Rainbow Trend Options: Valuation and Applications

Rainbow trend options: valuation and applications

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

Review of Derivatives Research

ISSN 1380-6645 Volume 20 Number 2

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

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1 23 Author's personal copy

Rev Deriv Res (2017) 20:91–133 DOI 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. To resolve these issues simultaneously, a new class of rainbow trend options is proposed. The diversiﬁcation effect of rainbow options can reduce the importance of asset selection decisions and trend options can mitigate unfavorable effects on market entry and exit decisions. We consider a general framework to facilitate the derivation of analytic pricing formulas for simple, pure, and Asian rainbow trend options using the martingale pricing method. The properties of these options and their Greeks are analyzed. We also investigate the performance of the dynamic delta hedging strategy for issuers of rainbow trend options. Last, this paper explores the applications of rainbow trend options for hedging price risks, designing executive stock options, modifying countercyclical capital buffer proposed by Basel Committee, and acting as control variates of the Monte Carlo simulation.

Keywords Rainbow option · Trend option · Timing risk · Asset selection · Martingale pricing method

B Jr-Yan Wang [email protected] Hsiao-Chuan Wang [email protected] Yi-Chen Ko [email protected] Mao-Wei Hung [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 123 Author's personal copy

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JEL Classiﬁcation G13

1 Introduction

This paper proposes and examines a new class of rainbow trend options (RTOs) by generalizing the simple, pure, and Asian trend options introduced by Leippold and Syz (2007). The motive for studying RTOs is their ability to simultaneously resolve the issues of how to choose a better-performing asset and how to time the market, the two central concerns for all investors. It is well known that rainbow options, defined as options with payoffs involving several asset prices, can solve the asset selection problem, especially if the maximum or minimum payoff function is considered. In addition, trend options can make the timing decision less important by linking the payoffs with the average trend of the asset price instead of the final value of the asset price at maturity. To meet investors’ demands, the combination of rainbow and trend options offers a promising solution and the resulting product is the RTO. In practice, financial institutions can incorporate RTOs into their structured-note products to satisfy the risk-return appetites of individual investors. Furthermore, the introduction of this new class of derivatives also can enhance the completeness and efficiency of a financial market. RTOs are useful not only for investment purposes but also in many other applications. For example, they can be employed to hedge the price risk of multiple substitutions or to design more effective executive compensation plans. For example, when designing executive compensation plans, multi-index measurement approaches are widely used to evaluate a manager’s overall performance. Traditionally, a minimum option can be adopted such that, as long as the manager can improve all evaluating indexes by a minimum target, the manager can receive a bonus. However, to examine the evaluating indexes at only one time point could lead to manipulation problems and thus cannot reflect the firm’s true improvement. Therefore, a minimum option on the average trends of, for example, the firm’s equity price, sales revenue, and market share can be employed to measure a manager’s true performance and thus design a more effective compensation plan. Section 4 discusses several practical as well as academic applications of RTOs. From a theoretical point of view, RTOs are desirable to investors due to their diversification effects over different assets and time. However, maybe due to the complexity of combining the pricing techniques for rainbow and trend options, RTOs are not explored in the literature. Stulz (1982) first derives the analytic pricing formula for rainbow options on the maximum or minimum of two assets by solving partial differential equations. Options on the maximum or minimum of multiple assets are evaluated by Johnson (1987), who reinterprets Margrabe’s (1978) method for pricing exchange options to derive the analytic pricing formulas for rainbow options. Ouwehand and West (2006) use the change-of-numeraire technique to derive pricing formulae for various rainbow options. Regarding trend options, Leippold and Syz (2007)usethe technique of changing probability measures to develop the pricing formulas for three types of trend-based derivatives, namely, simple, pure, and Asian trend options. This paper employs the martingale pricing method proposed by Harrison and Kreps (1979) to unify the pricing models for rainbow and trend options. We consider a 123 Author's personal copy

Rainbow trend options: valuation and applications 93 highly general framework for the payoffs of RTOs and develop an option pricing formula under this general framework. Therefore, the proposed pricing formula can be employed to evaluate many trend/average options under single- and multiple-asset cases. Moreover, due to the analytic property of the closed-form solution, we can derive the formulas for the Greeks of RTOs and analyze their properties. Knowledge of the behavior of the Greeks of RTOs is indispensable in the risk management of issuers of RTOs. Wu and Zhang (1999) also try to incorporate path-dependent features into rainbow options. They study options on the minimum/maximum of two geometric average prices. 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 than those of the geometric average price over the same period, which implies that the trend option offers more upside potential than the geometric average option and is thus more suited for investment purposes. Second, Leippold and Syz (2007) argue that the trend price is an attractive alternative to the terminal asset price in option payoffs to mitigate issues of when to enter and exit ﬁnancial markets. This is because the mean of the trend price is identical to that of the terminal asset price at issuance and the variance of the trend price will gradually decrease with time. Since our paper is motivated by the idea of using options to simultaneously resolve the issues of asset and timing selection, we choose the trend option rather than the average option to incorporate with the multiple-asset rainbow option. Last, Wu and Zhang (1999) derive the corresponding option pricing formulas by solving bivariate partial differential equations. Nevertheless, it is difﬁcult to extend their method to evaluate multiple- asset rainbow options or other types of average/trend options. In contrast, our pricing formula for RTOs is highly general and can encompass Wu and Zhang’s formulas for options on the minimum/maximum of two geometric average prices as a special case. This paper is organized as follows. Section 2 evaluates options with both rainbow and trend/average features and derives a general option pricing formula. It also presents the formulas for the three proposed types of RTOs, namely, the simple, pure, and Asian RTOs, and their Greeks. The numerical results in Sect. 3 analyze the properties of the RTO values and their Greeks and investigate the empirical hedging performance based on our option pricing formulas from the viewpoint of issuers of RTOs. Section 4 presents practical and academic applications based on RTOs. Section 5 concludes this paper.

2 Valuation of RTOs

2.1 Basic settings

This paper derives the pricing formula in the complete market setting speciﬁed by Black and Scholes (1973). We consider RTOs on m risky underlying assets. By denot- Q( ) ing Wi t as the standard Brownian motion for the ith underlying asset under the risk-neutral probability measure Q, the processes of the underlying asset prices under Q are postulated as 123 Author's personal copy

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( ) = ( − ) ( ) + σ ( ) Q( ), = ,..., , dSi t r qi Si t dt i Si t dWi t for i 1 m (1) where r is the risk-free interest rate and qi and σi are the dividend yield and the price Q( ) volatility of the ith asset, respectively. The standard Brownian motions Wi t and Q( ) ρ W j t are correlated with the coefﬁcient ij. 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, + arg max V S(T ) = Sˆ (T ) − X , where m = Sˆ (T ); (2) m 1 ≤ i ≤ m i for pure RTOs, + arg max V P (T ) = Sˆ (T ) − S (T ) , where m = Sˆ (T ); (3) m m 1 ≤ i ≤ m i and for Asian RTOs, + arg max V A(T ) = Sˆ (T ) − S¯ (T ) , where m = Sˆ (T ), (4) m m 1 ≤ i ≤ m i

ˆ ¯ with X representing the strike price and Si (T ) and Si (T ) representing the trend and geometric average variables, respectively, of the ith underlying asset over the option ˆ ¯ life, [0, T ]. The exact deﬁnitions of Si (T ) and Si (T ) are introduced later. To evaluate these 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 rainbow trend/average options. In it, the option life, [0, T ], is partitioned into n time intervals, = T each of length t n . In addition, the sampling dates for the trend/average options include t0(= 0), t1(= t),...,tn(= nt = T ). According to Leippold and Syz (2007), the trend variable of the ith asset at maturity can be obtained by solving the following exponential regression model

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

ε where is a standard white noise. Next, the ordinary least squares method is employed 2 ˆ n Si (th ) to solve the estimator B = arg min ∈ = ln − B (t − t ) . Equipped i Bi R h 0 Si (t0) i h 0 ˆ with Bi , the exponential trend at maturity can be expressed as 123 Author's personal copy

Rainbow trend options: valuation and applications 95 ˆ ( − ) n Si (th) Sˆ (T ) = S (t )eBi T t0 = S (t ) c , i i 0 i 0 exp = h ln (6) h 1 Si (th−1)

n − 1 12 k=h k t T ( − + ) where c = (T − t ) n b = 2 T = 6h n h 1 , for h = h 0 k=h k n(n+1)(n+2)(t)2 (n+1)(n+2) 1, 2,...,n. To evaluate different types of rainbow trend/average options, we consider a general payoff function

+ arg max V (T ) = X (T ) − K · Y (T ) , where m = X (T ), (7) m m 1 ≤ i ≤ m i

(α R , +α R , +α R , +···+α R , ) K is a constant, and for i = 1,...,m, Xi (T ) = e 0 i 0 1 i 1 2 i 2 n i n and (β R , +β R , +β R , +···+β R , ) Yi (T ) = e 0 i 0 1 i 1 2 i 2 n i n with arbitrary real numbers αh and βh that are not all zero for different values of h. The variable Ri,0 is deﬁned as ln Si (t0) and Si (th ) Ri,h ≡ ln is the logarithm of the return of the ith underlying asset for the Si (th−1) interval (th−1 , th] with the posited normal distribution 2 1 2 2 Ri,h∼N μi t,σ t = N r − qi − σ t,σ t , for h = 1, 2,...,n. i 2 i i

Based on the above deﬁnitions, at time ta = at, for 0 ≤ a ≤ n, we obtain ln X (T )∼N μ ,σ2 and ln Y (T )∼N μ ,σ2 , where i Xi Xi i Yi Yi a S (t ) n μ = α S (t ) + α i h + μ t α , Xi 0 ln i 0 = h ln i = + h (8) h 1 Si (th−1) h a 1 n σ 2 = σ 2t α2, (9) Xi i h=a+1 h a S (t ) n μ = β S (t )+ β i h + μ t β , Yi 0 ln i 0 = h ln i = + h (10) h 1 Si (th−1) h a 1 n σ 2 = σ 2t β2. (11) Yi i h=a+1 h

We also deﬁne ρX,ij as the correlation between ln Xi (T ) and ln X j (T ), ρY,ij as the correlation 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,ij = ρY,ij = ρij, (12) and n α β h=a+1 h h ρXY,i = . (13) n α2 n β2 h=a+1 h h=a+1 h

Appendix 1 shows the proofs of Eqs. (12) and (13). 123 Author's personal copy

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According to the risk-neutral valuation argument, the arbitrage-free price of the rainbow trend/average option with the general payoff function in Eq. (7) is the expected present value of the payoff at maturity under the risk-neutral measure. We can therefore express the option value with time to maturity τa = T − ta as

