 ω2

!

.

(32)

Therefore, we should observe identical prices between the calls (resp., puts) of Panel A and the puts (resp., calls) of Panel B. As before, the calls and puts are priced via both the FST and the COS approaches in the last four columns of Table 3. To enhance the accuracy of the option prices, the FST method is implemented using an extremely large number of space and time steps: N = M = 218 (and a maximum log asset return of 7.5). Nevertheless, we observe significative differences between the eleventh (resp., twelfth) column of Panel A and the twelfth (resp., eleventh) column of Panel B: the mean percentage absolute differences between calls (resp., puts) from Panel A and puts (resp., calls) from Panel B is equal to 1.77%. For instance, put #12 is worth 5.760 but the dual call #24 is (wrongly) priced well above (6.234), which contradicts our Corollary 1.

23

24

95 95 95 95 100 100 100 100 105 105 105 105

1 2 3 4 5 6 7 8 9 10 11 12

100 100 100 100 100 100 100 100 100 100 100 100

K

0.0549 0.0549 0.0549 0.0549 0.0549 0.0549 0.0549 0.0549 0.0549 0.0549 0.0549 0.0549

r

σ

θ

L

U

T − t0

Double knock-out options FST COS Call Put Call Put

0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011 0.011

Panel A: Option features in the original L´ evy market 0.19071 -0.28113 80 120 0.25 1.864 5.980 1.855 5.979 0.19071 -0.28113 80 120 0.5 4.623 7.923 4.602 7.921 0.19071 -0.28113 80 120 0.75 6.645 9.198 6.588 9.196 0.19071 -0.28113 80 120 1 8.064 10.034 7.971 10.033 0.19071 -0.28113 80 120 0.25 5.019 4.059 5.001 4.058 0.19071 -0.28113 80 120 0.5 7.735 5.950 7.702 5.949 0.19071 -0.28113 80 120 0.75 9.520 7.088 9.461 7.087 0.19071 -0.28113 80 120 1 10.655 7.784 10.597 7.784 0.19071 -0.28113 80 120 0.25 8.889 2.870 8.833 2.869 0.19071 -0.28113 80 120 0.5 11.373 4.414 11.105 4.413 0.19071 -0.28113 80 120 0.75 12.780 5.251 12.399 5.252 0.19071 -0.28113 80 120 1 13.603 5.760 13.232 5.762 Panel B: Option features in the auxiliary or dual market 0.0549 0.179479 0.216782 79.16667 118.75 0.25 5.989 1.855 5.979 1.855 0.0549 0.179479 0.216782 79.16667 118.75 0.5 7.947 4.604 7.921 4.603 0.0549 0.179479 0.216782 79.16667 118.75 0.75 9.276 6.592 9.196 6.588 0.0549 0.179479 0.216782 79.16667 118.75 1 10.176 7.984 10.033 7.971 0.0549 0.179479 0.216782 83.33333 125 0.25 4.074 5.004 4.058 5.004 0.0549 0.179479 0.216782 83.33333 125 0.5 5.984 7.705 5.949 7.703 0.0549 0.179479 0.216782 83.33333 125 0.75 7.159 9.467 7.087 9.458 0.0549 0.179479 0.216782 83.33333 125 1 7.870 10.634 7.784 10.598 0.0549 0.179479 0.216782 87.5 131.25 0.25 2.926 8.836 2.869 8.833 0.0549 0.179479 0.216782 87.5 131.25 0.5 4.699 11.109 4.413 11.106 0.0549 0.179479 0.216782 87.5 131.25 0.75 5.674 12.419 5.252 12.400 0.0549 0.179479 0.216782 87.5 131.25 1 6.234 13.329 5.762 13.233 51,694 18,902 23,198 duality errors 1.770% 0.007% call errors (FST vs COS) 1.850% put errors (FST vs COS) 0.078%

