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Review of Derivatives Research ISSN 1380-6645Volume 20Number 2 Rev Deriv Res (2017) 20:91-133DOI 10.1007/s11147-016-9125-z

Rainbow trend options: valuation andapplications

Jr-Yan Wang, Hsiao-Chuan Wang, Yi-Chen Ko & Mao-Wei Hung

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Rev Deriv Res (2017) 20:91–133DOI 10.1007/s11147-016-9125-z

Rainbow trend options: valuation and applications

Jr-Yan Wang1 · Hsiao-Chuan Wang2 ·Yi-Chen Ko1 · Mao-Wei Hung1

Published online: 5 October 2016© Springer Science+Business Media New York 2016

Abstract Asset selection and timing decisions are major investment concerns. Toresolve these issues simultaneously, a new class of rainbow trend options is proposed.The diversification effect of rainbow options can reduce the importance of asset selec-tion decisions and trend options can mitigate unfavorable effects on market entry andexit decisions. We consider a general framework to facilitate the derivation of analyticpricing formulas for simple, pure, and Asian rainbow trend options using the martin-gale pricing method. The properties of these options and their Greeks are analyzed.We also investigate the performance of the dynamic delta hedging strategy for issuersof rainbow trend options. Last, this paper explores the applications of rainbow trendoptions for hedging price risks, designing executive stock options, modifying coun-tercyclical capital buffer proposed by Basel Committee, and acting as control variatesof the Monte Carlo simulation.

Keywords Rainbow option · Trend option · Timing risk ·Asset selection ·Martingalepricing method

B Jr-Yan [email protected]

Hsiao-Chuan [email protected]

Yi-Chen [email protected]

Mao-Wei [email protected]

1 Department of International Business, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd.,Taipei 106, Taiwan

2 Department of Finance, National Taiwan University, No. 1, Sec. 4, Roosevelt Rd.,Taipei 106, Taiwan

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92 J.-Y. Wang et al.

JEL Classification G13

1 Introduction

This paper proposes and examines a new class of rainbow trend options (RTOs) bygeneralizing the simple, pure, and Asian trend options introduced by Leippold andSyz (2007). The motive for studying RTOs is their ability to simultaneously resolvethe issues of how to choose a better-performing asset and how to time the market, thetwo central concerns for all investors. It is well known that rainbow options, definedas options with payoffs involving several asset prices, can solve the asset selectionproblem, especially if the maximum or minimum payoff function is considered. Inaddition, trend options can make the timing decision less important by linking thepayoffs with the average trend of the asset price instead of the final value of the assetprice at maturity. To meet investors’ demands, the combination of rainbow and trendoptions offers a promising solution and the resulting product is the RTO. In practice,financial institutions can incorporateRTOs into their structured-note products to satisfythe risk-return appetites of individual investors. Furthermore, the introduction of thisnew class of derivatives also can enhance the completeness and efficiency of a financialmarket.

RTOs are useful not only for investment purposes but also in many other appli-cations. For example, they can be employed to hedge the price risk of multiplesubstitutions or to design more effective executive compensation plans. For example,when designing executive compensation plans, multi-index measurement approachesarewidely used to evaluate amanager’s overall performance. Traditionally, aminimumoption can be adopted such that, as long as the manager can improve all evaluatingindexes by a minimum target, the manager can receive a bonus. However, to examinethe evaluating indexes at only one time point could lead to manipulation problems andthus cannot reflect the firm’s true improvement. Therefore, a minimum option on theaverage trends of, for example, the firm’s equity price, sales revenue, and market sharecan be employed to measure a manager’s true performance and thus design a moreeffective compensation plan. Section 4 discusses several practical as well as academicapplications of RTOs.

From a theoretical point of view, RTOs are desirable to investors due to their diversi-fication effects over different assets and time. However, maybe due to the complexityof combining the pricing techniques for rainbow and trend options, RTOs are notexplored in the literature. Stulz (1982) first derives the analytic pricing formula forrainbow options on the maximum or minimum of two assets by solving partial differ-ential equations.Options on themaximumorminimumofmultiple assets are evaluatedby Johnson (1987), who reinterprets Margrabe’s (1978) method for pricing exchangeoptions to derive the analytic pricing formulas for rainbow options. Ouwehand andWest (2006) use the change-of-numeraire technique to derive pricing formulae forvarious rainbow options. Regarding trend options, Leippold and Syz (2007) use thetechnique of changing probability measures to develop the pricing formulas for threetypes of trend-based derivatives, namely, simple, pure, and Asian trend options.

This paper employs the martingale pricingmethod proposed by Harrison and Kreps(1979) to unify the pricing models for rainbow and trend options. We consider a

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Rainbow trend options: valuation and applications 93

highly general framework for the payoffs of RTOs and develop an option pricingformula under this general framework. Therefore, the proposed pricing formula canbe employed to evaluate many trend/average options under single- and multiple-assetcases. Moreover, due to the analytic property of the closed-form solution, we canderive the formulas for the Greeks of RTOs and analyze their properties. Knowledgeof the behavior of the Greeks of RTOs is indispensable in the risk management ofissuers of RTOs.

Wu and Zhang (1999) also try to incorporate path-dependent features into rainbowoptions. They study options on the minimum/maximum of two geometric averageprices. Our paper differs in several critical ways from theirs. First, according to Leip-pold and Syz (2007), both the mean and variance of the trend price are higher thanthose of the geometric average price over the same period, which implies that thetrend option offers more upside potential than the geometric average option and isthus more suited for investment purposes. Second, Leippold and Syz (2007) arguethat the trend price is an attractive alternative to the terminal asset price in optionpayoffs to mitigate issues of when to enter and exit financial markets. This is becausethe mean of the trend price is identical to that of the terminal asset price at issuanceand the variance of the trend price will gradually decrease with time. Since our paperis motivated by the idea of using options to simultaneously resolve the issues of assetand timing selection, we choose the trend option rather than the average option toincorporate with the multiple-asset rainbow option. Last, Wu and Zhang (1999) derivethe corresponding option pricing formulas by solving bivariate partial differentialequations. Nevertheless, it is difficult to extend their method to evaluate multiple-asset rainbow options or other types of average/trend options. In contrast, our pricingformula for RTOs is highly general and can encompass Wu and Zhang’s formulasfor options on the minimum/maximum of two geometric average prices as a specialcase.

This paper is organized as follows. Section 2 evaluates options with both rainbowand trend/average features and derives a general option pricing formula. It also presentsthe formulas for the three proposed types of RTOs, namely, the simple, pure, andAsianRTOs, and their Greeks. The numerical results in Sect. 3 analyze the properties ofthe RTO values and their Greeks and investigate the empirical hedging performancebased on our option pricing formulas from the viewpoint of issuers of RTOs. Section 4presents practical and academic applications based on RTOs. Section 5 concludes thispaper.

2 Valuation of RTOs

2.1 Basic settings

This paper derives the pricing formula in the complete market setting specified byBlack and Scholes (1973). We consider RTOs onm risky underlying assets. By denot-ing WQ

i (t) as the standard Brownian motion for the i th underlying asset under therisk-neutral probability measure Q, the processes of the underlying asset prices underQ are postulated as

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dSi (t) = (r − qi )Si (t)dt + σi Si (t)dWQi (t), for i = 1, . . . ,m, (1)

where r is the risk-free interest rate and qi and σi are the dividend yield and the pricevolatility of the i th asset, respectively. The standard Brownian motions WQ

i (t) and

WQj (t) are correlated with the coefficient ρi j . All of these parameters are assumed to

be constant during the option’s lifetime.Based on the trend derivatives introduced by Leippold and Syz (2007), this paper

considers three types of RTOs and their payoffs at maturity T are specified as follows:For simple RTOs,

V S(T ) =(Sm′ (T ) − X

)+, where m

′ = arg max1 ≤ i ≤ m

Si (T ); (2)

for pure RTOs,

V P (T ) =(Sm′ (T ) − Sm′ (T )

)+, where m


Si (T ); (3)

and for Asian RTOs,

V A(T ) =(Sm′ (T ) − Sm′ (T )

)+, where m


Si (T ), (4)

with X representing the strike price and Si (T ) and Si (T ) representing the trend andgeometric average variables, respectively, of the i th underlying asset over the optionlife, [0, T ]. The exact definitions of Si (T ) and Si (T ) are introduced later. To evaluatethese three types of RTOs, a general pricing formula is introduced next.

2.2 General pricing framework

This paper proposes a pricing framework that can evaluate several types of rainbowtrend/average options. In it, the option life, [0, T ], is partitioned into n time intervals,each of length �t = T

n . In addition, the sampling dates for the trend/average optionsinclude t0(= 0), t1(= �t), . . . , tn(= n�t = T ).

According to Leippold and Syz (2007), the trend variable of the i th asset at maturitycan be obtained by solving the following exponential regression model

Si (th)

Si (t0)= eBi (th−t0)+ε, for h = 0, . . . , n, (5)

where ε is a standard white noise. Next, the ordinary least squares method is employed

to solve the estimator Bi = argminBi∈R∑n

h=0

(ln Si (th)

Si (t0)− Bi (th − t0)

)2. Equipped

with Bi , the exponential trend at maturity can be expressed as

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Si (T ) = Si (t0)eBi (T−t0) = Si (t0)exp

(∑n

h=1ch ln

Si (th)

Si (th−1)

), (6)

where ch = (T − t0)∑n

k=h bk = 12∑n

k=h

(k�t− 1

2 T)

n(n+1)(n+2)(�t)2T = 6h(n−h+1)

(n+1)(n+2) , for h =1, 2, . . . , n.

To evaluate different types of rainbow trend/average options, we consider a generalpayoff function

V (T ) = (Xm′ (T ) − K · Ym′ (T ))+

, where m′ = arg max

1 ≤ i ≤ mXi (T ), (7)

K is a constant, and for i = 1, . . . ,m, Xi (T ) = e(α0Ri,0+α1Ri,1+α2Ri,2+···+αn Ri,n) andYi (T ) = e(β0Ri,0+β1Ri,1+β2Ri,2+···+βn Ri,n) with arbitrary real numbers αh and βh thatare not all zero for different values of h. The variable Ri,0 is defined as ln Si (t0) andRi,h ≡ ln Si (th)

Si (th−1)is the logarithm of the return of the i th underlying asset for the

interval (th−1 , th] with the posited normal distribution

Ri,h∼N(μi�t, σ 2

i �t)

= N

((r − qi − 1

2σ 2i

)�t, σ 2

i �t

), for h = 1, 2, . . . , n.

Based on the above definitions, at time ta = a�t, for 0 ≤ a ≤ n, we obtain

ln Xi (T )∼N(μXi , σ

2Xi

)and ln Yi (T )∼N

(μYi , σ

2Yi

), where

μXi = α0 ln Si (t0) +∑a

h=1αh ln

Si (th)

Si (th−1)+ μi�t

∑n

h=a+1αh, (8)

σ 2Xi

= σ 2i �t

∑n

h=a+1α2h, (9)

μYi = β0 ln Si (t0)+∑a

h=1βh ln

Si (th)

Si (th−1)+ μi�t

∑n

h=a+1βh, (10)

σ 2Yi = σ 2

i �t∑n

h=a+1β2h . (11)

We also defineρX,i j as the correlation between ln Xi (T ) and ln X j (T ), ρY,i j as the cor-relation between ln Yi (T ) and ln Y j (T ), and ρXY,i as the correlation between ln Xi (T )

and ln Yi (T ). It is straightforward to obtain

ρX,i j = ρY,i j = ρi j , (12)

and

ρXY,i =∑n

h=a+1 αhβh√∑nh=a+1 α2

h

√∑nh=a+1 β2

h

. (13)

Appendix 1 shows the proofs of Eqs. (12) and (13).

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According to the risk-neutral valuation argument, the arbitrage-free price of therainbow trend/average optionwith the general payoff function in Eq. (7) is the expectedpresent value of the payoff atmaturity under the risk-neutralmeasure.We can thereforeexpress the option value with time to maturity τa = T − ta as

V (ta) = e−rτa EQ [V (T )]

= e−rτa EQ[ (

Xm′ (T ) − K · Ym′ (T )) ·

I{Xm

′ (T )≥K ·Ym

′ (T )}∣∣∣∣m


Xi (T )

]

= e−rτa EQ[Xm′ (T )·I{

Xm

′ (T )≥K ·Ym

′ (T )}∣∣∣∣m


Xi (T )

]

− e−rτa EQ[KYm′ (T )·I{

Xm

′ (T )≥K ·Ym

′ (T )}∣∣∣∣m


Xi (T )

]

= e−rτa∑m

i=1EQ[Xi (T )·I{Xi (T )≥X j (T )}1≤ j �=i≤m

⋂{Xi (T )≥K ·Yi (T )}]

− Ke−rτa∑m

i=1EQ[Yi (T )·I{Xi (T )≥X j (T )}1≤ j �=i≤m

⋂{Xi (T )≥K ·Yi (T )}],

(14)

where I{·} is the indicator function and returns unity only when the event specified inbraces is true. The summation over i represents different scenarios in which the trendprice of the i th asset is realized as the maximum among the m assets.