− τ ( ) = r a Q ( ) V ta e E [V T ] − τ = r a Q ( ) − · ( ) · e E Xm T K Ym T arg max I m = Xi (T ) X (T )≥K ·Y (T ) 1 ≤ i ≤ m m m −rτa Q arg max = e E X (T )·I m = Xi (T ) m X (T )≥K ·Y (T ) 1 ≤ i ≤ m m m −rτa Q arg max − e E KY (T )·I m = Xi (T ) m X (T )≥K ·Y (T ) 1 ≤ i ≤ m m m − τ m = r a Q ( )· e E Xi T I{X (T )≥X (T )} {X (T )≥K ·Y (T )} i=1 i j 1≤ j=i≤m i i m −rτa Q − Ke E Yi (T )·I{ ( )≥ ( )} { ( )≥ · ( )} , i=1 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 in braces is true. The summation over i represents different scenarios in which the trend price of the ith 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

m μ + 1 σ 2 Q Q −rτa Xi 2 X Xi Xi Q X V (ta) = e e i Nm d , , d , ; R i X j Xi 1≤ j=i≤m Yi Xi i=1 m μ + 1 σ 2 Q Q −rτa Yi 2 Y Yi Yi QY −Ke e i Nm d , , d , ; R i , (15) i=1 X j Xi 1≤ j=i≤m Yi Xi • • 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 Q Xi 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

2 2 2 σ ≡ σ − 2ρ , σ σ + σ (16) X,ij Xi X ij Xi X j X j and

2 2 2 σ ≡ σ − 2ρ , σ σ + σ (17) XY,i Xi XY i Xi Yi Yi 123 Author's personal copy

Rainbow trend options: valuation and applications 97 and employ the terms μX ,σX ,μY ,σY ,ρX,ij,ρY,ij, and ρXY,i deﬁned in Eqs. (8)– i i• i i• (13), respectively, to express d•,• and R as μ + σ 2 − μ + ρ , σ σ Q Xi X X j X ij Xi X j Xi = i , ≤ = ≤ , dX ,X for 1 j i m j i σX,ij μ + σ 2 − μ + ρ σ σ − Xi X Yi XY,i Xi Yi ln K Q Xi i d , = , Yi Xi σ XY,i Q μX + ρXY,i σY σX − μX + ρX,ijρXY,i σY σX Yi = i i i j i j , ≤ = ≤ , dX ,X for 1 j i m j i σX,ij μ + ρ , σ σ − μ + σ 2 − Q Xi XY i Yi Xi Yi Y ln K d Yi = i , Yi, Xi σ , XY i ( ) ( ) Q Q I (m−1)×(m−1) II (m−1)×1 R Xi = R Yi = , (II) 1 1×(m−1) m×m where σ 2 − ρ , σ σ − ρ , σ σ + ρ , σ σ Xi X il Xi Xl X ij Xi X j X jl X j Xl (I )(m−1)×(m−1) = , σX,ijσX,il (m−1)×(m−1) ≤ = ≤ ≤ = ≤ , for 1 j i m and 1 l i m σ 2 − ρ , σ σ − ρ , σ σ + ρ , ρ , σ σ Xi XY i Xi Yi X ij Xi X j X ij XY i X j Yi (II)(m−1)×1 = , σX,ijσXY,i (m−1)×1 for 1 ≤ j = i ≤ m.

We also derive the formulas for the Greeks of the rainbow trend/average options based on the general pricing formula in Eq. (15). The delta with respect to the ith underlying asset at time ta is ∂ ( ) α 1 2 V ta a −rτa +μX + σ Q X Q X ( ) = = i 2 Xi i , i ; Q Xi i ta e Nm dX ,X dY ,X R ∂ Si (ta) Si (ta) j i 1≤ j=i≤m i i β 1 2 K a −rτa +μY + σ QY QY − i 2 Yi i , i ; QYi . e Nm dX ,X dY ,X R (18) Si (ta) j i 1≤ j=i≤m i i

Equipped with the formula for delta, the derivation of the formulas of gamma and cross gamma can be continued. The gamma of the rainbow trend/average option with respect to the ith underlying asset at time ta is given by

∂ ( ) α (α − ) 1 2 i ta a a 1 −rτa +μX + σ ( ) = = i 2 Xi i ta 2 e ∂ Si (ta) S (t ) i a Q Q Xi Xi Q X ×Nm d , , d , ; R i X j Xi 1≤ j=i≤m Yi Xi 123 Author's personal copy

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α 1 2 a −rτa +μX + σ + e i 2 Xi S (t ) i a ∂ Q X Q X × i , i ; Q Xi Nm dX ,X dY ,X R ∂ Si (ta) j i 1≤ j=i≤m i i β (β − ) 1 2 K a a 1 −rτa +μY + σ − e i 2 Yi S (t )2 i a Q Q Yi Yi QY ×Nm d , , d , ; R i X j Xi 1≤ j=i≤m Yi Xi β 1 2 K a −rτa +μY + σ − e i 2 Yi S (t ) i a ∂ QY QY × i , i ; QYi . Nm dX ,X dY ,X R (19) ∂ Si (ta) j i 1≤ j=i≤m i i

Note that the second and fourth terms of this formula need the calculation of the derivatives of the MSNCDFs. This paper adopts the integral reduction technique of Curnow and Dunnett (1962) to complete the calculation. Consider a MSNCDF Nm (d1(x), d2(x),...,dm(x); R(x)) with upper limits d1(x), d2(x),...,dm(x) and ( ) ≡ ρ ( ) a non-singular correlation matrix R x pq 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 x can represent Si (ta), σ, τa,orr for calculating the different Greeks of RTOs. Follow- ing Curnow and Dunnett, the derivative of Nm (d1(x), d2(x), ··· , dm(x); R(x)) with respect to x is

∂ m ∂ N ∂d (x) N (d (x), d (x), ··· , d (x); R(x)) = m k m 1 2 m = ∂x k 1 ∂dk(x) ∂x m m ∂ ∂ρ (x) + Nm pq , = = p 1 q 1 ∂ρpq(x) ∂x where

∂ Nm ˆ ˆ ˆ ˆ ˆ = Nm−1 d1,k(x), ··· , dk−1,k(x), dk+1,k(x), ··· , dm,k(x); Rk(x) n(dk(x)), ∂dk(x) with

( ) − ρ ( ) ( ) ˆ di x ik x dk x di,k(x) = , for i = 1,...,k − 1, k + 1,...,m, 1 − (ρ (x))2 ik ˆ Rk(x) = ρˆij(x) ⎛ (m−1)×(m−1) ⎞ ρ ( ) − ρ ( )ρ ( ) ⎝ ij x ik x jk x ⎠ = , 2 2 1 − (ρik(x)) 1 − ρ jk(x) 1 ≤ i = k ≤ m 1 ≤ j = k ≤ m

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Rainbow trend options: valuation and applications 99

1 − z2 n(z) = √ e 2 , 2π ∂ Nm = nm (d1(x),d2(x), ··· ,dm(x); R(x)) . ∂ρpq(x)

The notation of nm (d1(x),d2(x), ··· ,dm(x); R(x)) represents the multivariate standard normal probability density function. To evaluate the second and fourth terms in Eq. (19), we simply choose x = S (t ) and let {d (x)} ≤ ≤ ≡ i a u 1 u m Q Q xi xi Q X d , , d , and R(x) ≡ R i or {du(x)} ≤ ≤ ≡ X j Xi ≤ = ≤ Yi Xi 1 u m 1 j i m Q Q Yi Yi QY d , , d , and R(x) ≡ R i to calculate the respective partial X j Xi 1≤ j=i≤m Yi Xi derivatives of the MSNCDFs. This general method for calculating the partial derivative of the MSNCDF can also ∂ ( ) be applied to calculate the cross gamma ( (t ) = i ta ) and the vega (ν (t ) = ij a ∂ S j (ta ) l a ∂ ( ) V ta ) of the rainbow trend/average option. We omit the derivation of the formulas ∂σl of the cross gamma and vega to streamline this paper. Nevertheless, the complete formulas 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 of the three types of RTOs introduced in Sect. 2.1: simple, pure, and Asian RTOs. We first evaluate the simple RTO whose payoff is defined as the difference between the 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

n S (t ) Sˆ (T ) = S (t ) + c i h i exp ln i 0 = h ln h 1 Si (th−1) Si (t1) Si (tn) = exp ln Si (t0) + c1 ln +···+cn ln Si (t0) Si (tn−1) = exp Ri,0 + c1 Ri,1 +···+cn Ri,n ,

α = ,α = = 6h(n−h+1) = ,..., ,β = = we set 0 1 h ch (n+1)(n+2) for h 1 n h 0 for all h, and K X ( ) ( ) ˆ ( ) in Eq. (7) to change Xi T and KYi T into Si T and X, respectively. Therefore, + S( ) = ˆ ( ) − , = the payoff function in Eq. (7) becomes V T Sm T X where m arg max Sˆ (T ), which is identical to the payoff function of the simple RTO men- 1 ≤ i ≤ m i tioned in Eq. (2). According to the general pricing formula in Eq. (15), the pricing formula for the simple RTO can be derived as 123 Author's personal copy

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2 n 1 S − τ m μi t = + ch + σ V S(t ) = e r a Sˆ (t ) e h a 1 2 Xi a = i a i 1 Q ,S Q ,S Xi Xi Q X ,S ×Nm d , , d , ; R i X j Xi 1≤ j=i≤m Yi Xi m Q ,S Q ,S −rτa Yi Yi QY ,S −Xe Nm d , , d , ; R i , (20) i=1 X j Xi 1≤ j=i≤m Yi Xi

( ) ( ) Si t1 +···+ Si ta ˆ c1 ln S (t ) ca ln S (t − ) where Si (ta) = Si (t0)e i 0 i a 1 is the realized trend price for the ith , , , , Q X S Q X S QY S QY S Q ,S Q ,S asset up to time t and d i , d i , d i , d i , and R Xi = R Yi are a X j ,Xi Yi ,Xi X j ,Xi Yi ,Xi ˆ ( ) 2 ln Si ta + μ − μ t n c + σ S − ρS σ S σ S , ˆ ( ) i j h=a+1 h Xi X,ij Xi X j Q Xi S S j ta d , = , X j Xi σ S X,ij for 1 ≤ j = i ≤ m, (21) ˆ ( ) 2 Si ta + μ n + σ S , ln i t h=a+1 ch X Q Xi S X i d , = , (22) Yi Xi σ S Xi ˆ ( ) Si ta + μ − μ n , ln ˆ ( ) i j t h=a+1 ch QYi S S j ta d , = ,for1≤ j = i ≤ m, (23) X j Xi σ S X,ij Sˆ (t ) n , i a + μ t c QYi S ln X i h=a+1 h d , = , (24) Yi Xi σ S Xi ( ) ( ) Q ,S Q ,S I (m−1)×(m−1) II (m−1)×1 R Xi = R Yi = , (25) (II) 1 1×(m−1) m×m with ⎛ ⎞ 2 σ S − ρ S σ S σ S − ρ S σ S σ S + ρ S σ S σ S ⎜ Xi X,il Xi Xl X,ij Xi X j X, jl X j Xl ⎟ (I )( − )×( − ) = ⎝ ⎠ , m 1 m 1 σ S σ S X,ij X,il (m−1)×(m−1) for 1 ≤ j = i ≤ m and 1 ≤ l = i ≤ m, σ S − ρS σ S Xi X,ij X j (II)( − )× = , for 1 ≤ j = i ≤ m. m 1 1 σ S X,ij (m−1)×1 2 S 2 n 2 S Note that in the above formula, σ = σ t = + c ,ρ , = ρij, and Xi i h a 1 h X ij 2 2 2 σ S = σ S − 2ρS σ S σ S + σ S are deﬁned according to Eqs. (9), X,ij Xi X,ij Xi X j X j (12), and (16), respectively.