q

This table reports prices of American-style double knock-out call and put options under the VG model of Madan et al. (1998). The first ten columns show, respectively, the contract number, the spot price, the strike price, the interest rate, the dividend yield, the instantaneous volatility, the drift of the time-changed Brownian motion, the lower barrier level, the upper barrier level, and the time to maturity. The variance rate of the gamma process is set at ω = 0.49083. The parameter constellations of the first twelve calls and puts (in Panel A) are based on the “L´evy market” adopted by Jackson et al. (2008, Table 10), whereas the option contracts in Panel B correspond to the “dual market” defined through equations (25) and (32). Columns eleventh and twelfth price double knock-out calls and puts, respectively, through the Fourier space time-stepping (FST) algorithm of Jackson et al. (2008), using 218 space and time steps, and a maximum log asset return of 7.5. The Fourier-cosine (COS) approximation of Fang and Oosterlee (2009) is implemented in the last two columns using 210 time steps and 217 Fourier series terms, a range of integration defined through the first four cumulants of the log-rate of return, and the Fourier-cosine series coefficients given by equations (C.1) and (C.3). The last four lines report CPU times and average absolute percentage errors between symmetric contracts and different pricing schemes.

13 100 95 0.011 14 100 95 0.011 15 100 95 0.011 16 100 95 0.011 17 100 100 0.011 18 100 100 0.011 19 100 100 0.011 20 100 100 0.011 21 100 105 0.011 22 100 105 0.011 23 100 105 0.011 24 100 105 0.011 CPU (seconds) Mean absolute percentage Mean absolute percentage Mean absolute percentage

St0

#

Table 3: American-style double knock-out call and put options under the VG model of Madan et al. (1998).

To verify whether the mispricing arises from the calls—as suspected, given its unbounded payoff—or the puts, all the 24 calls and puts were repriced through the COS approximation, which is implemented with a smaller number of time steps (M = 210 ) and a much larger number of Fourier series terms (N = 217 ), in order to ensure stability (that is not achieved, for the parameter set used in Table 3, with, for instance, M = 215 ).3 The range of integration is still defined through the first four cumulants of the log-rate of return, which are retrieved from Fang and Oosterlee (2009, Table 9), and the Fourier-cosine series coefficients are computed through equations (C.1) and (C.3). Under the COS approach, the antepenultimate line of Table 3 shows that the put-call duality prescribed in Corollary 1 is now satisfied: the average percentage and absolute differences between calls (resp., puts) from Panel A and puts (resp., calls) from Panel B is equal to only 0.7 basis points—in opposition to the 177 basis points error observed under the FST approach. Columns 12 and 14 of Table 3 show that both valuation methods—FST and COS—offer roughly the same prices for the double knock-out puts: the mean absolute percentage difference between the twelfth and fourteenth columns of Table 3 is equal to 7.8 basis points. In opposition, columns 11 and 13 show that the prices of most of the 24 call options differ significantly between the FST and the COS approaches: the average absolute percentage difference between the eleventh and thirteenth columns of Table 3 is equal to 1.85%. Therefore, we may conclude that the nonobservance—under the FST method—of the put-call duality relation offered in Corollary 1 is mainly due to the lack of accuracy in pricing call options. In these cases, Corollary 1 is useful, since it can be used to price (accurately and more efficiently) such calls through the corresponding dual put options.

6. Expansion and abandonment options Even though the previous analysis has been cast only in the context of financial option contracts, this section provides an application of our main results—Propositions 1 and 2—in operations research; more specifically, to firm investment and divestment decisions made under uncertainty. Capital budgeting under uncertainty gives rise to several strategic (real) options—such as waiting, expansion, or abandonment options—that may enhance the value of the project under analysis. It 3

As noted by Fang and Oosterlee (2009, Remark 4.1 and Appendix A), and shown in Appendix F of the

supplementary file to this paper, accuracy may deteriorate for too large values of M (with N fixed) because the transition probability density function between (too close) exercise dates tends to become highly peaked.