We next employ the martingale pricing method proposed by Harrison and Kreps(1979) to evaluate Eq. (14). Appendix 2 provides the detailed derivation. The closed-form formula for this general rainbow trend/average option is expressed as

V (ta) = e−rτam∑i=1

eμXi + 1

2 σ 2Xi Nm

({dQXiX j ,Xi

}1≤ j �=i≤m

, dQXiYi ,Xi

; RQXi

)

−Ke−rτa∑m

i=1eμYi + 1

2 σ 2Yi Nm

({dQYiX j ,Xi

}1≤ j �=i≤m

, dQYiYi ,Xi

; RQYi

), (15)

where Nm(d••,•; R•) denotes a multivariate standard normal cumulative distribution

function (MSNCDF) with m parameters d••,• and an m × m correlation matrix R•.The terms QXi and QYi represent probability measures that are equivalent to the risk-neutral probability measure Q and can facilitate the derivation of the pricing formula.We also introduce

σ 2X,i j ≡ σ 2

Xi− 2ρX,i jσXi σX j + σ 2

X j(16)

and

σ 2XY,i ≡ σ 2

Xi− 2ρXY,iσXi σYi + σ 2

Yi (17)

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and employ the terms μXi , σXi , μYi , σYi , ρX,i j , ρY,i j , and ρXY,i defined in Eqs. (8)–(13), respectively, to express d••,• and R• as

dQXiX j ,Xi

=(μXi + σ 2

Xi

)− (μX j + ρX,i jσXi σX j

)

σX,i j, for 1 ≤ j �= i ≤ m,

dQXiYi ,Xi

=(μXi + σ 2

Xi

)− (μYi + ρXY,iσXi σYi

)− ln K

σXY,i,

dQYiX j ,Xi

=(μXi + ρXY,iσYi σXi

)− (μX j + ρX,i jρXY,iσYi σX j

)

σX,i j, for 1 ≤ j �= i ≤ m,

dQYiYi,Xi

=(μXi + ρXY,iσYi σXi

)−(μYi + σ 2

Yi

)− ln K

σXY,i,

RQXi = RQYi =(

(I )(m−1)×(m−1) (I I )(m−1)×1

(I I )′1×(m−1) 1

)

m×m

,

where

(I )(m−1)×(m−1) =(

σ 2Xi

− ρX,ilσXi σXl − ρX,i jσXi σX j + ρX, jlσX j σXl

σX,i jσX,il

)

(m−1)×(m−1)

,

for 1 ≤ j �= i ≤ m and 1 ≤ l �= i ≤ m,

(I I )(m−1)×1 =(

σ 2Xi

− ρXY,iσXi σYi − ρX,i jσXi σX j + ρX,i jρXY,iσX j σYi

σX,i jσXY,i

)

(m−1)×1

,

for 1 ≤ j �= i ≤ m.

We also derive the formulas for the Greeks of the rainbow trend/average optionsbased on the general pricing formula in Eq. (15). The delta with respect to the i thunderlying asset at time ta is

�i (ta) = ∂V (ta)

∂Si (ta)= αa

Si (ta)e−rτa+μXi + 1

2 σ 2Xi Nm

({dQXiX j ,Xi

}1≤ j �=i≤m

, dQXiYi ,Xi

; RQXi

)

− Kβa

Si (ta)e−rτa+μYi + 1

2 σ 2Yi Nm

({dQYiX j ,Xi

}1≤ j �=i≤m

,dQYiYi ,Xi

; RQYi

). (18)

Equipped with the formula for delta, the derivation of the formulas of gamma andcross gamma can be continued. The gamma of the rainbow trend/average option withrespect to the i th underlying asset at time ta is given by

i (ta) = ∂�i (ta)

∂Si (ta)= αa (αa − 1)

Si (ta)2e−rτa+μXi + 1

2 σ 2Xi

×Nm

({dQXiX j ,Xi

}1≤ j �=i≤m

, dQXiYi ,Xi

; RQXi

)

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+ αa

Si (ta)e−rτa+μXi + 1

2 σ 2Xi

× ∂

∂Si (ta)Nm

({dQXiX j ,Xi

}1≤ j �=i≤m

, dQXiYi ,Xi

; RQXi

)

−Kβa (βa − 1)

Si (ta)2e−rτa+μYi + 1

2 σ 2Yi

×Nm

({dQYiX j ,Xi

}1≤ j �=i≤m

, dQYiYi ,Xi

; RQYi

)

− Kβa

Si (ta)e−rτa+μYi + 1

2 σ 2Yi

× ∂

∂Si (ta)Nm

({dQYiX j ,Xi

}1≤ j �=i≤m

, dQYiYi ,Xi

; RQYi

). (19)

Note that the second and fourth terms of this formula need the calculation of thederivatives of the MSNCDFs. This paper adopts the integral reduction techniqueof Curnow and Dunnett (1962) to complete the calculation. Consider a MSNCDFNm (d1(x), d2(x), . . . , dm(x); R(x)) with upper limits d1(x), d2(x), . . . , dm(x) anda non-singular correlation matrix R(x) ≡ (

ρpq(x))1≤p,q≤m , where the values of all

the parameters di and ρpq can be expressed as functions of a variable x . Note that xcan represent Si (ta), σ, τa , or r for calculating the different Greeks of RTOs. Follow-ing Curnow and Dunnett, the derivative of Nm (d1(x), d2(x), · · · , dm(x); R(x)) withrespect to x is

∂

∂xNm (d1(x), d2(x), · · · , dm(x); R(x)) =

∑m

k=1

∂Nm

∂dk(x)

∂dk(x)

∂x

+∑m

p=1

∑m

q=1

∂Nm

∂ρpq(x)

∂ρpq(x)

∂x,

where

∂Nm

∂dk(x)= Nm−1

(d1,k(x), · · · , dk−1,k(x), dk+1,k(x), · · · , dm,k(x); Rk(x)

)n(dk(x)),

with

di,k(x) = di (x) − ρik(x)dk(x)√1 − (ρik(x))2

, for i = 1, . . . , k − 1, k + 1, . . . ,m,

Rk(x) = (ρi j (x))(m−1)×(m−1)

=⎛⎝ ρi j (x) − ρik(x)ρ jk(x)√

1 − (ρik(x))2√1 − (ρ jk(x)

)2

⎞⎠

1 ≤ i �= k ≤ m1 ≤ j �= k ≤ m

,

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n(z) = 1√2π

e− z22 ,

∂Nm

∂ρpq(x)= nm (d1(x),d2(x), · · · ,dm(x); R(x)) .

The notation of nm (d1(x),d2(x), · · · ,dm(x); R(x)) represents the multivariate stan-dard normal probability density function. To evaluate the second and fourthterms in Eq. (19), we simply choose x = Si (ta) and let {du(x)}1≤u≤m ≡{{

dQxiX j ,Xi

}1≤ j �=i≤m

, dQxiYi ,Xi

}and R(x) ≡ RQXi or {du(x)}1≤u≤m ≡

{{dQYiX j ,Xi

}1≤ j �=i≤m

, dQYiYi ,Xi

}and R(x) ≡ RQYi to calculate the respective partial

derivatives of the MSNCDFs.This general method for calculating the partial derivative of the MSNCDF can also

be applied to calculate the cross gamma (i j (ta) = ∂�i (ta)∂S j (ta)

) and the vega (νl(ta) =∂V (ta)

∂σl) of the rainbow trend/average option. We omit the derivation of the formulas

of the cross gamma and vega to streamline this paper. Nevertheless, the completeformulas of the Greeks of the simple RTO are presented next.

2.3 Pricing RTOs

This section employs the above general pricing formula to generate the formulas ofthe three types of RTOs introduced in Sect. 2.1: simple, pure, and Asian RTOs.

We first evaluate the simple RTOwhose payoff is defined as the difference betweenthe maximum realized trend of m underlying assets and a fixed strike price X , that is,the payoff function in Eq. (2). Since the trend price in Eq. (6) can be rewritten as

Si (T ) = exp

[ln Si (t0) +

∑n

h=1ch ln

Si (th)

Si (th−1)

]

= exp

[ln Si (t0) + c1 ln

Si (t1)

Si (t0)+ · · · + cn ln

Si (tn)

Si (tn−1)

]

= exp[Ri,0 + c1Ri,1 + · · · + cn Ri,n

],

we set α0 = 1, αh = ch = 6h(n−h+1)(n+1)(n+2) for h = 1, . . . , n, βh = 0 for all h, and K = X

in Eq. (7) to change Xi (T ) and KYi (T ) into Si (T ) and X , respectively. Therefore,

the payoff function in Eq. (7) becomes V S(T ) =(Sm′ (T ) − X

)+, where m

′ =arg max

1 ≤ i ≤ mSi (T ), which is identical to the payoff function of the simple RTO men-

tioned in Eq. (2). According to the general pricing formula in Eq. (15), the pricingformula for the simple RTO can be derived as

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V S(ta) = e−rτa∑m

i=1Si (ta) e

μi�t∑n

h=a+1 ch+ 12

(σ SXi

)2

×Nm

({dQXi ,SX j ,Xi

}1≤ j �=i≤m

, dQXi ,SYi ,Xi

; RQXi ,S)

−Xe−rτa∑m

i=1Nm

({dQYi ,SX j ,Xi

}1≤ j �=i≤m

, dQYi ,SYi ,Xi

; RQYi ,S)

, (20)

where Si (ta) = Si (t0)ec1 ln

Si (t1)

Si (t0)+···+ca ln

Si (ta )

Si (ta−1) is the realized trend price for the i th

asset up to time ta and dQXi ,SX j ,Xi

, dQXi ,SYi ,Xi

, dQYi ,SX j ,Xi

, dQYi ,SYi ,Xi

, and RQXi ,S = RQYi ,S are

dQXi ,SX j ,Xi

=ln Si (ta)

S j (ta)+ (μi − μ j

)�t∑n

h=a+1 ch +(σ SXi

)2 − ρSX,i jσ

SXi

σ SX j

σ SX,i j

,

for 1 ≤ j �= i ≤ m, (21)

dQXi ,SYi ,Xi

=ln Si (ta)

X + μi�t∑n

h=a+1 ch +(σ SXi

)2

σ SXi

, (22)

dQYi ,SX j ,Xi

=ln Si (ta)

S j (ta)+ (μi − μ j

)�t∑n

h=a+1 ch

σ SX,i j

, for 1 ≤ j �= i ≤ m, (23)

dQYi ,SYi ,Xi

= ln Si (ta)X + μi�t

∑nh=a+1 ch

σ SXi

, (24)

RQXi ,S = RQYi ,S =(

(I )(m−1)×(m−1) (I I )(m−1)×1

(I I )′1×(m−1) 1

)

m×m

, (25)

with

(I )(m−1)×(m−1) =⎛⎜⎝

(σ SXi

)2 − ρSX,ilσ

SXi

σ SXl

− ρSX,i jσ

SXi

σ SX j

+ ρSX, jlσ

SX j

σ SXl

σ SX,i jσ

SX,il

⎞⎟⎠

(m−1)×(m−1)

,

for 1 ≤ j �= i ≤ m and 1 ≤ l �= i ≤ m,

(I I )(m−1)×1 =(

σ SXi

− ρSX,i jσ

SX j

σ SX,i j

)

(m−1)×1

, for 1 ≤ j �= i ≤ m.

Note that in the above formula,(σ SXi

)2 = σ 2i �t

∑nh=a+1 c

2h, ρ

SX,i j = ρi j , and(

σ SX,i j

)2 =(σ SXi

)2 − 2ρSX,i jσ

SXi

σ SX j

+(σ SX j

)2are defined according to Eqs. (9),

(12), and (16), respectively.

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Similarly, by setting α0= 1, αh = ch = 6h(n−h+1)(n+1)(n+2) , βh = 0 for all h, and K = X

in Eqs. (18) and (19), we can derive the delta and gamma of the simple RTO withrespect to Si (ta):

�Si (ta) = ca

Si (ta)

Si (ta)e−rτa+μi�t

∑nh=a+1 ch+ 1

2

(σ SXi

)2

×Nm

({dQXi ,SX j ,Xi

}1≤ j �=i≤m

, dQXi ,SYi ,Xi

; RQXi ,S)

, (26)

Si (ta) = ca (ca − 1)

Si (ta)

[Si (ta)]2e−rτa+μi�t

∑nh=a+1 ch+ 1

2

(σ SXi

)2

×Nm

({dQXi ,SX j ,Xi

}1≤ j �=i≤m

, dQXi ,SYi ,Xi

; RQXi ,S)

+caSi (ta)


∑nh=a+1 ch+ 1

2

(σ SXi

)2

× ∂

∂Si (ta)Nm

({dQXi ,SX j ,Xi

}1≤ j �=i≤m

, dQXi ,SYi ,Xi

; RQXi ,S)

, (27)

where

∂

∂Si (ta)Nm

({dQXi ,SX j ,Xi

}1≤ j �=i≤m

, dQXi ,SYi ,Xi

; RQXi ,S)

=∑

1≤ j �=i≤m

∂Nm

∂dQXi ,SX j ,Xi

∂dQXi ,SX j ,Xi

∂Si (ta)+ ∂Nm

∂dQXi ,SYi ,Xi

∂dQXi ,SYi ,Xi

∂Si (ta)

=∑

1≤ j �=i≤m

∂Nm

∂dQXi ,SX j ,Xi

caSi (ta)σ S

X,i j

+ ∂Nm

∂dQXi ,SYi ,Xi

caSi (ta)σ S

Xi

.