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α = ,α = = 6h(n−h+1) ,β = = Similarly, by setting 0 1 h ch (n+1)(n+2) h 0 for all h, and K X in Eqs. (18) and (19), we can derive the delta and gamma of the simple RTO with respect to Si (ta):

2 ˆ ( ) n 1 S Si ta −rτa +μi t = + ch + σ S(t ) = c e h a 1 2 Xi i a a S (t ) i a Q ,S Q ,S Xi Xi Q X ,S ×Nm d , , d , ; R i , (26) X j Xi 1≤ j=i≤m Yi Xi 2 ˆ ( ) − τ +μ n + 1 σ S S Si ta r a i t h=a+1 ch 2 X (ta) = ca (ca − 1) e i i [S (t )]2 i a Q ,S Q ,S Xi Xi Q X ,S ×Nm d , , d , ; R i X j Xi 1≤ j=i≤m Yi Xi 2 ˆ ( ) n 1 S Si ta −rτa +μi t = + ch + σ +c e h a 1 2 Xi a S (t ) i a ∂ Q X ,S Q X ,S , × i , i ; Q Xi S , Nm dX ,X dY ,X R (27) ∂ Si (ta) j i 1≤ j=i≤m i i where

∂ Q X ,S Q X ,S , i , i ; Q Xi S Nm dX ,X dY ,X R ∂ Si (ta) j i 1≤ j=i≤m i i Q X ,S Q ,S ∂ i Xi ∂ N dX ,X ∂ N ∂d , = m j i + m Yi Xi ≤ = ≤ Q X ,S ∂ ( ) Q X ,S ∂ ( ) 1 j i m ∂d i Si ta ∂d i Si ta X j ,Xi Yi ,Xi ∂ ∂ = Nm ca + Nm ca . ≤ = ≤ Q ,S S Q ,S S 1 j i m Xi S (t )σ Xi S (t )σ ∂d i a X,ij ∂d i a Xi X j ,Xi Yi ,Xi

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

∂S S ( ) = i ij ta ∂ S j (ta) 2 ˆ ( ) n 1 S Si ta −rτa +μi t = + ch + σ = c e h a 1 2 Xi a S (t ) i a ∂ Q X ,S Q X ,S , × i , i ; Q Xi S , Nm dX ,X dY ,X R (28) ∂ S j (ta) j i 1≤ j=i≤m i i 123 Author's personal copy

102 J.-Y. Wang et al. where

Q X ,S ∂ i ∂ Q ,S Q ,S ∂ N dX ,X Xi Xi Xi ,S m j i Nm d , , d , ; R = ∂ ( ) X j Xi ≤ = ≤ Yi Xi ≤ = ≤ Q X ,S ∂ ( ) S j ta 1 j i m 1 j i m ∂d i S j ta X j ,Xi ∂ − = Nm ca , , 1≤ j=i≤m Q X S ( )σ S ∂d i S j ta X,ij X j ,Xi

, Q X S Q ,S because neither d i nor the entries in R Xi are functions of S (t ). Finally, the Yi ,Xi j a vega of the simple RTO with respect to the lth underlying asset is

∂ S ( ) νS( ) = V ta l ta ∂σl 2 − τ +μ n + 1 σ S n ˆ r a l t h=a+1 ch 2 X 2 = Sl (ta) e l σl t c − ch h=a+1 h Q ,S Q ,S Xl Xl Q X ,S ×Nm d , , d , ; R l X j Xl 1≤ j=l≤m Yl Xl 2 n 1 S m −rτa +μi t = + ch + σ + Sˆ (t )e h a 1 2 Xi = i a i 1 ∂ Q X ,S Q X ,S , × i , i ; Q Xl S Nm dX ,X dY ,X R ∂σl j i 1≤ j=i≤m i i ∂ , , −rτ m QY S QY S Q ,S −Xe a N d i , d i ; R Yi . (29) = m X j ,Xi Yi ,Xi i 1 ∂σl 1≤ j=i≤m

∂ ∂ ∂ Nm , Nm Nm The details for calculating Q ,S Q ,S , and ∂σ in Eqs. (27)–(29) are available Xi Xi l ∂d , ∂d , X j Xi Yi Xi 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 trend options on a single asset of Leippold and Syz (2007):

2 n 1 S , −rτa +μ1t = + ch + σ Q X S S ( ) = h a 1 2 X1 ˆ ( ) 1 V(m=1) ta e S1 ta N dY ,X 1 1 − τ QY ,S − Xe r a N d 1 , (30) Y1,X1 2 ˆ ( ) n 1 S , S1 ta −rτa +μ1t = + ch + σ Q X S S ( ) = h a 1 2 X1 1 , (m=1) ta ca e N dY ,X S1(ta) 1 1 2 ˆ ( ) n 1 S , S1 ta −rτa +μ1t = + ch + σ Q X S S (t ) = c (c − 1) e h a 1 2 X1 N d 1 (m=1) a a a 2 Y1,X1 [S1(ta)] 2 ˆ 2 ( ) − τ +μ n + 1 σ S Q ,S ca S1 ta r a 1 t h=a+1 ch 2 X X1 + e 1 n d , , σ S [S (t )]2 Y1 X1 X1 1 a 123 Author's personal copy

Rainbow trend options: valuation and applications 103

2 − τ +μ n + 1 σ S S ˆ r a 1 t h=a+1 ch 2 X ν( = ) (ta) = S1(ta)e 1 σ1t m 1 n Q ,S × 2 − X1 ch ch N dY ,X h=a+1 1 1 Q ,S n −rτa Y1 2 + Xe n d , t c , Y1 X1 h=a+1 h where ˆ ( ) 2 S1 ta + μ n + σ S , ln X 1 t h=a+1 ch X , , Q X1 S 1 QY1 S Q X1 S S d , = , d , = d , − σ , Y1 X1 σ S Y1 X1 Y1 X1 X1 X1 and 2 n σ S = σ 2t c2. X1 1 h=a+1 h

Based on the same argument, the general formula in Eq. (15) can be applied to price pure and Asian RTOs and their Greeks. For pure RTOs, the terminal payoff is the difference between the trend and the actual stock price at maturity [see Eq. (3)]. With α = ,α = = 6h(n−h+1) = ,..., ,β = = , ( ) 0 1 h ch (n+1)(n+2) for h 1 n h 1 for all h, and K 1 Xi T ˆ 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 for the pure RTO and its Greeks. The terminal payoff for the Asian RTO is determined by the difference between the trend price and the geometric average price at maturity [see α = ,α = = 6h(n−h+1) = ,..., ,β = (n+1)−h 1 Eq. (4)], so we set 0 1 h ch (n+1)(n+2) for h 1 n h n+1 ˆ for all h, and K = 1 such that Xi (T ) and Yi (T ) can express Si (T ) and the geometric ¯ average price Si (T ), respectively. Note that the general pricing formula in Eq. (15) can be applied to evaluate not only RTOs but also other types of rainbow options, such as Asian rainbow call options (on the maximum geometric average price) and rainbow call options (on the maximum ﬁnal + arg max asset price) with payoff functions S¯ (T ) − X , where m = S¯ (T ), and m 1 ≤ i ≤ m i + arg max S (T ) − X , where m = S (T ), respectively. Table 1 summarizes the m 1 ≤ i ≤ m i parameter settings used to price different types of options.

3 Numerical results

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

1 β = (n+1)−h = ,..., According to Eq. (24) and Fig. 3 of Leippold and Syz (2007), the weights h n+1 for h 0 n ¯ are the weights for logarithmic returns to generate the geometric average price Si (T ). 123 Author's personal copy

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

Options contracts that can be α0 αh, 1 ≤ h ≤ n β0 βh, 1 ≤ h ≤ nK priced using our general pricing formula

6h(n−h+1) Simple (rainbow) trend option 1 (n+1)(n+2) 00 X 6h(n−h+1) Pure (rainbow) trend option 1 (n+1)(n+2) 11 1 6h(n−h+1) (n+1)−h Asian (rainbow) trend option 1 (n+1)(n+2) 1 n+1 1 (n+1)−h Asian (rainbow) call option 1 n+1 00 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 trend options, Asian call options (with the geometric average method), and plain vanilla calls. Moreover, the most important contribution of our general option pricing formula is its ability to evaluate the proposed three types of RTOs, as well as the Asian rainbow call options (on the maximum geometric average price) and rainbow 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 can be examined. Moreover, to explain the option premium for the diversification effect over time, we analyze the differences between the prices of simple RTOs and rainbow call 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 have the rights to pick assets and to do timing, the liability of sellers of RTOs is manifest and needs to be managed carefully. This section uses the actual historical asset returns to 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 and time. We consider RTOs with two underlying assets (i.e., m = 2) as examples.2 The parameter values examined are summarized as follows: the interest rate r = 0.05, the dividend yields q1 = q2 = 0, the time to maturity T = 1 year with 252 sampling time 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}}.For simple RTOs, X = 100. Note that the symmetry of the two underlying assets is intentionally maintained such that, given ρ12= 1, the values of RTOs generated by our 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 be time-consuming. We test the value of m up to 7 and compare the accuracy and computational time between our pricing formulas and the Monte Carlo simulation, which is commonly acknowledged as the standard approach for pricing path-dependent rainbow options. The results show that our pricing formulas are more efﬁcient to generate accurate option values of RTOs than the Monte Carlo simulation. This part of analyses is available upon request from the authors. 123 Author's personal copy

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Table 2 Comparison of different RTOs