25

is well known that expansion and abandonment options can be valued as calls or puts, respectively, on the present value of the cash flows from the investment project and with a strike price equal to the additional investment or to the recovery value (i.e. divestment proceeds), respectively. These options are usually of American-style—since the expansion or divestment decisions can be formulated during some time-horizon—and can be valued using different stochastic processes for the underlying real asset. Under a complete market and “single-firm” setting—in the sense of Trigeorgis and Tsekrekos (2018, Page 12)—, the most common assumption is to model the present value of the project cash flows through a GBM process. One notable exception is the arithmetic Brownian motion (hereafter, ABM) assumption adopted by Alexander et al. (2012) that possesses two advantages over the usual GBM setting for valuing isolated projects or divisions of a single firm. First, the ABM process domain is the whole real line, and, hence, admits negative project values; and, even though the overall firm market value must be nonnegative, the value of an isolated project or division can always, in practice, become negative. Second, the ABM models free cash flows (like dividends from a stock) as proportions of the project value, and, therefore, takes into account that projects with a negative value are associated to free cash inflows (i.e. to cash infusions—through loans or equity—required to maintain operations, at least in the short run). However, and simply based on economic and physical arguments, there are usually upper (U ) and lower (L) bounds on the project or division admissible values. For instance, sales revenues are naturally bounded above by both the overall market size and by the firm production capacity; or, for example, a maximum cap level in feed-in tariffs of renewable energy projects might be imposed by governments and regulators—as argued by Couture and Gagnon (2010). Similarly, (negative) cash flows are often bounded below by (finite) fixed costs, and additional real options applications with lower knock-out barriers may be found in Engelen et al. (2016) or Liu et al. (2018), as well as in the references contained therein. Therefore, the usual approach of treating these real options as (American-style) plain-vanilla contracts (with unbounded positive payoffs) is likely to overstate their values by assuming exercise opportunities over unfeasible levels of the underlying real asset. Alternatively, some real options should be priced over the admissible domain [L, U ] for the underlying project value, not as a plain-vanilla option, but rather as a double knock-out American-style contract. 26

More specifically, an expansion option shall be treated as an American-style double knock-out call that is knocked-out (with a zero payoff) if the lowest possible (and unrecoverable) scenario for the underlying project value (L) is reached or, according to Gao et al. (2000, Footnote 15), is immediately exercised (with a positive payoff) if the best possible project value (U ) is ever attained. Likewise, an abandonment option corresponds to an American-style double knock-out put that is knocked-out if the highest admissible project value (U ) is reached or is exercised at the worst admissible scenario for the project value (L). Mathematically, the alternative valuation approach now proposed is equivalent to stopping the underlying real asset price process at the upper or lower levels of its admissible domain [L, U ], which precludes (upward) biases to the real option value caused by unrealistic (i.e. too high or too low) underlying project levels. To illustrate the analysis, Example 2 of Alexander et al. (2012, Section 4.1) is now reanalyzed: A franchise with an up-front investment of $10 million yields expected net cash flows of $2 million per year (that are modelled as a perpetuity and with a standard deviation of $1.824 million). Using $2 a cost of capital of 20% per year, the static net present value is zero (i.e. -$10 million + 0.2 million),

and, therefore, the project is merely marginal. Nevertheless, over the first two years of operations, not only the uncertainty about the long run level of cash flows should be resolved but also the abandonment of the franchise allows the recovery of $5 million from the initial capital cost; and, as stated by Keswani and Shackleton (2006), managerial consideration of exit options at the time of project initiation can add value. Assuming that the underlying franchise value (S) follows, under the risk-neutral measure Q, the stochastic differential equation dSt = (r − q) St dt + βdWt ,

(33)

with r = 8% (riskless interest rate), q = ln 1.2 (constant dividend rate), and β = $8.6 million, Alexander et al. (2012) price the abandonment option as a plain-vanilla American-style put on a real asset with initial value of $10 million (i.e.

$2 0.2

million), with a strike equal to the recovery value

of $5 million, and with a time to maturity of two years. Panel A of Table 4 implements (in the columns 2-5) the binomial tree of Alexander et al. (2012, Figure 3), for four different maximum time step size values (from 0.001 to 10−6 years), and the same abandonment option value is always obtained: around $2.725 million. 27