Note that in this last equation, it is not necessary to consider partial differentiation withrespect to the entries in RQXi ,S because they are all independent of Si (ta). Moreover,the cross gamma of the simple RTO is derived as

Si j (ta) = ∂�S

i

∂S j (ta)

= caSi (ta)


∑nh=a+1 ch+ 1

2

(σ SXi

)2

× ∂

∂S j (ta)Nm

({dQXi ,SX j ,Xi

}1≤ j �=i≤m

, dQXi ,SYi ,Xi

; RQXi ,S)

, (28)

123



where

∂

∂S j (ta)Nm

({dQXi ,SX j ,Xi

}1≤ j �=i≤m

, dQXi ,SYi ,Xi

; RXi ,S)

=∑

1≤ j �=i≤m

∂Nm

∂dQXi ,SX j ,Xi

∂dQXi ,SX j ,Xi

∂S j (ta)

=∑

1≤ j �=i≤m

∂Nm

∂dQXi ,SX j ,Xi

−caS j (ta)σ S

X,i j

,

because neither dQXi ,SYi ,Xi

nor the entries in RQXi ,S are functions of S j (ta). Finally, thevega of the simple RTO with respect to the lth underlying asset is

νSl (ta) = ∂V S (ta)

∂σl

= Sl (ta) e−rτa+μl�t

∑nh=a+1 ch+ 1

2

(σ SXl

)2σl�t

∑n

h=a+1

(c2h − ch

)

×Nm

({dQXl ,SX j ,Xl

}1≤ j �=l≤m

, dQXl ,SYl ,Xl

; RQXl ,S)

+∑m

i=1Si (ta)e

−rτa+μi�t∑n

h=a+1 ch+ 12

(σ SXi

)2

× ∂

∂σlNm

({dQXi ,SX j ,Xi

}1≤ j �=i≤m

, dQXi ,SYi ,Xi

; RQXl ,S)

−Xe−rτa∑m

i=1

∂

∂σlNm

({dQYi ,SX j ,Xi

}1≤ j �=i≤m

, dQYi ,SYi ,Xi

; RQYi ,S)

. (29)

The details for calculating ∂Nm

∂dQXi

,S

X j ,Xi

, ∂Nm

∂dQXi

,S

Yi ,Xi

, and ∂Nm∂σl

in Eqs. (27)–(29) are available

upon request.Finally, if there is only one asset, that is, m = 1, the formulas for the simple RTO

and its Greeks can be reduced to the corresponding formulas for the simple trendoptions on a single asset of Leippold and Syz (2007):

V S(m=1) (ta) = e

−rτa+μ1�t∑n

h=a+1 ch+ 12

(σ SX1

)2S1(ta)N

(dQX1 ,SY1,X1

)

− Xe−rτa N(dQY1 ,SY1,X1

), (30)

�S(m=1) (ta) = ca

S1(ta)

S1(ta)e−rτa+μ1�t

∑nh=a+1 ch+ 1

2

(σ SX1

)2N(dQX1 ,SY1,X1

),

S(m=1) (ta) = ca (ca − 1)

S1(ta)

[S1(ta)]2e−rτa+μ1�t

∑nh=a+1 ch+ 1

2

(σ SX1

)2N(dQX1 ,SY1,X1

)

+ c2a S1(ta)

σ SX1

[S1(ta)]2e−rτa+μ1�t

∑nh=a+1 ch+ 1

2

(σ SX1

)2n(dQX1 ,SY1,X1

),

123



νS(m=1) (ta) = S1(ta)e

−rτa+μ1�t∑n

h=a+1 ch+ 12

(σ SX1

)2σ1�t

×[∑n

h=a+1

(c2h − ch

)N(dQX1 ,SY1,X1

)]

+ Xe−rτa n(dQY1 ,SY1,X1

)√�t∑n

h=a+1c2h,

where

dQX1 ,SY1,X1

=ln S1(ta)

X + μ1�t∑n

h=a+1 ch +(σ SX1

)2

σ SX1

, dQY1 ,SY1,X1

= dQX1 ,SY1,X1

− σ SX1

,

and

(σ SX1

)2 = σ 21 �t

∑n

h=a+1c2h .

Based on the same argument, the general formula in Eq. (15) can be applied toprice pure and Asian RTOs and their Greeks. For pure RTOs, the terminal payoff is thedifference between the trend and the actual stock price at maturity [see Eq. (3)]. Withα0 = 1, αh = ch = 6h(n−h+1)

(n+1)(n+2) for h = 1, . . . , n, βh = 1 for all h, and K = 1, Xi (T )

and Yi (T ) in the general payoff function in Eq. (7) can represent Si (T ) and Si (T ),respectively. Thus these parameters can be applied to derive the pricing formulas forthe pure RTO and its Greeks. The terminal payoff for the Asian RTO is determined bythe difference between the trend price and the geometric average price at maturity [seeEq. (4)], so we set α0 = 1, αh = ch = 6h(n−h+1)

(n+1)(n+2) for h = 1, . . . , n, βh = (n+1)−hn+1

1

for all h, and K= 1 such that Xi (T ) and Yi (T ) can express Si (T ) and the geometricaverage price Si (T ), respectively.

Note that the general pricing formula in Eq. (15) can be applied to evaluate not onlyRTOs but also other types of rainbow options, such as Asian rainbow call options (onthemaximumgeometric averageprice) and rainbowcall options (on themaximumfinal

asset price) with payoff functions(Sm′ (T ) − X

)+, wherem


Si (T ), and

(Sm′ (T ) − X

)+, where m ′ = arg max1 ≤ i ≤ m

Si (T ), respectively. Table 1 summarizes the

parameter settings used to price different types of options.

3 Numerical results

This section conducts several numerical analyses to obtain insights on the differenttypes of RTOs. First, we compute the theoretical values of the three types of RTOs, aswell as Asian rainbow call options and rainbow call options, under several posited sets

1 According to Eq. (24) and Fig. 3 of Leippold and Syz (2007), the weights βh = (n+1)−hn+1 for h = 0, . . . , n

are the weights for logarithmic returns to generate the geometric average price Si (T ).

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Table 1 Variants of our general pricing formula under different parameter settings

Options contracts that can bepriced using our generalpricing formula

α0 αh , 1 ≤ h ≤ n β0 βh , 1 ≤ h ≤ n K

Simple (rainbow) trend option 1 6h(n−h+1)(n+1)(n+2) 0 0 X

Pure (rainbow) trend option 1 6h(n−h+1)(n+1)(n+2) 1 1 1

Asian (rainbow) trend option 1 6h(n−h+1)(n+1)(n+2) 1 (n+1)−h

n+1 1

Asian (rainbow) call option 1 (n+1)−hn+1 0 0 X

Vanilla (rainbow) call option 1 1 0 0 X

This table lists several variants of our general option pricing formula under different parameter settings.If m = 1, that is, in the single-asset case, our pricing formula can evaluate simple, pure, and Asian trendoptions, Asian call options (with the geometric average method), and plain vanilla calls. Moreover, the mostimportant contribution of our general option pricing formula is its ability to evaluate the proposed threetypes of RTOs, as well as the Asian rainbow call options (on the maximum geometric average price) andrainbow call options (on the maximum final asset price)

of parameter values. By contrasting option prices in the single- and multi-asset cases,the premiums of RTOs regarding the diversification effect over different assets canbe examined. Moreover, to explain the option premium for the diversification effectover time, we analyze the differences between the prices of simple RTOs and rainbowcall options. Second, this section also investigates the properties of the delta, gamma,cross gamma, and vega for the three types of RTOs. Last, since holders of RTOs havethe rights to pick assets and to do timing, the liability of sellers of RTOs is manifestand needs to be managed carefully. This section uses the actual historical asset returnsto examine the hedging performance of our option pricing formulas.

3.1 Diversification effects over different assets and time

This section examines the diversification effects of RTOs over different assets andtime. We consider RTOs with two underlying assets (i.e., m = 2) as examples.2 Theparameter values examined are summarized as follows: the interest rate r = 0.05, thedividend yields q1 = q2 = 0, the time to maturity T = 1 year with 252 samplingtime points, the volatilities σ1 = σ2 = 0.2, and the initial stock prices {S1(t0), S2(t0)}are chosen from the set {{80, 80}, {90, 90}, {100, 100}, {110, 110}, {120, 120}}. Forsimple RTOs, X = 100. Note that the symmetry of the two underlying assets isintentionally maintained such that, given ρ12= 1, the values of RTOs generated byour formulas converge to the values of the single-asset trend options of Leippold and

2 It is well known that evaluating MSNCDFs in our pricing formulas for larger values of m could betime-consuming. We test the value of m up to 7 and compare the accuracy and computational time betweenour pricing formulas and the Monte Carlo simulation, which is commonly acknowledged as the standardapproach for pricing path-dependent rainbow options. The results show that our pricing formulas are moreefficient to generate accurate option values of RTOs than the Monte Carlo simulation. This part of analysesis available upon request from the authors.

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Table 2 Comparison of different RTOs

Underlyingasset pricesat t0

Simple RTO Pure RTO Asian RTO

Price Premium fordiversificationeffect over assets



Panel A:ρ12 = 1 (equivalent to the single-asset case)

(80, 80) 2.3963 – 2.9847 - 5.9648 –

(90, 90) 5.9098 – 3.3578 – 6.7104 –

(100, 100) 11.4011 – 3.7309 – 7.4560 –

(110, 110) 18.5950 – 4.1040 – 8.2017 –

(120, 120) 27.0098 – 4.4771 – 8.9473 –

Panel B: ρ12 = 0.5

(80, 80) 4.0557 40.92% 3.8073 21.61% 8.6686 31.19%

(90, 90) 9.3687 36.92% 4.2832 21.61% 9.7521 31.19%

(100, 100) 16.9763 32.84% 4.7591 21.61% 10.8357 31.19%

(110, 110) 26.1976 29.02% 5.2350 21.61% 11.9193 31.19%

(120, 120) 36.3415 25.68% 5.7109 21.61% 13.0028 31.19%

Panel C: ρ12 = 0

(80, 80) 4.5549 47.39% 4.1368 27.85% 9.7541 38.85%

(90, 90) 10.6525 44.52% 4.6539 27.85% 10.9733 38.85%

(100, 100) 19.2620 40.81% 5.1710 27.85% 12.1926 38.85%

(110, 110) 29.4221 36.80% 5.6881 27.85% 13.4119 38.85%

(120, 120) 40.3067 32.99% 6.2052 27.85% 14.6311 38.85%

Panel D: ρ12 = −0.5

(80, 80) 4.7655 49.72% 4.3819 31.89% 10.5624 43.53%

(90, 90) 11.4736 48.49% 4.9297 31.89% 11.8827 43.53%

(100, 100) 21.0188 45.76% 5.4774 31.89% 13.2030 43.53%

(110, 110) 32.0125 41.91% 6.0251 31.89% 14.5233 43.53%

(120, 120) 43.4557 37.85% 6.5729 31.89% 15.8436 43.53%

This table compares the values of different RTOs given different initial stock prices (S1(t0),S2(t0)) and thecorrelation ρ12 givenm= 2. The initial stock prices (S1(t0),S2(t0)) range from (80, 80) to (120, 120) , r =0.05, q1 = q2 = 0, T = 1 year, σ1 = σ2 = 0.2, and X = 100 for simple RTOs. Note that we intentionallymaintain the symmetry of the two underlying assets such that, given ρ12= 1, our pricing formula generatesthe pricing results for single-asset trend options in Panel A. The premium for the diversification effect overassets is defined as the percentage difference between the rainbow option cases (ρ12 �= 1) and single-assetcases (ρ12= 1). For example, for the simple RTO with (S1(t0), S2(t0)) = (80, 80), the premium for thediversification effect over assets between ρ12= 0.5 and ρ12= 1 is 4.0557−2.3963

4.0557 = 40.92%

Syz (2007). Therefore, the benefit of incorporating rainbow options with trend optionscan be analyzed by comparing the option prices given different values of ρ12 with theoption values givenρ12= 1.Table 2 reports the option prices of different types ofRTOs.

First, Panel A of Table 2 (given ρ12= 1) shows that the prices of the RTOs cal-culated based on Eq. (15) can converge to those of the single-asset trend options ofLeippold and Syz (2007). For example, when S1 (t0) = S2 (t0) = 100 in Panel A, the

123



simple RTO based on our pricing formula is worth 11.4011, which is slightly lowerthan the price of the simple trend option (11.4065) reported by Leippold and Syz(2007, Table II). However, note that this paper assumes 252 business days in one year.If 365 days are assumed in 1 year, our pricing result converges to 11.4065.

The second finding is that the prices of RTOs (Panels B to D of Table 2) arehigher than the prices of single-asset trend options (Panel A of Table 2) due to thediversification effect of the maximum function in the payoff function of the RTOs.These differences can be attributed to option premiums from the diversification effectover the underlying assets. These option premiums are also found to increase as thecorrelation ρ12 decreases, which is consistent with the general understanding, sincethe magnitude of the diversification effect is negatively correlated with the correla-tion ρ12. Moreover, by comparing the percentage difference between the rainbow andsingle-asset trend options, we note this option premium for the diversification effectover assets is generally more pronounced for simple and Asian RTOs than for pureRTOs. For instance, the value and percentage proportion of the option premium for thediversification effect of the simple RTO with S1 (t0) = S2 (t0) = 100 and ρ12= 0.5are (16.9763 − 11.4011) = 5.5752 and 16.9763−11.4011

16.9763 = 32.84%, both of which arehigher than (4.7591 − 3.7309) = 1.0282 and 4.7591−3.7309

4.7591 = 21.61% of the counter-parts for the pure RTO.