Underlying Simple RTO Pure RTO Asian RTO asset prices Price Premium for Price Premium for Price Premium for at t0 diversification diversification diversification effect over assets effect over assets effect 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 the correlation ρ12 given m= 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 intentionally maintain the symmetry of the two underlying assets such that, given ρ12= 1, our pricing formula generates the pricing results for single-asset trend options in Panel A. The premium for the diversification effect over assets is defined as the percentage difference between the rainbow option cases (ρ12= 1) and single-asset cases (ρ12= 1). For example, for the simple RTO with (S1(t0), S2(t0)) = (80, 80), the premium for the ρ = . ρ = 4.0557−2.3963 = . diversification effect over assets between 12 0 5and 12 1is 4.0557 40 92 %

Syz (2007). Therefore, the beneﬁt of incorporating rainbow options with trend options can be analyzed by comparing the option prices given different values of ρ12 with the option values given ρ12= 1. Table 2 reports the option prices of different types of RTOs. First, Panel A of Table 2 (given ρ12= 1) shows that the prices of the RTOs calculated based on Eq. (15) can converge to those of the single-asset trend options of Leippold and Syz (2007). For example, when S1 (t0) = S2 (t0) = 100 in Panel A, the 123 Author's personal copy

106 J.-Y. Wang et al. simple RTO based on our pricing formula is worth 11.4011, which is slightly lower than 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)are higher than the prices of single-asset trend options (Panel A of Table 2) due to the diversification effect of the maximum function in the payoff function of the RTOs. These differences can be attributed to option premiums from the diversification effect over the underlying assets. These option premiums are also found to increase as the correlation ρ12 decreases, which is consistent with the general understanding, since the magnitude of the diversification effect is negatively correlated with the correlation ρ12. Moreover, by comparing the percentage difference between the rainbow and single-asset trend options, we note this option premium for the diversification effect over assets is generally more pronounced for simple and Asian RTOs than for pure RTOs. For instance, the value and percentage proportion of the option premium for the diversification effect of the simple RTO with S1 (t0) = S2 (t0) = 100 and ρ12= 0.5 ( . − . ) = . 16.9763−11.4011 = . are 16 9763 11 4011 5 5752 and 16.9763 32 84 %, both of which are ( . − . ) = . 4.7591−3.7309 = . higher than 4 7591 3 7309 1 0282 and 4.7591 21 61 % of the counter- parts for the pure RTO. Another interesting observation is that, given different values of ρ12, the percentage option premiums for the diversification effect of pure RTOs and Asian RTOs are ˆ ( ), ( ) ¯ ( ) stable across different initial asset prices. Since Sm T Sm T , and Sm T are all homogeneous functions of degree one with respect to the initial prices of the underlying assets, the values of pure and Asian RTOs should be proportional to the value of the initial asset prices. For example, when ρ12 = 0.5(ρ12 = 0), the ratio between the values of the pure RTOs given S1(t0) = S2(t0) = 90 and S1(t0) = S2(t0) = 80 is 4.2832 = . ( 4.6539 = . ) 90 = . 3.8073 1 125 4.1368 1 125 , which is identical to the ratio of 80 1 125. This characteristic results in stable percentage option premiums for pure and Asian RTOs. Table 3 compares the simple RTO with the rainbow call option. It is well known that the 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 has diversification effects over both different underlying assets and time. Investors would like 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 the percentage difference between the values of simple RTOs and rainbow call options. Consistent with the single-asset results of Leippold and Syz (2007), the premium of the diversification effect over time is found to decrease with moneyness. This is because if the simple RTO is initially deep in the money (ITM) with respect to all the underlying assets, the diversification effect over time can contribute a little to the option value by slightly enhancing the ITM probability at maturity. Moreover, the premiums of the diversification effect over time show little sensitivity to ρ12, which implies the effect of incorporating the trend option into the rainbow option is persistent, regardless of the 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 by comparing the prices of simple RTOs with those of rainbow call options. All parameter values are the same as those in Table 2. A rainbow call option has a diversification effect over different assets; a simple RTO has a diversification effect over both different assets and time. Thus, the premium for the diversification effect over time can be defined as the percentage price difference between the simple RTO and the rainbow call options. For example, for the simple RTO with (S1 (t0) , S2 (t0)) = (80, 80) and ρ12= 1, the premium 2.3963−1.8594 = . for the diversification effect over time is 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 of 252 per year. We also examine four values of ta: ta = T/8, ta = T/2, ta = 7T/8, and ta → T , corresponding to a= 31, a= 126, a= 220, and a= 251, respectively. 123 Author's personal copy

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S( ) Fig. 1 The delta of the simple RTO with respect to the price of the first asset, 1 ta . The parameter values ˆ ˆ 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, ρ = . S( ) and 12 0 5. Positive deltas are in blue and zero deltas are in gray. The values of 1 ta are found S( ) = / to be non-negative. The values of 1 ta could be higher than one when ta T 2. In addition, when → ,S( ) ( )< ( ) ( )< ta T 1 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 we generate the diagrams of the delta, gamma, and cross gamma of the three types of RTOs, we control the realized information up to time ta−1. Taking the simple RTO ˆ ˆ for 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 ˆ ˆ (S1(ta−1), S2(ta−1)) and the values of (S1(ta), S2(ta)) of that point, we can obtain ˆ ˆ the corresponding realized trend prices (S1(ta), S2(ta)) up to ta and ﬁnally employ Eqs. (26)–(28) to compute the delta, gamma, and cross gamma of the simple RTO, respectively.3 S( ) Figure 1 shows that the delta of the simple RTO, 1 ta , is always positive and S( ) decreasing with the initial strike price X. By analyzing the formula of 1 ta in Eq. S( ) (26), it is straightforward to infer that 1 ta is non-negative, since it is a product of several non-negative terms. Intuitively speaking, because an increase in S1(ta) results ˆ ( ) S( ) in an increase in S1 ta and thus enhances the ITM probability of simple RTOs, 1 ta

3 ˆ ˆ The reason for fixing the information up to ta−1 (e.g., S1(ta−1) and S2(ta−1)) rather than up to the ˆ ˆ present time ta (e.g., S1(ta) and S2(ta)) is to enhance the readability of the diagrams. To understand the diagrams in this section, note merely that we generate the values of delta, gamma, and cross gamma based on the same historical information until ta−1 for each node. In contrast, suppose we fix the information up ˆ ˆ to ta by specifying the levels of, for example, the realized trend prices S1(ta) and S2(ta) when computing the Greeks. Then, each point in a diagram (corresponding to different combinations of (S1(ta), S2(ta))) ˆ ˆ implies a different combination of (S1(ta−1), S2(ta−1)), which reflects the different information until ta−1 of each node. Consequently, further comparisons of the Greeks between different points are difficult to analyze. 123 Author's personal copy

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P ( ) Fig. 2 The delta of the pure RTO with respect to the price of the first asset, 1 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 = ˆ 0.2, S2(ta−1)= 100, and ρ12 = 0.5. Positive deltas are in blue, negative deltas are in red, and zero deltas P ( ) are in gray. This figure shows that the values of 1 ta could be negative, especially when ta is relatively → P ( ) small or close to maturity. In addition, when ta T , the values of 1 ta could be significantly negative ˆ ˆ ˆ ˆ if S1(ta−1)>S2(ta−1) or S1(ta−1) = S2(ta−1) ≥ S1(ta)>S2(ta) (Color figure online)

A( ) Fig. 3 The delta of the Asian RTO with respect to the price of the first asset, 1 ta . The parameter values ˆ examined in this figure are m = 2, T = 1 year, r = 0.05, q1 = q2 = 0,σ1 = σ2 = 0.2, S1(ta−1) = ˆ ¯ S2(ta−1) = S2(ta−1) = 100, and ρ12 = 0.5. Positive deltas are in blue, negative deltas are in red, and zero A( ) deltas are in gray. This figure shows that the values of 1 ta can be negative only when ta is relatively A( ) → ( )< ( ) small. In addition, the values of 1 ta approaches zero when ta T , particularly if S1 ta S2 ta or ¯ S1(ta)

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S( ) = / of 1 ta could be higher than unity, particularly when ta T 2. This is because Eq. S( ) (26) shows that the value of 1 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 its = / = / ,S( ) maximum when ta T 2. Due to the large value of ca when ta T 2 1 ta of the simple RTO could be higher than unity.4 Finally, as shown in Fig. 1, when → ,S( ) → , ta T 1 ta is close to zero. This is because when ta T ca converges to zero, S( ) as shown in Fig. 4a, and thus 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 in Fig. 1, there could be some scenarios in which the deltas of pure and Asian RTOs are ˆ negative. Note that for the value of a pure RTO, it is a function of both S1(ta) and S1(ta) ∂ P ( ) ∂ ˆ ( ) ∂ P ( ) ∂ ( ) ( ) P ( ) = V ta S1 ta + V ta S1 ta = and thus its delta with respect to S1 ta is 1 ta ˆ ∂ ( ) ∂ ( ) ∂ ( ) ∂ S1(ta ) S1 ta S1 ta S1 ta ∂ P ( ) ˆ ( ) ∂ P ( ) ∂ P ( ) ∂ P ( ) V ta S1 ta + V ta V ta > V ta < 5 ˆ ca ( ) ∂ ( ) 1. Since ˆ 0 and ∂ ( ) 0, we can deduce that the ∂ S1(ta ) S1 ta S1 ta ∂ S1(ta ) S1 ta ˆ ( ) sign of P (t ) depends on the comparative levels of c S1 ta and one. To demonstrate 1 a a S1(ta ) P ( ) this argument, we consider the delta of the single-asset pure trend option, (m=1) ta , ˆ in Fig. 4c, where S1(ta) = S1 (ta) = 100 is intentionally set such that we can focus on P how the comparative levels of ca and one determine the sign of (m=1). By comparing Fig. 4a and c, we find that when ca (dashed curve in Fig. 4a) is smaller than one (solid ,P ( ) straight line in Fig. 4a), that is, when ta is small or ta is near T (m=1) ta is inclined to be negative. Note that this effect still holds for the pure RTO in Fig. 2.InFig.2, P ( ) = / ) = / 1 ta could be negative when ta is small (ta T 8 or ta is near T (ta 7T 8 → ) → P ( ) and ta T . In addition, note that when ta T in Fig. 2,thevalueof 1 ta is ˆ ˆ ˆ ˆ nonzero only when S1(ta−1)>S2(ta−1) or S1(ta−1) = S2(ta−1) ≥ S1(ta)>S2(ta). ˆ This is because, in these scenarios, the trend price of the first asset, S1(ta), is likely to appear in the final option payoff and be larger than S1(ta). The present option value ( ) P ( ) should therefore be sensitive to S1 ta and thus 1 ta is nonzero. Moreover, these ∂ P ( ) nonzero P (t ) should be negative and can be approximated by V ta , which is 1 a ∂ S1(ta ) around −0.93 to −0.95 in Fig. 2. This is because when ta → T, ca → 0 such that ∂ P ( ) ˆ ( ) ∂ P ( ) ∂ P ( ) P ( ) = V ta S1 ta + V ta ≈ V ta 1 ta ˆ ca ( ) ∂ ( ) 1 ∂ ( ) . ∂ S1(ta ) S1 ta S1 ta S1 ta A similar argument can explain the negative delta of the Asian RTO in Fig. 3. Since ∂ A( ) ∂ ˆ ( ) ∂ A( ) ∂ ¯ ( ) ∂ A( ) ˆ ( ) ∂ A( ) ( + )− ¯ ( ) A( ) = V ta S1 ta + V ta S1 ta = V ta S1 ta + V ta n 1 a S1 ta 1 ta ˆ ∂ ( ) ∂ ¯ ( ) ∂ ( ) ˆ ca ( ) ∂ ¯ ( ) + ( ) ∂ S1(ta ) S1 ta S1 ta S1 ta ∂ S1(ta ) S1 ta S1 ta n 1 S1 ta