28

Binomial 0.0001 0.00001 0.000001

2.7248 0.09 -0.016%

2.7249 4.66 -0.011%

2.7249 460.87 -0.012%

2.7249 45,853.05 -0.012%

2.7251 0.61 -0.005%

0.02

2.7212 0.07 -0.028%

2.7218 5.97 -0.005%

2.7219 598.04 -0.001%

2.7220 59,823.94 0.001% 0.0029

2.7221 0.25 0.006%

2.7220 9.82 0.001%

2.6910 0.11 -0.101%

2.6929 5.98 -0.028%

2.6934 597.91 -0.009%

2.6936 59,817.30 -0.002% 0.0313

2.6937 0.28 0.002%

1.7031 0.06 1.303%

1.6885 5.96 0.435%

1.6835 597.86 0.139%

1.6819 59,812.53 0.045% 1.0429

1.6808 0.18 -0.024%

1.6812 7.64 0.000%

2.6937 8.45 0.001%

1.6812 61.67 0.000%

2.6937 66.10 0.000%

2.7220 71.42 0.001%

1.0440

1.6812 166.86

0.0315

2.6937 178.92

0.0032

2.7220 195.66

2.7252 8.40

0.0004

This table prices an abandonment option with a strike price of $5 million (in recoverable capital costs), with a time to maturity of two years, and on a franchise whose present value of future cash flows is equal to $10 million and is assumed to follow the stochastic process (33) with r = 8% (risk-free interest rate), dividend yield q = ln (1.2), and volatility β = $8.6 million. The abandonment option is priced using both the binomial method offered by Alexander et al. (2012, Figure 3) and the novel SHP scheme proposed, both implemented using different lengths for the maximum admissible size of each time step in which the option life is divided. All percentage errors are computed with respect to the most accurate SHP price that is computed using 5,000 time steps (of 0.0004 years each). The overpricing error measures the overstatement of the abandonment option value when adopting the standard American-style put valuation approach that wrongly considers unfeasible levels of the underlying project value.

Abandonment option value ($ millions) CPU (seconds) Percentage errors Overpricing error ($ millions)

Panel D: Double knock-out American-style put (L = −$2 million and U = $14 million)

Abandonment option value ($ millions) CPU (seconds) Percentage errors Overpricing error ($ millions)

Panel C: Double knock-out American-style put (L = −$5.002 million and U = $25.002 million)

Abandonment option value ($ millions) CPU (seconds) Percentage errors Overpricing error ($ millions)

2.7249 4.35 -0.010%

SHP 0.002 0.000667

2.7249 1.45 -0.011%

Panel B: Double knock-out American-style put (L = −$6.670 million and U = $33.337 million)

Abandonment option value ($ millions) CPU (seconds) Percentage errors

0.001

Panel A: Standard American-style put (L = −∞ and U = +∞)

Maximum time step length (in years):

Table 4: Abandonment options under the Alexander et al. (2012) setup.

The problem is that the $2.725 million estimate is almost surely overstated because it (wrongly) assumes that the franchise value can be arbitrarily positive or negative. To gauge the magnitude of such overstatement, three alternative domains are also tested in Table 4. Panel B considers a 90% confidence interval, using the 15% per year cost of capital for an established business of the same type: L (resp., U ) =

$2 0.15

∓ 1.645 ×

$1.824 0.15

= −$6.670 (resp., $33.337). Panel C recomputes

the confidence interval but using the 20% per year hurdle rate for the business in place: L (resp., U )=

$2 0.2

∓ 1.645 ×

$1.824 0.2

= −$5.002 (resp., $25.002). Finally, Panel D imposes a more realistic

upper bound U of $14 million (associated to an extreme internal rate of return of

14×20% 10

= 28%)

that would determine the definite continuation of the franchise as well as a lower bound L of −$2 (corresponding to an internal rate of return of

−2×20% 10

= −4%) that is considered too negative to

be sustainable or reversible in the long run, thus yielding an immediate abandonment decision. Columns 2-5 of Panels B-D reprice the abandonment option as a double knock-out Americanstyle put, and using a modified version of the binomial scheme proposed by Alexander et al. (2012, Figure 3).4 Based on the smallest maximum time step length (of 10−6 years), the $2.725 million estimate is, as expected, shown to be upward biased at about $2,900 (Panel B), $31,300 (panel C), and $1.043 million (Panel D), being the latter figure economically relevant. Note that, specially in Panel D, there are significant differences amongst the option values computed for different time step lengths (in the order of tens of thousands of dollars), which questions the accuracy of the binomial pricing method adopted. This should come as no surprise because binomial lattices can yield significantly biased estimates for barrier option prices, even if the number of time steps used is large—see, for instance, Boyle and Lau (1994). Unfortunately, the binomial tree of Alexander et al. (2012, Figure 3) recombines only in a transformed time scale, and, hence, the stretch parameter technique of Ritchken (1995) can not be applied in the original time scale (to ensure the coincidence of each barrier level with a layer of nodes). Moreover, the process (33) is not a L´evy process, and, therefore, the FST and COS valuation methods are not readily available as well. Fortunately, and as detailed in Appendix G of the supplementary file to this paper, Propositions 1 and 2 allow the SHP approach of Chung et al. (2013) to be extended from single barrier knock4

The only modification is the multiplication of the option’s intrinsic value by zero, if the underlying real asset

value S is outside the domain [L, U ].