Another interesting observation is that, given different values of ρ12, the percentageoption premiums for the diversification effect of pure RTOs and Asian RTOs arestable across different initial asset prices. Since Sm′ (T ), Sm′ (T ), and Sm′ (T ) are allhomogeneous functions of degree onewith respect to the initial prices of the underlyingassets, the values of pure and Asian RTOs should be proportional to the value of theinitial asset prices. For example, when ρ12 = 0.5 (ρ12 = 0), the ratio between thevalues of the pure RTOs given S1(t0) = S2(t0) = 90 and S1(t0) = S2(t0) = 80 is4.28323.8073 = 1.125( 4.65394.1368 = 1.125), which is identical to the ratio of 90

80 = 1.125. Thischaracteristic results in stable percentage option premiums for pure and Asian RTOs.

Table 3 compares the simple RTOwith the rainbow call option. It is well known thatthe rainbow call option has a diversification effect over different underlying assets.However, due to the combination of the rainbow and trend options, the simple RTO hasdiversification effects over both different underlying assets and time. Investors wouldlike to pay more for the diversification effect over time provided by simple RTOs.Consequently, the premium of the diversification effect over time is defined as thepercentage difference between the values of simple RTOs and rainbow call options.Consistent with the single-asset results of Leippold and Syz (2007), the premiumof thediversification effect over time is found to decrease with moneyness. This is because ifthe simple RTO is initially deep in the money (ITM) with respect to all the underlyingassets, the diversification effect over time can contribute a little to the option valueby slightly enhancing the ITM probability at maturity. Moreover, the premiums of thediversification effect over time show little sensitivity to ρ12, which implies the effectof incorporating the trend option into the rainbow option is persistent, regardless ofthe magnitude of the diversification effect over different assets.

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Table 3 Comparison between simple RTOs and rainbow call options

Underlying asset prices at t0 Rainbow call option Simple RTO

Price Price Premium for diver-sification over time

Panel A: ρ12 = 1

(80, 80) 1.8594 2.3963 22.4032%

(90, 90) 5.0912 5.9098 13.8517%

(100, 100) 10.4506 11.4011 8.3369%

(110, 110) 17.6630 18.5950 5.0121%

(120, 120) 26.1690 27.0098 3.1129%

Panel B: ρ12 = 0.5

(80, 80) 3.1750 4.0557 21.7155%

(90, 90) 8.8987 9.3687 13.5566%

(100, 100) 15.5185 16.9763 8.5873%

(110, 110) 24.6864 26.1976 5.7685%

(120, 120) 34.8057 36.3415 4.2260%

Panel C: ρ12 = 0

(80, 80) 3.5553 4.5549 21.9451%

(90, 90) 9.2074 10.6525 13.5657%

(100, 100) 17.6058 19.2620 8.5983%

(110, 110) 27.6780 29.4221 5.9279%

(120, 120) 38.4780 40.3067 4.5370%

Panel D: ρ12 = −0.5

(80, 80) 3.7034 4.7655 22.2879%

(90, 90) 9.9056 11.4736 13.6665%

(100, 100) 19.2192 21.0188 8.5619%

(110, 110) 30.0884 32.0125 6.0105%

(120, 120) 41.3865 43.4557 4.7616%

This table examines the option premium for the diversification effect over time of the simple RTO bycomparing the prices of simple RTOs with those of rainbow call options. All parameter values are the sameas those in Table 2. A rainbow call option has a diversification effect over different assets; a simple RTOhas a diversification effect over both different assets and time. Thus, the premium for the diversificationeffect over time can be defined as the percentage price difference between the simple RTO and the rainbowcall options. For example, for the simple RTO with (S1 (t0) , S2 (t0)) = (80, 80) and ρ12= 1, the premiumfor the diversification effect over time is 2.3963−1.8594

2.3963 = 22.4032%

3.2 The Greeks of RTOs

This section analyzes the delta, gamma, cross gamma, and vega of the simple, pure,and Asian RTOs. The basic set of parameter values includes m= 2, r = 0.05, q1 =q2 = 0, σ1 = σ2 = 0.2, ρ12 = 0.5, and T = 1 year, with a sampling frequency of252 per year. We also examine four values of ta : ta = T/8, ta = T/2, ta = 7T/8, andta → T , corresponding to a= 31, a= 126, a= 220, and a= 251, respectively.

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Fig. 1 The delta of the simple RTOwith respect to the price of the first asset,�S1 (ta). The parameter values

in this figure are m = 2, T = 1 year, r = 0.05, q1 = q2 = 0, σ1 = σ2 = 0.2, S1(ta−1) = S2(ta−1)= 100,and ρ12 = 0.5. Positive deltas are in blue and zero deltas are in gray. The values of �S

1 (ta) are found

to be non-negative. The values of �S1 (ta) could be higher than one when ta = T/2. In addition, when

ta → T,�S1 (ta) approaches zero, particularly when S1(ta) < S2(ta) or S1(ta) < X (Color figure online)

Figures 1, 2 and 3 show the delta of the three types of RTOs with respect to S1(ta),given different combinations of S1(ta) and S2(ta) from 80 to 120. Note that when wegenerate the diagrams of the delta, gamma, and cross gamma of the three types ofRTOs, we control the realized information up to time ta−1. Taking the simple RTOfor example, we fix the realized trend prices up to ta−1, that is, S1(ta−1) and S2(ta−1),both at 100. Next, for each point in these diagrams, based on the information of(S1(ta−1), S2(ta−1)) and the values of (S1(ta), S2(ta)) of that point, we can obtainthe corresponding realized trend prices (S1(ta), S2(ta)) up to ta and finally employEqs. (26)–(28) to compute the delta, gamma, and cross gamma of the simple RTO,respectively.3

Figure 1 shows that the delta of the simple RTO, �S1 (ta), is always positive and

decreasing with the initial strike price X . By analyzing the formula of �S1 (ta) in Eq.

(26), it is straightforward to infer that �S1 (ta) is non-negative, since it is a product of

several non-negative terms. Intuitively speaking, because an increase in S1(ta) resultsin an increase in S1(ta) and thus enhances the ITM probability of simple RTOs,�S

1 (ta)

3 The reason for fixing the information up to ta−1 (e.g., S1(ta−1) and S2(ta−1)) rather than up to thepresent time ta (e.g., S1(ta) and S2(ta)) is to enhance the readability of the diagrams. To understand thediagrams in this section, note merely that we generate the values of delta, gamma, and cross gamma basedon the same historical information until ta−1 for each node. In contrast, suppose we fix the information upto ta by specifying the levels of, for example, the realized trend prices S1(ta) and S2(ta) when computingthe Greeks. Then, each point in a diagram (corresponding to different combinations of (S1(ta), S2(ta)))

implies a different combination of (S1(ta−1), S2(ta−1)), which reflects the different information until ta−1of each node. Consequently, further comparisons of the Greeks between different points are difficult toanalyze.

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Fig. 2 The delta of the pure RTO with respect to the price of the first asset, �P1 (ta). All the diagrams in

this figure are generated for the parameter values m = 2, T = 1 year, r = 0.05, q1 = q2 = 0, σ1 = σ2 =0.2, S2(ta−1)= 100, and ρ12 = 0.5. Positive deltas are in blue, negative deltas are in red, and zero deltasare in gray. This figure shows that the values of �P

1 (ta) could be negative, especially when ta is relatively

small or close to maturity. In addition, when ta → T , the values of �P1 (ta) could be significantly negative

if S1(ta−1)>S2(ta−1) or S1(ta−1) = S2(ta−1) ≥ S1(ta) > S2(ta) (Color figure online)

Fig. 3 The delta of the Asian RTOwith respect to the price of the first asset,�A1 (ta). The parameter values

examined in this figure are m = 2, T = 1 year, r = 0.05, q1 = q2 = 0, σ1 = σ2 = 0.2, S1(ta−1) =S2(ta−1) = S2(ta−1) = 100, and ρ12 = 0.5. Positive deltas are in blue, negative deltas are in red, and zerodeltas are in gray. This figure shows that the values of �A

1 (ta) can be negative only when ta is relatively

small. In addition, the values of �A1 (ta) approaches zero when ta → T , particularly if S1(ta) < S2(ta) or

S1(ta) < S1(ta−1) (Color figure online)

should be positive. Moreover, this effect is weakened when the initial strike price Xis relatively high (i.e., the simple RTO is deeply out of the money), so the value of�S

1 (ta) decreases with X . Another interesting phenomenon in Fig. 1 is that the values

123



of �S1 (ta) could be higher than unity, particularly when ta = T/2. This is because Eq.

(26) shows that the value of �S1 (ta) is remarkably influenced by the value of ca . The

value of ca , given different values of a, is depicted by the dashed curve in of Fig. 4a.Similar to what is addressed by Leippold and Syz (2007), the value of ca attains itsmaximum when ta = T/2. Due to the large value of ca when ta = T/2,�S

1 (ta)of the simple RTO could be higher than unity.4 Finally, as shown in Fig. 1, whenta → T,�S

1 (ta) is close to zero. This is because when ta → T, ca converges to zero,as shown in Fig. 4a, and thus �S

1 (ta) in Eq. (26) will approach zero.Figures 2 and 3 plot the deltas of the pure and Asian RTOs with respect to S1(ta)

given different (S1(ta), S2(ta)). In contrast to the positive delta of the simple RTO inFig. 1, there could be some scenarios in which the deltas of pure and Asian RTOs arenegative. Note that for the value of a pure RTO, it is a function of both S1(ta) and S1(ta)

and thus its delta with respect to S1(ta) is �P1 (ta) = ∂V P (ta)

∂ S1(ta)∂ S1(ta)∂S1(ta)

+ ∂V P (ta)∂S1(ta)

∂S1(ta)∂S1(ta)

=∂V P (ta)∂ S1(ta)

caS1(ta)S1(ta)

+ ∂V P (ta)∂S1(ta)

1. Since ∂V P (ta)∂ S1(ta)

> 0 and ∂V P (ta)∂S1(ta)

< 0,5 we can deduce that the

sign of �P1 (ta) depends on the comparative levels of ca

S1(ta)S1(ta)

and one. To demonstrate

this argument, we consider the delta of the single-asset pure trend option, �P(m=1)(ta),

in Fig. 4c, where S1(ta) = S1 (ta) = 100 is intentionally set such that we can focus onhow the comparative levels of ca and one determine the sign of�P

(m=1). By comparingFig. 4a and c, we find that when ca (dashed curve in Fig. 4a) is smaller than one (solidstraight line in Fig. 4a), that is, when ta is small or ta is near T,�P

(m=1)(ta) is inclinedto be negative. Note that this effect still holds for the pure RTO in Fig. 2. In Fig. 2,�P

1 (ta) could be negative when ta is small (ta = T/8) or ta is near T (ta = 7T/8and ta → T ). In addition, note that when ta → T in Fig. 2, the value of �P

1 (ta) isnonzero only when S1(ta−1)>S2(ta−1) or S1(ta−1) = S2(ta−1) ≥ S1(ta) > S2(ta).This is because, in these scenarios, the trend price of the first asset, S1(ta), is likely toappear in the final option payoff and be larger than S1(ta). The present option valueshould therefore be sensitive to S1(ta) and thus �P

1 (ta) is nonzero. Moreover, these

nonzero �P1 (ta) should be negative and can be approximated by ∂V P (ta)

∂S1(ta), which is

around −0.93 to −0.95 in Fig. 2. This is because when ta → T, ca → 0 such that

�P1 (ta) = ∂V P (ta)

∂ S1(ta)ca

S1(ta)S1(ta)

+ ∂V P (ta)∂S1(ta)

1≈ ∂V P (ta)∂S1(ta)

.

A similar argument can explain the negative delta of the Asian RTO in Fig. 3. Since

�A1 (ta) = ∂V A(ta)

∂ S1(ta)∂ S1(ta)∂S1(ta)

+ ∂V A(ta)∂ S1(ta)

∂ S1(ta)∂S1(ta)

= ∂V A(ta)∂ S1(ta)

caS1(ta)S1(ta)

+ ∂V A(ta)∂ S1(ta)

((n+1)−a

n+1

)S1(ta)S1(ta)

4 As illustrated in Fig. 4b, the delta of the single-asset simple trend option also peaks at ta = T/2. Thisphenomenon demonstrates that the value of ca should be a determining factor in influencing the delta of thesingle-asset simple trend option. A similar result is also mentioned by Leippold and Syz (2007, Table II).5 Since ∂V P (ta )

∂ S1(ta )( ∂V P (ta )

∂S1(ta )) measures the sensitivity of V P (ta) with respect to S1(ta) (S1(ta)) under the

assumption that S2(ta), S1(ta), and S2(ta) (S1(ta), S2(ta), and S2(ta)) are held as constants, an increase inS1(ta) (S1(ta)) results in an increase of the expected value of S

m′ (T ) (Sm′ (T )) and thus causes an increase

(decrease) of the expected value of the payoff of (Sm′ (T ) − S

m′ (T ))+of the pure RTO. Consequently, it

can be inferred that ∂V P (ta )

∂ S1(ta )> 0 ( ∂V P (ta )

∂S1(ta )< 0).