4 As illustrated in Fig. 4b, the delta of the single-asset simple trend option also peaks at ta = T/2. This phenomenon demonstrates that the value of ca should be a determining factor in inﬂuencing the delta of the single-asset simple trend option. A similar result is also mentioned by Leippold and Syz (2007, Table II). P P 5 ∂V (ta ) ∂V (ta ) ) P ( ) ˆ ( ) ( )) Since ˆ ( ∂ S (t ) measures the sensitivity of V ta with respect to S1 ta (S1 ta under the ∂ S1(ta ) 1 a ˆ ˆ ˆ assumption that S2(ta), S1(ta),andS2(ta) (S1(ta), S2(ta),andS2(ta)) are held as constants, an increase in Sˆ (t ) (S (t )) results in an increase of the expected value of Sˆ (T ) (S (T )) and thus causes an increase 1 a 1 a m m + (decrease) of the expected value of the payoff of (Sˆ (T ) − S (T )) of the pure RTO. Consequently, it m m P P ∂V (ta ) > ∂V (ta ) < ) can be inferred that ˆ 0(∂ S (t ) 0 . ∂ S1(ta ) 1 a 123 Author's personal copy

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(a)

(b)

(c)

(d)

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¯ ˆ Fig. 4 Weights for the logarithmic returns of the underlying asset to generate Si (T ), Si (T ),andSi (T ) and the evolution of the deltas of single-asset trend options over the option lifespan. The examined parameter ˆ ¯ values are T = 1 year, r = 0.05, q1 = 0,σ1 = 0.2, S1(ta) = S1(ta) = S1(ta) = 100, and X = 100 S ( ) for the simple trend option. For the simple trend option, (m=1) ta is always positive. For the pure trend P ( ) ˆ ( ) option, (m=1) ta is negative when the weights required to generate S1 T are lower than those required ( ) A ( ) to generate S1 T . For the Asian trend option, (m=1) ta is negative when the weights required to generate ˆ ¯ S1(T ) are lower than those required to generate S1(T )

∂ A( ) ∂ A( ) ˆ ( ) V ta > V ta < 6 S1 ta with ˆ 0 and ∂ ¯ ( ) 0, we can deduce that the comparative levels of ca S (t ) ∂ S1(ta ) S1 ta 1 a ( + )− ¯ ( ) and n 1 a S1 ta can influence the sign of A(t ). By comparing Fig. 4a and d, n+1 S1(ta ) 1 a ˆ ¯ where S1(ta) = S1(ta) = S1(ta) = 100 is deliberately posited such that we can (n+1)−a focus on the impact from comparable levels of ca and n+1 on the delta of the A ( ) single-asset Asian trend option, (m=1) ta , one sees that when ca (dashed curve in (n+1)−a Fig. 4a) is smaller than n+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 A( ) = / ) the delta of the Asian RTO, 1 ta , is negative only when ta is small (ta T 8 . → A( ) Moreover, when ta T in Fig. 3,thevalueof 1 ta approaches zero because (n+1)−a → both ca and n+1 approach zero given ta T (as illustrated in Fig. 4a) and thus ∂ A( ) ˆ ( ) ∂ A( ) ( + )− ¯ ( ) A( ) = V ta S1 ta + V ta n 1 a S1 ta ≈ 1 ta ˆ ca ( ) ∂ ¯ ( ) + ( ) 0. ∂ S1(ta ) S1 ta S1 ta n 1 S1 ta The gammas of simple, pure, and Asian RTOs with respect to the price of the first S ( ) , P ( ) A ( ) underlying asset, denoted as 1 ta 1 ta , and 1 ta , respectively, are depicted in Figs. 5, 6 and 7. All results are generated based on Eq. (19) with the proper parameter S ( ) , P ( ) settings. One can also understand intuitively the behaviors of 1 ta 1 ta , and A ( ) 1 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, when = / S ( ) , P ( ) ta T 2 and the weight for the trend price, ca, reaches its maximum, 1 ta 1 ta , A ( ) and 1 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. 56 and 7, positive gammas for RTOs are not always available, so the risk management for RTOs should consider the risk of a negative gamma.7

A A 6 ∂V (ta ) > ∂V (ta ) < One can obtain ˆ 0and ∂ ¯ ( ) 0 following similar arguments as those in the previous ∂ S1(ta ) S1 ta footnote. 7 S ( ) , P ( ) We also generate the cross gammas of simple, pure, and Asian RTOs, denoted as 12 ta 12 ta ,and A ( ) 12 ta , according to our analytic cross gamma formula with the proper parameter settings. The cross S ( ) P ( ) gamma of the simple RTO, 12 ta , is non-positive. The cross gamma of the pure RTO, 12 ta , could be positive as well as negative. When ta is relatively small (i.e., ta = T/8) or ta is near maturity (i.e., = / → ) P ( ) ta 7T 8andta T , the value of 12 ta tends to be non-negative. As for the cross gamma of the A ( ) A ( ) Asian RTO, 12 ta , positive and negative values are both admitted, but the positive values of 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 are S ( ) , P ( ) A ( ) omitted to streamline this paper. Nevertheless, the behaviors of 12 ta 12 ta ,and 12 ta can also be 123 Author's personal copy

Rainbow trend options: valuation and applications 113

S( ) Fig. 5 The gamma of the simple RTO with respect to the Price of the first asset, 1 ta . The parameters ˆ ˆ 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. S( ) = / → Our results show that the values of 1 ta could be slightly negative when ta 7T 8orta T (Color figure online)

P ( ) P ( ) Fig. 6 The gamma of the pure RTO with respect to the price of the first asset, 1 ta . We examine 1 ta ˆ in this figure, given m = 2, T = 1 year, r = 0.05, q1 = q2 = 0,σ1 = σ2 = 0.2, 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 P ( ) = / values of 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 Asian νS( ), νP ( ) νA( ) RTOs regarding the volatility of the ﬁrst asset, denoted as 1 ta 1 ta , and 1 ta ,

Footnote 7 continued derived 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. 123 Author's personal copy

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A( ) Fig. 7 The gamma of the Asian RTO with respect to the price of the first asset, 1 ta . In this figure, ˆ ˆ ¯ 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. A( ) = / The values of 1 ta could be negative, except when ta T 2 (Color figure online)

νS( ), νP ( ) νA( ) Fig. 8 The vegas of RTOs, 1 ta 1 ta ,and 1 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 = 1 ¯ ¯ year, r = 0.05, q1 = q2 = 0,σ1 = σ2 = 0.2, S1(ta) = S2(ta) = 100, S1(ta) = S2(ta) = 100, X = 100 νS( ) ˆ ( ) = ˆ ( ) for 1 ta ,andS1 ta S2 ta issettobe80(the dashed line), 100 (the dotted line), and 120 (the solid line). In general, the vegas of RTOs are positive and increase with the decreasing correlation between the underlying 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 with the dimensions of both ta and ρ12. Several interesting phenomena are elaborated as follows. First, Leippold and Syz (2007) have proven that, in the single-asset case, the vega of the simple trend option can be negative when the option is deeply ITM and close to maturity. This phenomenon can be explained more intuitively. For an increase in the volatility of the first asset, σ1, there are two countervailing impacts on the price 123 Author's personal copy

Rainbow trend options: valuation and applications 115 of trend options. Take the single-asset simple trend option for example. Its pricing formulainEq.(30) can be rewritten as − τ + ˆ τ Q X ,S − τ QY ,S V S (t ) = e r a M1 a Sˆ (t )N d 1 − Xe r a N d 1 , (31) (m=1) a 1 a Y1,X1 Y1,X1 where ˆ ( ) 2 S1 ta + μ n + σ S , ln X 1 t h=a+1 ch X Q X1 S 1 d , = , Y1 X1 σ S X1 Q ,S Q ,S Y1 = X1 − σ S , d , d , X Y1 X1 Y1 X1 1 2 n σ S = σ 2t c2 X1 1 h=a+1 h and ! 2 ˆ n 1 S M1 = μ1t ch + σ τa h=a+1 2 X1 ! 2 1 2 n 1 S = r − q1 − σ t ch + σ τa. 2 1 h=a+1 2 X1

ˆ Note that M1 can be interpreted as the growth rate of the trend price because Mˆ τ ˆ ˆ ˆ e 1 a S1(ta) is the expected value of S1(T ) conditional on S1(ta). When σ1 increases, ˆ its ﬁrst effect is to enhance the volatility of the conditional distribution of S1(T ), i.e., σ S , and thus contribute an extra premium to the simple trend option. Its second effect X1 ˆ − 1 σ 2 is to reduce M1 via the term 2 1 , leading to a decrease in the expected value of ˆ S1(T ) and the option premium of simple trend options. Figure 9 illustrates the evolu- ˆ ˆ tion of ∂ M1/∂σ1 over time. It is apparent that ∂ M1/∂σ1 could be negative, particularly when it is close to maturity. Due to these two opposite impacts corresponding to an increase 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 formulas can be derived, respectively, as − τ + ˆ τ Q X ,P P ( ) = r a M1 a ˆ ( ) 1 V(m=1) ta e S1 ta N dY ,X 1 1 − τ QY ,P − e q1 a S (t )N d 1 (32) 1 a Y1,X1 and − τ + ˆ τ Q X ,A A ( ) = r a M1 a ˆ ( ) 1 V(m=1) ta e S1 ta N dY ,X 1 1 − τ + ¯ τ QY ,A − e r a M1 a S¯ (t )N d 1 , (33) 1 a Y1,X1 123 Author's personal copy