29

out contracts to double barrier knock-out American-style options. For an American-style double barrier put (call, resp.), and while the early exercise boundary is above (below, resp.) the lower (upper, resp.) knock-out barrier, the static hedge portfolio is set exactly as in Chung et al. (2013) for a single barrier up-and-out put (down-and-out call, resp.). Otherwise, equations (12) and (16) imply that the early exercise boundary of the double barrier option must be set at the lower (upper, resp.) barrier level, and a smooth-pasting condition is no longer required. Hence, the valuation of double knock-out options can be even faster than the pricing of the corresponding single barrier contracts since the early exercise boundary becomes known after crossing the lower (for double knock-out puts) or the upper (for double knock-out calls) barrier level. Even though the process (33) can take negative values, and as shown in Appendix H of the supplementary file, Propositions 1 and 2 are still valid if (St )t∈T is defined over all the real line. Therefore, the last four columns of Table 4 reprice the abandonment put through the novel SHP approach, by dividing the (2 years) time to maturity by different numbers of time steps (100, 1,000, 3,000 and 5,000), and three conclusions emerge. First, the fastest SHP discretization (with only 100 time steps of 0.02 years each) is enough to obtain accurate results: the maximum percentage error (with respect to the finest discretization with 5,000 time steps) is equal to only -0.024% (in Panel D). Second, the binomial scheme of Alexander et al. (2012, Figure 3) converges to the SHP prices: the percentage errors decline as the time step length decreases. Third, and more important, even the most time consuming binomial scheme (with a maximum time step length of 10−6 years and a CPU time over 59,800 seconds) yields biased results: in Panel D, the percentage error associated to the binomial double knock-out put (and with respect to the 5,000 time steps SHP scheme) is equal to 0.045%. In summary, the previous results show that, in some circumstances, expansion or abandonment options should be valued, not as plain-vanilla, but as double knock-out American-style options, and, for this purpose, the proposed SHP scheme offers the best speed-accuracy trade-off (under single factor diffusion models).

7. Conclusions This paper provides five contributions to the existent literature on option pricing. First, and more importantly, a novel representation is proposed for the early exercise boundary of American30

style double knock-out options in terms of its “in-the-money” barrier level and as a function of the simpler optimal stopping boundary of the nested single barrier American-style option associated to the “out-of-the-money” barrier. Hence, the valuation of American-style double barrier options is reduced to the same complexity level as the one faced to price single barrier contracts, as long as the corresponding exercise boundary exists and is a monotonic and continuous function of time alone. The existence, uniqueness, continuity and monotonicity of such boundary is proved under the whole class of exponential-L´evy processes, which constitutes our second contribution. Third, and still under the same class of L´evy processes, a novel put-call duality relation is provided for American-style double knock-out options. Such new duality relation has enabled us to overcome several convergence issues when pricing American-style knock-out calls under the VG model. Fourth, and to illustrate the practical relevance of the novel representation offered for the early exercise boundary of double knock-out options, both the FST and the COS methods were adapted to the valuation of American-style double barrier knock-out options. As a final contribution, it was shown that the usual approach of treating expansion or abandonment options as (American-style) plain-vanilla contracts (with unbounded positive payoffs) can significantly overstate their value. To avoid such upward bias, and under a single-factor diffusion setup, a new SHP scheme is proposed to price such real options as American-style double knock-out calls or puts. Appendix A. Proof of Proposition 3 The hitting time definitions (1), (4), and (7) imply that τLU = τL ∧ τU .

(A.1)

Additionally, and for double barrier puts, equations (5), (7), (10) and (12) yield n o τeko(1) = inf t > t0 : St ≤ L ∨ E uo(1) (t) = τL ∧ τeuo(1) ,

(A.2)

because it is assumed that the double barrier option has not been knocked-out or exercised yet, i.e. St0 > L ∨ E uo(1) (t0 ). Consequently, equation (17) arises because equations (11), (A.1), and (A.2) can be combined into       uo(1) −r T ∧τL ∧τe ∧τU −t0 ko ko(1) Vt0 St0 , K, L, U, T, E ; 1 = EQ e G1 ST ∧τ 31

  F , uo(1) ∧τU t0 L ∧τe

uo(φ)

which is exactly equivalent to equation (6) when the first hitting time τe

uo(1)

is replaced by τL ∧τe

,

i.e. when the American-style up-and-out put is evaluated at the early exercise boundary t 7−→ L ∨ E uo(1) (t).