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(a)

(b)

(c)

(d)

123



� Fig. 4 Weights for the logarithmic returns of the underlying asset to generate Si (T ), Si (T ), and Si (T ) andthe evolution of the deltas of single-asset trend options over the option lifespan. The examined parametervalues are T = 1 year, r = 0.05, q1 = 0, σ1 = 0.2, S1(ta) = S1(ta) = S1(ta) = 100, and X = 100for the simple trend option. For the simple trend option, �S

(m=1)(ta) is always positive. For the pure trend

option, �P(m=1)(ta) is negative when the weights required to generate S1(T ) are lower than those required

to generate S1(T ). For the Asian trend option,�A(m=1)(ta) is negative when the weights required to generate

S1(T ) are lower than those required to generate S1(T )

with ∂V A(ta)∂ S1(ta)

> 0 and ∂V A(ta)∂ S1(ta)

<0,6 we can deduce that the comparative levels of caS1(ta)S1(ta)

and(

(n+1)−an+1

)S1(ta)S1(ta)

can influence the sign of �A1 (ta). By comparing Fig. 4a and d,

where S1(ta) = S1(ta) = S1(ta) = 100 is deliberately posited such that we canfocus on the impact from comparable levels of ca and (n+1)−a

n+1 on the delta of thesingle-asset Asian trend option, �A

(m=1)(ta), one sees that when ca (dashed curve in

Fig. 4a) is smaller than (n+1)−an+1 (dotted curve in Fig. 4a)—that is, when ta is small—

�A(m=1)(ta) is inclined to be negative. A similar phenomenon is verified in Fig. 3, where

the delta of the Asian RTO, �A1 (ta), is negative only when ta is small (ta = T/8).

Moreover, when ta → T in Fig. 3, the value of �A1 (ta) approaches zero because

both ca and (n+1)−an+1 approach zero given ta → T (as illustrated in Fig. 4a) and thus

�A1 (ta) = ∂V A(ta)

∂ S1(ta)ca

S1(ta)S1(ta)

+ ∂V A(ta)∂ S1(ta)

((n+1)−a

n+1

)S1(ta)S1(ta)

≈ 0.

The gammas of simple, pure, and Asian RTOs with respect to the price of the firstunderlying asset, denoted asS

1 (ta) , P1 (ta), andA

1 (ta), respectively, are depicted inFigs. 5, 6 and 7. All results are generated based on Eq. (19) with the proper parametersettings. One can also understand intuitively the behaviors of S

1 (ta) , P1 (ta), and

A1 (ta) by examining the movement of the deltas in response to a small change in

S1(ta) in Figs. 1, 2 and 3, respectively. It is found that the gammas of simple, pure,and Asian RTOs could be negative in Figs. 5, 6 and 7, respectively. Nevertheless, whenta = T/2 and theweight for the trend price, ca , reaches itsmaximum,S

1 (ta) , P1 (ta),

and A1 (ta) are all positive under our examined parameter values. It is well known

that, for many options, e.g., plain vanilla or Asian options, their gammas are positive,a desirable property for option holders. However, according to our results in Figs. 5 6and 7, positive gammas for RTOs are not always available, so the risk managementfor RTOs should consider the risk of a negative gamma.7

6 One can obtain ∂V A(ta )

∂ S1(ta )> 0 and ∂V A(ta )

∂ S1(ta )<0 following similar arguments as those in the previous

footnote.7 We also generate the cross gammas of simple, pure, and Asian RTOs, denoted as S

12 (ta) , P12 (ta), and

A12 (ta), according to our analytic cross gamma formula with the proper parameter settings. The cross

gamma of the simple RTO, S12 (ta), is non-positive. The cross gamma of the pure RTO, P

12 (ta), couldbe positive as well as negative. When ta is relatively small (i.e., ta = T/8) or ta is near maturity (i.e.,ta = 7T/8 and ta → T ), the value of P

12 (ta) tends to be non-negative. As for the cross gamma of the

Asian RTO, A12 (ta), positive and negative values are both admitted, but the positive values of A

12 (ta)

occur only when ta is relatively small (i.e., ta = T/8). Finally, all of the cross gammas of simple, pure,and Asian RTOs tend to be negative, especially when ta = T/2. Note that the figures of cross gammas areomitted to streamline this paper. Nevertheless, the behaviors of S

12 (ta) , P12 (ta), and A

12 (ta) can also be

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Fig. 5 The gamma of the simple RTO with respect to the Price of the first asset, S1 (ta). The parameters

in this figure are m= 2, T = 1 year, r = 0.05, q1 = q2 = 0, σ1 = σ2 = 0.2, S1(ta−1) = S2(ta−1)= 100,and ρ12 = 0.5. Positive gammas are in blue, negative gammas are in red, and zero gammas are in gray.Our results show that the values of S

1 (ta) could be slightly negative when ta = 7T/8 or ta → T (Colorfigure online)

Fig. 6 The gamma of the pure RTO with respect to the price of the first asset, P1 (ta). We examine P

1 (ta)

in this figure, given m = 2, T = 1 year, r = 0.05, q1 = q2 = 0, σ1 = σ2 = 0.2, S2(ta−1) = 100, andρ12 = 0.5. Positive gammas are in blue, negative gammas are in red, and zero gammas are in gray. Thevalues of P

1 (ta) could be negative, except when ta = T/2 (Color figure online)

This section also investigates the values of the vega of the simple, pure, and AsianRTOs regarding the volatility of the first asset, denoted as νS1 (ta), ν

P1 (ta), and νA

1 (ta),

Footnote 7 continuedderived by examining the change of the deltas in response to a small change in S2(ta) in Figs. 1, 2 and 3,respectively. For readers interested in the details, these figures of cross gammas are available upon request.

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Fig. 7 The gamma of the Asian RTO with respect to the price of the first asset, A1 (ta). In this figure,

m = 2, T = 1 year, r = 0.05, q1 = q2 = 0, σ1 = σ2 = 0.2, S1(ta−1) = S2(ta−1) = S2(ta−1) = 100,and ρ12 = 0.5. Positive gammas are in blue, negative gammas are in red, and zero gammas are in gray.The values of A

1 (ta) could be negative, except when ta = T/2 (Color figure online)

Fig. 8 The vegas of RTOs, νS1 (ta), νP1 (ta), and νA1 (ta), for different correlations. The vegas of simple, pure,and Asian RTOs are depicted in the first, second, and third rows, respectively. In this figure, m = 2, T = 1year, r = 0.05, q1 = q2 = 0, σ1 = σ2 = 0.2, S1(ta) = S2(ta) = 100, S1(ta) = S2(ta) = 100, X = 100for νS1 (ta), and S1(ta) = S2(ta) is set to be 80 (the dashed line), 100 (the dotted line), and 120 (the solidline). In general, the vegas of RTOs are positive and increase with the decreasing correlation between theunderlying assets. The only exception is the pure RTO near maturity

respectively, in Fig. 8. We examine the vegas of these three types of RTOs along withthe dimensions of both ta and ρ12. Several interesting phenomena are elaborated asfollows.

First, Leippold and Syz (2007) have proven that, in the single-asset case, the vegaof the simple trend option can be negative when the option is deeply ITM and closeto maturity. This phenomenon can be explained more intuitively. For an increase inthe volatility of the first asset, σ1, there are two countervailing impacts on the price

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of trend options. Take the single-asset simple trend option for example. Its pricingformula in Eq. (30) can be rewritten as

V S(m=1) (ta) = e−rτa+M1τa S1(ta)N

(dQX1 ,SY1,X1

)− Xe−rτa N

(dQY1 ,SY1,X1

), (31)

where

dQX1 ,SY1,X1

=ln S1(ta)

X + μ1�t∑n

h=a+1 ch +(σ SX1

)2

σ SX1

,

dQY1 ,SY1,X1

= dQX1 ,SY1,X1

− σ SX1

,(σ SX1

)2 = σ 21 �t

∑n

h=a+1c2h

and

M1 =(

μ1�t∑n

h=a+1ch + 1

2

(σ SX1

)2)/τa

=((

r − q1 − 1

2σ 21

)�t∑n

h=a+1ch + 1

2

(σ SX1

)2)/τa .

Note that M1 can be interpreted as the growth rate of the trend price because

eM1τa S1(ta) is the expected value of S1(T ) conditional on S1(ta). When σ1 increases,its first effect is to enhance the volatility of the conditional distribution of S1(T ), i.e.,σ SX1, and thus contribute an extra premium to the simple trend option. Its second effect

is to reduce M1 via the term(− 1

2σ21

), leading to a decrease in the expected value of

S1(T ) and the option premium of simple trend options. Figure 9 illustrates the evolu-tion of ∂ M1/∂σ1 over time. It is apparent that ∂ M1/∂σ1 could be negative, particularlywhen it is close to maturity. Due to these two opposite impacts corresponding to anincrease in σ1, the vega of the simple trend option could be positive as well as negative.As for the pure and Asian trend options in the single-asset case, their pricing formulascan be derived, respectively, as

V P(m=1) (ta) = e−rτa+M1τa S1(ta)N

(dQX1 ,PY1,X1

)

− e−q1τa S1(ta)N(dQY1 ,PY1,X1

)(32)

and

V A(m=1) (ta) = e−rτa+M1τa S1(ta)N

(dQX1 ,AY1,X1

)

− e−rτa+M1τa S1(ta)N(dQY1 ,AY1,X1

), (33)

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Fig. 9 The partial derivative of the average growth rate of the trend price of the first asset, M1,with respect to

the volatility σ1. The average growth rate of S1 is defined as M1 = (μ1�t∑n

h=a+1 ch + 12

(σ SX1

)2)/τa =

((r − q1 − 12σ 2

1 )�t∑n

h=a+1 ch + 12

(σ SX1

)2)/τa in Sect. 3.2. This figure exhibits the sensitivity of M1

regarding σ1 given different time points ta . The parameters examined in this figure are T = 1 year,

r = 0.05, q1 = 0, and σ1 = 0.2. It can be found that, with the passage of time, ∂ M1∂σ1

first increases, thendecreases, and finally becomes negative near maturity

where

M1 =(

μ1�t(n − a) (n − a + 1)

2 (n + 1)+ 1

2

(σ AY1

)2)/τa .

It is also true for pure and Asian trend options that an increase in σ1 could reduce M1and next generate a negative impact on the option price. Therefore, the signs of thevegas of the pure and Asian trend options in the single-asset case are also uncertain.

Second, the diversification effect in the maximum option of simple, pure, andAsian RTOs can generally increase their vegas. Due to the diversification effect ofmax(S1(T ), S2(T ), . . . , Sm(T )) in the payoff function, the negative impact on thegrowth rate of M1 and more negative returns of the first asset due to an increase of σ1could be partially eliminated because S1(T ) is less likely to be the highest trend pricein these scenarios and thus does not affect the final payoff of RTOs. Therefore, themore positive returns of the first asset due to an increase of σ1 dominate and enhancethe vegas of RTOs upward. Consistent with this inference, most vegas in Fig. 8 varyinversely with ρ12, because a more negative ρ12 represents a scenario with a strongerdiversification effect. In addition, when ta approachesmaturity, the impact of the diver-sification effect on the vegas is weakened. For example, this phenomenon is clearlyobserved in the first row in Fig. 8, which shows the vegas of simple RTOs decreasewith ta . This is because the period for the diversification effect to be in effect shortenswhen ta is near the maturity date.

Third, when ta approaches maturity, the diversification effect, however, could hurtthe vega of the pure RTO, as illustrated in the second row of Fig. 8, while ta = 7T/8

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and ta → T . From the previous paragraph, we know the diversification effect canscreen out the negative impacts on M1 and more negative returns due to an increase ofσ1, but the remainingmore positive returns due to an increase of σ1 will enhance S1(T )

by an amount larger than S1(T ) when ta approaches maturity. This is because ca (theweights for the logarithmic returns to generate S1(T )) is less than one (the weightsfor the logarithmic returns to generate S1(T )) when ta approaches maturity, as shownin Fig. 4a. Consequently, the diversification effect (measured by a more negative ρ12)

has a negative impact on the vegas of the pure RTO when ta = 7T/8 and ta → T . Incontrast, the diversification effect does not hurt the Asian RTO because ca (the weightsfor the logarithmic returns to generate S1(T )) is larger than (n+1)−a

n+1 (the weights for

the logarithmic returns to generate S1(T )) when ta approaches maturity.Last, from the second row of Fig. 8, one can observe that when we consider a

time ta near maturity, for example, ta = 7T/8 or ta → T , the vegas of deeply ITMsingle-asset pure trend options, which can be approximated by the vegas of the deeplyITM pure RTO at ρ12 = 1, can be negative. It is intuitive to obtain this result. Whenσ1 increases, the future movements of S1 should be more volatile than those of S1due to one (the weights for logarithmic returns to generate S1(T )) being larger thanca (the weights for logarithmic returns to generate S1(T )). The excess volatility of S1can result in a higher probability of the ITM single-asset pure trend options turning tobe out of the money and thus reduce the option values. Consequently, the vegas of thedeeply ITM pure RTOs can be negative at ρ12= 1 when ta approaches maturity in thesecond row of Fig. 8. In addition, combining with the fact explained above that thediversification effect could hurt the vega of the pure RTO, we can obtain the negative,upward-sloping vegas of deeply ITM pure RTOs when ta = 7T/8 and ta → T in thesecond row of Fig. 8.