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ˆ Fig. 9 The partial derivative of the average growth rate of the trend price of the first asset, M1, with respect to 2 ˆ ˆ n 1 S the volatility σ1. The average growth rate of S1 is defined as M1 = (μ1t = + ch + σ )/τa = h a 1 2 X1 2 1 2 n 1 S ˆ ((r − q − σ )t = + c + σ )/τa in Sect. 3.2. This figure exhibits the sensitivity of M 1 2 1 h a 1 h 2 X1 1 regarding σ1 given different time points ta. The parameters examined in this figure are T = 1 year, ∂ ˆ r = 0.05, q = 0, and σ = 0.2. It can be found that, with the passage of time, M1 first increases, then 1 1 ∂σ1 decreases, and finally becomes negative near maturity where ( − )( − + ) 2 ¯ n a n a 1 1 A M = μ t + σ /τa. 1 1 2 (n + 1) 2 Y1

ˆ It is also true for pure and Asian trend options that an increase in σ1 could reduce M1 and next generate a negative impact on the option price. Therefore, the signs of the vegas 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, and Asian RTOs can generally increase their vegas. Due to the diversification effect of ˆ ˆ ˆ max(S1(T ), S2(T ),...,Sm(T )) in the payoff function, the negative impact on the ˆ growth rate of M1 and more negative returns of the first asset due to an increase of σ1 ˆ could be partially eliminated because S1(T ) is less likely to be the highest trend price in these scenarios and thus does not affect the final payoff of RTOs. Therefore, the more positive returns of the first asset due to an increase of σ1 dominate and enhance the vegas of RTOs upward. Consistent with this inference, most vegas in Fig. 8 vary inversely with ρ12, because a more negative ρ12 represents a scenario with a stronger diversification effect. In addition, when ta approaches maturity, the impact of the diversification effect on the vegas is weakened. For example, this phenomenon is clearly observed in the first row in Fig. 8, which shows the vegas of simple RTOs decrease with ta. This is because the period for the diversification effect to be in effect shortens when ta is near the maturity date. Third, when ta approaches maturity, the diversification effect, however, could hurt the vega of the pure RTO, as illustrated in the second row of Fig. 8, while ta = 7T/8 123 Author's personal copy

Rainbow trend options: valuation and applications 117 and ta → T . From the previous paragraph, we know the diversification effect can ˆ screen out the negative impacts on M1 and more negative returns due to an increase of σ1, but the remaining more positive returns due to an increase of σ1 will enhance S1(T ) ˆ by an amount larger than S1(T ) when ta approaches maturity. This is because ca (the ˆ weights for the logarithmic returns to generate S1(T )) is less than one (the weights for the logarithmic returns to generate S1(T )) when ta approaches maturity, as shown in 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 .In contrast, the diversification effect does not hurt the Asian RTO because ca (the weights ˆ ( )) (n+1)−a for the logarithmic returns to generate S1 T is larger than 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/8orta → T , the vegas of deeply ITM single-asset pure trend options, which can be approximated by the vegas of the deeply ITM pure RTO at ρ12 = 1, can be negative. It is intuitive to obtain this result. When ˆ σ1 increases, the future movements of S1 should be more volatile than those of S1 due to one (the weights for logarithmic returns to generate S1(T )) being larger than ˆ ca (the weights for logarithmic returns to generate S1(T )). The excess volatility of S1 can result in a higher probability of the ITM single-asset pure trend options turning to be out of the money and thus reduce the option values. Consequently, the vegas of the deeply ITM pure RTOs can be negative at ρ12= 1 when ta approaches maturity in the second row of Fig. 8. In addition, combining with the fact explained above that the diversification 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 the second row of Fig. 8.

3.3 Hedging performance of the DDHS for RTOs

This section examines the hedging performance of our RTO pricing formulas, which are developed under the Black and Scholes (1973) framework, using the actual historical returns of underlying assets. We take simple RTOs for example and examine the 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), who investigates the performance of the DDHS for single-asset Asian options, to rebalance the delta-neutral portfolios and calculate the NHC. In our experiments, the transaction cost is ignored and the hedging portfolio is rebalanced at the daily frequency, which is the most common arrangement in practice. The three steps of the DDHS for RTOs are described as follows. For any issue day t0, suppose that the current and the subsequent daily prices of m underlying stock are denoted as Si (th), for i = 1,...,m, h = 0, 1,...,n. Note that when using our pricing formulas, we always shift t0 to be 0, and t1 = t,...,tn = nt = 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 the selling price of the examined RTO according to Eq. (15), and the hedging ratio for 123 Author's personal copy

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( ) 8 the ith underlying asset, i t0 , is computed by Eq. (18). We thus purchase the m ( ) ( ) η( ) = underlying assets with i=1 i t0 Si t0 and invest the remaining cash, t0 π( ) − m ( ) ( ) ( )9 t0 i=1 i t0 Si t0 , at the interest rate r t0 for the following t period. Step 2: At time point th, the value of hedging portfolio becomes π(th) = ( ) m ( ) ( )+η( ) r th−1 t . ( ) i=1 i th−1 Si th th−1 e The new hedging ratio, i th , is computed by Eq. (18) with the inputs of the underlying asset prices today Si (th). The money account which compounds at r(th) for the following t period is thus recalculated as η( ) = π( ) − m ( ) ( ) th th 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 issuing and hedging the RTO. Theoretically speaking, the NHC is close to zero (distributed with a nearly zero mean and a small standard deviation) given some ideal conditions. The first condition is that the dynamics of the actual underlying asset prices can be perfectly described by Eq. (1) and thus follow lognormal distributions. Second, one can predict exactly the values of parameters, like σi and ρij, for the future option life period. However, much literature confirms the existence of stochastic volatility and jumps in actual stock prices, so the first condition usually does not hold and the real-world NHC may deviate from its ideal distribution. We view this deviation as the model error of our framework for pricing RTOs. On the other hand, because one cannot foresee parameter values for future periods given historical information, estimation errors emerge. Specifically, for each issue day t0, we follow the common arrangement in practice to calculate the estimates of σi and ρij, denoted as σî and ρîj, with the returns of D days prior to t0. During the following option life [t0, tn], we always employ σî and ρîj to calculate option values and deltas based on Eqs. (15) and (18), respectively, for the DDHS. σˆ =σ ∗ ρˆ = ρ∗ σ ∗ ρ∗ However, the estimation error occurs because i i and ij ij, where i and ij 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 corresponding i (th) based on our RTO pricing formulas (with historical σî and ρîj as input parameter values) for the DDHS, we suffer both estimation and model errors. To distinguish the influences on the hedging performance from the above two errors, we construct three models to simulate or obtain the daily prices Si (th) after t0,asshown in Table 4. To capture the estimation error, both Models 1 and 2 assume that the log returns Si (th ) of the underlying assets (Ri,h ≡ ln ) of RTOs follow normal distributions. Si (th−1) In Model 1, we simulate the returns of the future n days based on σî , ρîj, and " ( ) = the average log return of√ the D days prior to t0, denoted as ERi , with Si th " Si (th−1)exp ERi t +ˆσi ti (th) , where i (th) ∼ N(0, 1), and i (th) and j (th)

8 When evaluating Eqs. (15)and(18), we employ the spline interpolation (based on daily term structures of the risk-free zero rates in OptionMetrics) to derive the level of r whose maturity matches the remaining option 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). 123 Author's personal copy

Rainbow trend options: valuation and applications 119

Table 4 Three models to distinguish estimation and model errors

Parameters used to simulate Error type stock prices during option life ˆ Model 1 ERi , σî , ρîj with lognormal Model 1 vs. 2: estimation distribution error ∗,σ∗,ρ∗ Model 2 ERi i ij with lognormal Model 2 vs. 3: model distribution error Model 3 Use actual log returns and Model 1 vs. 3: estimation Si (t0) to infer future Si (th) and model errors This table compares the three models designed to differentiate the estimation and model errors of the DDHS implemented based on our RTO pricing formulas. Given any issue date t0, the values of the expected log ˆ , σˆ ρˆ ∗,σ∗ ρ∗ ) ) return ERi i ,and ij (ERi i ,and ij 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 σî ,andρîj. In addition, to make the results of different t0 comparable, Si (t0) is normalized to 100 for all underlying assets, and the strike price X is always fixed at 100. The dividend yields of all underlying assets are assumed to be zero. The NHC differences between Models 1 and 3 reflect both the parameter estimation error and the model error of the DDHS based on our RTO pricing formulas, but the NHC differences between Models 1 and 2 (Models 2 and 3) capture the estimation error (model error) individually

are correlated with ρîj. By this means, the future stock prices after t0 follow the lognormal distribution and all the associated parameter values are the same as those used in our formulas to calculate option and delta values. As a result, both the model and estimation errors of Model 1 are minimized and thus Model 1 can be taken as a theoretical performance benchmark of the DDHS based on our pricing formulas, 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 on σ ∗,ρ∗ , and the average log return of the n days after t , denoted as ER∗, with i ij √ 0 i ( ) = ( ) ∗ + σ ∗ ξ ( ) ξ ( ) ∼ ( , ) ξ ( ) Si th Si th−1 exp ERi t i t i th , where i th N 0 1 , and i th ξ ( ) ρ∗ and j th are correlated with ij. 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 of the 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 empirical DDHS performance based on our pricing formulas. The differences between the NHCs of Models 1 and 3 capture both model and estimation errors, 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 to be 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), and Intel Corporation (INTC) listed on Nasdaq. The corresponding results are available upon request. 123 Author's personal copy

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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. The Model 1, Model 2, and Model 3 columns report respectively the NHC percentiles from 5 to 95 % for the two 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 that the NHC differences between Models 2 and 3 are relatively small, but the differences between the NHCs of Model 1 and those of Models 2 and 3 are more pronounced

σî and ρîj. The historical daily stock prices are collected from the Yahoo! Finance website covering 1996–2015, based on which we derive daily log returns of these five underlying assets. Across the nearly 2500-day sample, we randomly choose 250 time points as t0. In other words, to obtain the NHC distribution, we repeat the calculation 250 times. To make the NHC levels comparable among the 250 repetitions, all initial stock prices are normalized to 100, i.e., Si (t0) = 100, for i = 1,...,m., and the strike price X is always fixed at 100. In addition, the dividend yields of all underlying assets are assumed to be zero. The NHC percentiles from 5 to 95% under different models are reported in Table 5. First, we observe that the NHC distributions under Model 1 center on zero with small standard deviations for the simple RTO. We expect this phenomenon since in the ideal condition in Model 1, the cost of the DDHS converges to the initial option values and therefore the NHC should be near zero. Second, for Models 2 and 3, due 123 Author's personal copy