For the double barrier call (18), the proof is similar and is, therefore, omitted.

Appendix B. Proof of Corollary 1 Both sides of equation (22) can be understood as random maturity options. The left-hand side yields the time-t0 price of a call on the process (St )t≥t0 , with strike K, and in a “L´evy market”
where the process e−(r−q)(t−t0 ) St t≥t0 is a Q-martingale. The right-hand side corresponds to
a put on the dual process S¯t t≥t0 , with strike St0 , and in a “dual market” where the process
¯ e−(q−r)(t−t0 ) S¯t t≥t0 is a Q-martingale. That is, the roles of the argument pairs (St0 , K) and (r, q)

are interchanged in the two markets.

Therefore, equations (9) and (18) as well as Lemma 1 imply that   ko(−1) Vtko S , K, L, U, T, r, q, E ; −1; Ψ t0 0   do(−1) = Vtdo S , K, L, T, r, q, U ∧ E ; −1; Ψ t 0 0 h i = EQ e−r(T ∧¯τe ∧τL −t0 ) (ST ∧¯τe ∧τL − K)+ Ft0 h i
+ = EQ¯ e−q(T ∧¯τe ∧τL −t0 ) St0 − S¯T ∧¯τe ∧τL Ft0 ,

(B.1)

 where τL and τ¯e are given by equations (7) and (8), i.e. τ¯e = inf t > t0 : St ≥ U ∧ E do(−1) (t) .

Furthermore, and using equation (23), both hitting times can be rewritten in terms of S¯ (and not o n o n KSt0 KS . Hence, τL and τ¯e S), i.e. τL = inf t ≥ t0 : S¯t ≥ Lt0 , and τ¯e = inf t > t0 : S¯t ≤ U ∧E do(−1) (t)
can be also understood as the first passage times of the process S¯t t≥t0 through the up-barrier KSt0 L

and through the exercise boundary

equation (B.1) can be restated as

KSt0 , U ∧E do(−1) (t)

respectively, and Definition 1 implies that

  ko(−1) Vtko S , K, L, U, T, r, q, E ; −1; Ψ t 0 0   St0 K ¯ . ¯t , St , KSt0 , T, q, r, ; 1; Ψ = Vtuo S 0 0 0 L U ∧ E do(−1)

(B.2)

Since S¯t0 = K and St0 K St0 K St0 K = ∨ , U U ∧ E do(−1) E do(−1) 32

(B.3)

then equation (B.2) yields   ko(−1) , K, L, U, T, r, q, E ; −1; Ψ Vtko S t0 0   KSt0 St0 K St0 K uo ¯ = Vt0 K, St0 , , T, q, r, ∨ do(−1) ; 1; Ψ L U E   St0 K KSt0 St0 K St0 K ko ¯ = Vt0 K, St0 , , , T, q, r, ∨ do(−1) ; 1; Ψ , U L U E

(B.4)

where the second equality follows from equations (12) and (17). Finally, equation (25) arises from equation (B.4) after rewriting the early exercise boundary with the help of equations (16) and (B.3):

St0 K U

∨

St0 K E do(−1)

=

St0 K U ∧E do(−1)

=

St0 K . E ko(−1)

Appendix C. The COS method for American-style knock-out options Following Fang and Oosterlee (2009), a collection (t1 , . . . , tM ) of M exercise dates is considered, such that t0 < t1 < . . . < tM = T . We denote by w the log-ratio between the asset price and the strike price (K), by [a, b] the truncated range of integration—defined through the first four cumulants of w—, and represent by N the number of Fourier series terms used to approximate the

L U density of w. Similarly, we define hl := ln K , for the lower barrier level, and hu := ln K , for

the upper barrier.