3.3 Hedging performance of the DDHS for RTOs

This section examines the hedging performance of our RTO pricing formulas, whichare developed under the Black and Scholes (1973) framework, using the actual his-torical returns of underlying assets. We take simple RTOs for example and examinethe distribution of the net hedging cost (NHC) of the dynamic delta hedging strategy(DDHS) from the issuer viewpoint. We extend the approach of Jacques (1996), whoinvestigates the performance of the DDHS for single-asset Asian options, to rebalancethe delta-neutral portfolios and calculate the NHC. In our experiments, the transactioncost is ignored and the hedging portfolio is rebalanced at the daily frequency, whichis the most common arrangement in practice.

The three steps of the DDHS for RTOs are described as follows. For any issue dayt0, suppose that the current and the subsequent daily prices of m underlying stockare denoted as Si (th), for i = 1, . . . ,m, h = 0, 1, . . . , n. Note that when using ourpricing formulas, we always shift t0 to be 0, and t1 = �t, . . . , tn = n�t = T , where�t ≡ 1/252, and T equals the time to maturity of the examined RTO.

Step 1: At time t0, the value of hedging portfolio, π(t0), is initialized to be theselling price of the examined RTO according to Eq. (15), and the hedging ratio for

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the i th underlying asset, �i (t0), is computed by Eq. (18).8 We thus purchase theunderlying assets with

∑mi=1 �i (t0)Si (t0) and invest the remaining cash, η(t0) =

π(t0) −∑mi=1 �i (t0)Si (t0), at the interest rate r(t0)9 for the following �t period.

Step 2: At time point th , the value of hedging portfolio becomes π(th) =∑mi=1 �i (th−1)Si (th)+η(th−1)er(th−1)�t . The new hedging ratio,�i (th), is computed

by Eq. (18) with the inputs of the underlying asset prices today Si (th). The moneyaccount which compounds at r(th) for the following �t period is thus recalculated asη(th) = π(th) −∑m

i=1 �i (th)Si (th).Step 3: At time T , we first compute the intrinsic value of the RTO, i.e., V (T ) in

Eq. (7). Next the NHC is calculated to be − (π(T ) − V (T )). A positive (negative)value of NHC means that the issuer loses (earns) money from this round of issuingand hedging the RTO.

Theoretically speaking, the NHC is close to zero (distributed with a nearly zeromean and a small standard deviation) given some ideal conditions. The first conditionis that the dynamics of the actual underlying asset prices can be perfectly describedby Eq. (1) and thus follow lognormal distributions. Second, one can predict exactlythe values of parameters, like σi and ρi j , for the future option life period. However,much literature confirms the existence of stochastic volatility and jumps in actual stockprices, so the first condition usually does not hold and the real-world NHCmay deviatefrom its ideal distribution. We view this deviation as the model error of our frameworkfor pricing RTOs. On the other hand, because one cannot foresee parameter valuesfor future periods given historical information, estimation errors emerge. Specifically,for each issue day t0, we follow the common arrangement in practice to calculate theestimates of σi and ρi j , denoted as σi and ρi j , with the returns of D days prior tot0. During the following option life [t0, tn], we always employ σi and ρi j to calculateoption values and deltas based on Eqs. (15) and (18), respectively, for the DDHS.However, the estimation error occurs because σi �=σ ∗

i and ρi j �= ρ∗i j , where σ ∗

i and ρ∗i j

are the actual volatilities and correlations in the option life [t0, tn].Consequently, when we use the actual daily prices Si (th) and calculate the corre-

sponding �i (th) based on our RTO pricing formulas (with historical σi and ρi j asinput parameter values) for the DDHS, we suffer both estimation and model errors. Todistinguish the influences on the hedging performance from the above two errors, weconstruct three models to simulate or obtain the daily prices Si (th) after t0, as shownin Table 4.

To capture the estimation error, both Models 1 and 2 assume that the log returnsof the underlying assets (Ri,h ≡ ln Si (th)

Si (th−1)) of RTOs follow normal distributions.

In Model 1, we simulate the returns of the future n days based on σi , ρi j , andthe average log return of the D days prior to t0, denoted as E Ri , with Si (th) =Si (th−1)exp

(E Ri�t + σi

√�tεi (th)

), where εi (th) ∼ N (0, 1), and εi (th) and ε j (th)

8 When evaluating Eqs. (15) and (18), we employ the spline interpolation (based on daily term structuresof the risk-free zero rates in OptionMetrics) to derive the level of r whose maturity matches the remainingoption life of the examined RTO.9 The risk-free zero rate with the maturity of one week, the shortest-maturity zero rate provided in Option-Metrics, is employed to approximate r(t).

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Table 4 Three models to distinguish estimation and model errors

Parameters used to simulatestock prices during option life

Error type

Model 1 ˆERi , σi , ρi j with lognormaldistribution

Model 1 vs. 2: estimationerror

Model 2 ER∗i , σ

∗i , ρ∗

i j with lognormaldistribution

Model 2 vs. 3: modelerror

Model 3 Use actual log returns andSi (t0) to infer future Si (th)

Model 1 vs. 3: estimationand model errors

This table compares the three models designed to differentiate the estimation and model errors of the DDHSimplemented based on our RTO pricing formulas. Given any issue date t0, the values of the expected logreturn ˆERi , σi , and ρi j (ER∗

i , σ∗i , and ρ∗

i j ) are estimated by the returns of the prior D (subsequent n) days.

In all three models, we compute the option values and deltas based on σi , and ρi j . In addition, to make theresults of different t0 comparable, Si (t0) is normalized to 100 for all underlying assets, and the strike priceX is always fixed at 100. The dividend yields of all underlying assets are assumed to be zero. The NHCdifferences between Models 1 and 3 reflect both the parameter estimation error and the model error of theDDHS based on our RTO pricing formulas, but the NHC differences between Models 1 and 2 (Models 2and 3) capture the estimation error (model error) individually

are correlated with ρi j . By this means, the future stock prices after t0 follow the log-normal distribution and all the associated parameter values are the same as thoseused in our formulas to calculate option and delta values. As a result, both themodel and estimation errors of Model 1 are minimized and thus Model 1 can betaken as a theoretical performance benchmark of the DDHS based on our pricingformulas, i.e., the average NHC of Model 1 converges to zero as the number of tri-als increases. In Model 2, we simulate the returns of the future n days based onσ ∗i , ρ∗

i j , and the average log return of the n days after t0, denoted as ER∗i , with

Si (th) = Si (th−1)exp(ER∗

i �t + σ ∗i

√�tξi (th)

), where ξi (th) ∼ N (0, 1), and ξi (th)

and ξ j (th) are correlated with ρ∗i j . Since Model 2 is based on the information on the

future returns in the option life but still adopts the lognormal distribution assumption,the differences between the NHCs of Models 1 and 2 represent the estimation error ofthe DDHS based on our pricing formulas.

In Model 3, we employ the actual returns of the n days after t0 to determine Si (th)and thus evaluate the empiricalDDHSperformance based on our pricing formulas. Thedifferences between the NHCs of Models 1 and 3 capture both model and estimationerrors, and the differences between the NHCs of Models 2 and 3 capture model errors.

Consider a hypothetic simple RTO, for which the underlying assets are five (m = 5)actively-traded stocks listed on New York Stock Exchange: American Eagle Outfit-ters, Inc. (AEO), Ford Motor Co. (F), Host Hotels & Resorts, Inc. (HST), KeyCorp.(KEY) and Marathon Oil Corporation (MRO).10 The days to maturity is assumed tobe 120, i.e., n = 120. We employ the prior 120 daily returns (D = 120) to obtain

10 For the robustness test, we conducted the same analysis and obtained similar results for CSX Corp.(CSX), Cisco System, Inc. (CSCO), Fifth Third Bancorp (FITB), Applied Materials, Inc. (AMAT), andIntel Corporation (INTC) listed on Nasdaq. The corresponding results are available upon request.

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Table 5 NHC distributions for 120-day simple RTOs

Percentile Model 1 Model 2 Model 3

95th 7.6096 22.5522 15.4713

90th 5.0861 13.7307 12.7950

85th 3.5286 9.4700 8.6714

80th 2.6529 7.7838 6.5394

75th 1.9789 5.4413 4.4874

70th 1.4804 4.2276 3.2697

65th 1.0211 3.1118 1.8950

60th 0.7169 1.4048 0.6023

55th 0.1887 0.7035 −0.3440

50th −0.2141 −0.0649 −0.9761

45th −0.4543 −1.0766 −2.4820

40th −0.7239 −2.4985 −3.3699

35th −1.1282 −4.3367 −4.3573

30th −1.5441 −5.1788 −6.1049

25th −2.1499 −6.8774 −8.4050

20th −2.4414 −8.4475 −9.5043

15th −2.9331 −10.4216 −12.3365

10th −3.6393 −13.9477 −15.6985

5th −6.9030 −19.2110 −21.2140

Mean 0.4310 0.4998 −1.4594

RMSE of the above 19 NHCs Model 1 versus 2 Model 1 versus 3 Model 2 versus 3

6.4220 6.2573 2.0114

This table shows the NHC distribution for the 120-day simple RTO by performing DDHS 250 times. TheModel 1, Model 2, and Model 3 columns report respectively the NHC percentiles from 5 to 95% for thetwo option types. We also calculate the NHC means. In addition, the root mean squared errors (RMSEs)between the 19 NHC percentile values for different models are also reported. The RMSE results show thatthe NHC differences between Models 2 and 3 are relatively small, but the differences between the NHCsof Model 1 and those of Models 2 and 3 are more pronounced

σi and ρi j . The historical daily stock prices are collected from the Yahoo! Financewebsite covering 1996–2015, based on which we derive daily log returns of these fiveunderlying assets. Across the nearly 2500-day sample, we randomly choose 250 timepoints as t0. In other words, to obtain the NHC distribution, we repeat the calculation250 times. To make the NHC levels comparable among the 250 repetitions, all initialstock prices are normalized to 100, i.e., Si (t0) = 100, for i = 1, . . . ,m., and thestrike price X is always fixed at 100. In addition, the dividend yields of all underlyingassets are assumed to be zero. The NHC percentiles from 5 to 95% under differentmodels are reported in Table 5.

First, we observe that the NHC distributions under Model 1 center on zero withsmall standard deviations for the simple RTO. We expect this phenomenon since inthe ideal condition in Model 1, the cost of the DDHS converges to the initial optionvalues and therefore the NHC should be near zero. Second, for Models 2 and 3, due

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Fig. 10 Cumulative distribution functions of the NHC of the 120-day simple RTO under different models.This figure plots the cumulative distribution functions of the NHCs generated by the three models specifiedin Table 4. We also conduct the two-sample Kolmogorov–Smirnov test (K–S test) to examine statisticallywhether a pair of samples comes from the same distribution. The null hypothesis is that the two samplescome from the same distribution. The p values of K–S test are 1.2041e-08 (Model 1 vs. 2), 3.6814e-10(Model 1 vs. 3) and 0.3861 (Model 2 vs. 3), respectively

to the estimation and model errors, the standard deviations of the NHC distributionsincrease substantially, and the means of the NHCs of Models 2 and 3 could deviatefrom zero, e.g., the means of the NHCs for Model 3 is −1.4594. Third, by comparingthe root mean square errors (RMSEs) between the reported 19 percentile levels ofdifferent models, it is apparent that the NHC distributions in Models 2 and 3 are close,but both deviate significantly from those in Model 1. To further confirm this result in astatistical sense, we plot the empirical cumulative distribution functions of all modelsin Fig. 10 and perform the two-sample Kolmogorov–Smirnov test (K–S test), whichexamines whether a pair of samples comes from the same distribution. The three p-value results of the K–S test in Fig. 10 show that we cannot reject the hypothesis thatthe NHCs based on Models 2 and 3 come from the same distribution, but Model 1’sNHC distribution differs from those of Models 2 and 3. According to the evidence inTable 5 and Fig. 10, we conclude that the impact of the model error resulting from thelognormal distribution assumption of our model is much smaller than the estimationerror associated with σi and ρi j .

Due to the fact that the values of RTOs are highly sensitive to the levels of σi and ρi j ,it is natural that estimation errorwould play a significant role. As to the relativelyminormodel error, one possibility is that the lognormal distribution assumption performs notso poorly for the examined underlying assets in 1996–2015. In addition, we conjecturethat the unique feature of the trend price, which is calculated based on the least-squaresregression, may partially explain this phenomenon. If there are stochastic volatilityor jumps in actual stock returns, the actual distribution should have fatter tails thanthe assumed normal distribution, which means the probability to meet extreme returnsincreases. However, the regression-based trend price is theoretically less sensitive toa small number of outliers (or said extreme returns). Therefore, if the fat-tail effect

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caused by stochastic volatility or jumps is not that excessive, one can expect that itsinfluence on the expectation of the trend price and thus the values of RTOs wouldbe minor. As a result, the pricing and hedging performance of our model based on alognormal distribution assumption should not hurt too much. Finally, we suggest thatwhen the DDHS based on our pricing formulas is applied to hedge RTOs, the issuercan pay less attention to the model error but should prepare a greater buffer to absorbestimation errors associated with σi and ρi j .

4 Applications of RTOs

To illustrate the great potential benefits of rainbow trend/average options, this sectionintroduces some of their practical applications, not only for investments but also forrisk management. We also investigate how to apply rainbow trend/average options todesigning effective compensation plans for managers of corporations. Moreover, ouridea of rainbow trend options can be applied to modify the countercyclical capitalbuffer proposed by Basel Committee. In addition to those practical applications, theacademic application of rainbow trend/average option formulas in option pricing isalso explored.