Rainbow trend options: valuation and applications 121

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 specified in Table 4. We also conduct the two-sample Kolmogorov–Smirnov test (K–S test) to examine statistically whether a pair of samples comes from the same distribution. The null hypothesis is that the two samples come 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 distributions increase substantially, and the means of the NHCs of Models 2 and 3 could deviate from zero, e.g., the means of the NHCs for Model 3 is −1.4594. Third, by comparing the root mean square errors (RMSEs) between the reported 19 percentile levels of different 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 a statistical sense, we plot the empirical cumulative distribution functions of all models in Fig. 10 and perform the two-sample Kolmogorov–Smirnov test (K–S test), which examines 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 that the NHCs based on Models 2 and 3 come from the same distribution, but Model 1’s NHC distribution differs from those of Models 2 and 3. According to the evidence in Table 5 and Fig. 10, we conclude that the impact of the model error resulting from the lognormal distribution assumption of our model is much smaller than the estimation error associated with σi and ρij. Due to the fact that the values of RTOs are highly sensitive to the levels of σi and ρij, it is natural that estimation error would play a significant role. As to the relatively minor model error, one possibility is that the lognormal distribution assumption performs not so poorly for the examined underlying assets in 1996–2015. In addition, we conjecture that the unique feature of the trend price, which is calculated based on the least-squares regression, may partially explain this phenomenon. If there are stochastic volatility or jumps in actual stock returns, the actual distribution should have fatter tails than the assumed normal distribution, which means the probability to meet extreme returns increases. However, the regression-based trend price is theoretically less sensitive to a small number of outliers (or said extreme returns). Therefore, if the fat-tail effect 123 Author's personal copy

122 J.-Y. Wang et al. caused by stochastic volatility or jumps is not that excessive, one can expect that its inﬂuence on the expectation of the trend price and thus the values of RTOs would be minor. As a result, the pricing and hedging performance of our model based on a lognormal distribution assumption should not hurt too much. Finally, we suggest that when the DDHS based on our pricing formulas is applied to hedge RTOs, the issuer can pay less attention to the model error but should prepare a greater buffer to absorb estimation errors associated with σi and ρij.

4 Applications of RTOs

To illustrate the great potential beneﬁts of rainbow trend/average options, this section introduces some of their practical applications, not only for investments but also for risk management. We also investigate how to apply rainbow trend/average options to designing effective compensation plans for managers of corporations. Moreover, our idea of rainbow trend options can be applied to modify the countercyclical capital buffer proposed by Basel Committee. In addition to those practical applications, the academic application of rainbow trend/average option formulas in option pricing is also explored.

4.1 Hedging timing risk and asset selection risk

Financial instruments offering both functions of hedging timing and asset selection risks are attractive to most investors, especially those who adopt a passive asset management style. Most of them traditionally choose their investment targets from various index-related products. The rainbow option, or more specifically the maximum option, on different indexes can resolve the asset selection issue. However, the decisions of when to enter into the rainbow option contract and its time to maturity could still influence the final payoff significantly. The maximum option on the trend-based payoff can mitigate this timing issue, since the payoff function is associated with the asset price ˆ that performs, on average, the best over the entire option life. Let Ii denote the trend price of the ith stock index calculated over the entire investment period. Then the RTO serving in the same period with payoff ˆ ˆ ˆ 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, especially when the purchase date can only be predicted approximately but cannot be known exactly. Suppose a firm is developing a new product. The design process is scheduled 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 123 Author's personal copy

Rainbow trend options: valuation and applications 123 will start manufacturing the new product immediately. Assume also that manufacturing the product requires L units of a major material, which is available from m different sources. To hedge in advance the price risk of purchasing the material, the + ˆ ˆ ˆ ˆ 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 ith source and X is the tolerable upper bound of the material’s unit price. The firm can receive some ˆ compensation when the minimum trend price Si at T rises beyond the upper bound X. The aforementioned RTO is more appropriate than a typical minimum rainbow + option 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 the material, the hedging performance of the minimum rainbow option could be poor if the material prices at the planned purchase time T and the actual purchasing time ˆ differ significantly. In contrast, the trend prices Si can better forecast the prevailing asset prices in the period around T . Consequently, the hedging performance of the RTO is relatively stable in most scenarios, even when the actual purchasing date is uncertain. Moreover, compared with a geometric average rainbow option (with a payoff ¯ ¯ ¯ + L× min S1,S2,...,Sm − X ), the RTO can again better forecast the prevailing ¯ asset prices around T , since the average price Si in [0, T ] is not a highly-correlated proxy for the prevailing asset price around T . The potential gap between the geometric average prices during the option life and the asset prices on the actual purchasing date could 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; Ingersoll 2006; Cooper et al. 2009). Two possible reasons are that the earnings or stock prices could be manipulated by managers or it is inappropriate to measure the performance of managers or firms using only stock prices. To address the above two issues, more ideal executive compensation plans could be achieved with RTOs. We provide two possible designs based on RTOs: one emphasizes a firm’s general improvement in different aspects and the other exploits inter-firm comparisons to identify a manager’s superior performance. The first design, to ensure overall improvement in a firm’s operation, proposes to measure managerial performance by multiple indexes, such as the stock price, sales revenue, or market shares of major products. The firm can consider the compensation plan

+ ˆ ˆ ˆ min I1/I1,0, I2/I2,0,...,Im/Im,0 − X ,

ˆ where Ii /Ii,0 represents the normalized trend of the ith evaluating index. The manager can receive a bonus if all the trends of these indexes are higher than the minimum growth rateX. 123 Author's personal copy

124 J.-Y. Wang et al.

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 can be considered: + Sˆ Sˆ Sˆ Sˆ 1 − max 2 , 3 ,..., m , S1,0 S2,0 S3,0 Sm,0

ˆ ˆ where S1/S1,0 and Si /Si,0 for 2 ≤i≤m denote the normalized trends of the stock prices of the firm and its competitors, respectively, in an industry. As long as the firm’s trend-based performance dominates that of its competitors, the manager can receive the bonus. We argue that this rainbow trend ESO is more capable of measuring a manager’s superior performance than the traditional indexed ESO,11 where the strike price is indexed to a benchmark to capture common risks beyond the manager’s control. First, similar to the index ESO, our rainbow trend ESO also eliminates common risks in an industry if a sufficiently large m is considered. Second, our rainbow trend ESO directly compares the performance of a firm and its main competitors, rather than the average performance of all firms (including better- and poorly performing ones) in an industry. If the goal of an ESO is to measure a manager’s superior performance, it is unreasonable 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 can avoid the manipulation problem and thus measure a manager’s true performance.

4.4 Modiﬁcation of the countercyclical capital buffer

Basel Committee on Banking Supervision (BCBS) proposed the scheme of the countercyclical capital buffer (CCB) to protect the banking sector in a nation from a downturn after excessive credit growth. In the proposal, banks are required to build up additional equity capital, i.e., the CCB, when the ratio of It = Creditt /GDPt deviates positively from its long-term trend, where GDPt and Creditt stand for, respectively, the gross domestic product and the aggregate credit granted to the household and private ˆ non-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 the credit risk faced by the bank in the following recession. The details regarding to the CCB can refer to BCBS (2010a, b). However, the cost to prepare the CCB is quite substantial and should be carried out only when the excessive growth is positively verified. On the other hand, it is well known that the use of GDP index alone may not be sufficient to reflect the true performance 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 improve the scheme of the CCB. For example, one can use multiple indexes to gauge the performance of an economy, such as the levels of retail sales (RSt ), the Industrial Pro- duction Index (IPIt ), or even the S&P 500 index (SPIt ), in addition to the GDP. By defining I1,t = Creditt /GDPt , I2,t = Creditt /RSt , I3,t = Creditt /IPIt , I4,t =

11 Indexed ESOs were ﬁrst proposed and discussed by Johnson and Tian (2000). 123 Author's personal copy

Rainbow trend options: valuation and applications 125

Table 6 Our option formulas as control variates for variance reduction

Linear simple RTO Arithmetic-average Asian rainbow 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 call options based on the Monte Carlo simulation with 50,000 paths and 20 repetitions. The examined values of parameters are m= 2, r = 0.05, q1 = q2 = 0, T = 1 year, S2 (t0) = 100,σ2 = 0.2, X = 100, and ρ12 = 0.5. In addition to option values, the corresponding standard errors of the simulation results are shown 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 variates for pricing the linear simple RTO and arithmetic-average Asian rainbow call option, respectively

Creditt /SPIt , one scheme that mitigates the above drawback can then be designed as follows. We use the event that − ˆ − ˆ − ˆ − ˆ I1,t I1,t , I2,t I2,t , I3,t I3,t , I4,t I4,t min ˆ ˆ ˆ ˆ I1,t I2,t I3,t I4,t is larger than 10%, i.e., when all ratios are higher than their individual trend levels by 10%, as a true signal of the excessive credit growth, instead of only using GDP. Once the excessive credit growth is veriﬁed, and banks should prepare the CCB amount equal 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 derive can be used to resolve several academic issues. For example, there are no closed-form 123 Author's personal copy

126 J.-Y. Wang et al. formulas for the linear RTO12 or the arithmetic-average Asian rainbow option and it is 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 for pricing them. However, its efﬁciency and accuracy depend crucially on the number of simulations, 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 and geometric-average Asian rainbow call options can be used as the control variates to reduce the variances of the simulation results of linear RTOs13 and arithmetic-average Asian rainbow call options,14 respectively. The results in Table 6 verify the efﬁciency of this control variates method. With the option values generated from our pricing formulainEq.(15) as the control variates, the standard errors can be reduced by 91.23 and 98.10%, on average, for pricing linear simple RTOs and arithmetic-average Asian rainbow call options, respectively. 5 Conclusion

This paper introduces a new class of option contracts by combining trend-based payoffs with rainbow options. RTOs can simultaneously solve investors’ asset selection and timing problems. We construct a pricing framework that can facilitate the derivation of the closed-form formulas for three different types of RTOs: simple, pure, and Asian RTOs. Our framework is general and able to evaluate rainbow options on the payoffs determined by different weighting averages of discrete-sampling asset returns. Due to the analytic property of the option pricing formulas, the formulas of the Greeks can be derived and analyzed as well. Although the lognormal distribution assumption in this 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 little affected by this inexact distribution assumption. On the other hand, issuers of RTOs should pay more attention to managing the estimation error for input parameters of our pricing formulas. This paper also explores possible applications of RTOs, including hedging the price risk of multiple substitutions, designing executive compensation plans, modifying countercyclical capital buffer proposed by Basel Committee, and acting as a control variate for pricing arithmetic average rainbow options or linear RTOs 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. ˇ Suppose the least-squares regression coefﬁcient is denoted as Bi . The linear trend price at T can be expressed ˇ ˇ as Si (T ) = Si (t0)+Bi (T − t0). The advantage of the linear trend is that it is more intuitive for the investing public to understand. + arg max 13 The payoff of a linear rainbow trend option is Sˇ (T ) − X ,wherem = Sˇ (T ).The m 1 ≤ i ≤ m i ˇ 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 + arg max ¯ rainbow call option can be expressed as S (T ) − X ,wherem = S¯ (T ). m 1 ≤ i ≤ m i 123 Author's personal copy

Rainbow trend options: valuation and applications 127 applicability of this new class of options, this paper contributes substantially to both the theoretical and practical literature.