To adapt the COS approximation of Fang and Oosterlee (2009) to the valuation of different types of early-exercisable barrier options, it is only necessary to redefine the formulae presented by Fang and Oosterlee (2009, Equations (16) and (17)) for the computation of the Fourier-cosine series coefficients of the option price. For instance, for American-style double knock-out options, and on the maturity date of the option contract, Definition 3 and Fang and Oosterlee (2009, Equation (8)) yield the following formula for the option’s Fourier-cosine series coefficients (at n = 0, . . . , N − 1):   Z b 2 w−a + ko(φ) w dw Vn (tM ) = [−φK (e − 1)] 11{hl ≤w≤hu } cos nπ b−a a b−a   Gn (a ∨ 0 ∨ h , b ∧ hu ; 1) ⇐= φ = −1 l = , (C.1)  G (a ∨ h , b ∧ 0 ∧ h ; −1) ⇐= φ = 1 n

u

l

where φ = −1 for a call, φ = 1 for a put, and the function Gn (·) is given by Fang and Oosterlee

(2009, Equation (22)). Before the maturity date of the option contract (i.e. for m = M − 1, . . . , 1), the time-tm double knock-out option value is given, not by Fang and Oosterlee (2009, Equation 33

(1)), but rather by
v (w, tm ) = max g (w, tm ) , c (w, tm ) 11{hl ≤w≤hu } ,

(C.2)

where c (w, tm ) and g (w, tm ) represent the time-tm continuation value and option payoff, respectively. Combining equation (C.2) with Fang and Oosterlee (2009, Equation (8)), and denoting by x∗m the time-tm early exercise point—i.e. the root of the equation v (x∗m , tm ) = 0—, the time-tm Fourier-cosine series coefficients (at n = 0, . . . , N − 1) for American-style double knock-out options are obtained: Vnko(φ) (tm )

  w−a = [−φK (e − 1)] 11{φw<φx∗m } cos nπ dw b−a a   Z b 2 w−a + c (w, tm ) 11{φw>φx∗m ,hl ≤w≤hu } cos nπ dw b−a a b−a   Gn (a ∨ x∗ , b; 1) + Cn (a ∨ h , b ∧ x∗ ∧ hu , tm ) ⇐= φ = −1 l m m , =  G (a, b ∧ x∗ ; −1) + C (a ∨ x∗ ∨ h , b ∧ h , t ) ⇐= φ = 1 2 b−a

Z

n

b

+

w

n

m

m

u

l

(C.3)

m

where the function Cn (·) is given by Fang and Oosterlee (2009, Equation (19)).

For American-style up-and-out options, the Fourier-cosine series coefficients follow from equations (C.1) and (C.3) by taking hl → −∞:   Gn (a ∨ 0, b ∧ hu ; 1) ⇐= φ = −1 , Vnuo(φ) (tM ) =  G (a, b ∧ 0 ∧ h ; −1) ⇐= φ = 1 n

and

  Gn (a ∨ x∗ , b; 1) + Cn (a, b ∧ x∗ ∧ hu , tm ) ⇐= φ = −1 m m uo(φ) Vn (tm ) = ,  G (a, b ∧ x∗ ; −1) + C (a ∨ x∗ , b ∧ h , t ) ⇐= φ = 1 n

(C.4)

u

n

m

u

m

(C.5)

m

for n = 0, . . . , N − 1, and where φ = −1 for an up-and-out call while φ = 1 for an up-and-out put. For American-style down-and-out options, the Fourier-cosine series coefficients also follow from equations (C.1) and (C.3) by letting hu → ∞:   Gn (a ∨ 0 ∨ h , b; 1) ⇐= φ = −1 l Vndo(φ) (tM ) = ,  G (a ∨ h , b ∧ 0; −1) ⇐= φ = 1 n

and

l

  Gn (a ∨ x∗ , b; 1) + Cn (a ∨ h , b ∧ x∗ , tm ) ⇐= φ = −1 l m m Vndo(φ) (tm ) = ,  G (a, b ∧ x∗ ; −1) + C (a ∨ x∗ ∨ h , b, t ) ⇐= φ = 1 n

(C.6)

n

m

m

l

(C.7)

m

for n = 0, . . . , N − 1, and where φ = −1 for a down-and-out call while φ = 1 for a down-and-out put. 34

Acknowledgements The authors thank the editor (Professor Emanuele Borgonovo) and all anonymous reviewers, whose suggestions and corrections have significantly improved this article. We also thank the participants of the 10th World Congress of the Bachelier Finance Society (Dublin, Ireland) for useful comments as well as Ant´ onio Barbosa, Lu´ıs Laureano and Rui Lopes for helpful discussions. Of course, all the remaining errors are the exclusive responsibility of the authors. This work was supported by FCT [grant number UID/GES/00315/2019].

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