4.1 Hedging timing risk and asset selection risk

Financial instruments offering both functions of hedging timing and asset selectionrisks are attractive to most investors, especially those who adopt a passive asset man-agement style. Most of them traditionally choose their investment targets from variousindex-related products. The rainbow option, or more specifically themaximum option,on different indexes can resolve the asset selection issue. However, the decisions ofwhen to enter into the rainbow option contract and its time to maturity could still influ-ence the final payoff significantly. The maximum option on the trend-based payoff canmitigate this timing issue, since the payoff function is associated with the asset pricethat performs, on average, the best over the entire option life. Let Ii denote the trendprice of the i th stock index calculated over the entire investment period. Then the RTOserving in the same period with payoff

max(I1, I2, · · · , Im

)

can satisfy the demand of investors adopting a passive asset management style.

4.2 Hedging price risks of multiple substitutions

RTOs can also be employed to hedge the price risks of different alternatives, espe-cially when the purchase date can only be predicted approximately but cannot beknown exactly. Suppose a firm is developing a new product. The design process isscheduled to be finished at time T , but the precise time of completion is not certain.To minimize the time to market, once the product development is complete, the firm

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will start manufacturing the new product immediately. Assume also that manufac-turing the product requires L units of a major material, which is available from mdifferent sources. To hedge in advance the price risk of purchasing the material, the

firm can consider a RTO with a payoff L ×(min

(S1,S2, . . . , Sm

)− X

)+, where Si

is the trend price of purchasing one unit of the material from the i th source and Xis the tolerable upper bound of the material’s unit price. The firm can receive somecompensation when the minimum trend price Si at T rises beyond the upper bound X .

The aforementioned RTO is more appropriate than a typical minimum rainbowoption with a payoff L × (min

(S1,T , S2,T , . . . , Sm,T

)− X)+. Due to the uncertain

product development completion time and thus the uncertain time for purchasing thematerial, the hedging performance of the minimum rainbow option could be poor ifthe material prices at the planned purchase time T and the actual purchasing timediffer significantly. In contrast, the trend prices Si can better forecast the prevailingasset prices in the period around T . Consequently, the hedging performance of theRTO is relatively stable in most scenarios, even when the actual purchasing date isuncertain.Moreover, comparedwith a geometric average rainbowoption (with a payoffL× (min

(S1,S2, . . . , Sm

)− X)+

), the RTO can again better forecast the prevailingasset prices around T , since the average price Si in [0, T ] is not a highly-correlatedproxy for the prevailing asset price around T . The potential gap between the geometricaverage prices during the option life and the asset prices on the actual purchasing datecould hurt the hedging performance of the geometric average rainbow option.

4.3 Designing executive compensation plans

Many studies find negative or insignificant relations between the adoption of equity-linked executive stock options (ESOs) and firm performance (e.g., Bizjak et al. 1993;Kohn 1993; Yermack 1995; Denis et al. 1997; Himmelberg et al. 1999; Ingersoll2006; Cooper et al. 2009). Two possible reasons are that the earnings or stock pricescould be manipulated by managers or it is inappropriate to measure the performanceof managers or firms using only stock prices. To address the above two issues, moreideal executive compensation plans could be achieved with RTOs. We provide twopossible designs based on RTOs: one emphasizes a firm’s general improvement indifferent aspects and the other exploits inter-firm comparisons to identify a manager’ssuperior performance.

The first design, to ensure overall improvement in a firm’s operation, proposes tomeasure managerial performance by multiple indexes, such as the stock price, salesrevenue, or market shares of major products. The firm can consider the compensationplan

(min

(I1/I1,0, I2/I2,0, . . . , Im/Im,0

)− X

)+,

where Ii/Ii,0 represents the normalized trend of the i th evaluating index. The managercan receive a bonus if all the trends of these indexes are higher than the minimumgrowth rateX .

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In the second design, if the board of directors is concerned about a firm’s com-petiveness with its major competitors, the following executive compensation plan canbe considered: (

S1S1,0

− max

(S2S2,0

,S3S3,0

, . . . ,SmSm,0

))+,

where S1/S1,0 and Si/Si,0 for 2 ≤i≤m denote the normalized trends of the stockprices of the firm and its competitors, respectively, in an industry. As long as the firm’strend-based performance dominates that of its competitors, the manager can receivethe bonus. We argue that this rainbow trend ESO is more capable of measuring amanager’s superior performance than the traditional indexed ESO,11 where the strikeprice is indexed to a benchmark to capture common risks beyond themanager’s control.First, similar to the index ESO, our rainbow trend ESO also eliminates common risksin an industry if a sufficiently large m is considered. Second, our rainbow trend ESOdirectly compares the performance of a firm and its main competitors, rather than theaverage performance of all firms (including better- and poorly performing ones) in anindustry. If the goal of an ESO is to measure a manager’s superior performance, it isunreasonable to compare the firm against poorly performing firms in the same industry.Last, using the trend performance rather than the stock price level at a future time canavoid the manipulation problem and thus measure a manager’s true performance.

4.4 Modification of the countercyclical capital buffer

BaselCommittee onBankingSupervision (BCBS)proposed the schemeof the counter-cyclical capital buffer (CCB) to protect the banking sector in a nation from a downturnafter excessive credit growth. In the proposal, banks are required to build up additionalequity capital, i.e., the CCB, when the ratio of It = Credit t/GDPt deviates posi-tively from its long-term trend, where GDPt and Credit t stand for, respectively, thegross domestic product and the aggregate credit granted to the household and privatenon-financial corporate sectors. For example, if It is higher than its trend It by 10%(usually occurring in a boom), a bank should prepare a CCB amount equal to 2.5%of its risk-weighted assets. This extra capital preparation can significantly reduce thecredit risk faced by the bank in the following recession. The details regarding to theCCB can refer to BCBS (2010a, b).

However, the cost to prepare the CCB is quite substantial and should be carriedout only when the excessive growth is positively verified. On the other hand, it iswell known that the use of GDP index alone may not be sufficient to reflect the trueperformance of an economy and thus the risk of false positive may be significant.To mitigate such a drawback, we can apply the idea of RTOs to further improvethe scheme of the CCB. For example, one can use multiple indexes to gauge theperformance of an economy, such as the levels of retail sales (RSt ), the Industrial Pro-duction Index (I P I t ), or even the S&P 500 index (SP I t ), in addition to the GDP. Bydefining I1,t = Credit t/GDPt , I2,t = Credit t/RSt , I3,t = Credit t/I P I t , I4,t =

11 Indexed ESOs were first proposed and discussed by Johnson and Tian (2000).

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Table 6 Our option formulas as control variates for variance reduction

Linear simple RTO Arithmetic-average Asianrainbow call option

σ1= 0.1 σ1= 0.2 σ1= 0.3 σ1= 0.1 σ1= 0.2 σ1= 0.3

Panel A: With control variates

S1(t0)= 90 11.5993 13.3153 16.0755 8.2273 9.7823 12.0100

(0.0043) (0.0038) (0.0118) (0.0007) (0.0010) (0.0015)

S1(t0)= 100 13.8562 16.5907 20.2064 8.5562 10.5248 13.1212

(0.0042) (0.0070) (0.0108) (0.0010) (0.0010) (0.0017)

S1(t0)= 110 19.0958 21.8088 25.731 8.3481 10.8533 13.9373

(0.0050) (0.0040) (0.0165) (0.0008) (0.0010) (0.0018)

Panel B: Without control variates

S1(t0)= 90 11.5961 13.3039 16.0755 8.2192 9.7814 12.0151

(0.0635) (0.0700) (0.0758) (0.0338) (0.0467) (0.0910)

S1(t0)= 100 13.8589 16.59 20.24 8.5447 10.5524 13.1323

(0.0640) (0.0892) (0.1015) (0.0532) (0.0473) (0.0972)

S1(t0)= 110 19.1076 21.8324 25.7083 8.3529 10.8394 13.9685

(0.0743) (0.0975) (0.1068) (0.0460) (0.0545) (0.0968)

Panel C: Reduction in standard errors

S1(t0)= 90 93.3071% 94.6429% 84.4884% 97.7778% 97.8610% 98.3516%

S1(t0)= 100 93.3594% 92.1569% 89.4089% 98.1221% 97.8836% 98.2005%

S1(t0)= 110 93.2660% 95.8974% 84.5433% 98.3696% 98.1651% 98.1912%

This table presents the pricing results of linear simple RTOs and arithmetic-average Asian rainbow calloptions based on the Monte Carlo simulation with 50,000 paths and 20 repetitions. The examined valuesof parameters are m= 2, r = 0.05, q1 = q2 = 0, T = 1 year, S2 (t0) = 100, σ2 = 0.2, X = 100, andρ12 = 0.5. In addition to option values, the corresponding standard errors of the simulation results areshown in parentheses. In Panel (A), we employ the values of the exponential simple RTO and geometric-average Asian rainbow call option generated from our pricing formula in Eq. (15) as the control variatesfor pricing the linear simple RTO and arithmetic-average Asian rainbow call option, respectively

Credit t/SP I t , one scheme that mitigates the above drawback can then be designedas follows. We use the event that

min

(I1,t − I1,t

I1,t,I2,t − I2,t

I2,t,I3,t − I3,t

I3,t,I4,t − I4,t

I4,t

)

is larger than 10%, i.e., when all ratios are higher than their individual trend levels by10%, as a true signal of the excessive credit growth, instead of only using GDP. Oncethe excessive credit growth is verified, and banks should prepare the CCB amountequal to 2.5% of their risk-weighted assets.

4.5 Constructing control variates

In addition to the above practical applications, the general pricing formula we derivecan be used to resolve several academic issues. For example, there are no closed-form

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formulas for the linear RTO12 or the arithmetic-average Asian rainbow option and itis almost infeasible to price these options with the lattice model due to the dimen-sionality curse. The Monte Carlo simulation seems to be the only feasible method forpricing them. However, its efficiency and accuracy depend crucially on the number ofsimulations, which implies it is time-consuming to obtain satisfactory results. Extend-ing the idea of Kemna and Vorst (1990), the option prices for exponential RTOs andgeometric-average Asian rainbow call options can be used as the control variates toreduce the variances of the simulation results of linear RTOs13 and arithmetic-averageAsian rainbow call options,14 respectively. The results in Table 6 verify the efficiencyof this control variates method. With the option values generated from our pricingformula in Eq. (15) as the control variates, the standard errors can be reduced by 91.23and 98.10%, on average, for pricing linear simple RTOs and arithmetic-average Asianrainbow call options, respectively.

5 Conclusion

This paper introduces a newclass of option contracts by combining trend-based payoffswith rainbow options. RTOs can simultaneously solve investors’ asset selection andtiming problems. We construct a pricing framework that can facilitate the derivationof the closed-form formulas for three different types of RTOs: simple, pure, and AsianRTOs. Our framework is general and able to evaluate rainbow options on the payoffsdetermined by different weighting averages of discrete-sampling asset returns. Due tothe analytic property of the option pricing formulas, the formulas of the Greeks canbe derived and analyzed as well. Although the lognormal distribution assumption inthis paper cannot count for stochastic volatilities or jumps in underlying asset prices,the performance of the dynamic delta hedging strategy based on our model is littleaffected by this inexact distribution assumption. On the other hand, issuers of RTOsshould paymore attention tomanaging the estimation error for input parameters of ourpricing formulas. This paper also explores possible applications of RTOs, includinghedging the price risk of multiple substitutions, designing executive compensationplans, modifying countercyclical capital buffer proposed by Basel Committee, andacting as a control variate for pricing arithmetic average rainbow options or linearRTOs based on the Monte Carlo simulation. Due to the desired properties and wide

12 Precisely speaking, this paper considers the exponential trend defined in Eq. (5). For the linear trend,the following regression equation should be considered:

Si (th) − Si (t0) = Bi (th − t0) + ε, for h = 0, . . . , n.

Suppose the least-squares regression coefficient is denoted as Bi . The linear trend price at T can be expressedas Si (T ) = Si (t0)+ Bi (T − t0). The advantage of the linear trend is that it ismore intuitive for the investingpublic to understand.

13 The payoff of a linear rainbow trend option is(Sm′ (T ) − X

)+, where m


Si (T ). The

previous footnote presents the definition of the linear trend price Si (T ).14 Denote the arithmetic average price of asset i as Si (T ). Then the payoff of an arithmetic-average Asian

rainbow call option can be expressed as(Sm′ (T ) − X

)+, where m


¯Si (T ).

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applicability of this new class of options, this paper contributes substantially to boththe theoretical and practical literature.

Acknowledgements The authors thank the Ministry of Science and Technology of Taiwan for financialsupport.

Appendix 1: Derivations of the correlations ρX,i j , ρY,i j , and ρXY,i

The evaluation of ρX,i j , ρY,i j , and ρXY,i uses the definition of the correlation coeffi-cient. For ρX,i j ,

ρX,i j = corr(ln Xi (T ), ln X j (T )

)

= cov(ln Xi (T ), ln X j (T )

)√Var (ln Xi (T ))

√Var(ln X j (T )

)

= cov(αa+1Ri,a+1 + · · · + αn Ri,n, αa+1R j,a+1 + · · · + αn R j,n

)√

σ 2i �t

∑nh=a+1 α2

h

√σ 2j �t

∑nh=a+1 α2

h

=∑n

h=a+1 α2hcov

(Ri,h, R j,h

)+∑a+1≤h1 �=h2≤n αh1αh2cov(Ri,h1 , R j,h2

)

σiσ j∑n

h=a+1 α2h�t

=∑n

h=a+1 α2hρi jσiσ j�t

σiσ j∑n

h=a+1 α2h�t

= ρi j .

Similarly, the correlation between ln Yi (T )and ln Y j (T ), ρY,i j , is also ρi j . As for thecorrelation between ln Xi (T ) and ln Yi (T ),

ρXY,i = corr (ln Xi (T ), ln Yi (T ))

= cov (ln Xi (T ), ln Yi (T ))√Var (ln Xi (T ))

√Var (ln Yi (T ))

= cov(αa+1Ri,a+1 + · · · + αn Ri,n, βa+1Ri,a+1 + · · · + βn Ri,n

)√

σ 2i �t

∑nh=a+1 α2

h

√σ 2i �t

∑nh=a+1 β2

h

=∑n

h=a+1 αhβhcov(Ri,h, Ri,h

)

σ 2i �t

√∑nh=a+1 α2

h

√∑nh=a+1 β2

h

=

n∑h=a+1

αhβhσ2i �t

σ 2i �t

√∑nh=a+1 α2

h

√∑nh=a+1 β2

h

=∑n

h=a+1 αhβh√n∑

h=a+1α2h

√∑nh=a+1 β2

h

.

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Appendix 2: Derivations of the general pricing formula

In Eq. (14), the arbitrage-free price of the examined option is

V (ta) = e−rτa∑m

i=1EQ[Xi (T ) · I{Xi (T )≥X j (T )}1≤ j �=i≤m

⋂{Xi (T )≥KYi (T )}]

︸︷︷︸I1

− Ke−rτa∑m

i=1EQ[Yi (T ) · I{Xi (T )≥X j (T )}1≤ j �=i≤m

⋂{Xi (T )≥KYi (T )}]

︸︷︷︸I2

. (34)

Next, we evaluate I1 and I2 separately.According to the definitions of ln Xi (T ) and ln Yi (T ) in Sect. 2, Xi (T ) and Yi (T )

can be expressed as

Xi (T ) = exp(μXi + σXi ε

QXi

)and Yi (T ) = exp

(μYi + σYi ε

QYi

),

where

μXi = α0 ln Si (t0) +∑a

h=1αh

ln Si (th)

ln Si (th−1)

+μi�t∑n

h=a+1αh,

σ 2Xi

= σ 2i �t

∑n

h=a+1α2h,

μYi = β0 ln Si (t0) +∑a

h=1βh

ln Si (th)

ln Si (th−1)

+μi�t∑n

h=a+1βh,

σ 2Yi = σ 2

i �t∑n

h=a+1β2h ,

and εQXi

and εQYi

follow the standard normal distribution. In addition, μi�t =(r − qi − σ 2

i2

)�t and σ 2

i �t are the mean and variance, respectively, of the loga-

rithm of the return of the i th underlying asset in each time interval. The correlationbetween ε

QXi

and εQX j

is ρX,i j , the correlation between εQYi

and εQY j

is ρY,i j , and the

correlation between εQXi

and εQYiis ρXY,i .

Evaluating I1 in Eq. (34)

We have

I1 = EQ[Xi (T )·I{Xi (T )≥X j (T )}1≤ j �=i≤m

⋂{Xi (T )≥KYi (T )}]

= EQ[eμXi +σXi ε

QXi ·I{Xi (T )≥X j (T )}1≤ j �=i≤m

⋂{Xi (T )≥KYi (T )}]

123



= EQ[eμXi + 1

2 σ 2Xi

+σXi εQXi

− 12 σ 2

Xi ·I{Xi (T )≥X j (T )}1≤ j �=i≤m

⋂{Xi (T )≥KYi (T )}]

= eμXi + 1

2 σ 2Xi EQ

[e− 1

2 σ 2Xi

+σXi εQXi ·I{Xi (T )≥X j (T )}1≤ j �=i≤m

⋂{Xi (T )≥KYi (T )}]

.

Next we apply the technique of changing measures by introducing the Radon–Nikodym derivative15

ηXi = dQXi

dQ= e

− 12 σ 2

Xi+σXi ε

QXi

with respect to the risk-neutral measure Q and an auxiliary measure QXi that isequivalent to the risk-neutral measure Q. Then we can write

I1 = eμXi + 1

2 σ 2Xi EQXi

[I{Xi (T )≥X j (T )}1≤ j �=i≤m

⋂{Xi (T )≥KYi (T )}]

= eμXi + 1

2 σ 2Xi PQXi

({Xi (T )≥X j (T )

}1≤ j �=i≤m

⋂{Xi (T ) ≥ KYi (T )}

),

where EQXi [·] is the expectation operator under the measure QXi . To evaluateEQXi [IA], where A denotes any set of events, we proceed with the distributional prop-erty of the underlying assets, that is, EQXi [IA] = PQXi (A). In addition, according tothe Girsanov theorem, we derive

εQXi

= εQXiXi

+ σXi , εQX j

= εQXiX j

+ ρX,i jσXi , εQYi

= εQXiYi

+ ρXY,iσXi ,

where εQXiXi

, εQXiX j

, and εQXiYi

follow the standard normal distribution under themeasureQXi and thus Xi , X j , and Yi can be expressed, respectively, under the measure QXi

as

Xi (T ) = exp(μXi + σ 2

Xi+ σXi ε

QXiXi

),

X j (T ) = exp(μX j + ρX,i jσXi σX j + σX j ε

QXiX j

),

Yi (T ) = exp(μYi + ρXY,iσYi σXi + σYi ε

QXiYi

).

Consequently, I1 can be evaluated as

I1 = eμXi + 1

2 σ 2Xi PQXi

({ZQXiX j ,Xi

≤ dQXiX j ,Xi

}1≤ j �=i≤m

, ZQXiYi ,Xi

≤ dQXiYi ,Xi

)

= eμXi + 1

2 σ 2Xi Nm

({dQXiX j ,Xi

}1≤ j �=i≤m

, dQXiYi ,Xi

; RQXi

),

15 This measure change technique was first proposed by Cameron and Martin (1944). Moreover, thistechnique is a special case of the Girsanov theorem, which is applied to changing measures for stochasticprocesses.

123



where

ZQXiX j ,Xi

≡σX j ε

QXiX j

− σXi εQXiXi√

var(σX j ε

QXiX j

− σXi εQXiXi

)

=σX j ε

QXiX j

− σXi εQXiXi

σX,i j, for 1 ≤ j �= i ≤ m,

ZQXiYi ,Xi

≡ σYi εQXiYi

− σXi εQXiXi√

var(σYi ε

QXiYi

− σXi εQXiXi

)

= σYi εQXiYi

− σXi εQXiXi

σXY,i,

dQXiX j ,Xi

=(μXi + σ 2

Xi

)− (μX j + ρX,i jσXi σX j

)

σX,i j, for 1 ≤ j �= i ≤ m,

dQXiYi ,Xi

=(μXi + σ 2

Xi

)− (μYi + ρXY,iσXi σYi

)− ln K

σXY,i,

and

RQXi =(

(I )(m−1)×(m−1) (I I )(m−1)×1

(I I )′1×(m−1) (I I I )1×1

)

m×m

,

with

(I )(m−1)×(m−1) =(corr

(ZQXiXl ,Xi

, ZQXiX j ,Xi

))

(m−1)×(m−1)

=(

σ 2Xi

− ρX,ilσXi σXl − ρX,i jσXi σX j + ρX, jlσX j σXl

σX,i jσX,il

)

(m−1)×(m−1)

,

for 1 ≤ l �= i ≤ m and 1 ≤ j �= i ≤ m,

(I I )(m−1)×1 =(corr

(ZQXiX j ,Xi

, ZQXiYi ,Xi

))

(m−1)×1

=(

σ 2Xi

− ρXY,iσXi σYi − ρX,i jσXi σX j + ρX,i jρXY,iσX j σYi

σX,i jσXY,i

)

(m−1)×1

,

for 1 ≤ j �= i ≤ m,

(I I I )1×1 =(corr

(ZQXiYi ,Xi

, ZQXiYi ,Xi

))

1×1= 1.

Note that we define σ 2X,i j ≡ σ 2

Xi− 2ρX,i jσXi σX j + σ 2

X jand σ 2

XY,i ≡ σ 2Xi

−2ρXY,iσXi σY j + σ 2

Yito obtain the above formulas.

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Evaluating I2 in Eq. (34)

We have

I2 = EQ[Yi (T )·I{Xi (T )≥X j (T )}1≤ j �=i≤m

⋂{Xi (T )≥KYi (T )}]

= EQ[eμYi +σYi ε

QYi ·I{Xi (T )≥X j (T )}1≤ j �=i≤m

⋂{Xi (T )≥KYi (T )}]

= EQ[eμYi + 1

2 σ 2Yi

+σYi εQYi

− 12 σ 2

Yi ·I{Xi (T )≥X j (T )}1≤ j �=i≤m

⋂{Xi (T )≥KYi (T )}]

= eμYi + 1

2 σ 2Yi EQ

[e− 1

2 σ 2Yi

+σYi εQYi ·I{Xi (T )≥X j (T )}1≤ j �=i≤m

⋂{Xi (T )≥KYi (T )}]

.

Similarly, we introduce a Radon–Nikodym derivative for the change from the risk-neutral measure Q to an auxiliary measure QYi that is equivalent to the risk-neutralmeasure Q:

ηYi = dQYi

dQ= e

− 12 σ 2

Yi+σYi ε

QYi .

Then we can write

I2 = eμYi + 1

2 σ 2Yi EQYi

[I{Xi (T )≥X j (T )}1≤ j �=i≤m

⋂{Xi (T )≥KYi (T )}]

= eμYi + 1

2 σ 2Yi PQYi

({Xi (T )≥X j (T )

}1≤ j �=i≤m

⋂{Xi (T ) ≥ KYi (T )}

),

where EQYi [·] and PQYi (·) are the expectation and probability operator, respectively,under the equivalent measure QYi . Using the transformations

εQXi

= εQYiXi

+ ρXY,iσYi , εQX j

= εQYiX j

+ ρXY,iρX,i jσYi , εQYi

= εQYiYi

+ σYi ,

based on theGirsanov theorem,where εQYiXi

, εQYiX j

, and εQYiYi

follow the standard normaldistribution, we can express Xi , X j , and Yi under the measure QYi as, respectively,

Xi (T ) = exp(μXi + ρXY,iσYi σXi + σXi ε

QYiXi

),

X j (T ) = exp(μX j + ρXY,iρX,i jσYi σX j + σX j ε

QYiX j

),

Yi (T ) = exp(μYi + σ 2

Yi + σYi εQYiYi

).

Consequently, I2 can be evaluated as

I2 = eμYi + 1

2 σ 2Yi PQYi

({ZQYiX j ,Xi

≤ dQYiX j ,Xi

}1≤ j �=i≤m

,ZQYiYi ,Xi

≤ dQYiYi ,Xi

)

= eμYi + 1

2 σ 2Yi Nm

({dQYiX j ,Xi

}1≤ j �=i≤m

,dQYiYi ,Xi

; RQYi

),

123



where

ZQYiX j ,Xi

≡σX j ε

QYiX j

− σXi εQYiXi√

var(σX j ε

QYiX j

− σXi εQYiXi

)

=σX j ε

QYiX j

− σXi εQYiXi

σX,i j, for 1 ≤ j �= i ≤ m,

ZQYiYi ,Xi

≡ σYi εQYiYi

− σXi εQYiXi√

var(σYi ε

QYiYi

− σXi εQYiXi

)

= σYi εQYiYi

− σXi εQYiXi

σXY,i,

dQYiX j ,Xi

=(μXi + ρXY,iσYiσXi

)− (μX j + ρX,i jρXY,iσYi σX j

)

σX,i j, for 1 ≤ j �= i ≤ m,

dQYiYi ,Xi

=(μXi + ρXY,iσYiσXi

)−(μYi + σ 2

Yi

)− ln K

σXY,i,

and

RQYi =(

(I )(m−1)×(m−1) (I I )(m−1)×1

(I I )′1×(m−1) (I I I )1×1

)

m×m

,

with

(I )(m−1)×(m−1) =(corr

(ZQYiXl ,Xi

,ZQYiX j ,Xi

))(m−1)×(m−1)

=(corr

(ZQXiXl ,Xi

, ZQXiX j ,Xi

))(m−1)×(m−1)

,

for 1 ≤ l �= i ≤ m and 1 ≤ j �= i ≤ m,

(I I )(m−1)×1 =(corr

(ZQYiX j ,Xi

, ZQYiYi ,Xi

))(m−1)×1

=(corr

(ZQXiX j ,Xi

, ZQXiYi ,Xi

))(m−1)×1

,

for 1 ≤ j �= i ≤ m,

(I I I )1×1 =(corr

(ZQYiYi ,Xi

, ZQYiYi ,Xi

))1×1

=(corr

(ZQXiYi ,Xi

, ZQXiYi ,Xi

))1×1

= 1.

123



Note that the correlation matrix RQYi in I2 is identical to the correlation matrix RQXi

in I1. Finally, we combine everything to derive the general pricing formula

V (ta) = e−rτa∑m

i=1eμXi + 1

2 σ 2Xi Nm

({dQXiX j ,Xi

}1≤ j �=i≤m

, dQXiYi ,Xi

; RQXi

)

− Ke−rτa∑m

i=1eμYi + 1

2 σ 2Yi Nm

({dQYiX j ,Xi

}1≤ j �=i≤m

, dQYiYi ,Xi

; RQYi

).

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