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

Appendix 1: Derivations of the correlations ρX,ij,ρY,ij,andρXY,i

The evaluation of ρX,ij,ρY,ij, and ρXY,i uses the deﬁnition of the correlation coefﬁ- cient. For ρX,ij, ρ , = corr ln X (T ), ln X (T ) X ij i j cov ln Xi (T ), ln X j (T ) = √ Var (ln Xi (T )) Var ln X j (T ) cov αa+1 Ri,a+1 +···+αn Ri,n,αa+1 R j,a+1 +···+αn R j,n = σ 2 n α2 σ 2 n α2 i t h=a+1 h j t h=a+1 h n 2 = + α cov Ri,h, R j,h + + ≤ = ≤ αh αh cov Ri,h , R j,h = h a 1 h a 1 h1 h2 n 1 2 1 2 σ σ n α2t i j h=a+1 h n α2ρ σ σ t = h=a+1 h ij i j = ρ . σ σ n α2 ij i j h=a+1 h t

Similarly, the correlation between ln Yi (T )and ln Y j (T ), ρY,ij,isalsoρij.Asforthe correlation between ln Xi (T ) and ln Yi (T ),

ρXY,i = corr (ln Xi (T ), ln Yi (T )) ( X (T ), Y (T )) = √ cov ln i √ ln i ( ( )) ( ( )) Var ln Xi T Var ln Yi T cov αa+1 Ri,a+1 +···+αn Ri,n,βa+1 Ri,a+1 +···+βn Ri,n = σ 2 n α2 σ 2 n β2 t h=a+1 t h=a+1 i h i h n α β , h=a+1 h hcov Ri,h Ri,h = σ 2 n α2 n β2 i t h=a+1 h h=a+1 h n α β σ 2 h h i t h=a+1 = 2 n 2 n 2 σ t = + α = + β i h a 1 h h a 1 h n α β h=a+1 h h = # . n α2 n β2 h h=a+1 h h=a+1

123 Author's personal copy

128 J.-Y. Wang et al.

Appendix 2: Derivations of the general pricing formula

In Eq. (14), the arbitrage-free price of the examined option is − τ m ( ) = r a Q ( ) · V ta e E Xi T I{X (T )≥X (T )} {X (T )≥KY (T )} i=1 $ i j %& 1≤ j=i≤m i i ' I 1 − τ m − r a Q ( ) · . Ke E Yi T I{X (T )≥X (T )} {X (T )≥KY (T )} (34) i=1 $ i j %& 1≤ j=i≤m i i '

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 X (T ) = exp μ + σ εQ and Y (T ) = exp μ + σ εQ , i Xi Xi Xi i Yi Yi Yi where a ln S (t ) μ = α S (t ) + α i h Xi 0 ln i 0 = h h 1 ln Si (th−1) n + μi t αh, h=a+1 n σ 2 = σ 2t α2, Xi i h=a+1 h a ln S (t ) μ = β S (t ) + β i h Yi 0 ln i 0 = h h 1 ln Si (th−1) n + μi t βh, h=a+1 n σ 2 = σ 2t β2, Yi i h=a+1 h and εQ and εQ follow the standard normal distribution. In addition, μ t = Xi Yi i σ 2 − − i σ 2 r qi 2 t and i t are the mean and variance, respectively, of the logarithm of the return of the ith underlying asset in each time interval. The correlation Q Q Q Q between ε and ε is ρ , , the correlation between ε and ε is ρ , , and the Xi X j X ij Yi Y j Y ij Q Q correlation between ε and ε is ρ , . Xi Yi XY i

Evaluating I1 in Eq. (34)

We have = Q ( )· I1 E Xi T I{X (T )≥X (T )} {X (T )≥KY (T )} i j 1≤ j=i≤m i i Q Q μX +σX ε = E e i i Xi ·I{ ( )≥ ( )} { ( )≥ ( )} Xi T X j T 1≤ j=i≤m Xi T KYi T 123 Author's personal copy

Rainbow trend options: valuation and applications 129 1 2 Q 1 2 Q μX + σ +σX ε − σ = E e i 2 Xi i Xi 2 Xi ·I{ ( )≥ ( )} { ( )≥ ( )} Xi T X j T 1≤ j=i≤m Xi T KYi T 1 2 1 2 Q μX + σ Q − σ +σX ε = e i 2 Xi E e 2 Xi i Xi ·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

1 2 Q dQXi − σ +σX ε η = = e 2 Xi i Xi Xi dQ with respect to the risk-neutral measure Q and an auxiliary measure Q Xi that is equivalent to the risk-neutral measure Q. Then we can write 1 2 μX + σ = i 2 Xi Q Xi I1 e E I{Xi (T )≥X j (T )} {Xi (T )≥KYi (T )} 1≤ j=i≤m 1 2 ( ) * μX + σ = i 2 Xi Q Xi ( )≥ ( ) { ( ) ≥ ( )} , e P Xi T X j T 1≤ j=i≤m Xi T KYi T

Q Xi [·] where E is the expectation operator under the measure Q Xi . To evaluate Q X E i [IA], where A denotes any set of events, we proceed with the distributional prop- Q X Q X erty of the underlying assets, that is, E i [IA]=P i (A). In addition, according to the Girsanov theorem, we derive

Q Q X Q Q X Q Q X ε = ε i + σ ,ε = ε i + ρ , σ ,ε = ε i + ρ , σ , Xi Xi Xi X j X j X ij Xi Yi Yi XY i Xi

Q Q Q where ε Xi ,ε Xi , and ε Xi follow the standard normal distribution under the measure Xi X j Yi , Q Xi and thus Xi X j , and Yi can be expressed, respectively, under the measure Q Xi as Q ( ) = μ + σ 2 + σ ε Xi , Xi T exp Xi X Xi X i i Q ( ) = μ + ρ σ σ + σ ε Xi , X j T exp X j X,ij Xi X j X j X j Q X Y (T ) = exp μ + ρ , σ σ + σ ε i . i Yi XY i Yi Xi Yi Yi

Consequently, I1 can be evaluated as μ + 1 σ 2 Q Q Q Q Xi 2 X Q X Xi Xi Xi Xi I1 = e i P i Z , ≤ d , , Z , ≤ d , X j Xi X j Xi 1≤ j=i≤m Yi Xi Yi Xi μ + 1 σ 2 Q Q Xi 2 X Xi Xi Q X = e i Nm d , , d , ; R i , X j Xi 1≤ j=i≤m Yi Xi

15 This measure change technique was ﬁrst proposed by Cameron and Martin (1944). Moreover, this technique is a special case of the Girsanov theorem, which is applied to changing measures for stochastic processes. 123 Author's personal copy

130 J.-Y. Wang et al. where Q Q σ ε Xi − σ ε Xi Q X j X Xi X Xi ≡ j i Z X ,X j i Q Q var σ ε Xi − σ ε Xi X j X j Xi Xi Q Q σ ε Xi − σ ε Xi X j X Xi X = j i , for 1 ≤ j = i ≤ m, σX,ij

Q Xi Q Xi Q σY ε − σX ε Xi ≡ i Yi i Xi ZY ,X i i Q Q var σ ε Xi − σ ε Xi Yi Yi Xi Xi

Q Xi Q Xi σY ε − σX ε = i Yi i Xi , σXY,i μ + σ 2 − μ + ρ , σ σ Q Xi X X j X ij Xi X j Xi = i , ≤ = ≤ , dX ,X for 1 j i m j i σX,ij μ + σ 2 − μ + ρ , σ σ − Q Xi X Yi XY i Xi Yi ln K Xi = i , dY ,X i i σXY,i and ( ) ( ) Q I (m−1)×(m−1) II (m−1)×1 R Xi = , (II) (III) × 1×(m−1) 1 1 m×m with Q X Q X (I )( − )×( − ) = corr Z i , Z i m 1 m 1 Xl ,Xi X j ,Xi (m− )×(m− ) 1 1 2 σ − ρX,ilσX σX − ρX,ijσX σX + ρX, jlσX σX = Xi i l i j j l , σ , σ , X ij X il (m−1)×(m−1) ≤ = ≤ ≤ = ≤ , for 1 l i m and 1 j i m Q X Q X (II)( − )× = corr Z i , Z i m 1 1 X j ,Xi Yi ,Xi (m− )× 1 1 2 σ − ρXY,i σX σY − ρX,ijσX σX + ρX,ijρXY,i σX σY = Xi i i i j j i , σ , σ , X ij XY i (m−1)×1 ≤ = ≤ , for 1 j i m Q X Q X (III) × = corr Z i , Z i = 1. 1 1 Yi ,Xi Yi ,Xi 1×1

Note that we deﬁne σ 2 ≡ σ 2 − 2ρ , σ σ + σ 2 and σ 2 ≡ σ 2 − X,ij Xi X ij Xi X j X j XY,i Xi 2ρ , σ σ + σ 2 to obtain the above formulas. XY i Xi Y j Yi 123 Author's personal copy

Rainbow trend options: valuation and applications 131

Evaluating I2 in Eq. (34)

We have = Q ( )· I2 E Yi T I{X (T )≥X (T )} {X (T )≥KY (T )} i j 1≤ j=i≤m i i Q Q μY +σY ε = E e i i Yi ·I{ ( )≥ ( )} { ( )≥ ( )} Xi T X j T 1≤ j=i≤m Xi T KYi T 1 2 Q 1 2 Q μY + σ +σY ε − σ = E e i 2 Yi i Yi 2 Yi ·I{ ( )≥ ( )} { ( )≥ ( )} Xi T X j T 1≤ j=i≤m Xi T KYi T 1 2 1 2 Q μY + σ Q − σ +σY ε = e i 2 Yi E e 2 Yi i Yi ·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-neutral measure Q:

1 2 Q dQYi − σ +σY ε η = = e 2 Yi i Yi . Yi dQ

Then we can write 1 2 μY + σ = i 2 Yi QYi I2 e E I{Xi (T )≥X j (T )} {Xi (T )≥KYi (T )} 1≤ j=i≤m 1 2 ( ) * μY + σ = i 2 Yi QYi ( )≥ ( ) { ( ) ≥ ( )} , e P Xi T X j T 1≤ j=i≤m Xi T KYi T

Q Q where E Yi [·] and P Yi (·) are the expectation and probability operator, respectively, under the equivalent measure QYi . Using the transformations

Q QY Q QY Q QY ε = ε i + ρ , σ ,ε = ε i + ρ , ρ , σ ,ε = ε i + σ , Xi Xi XY i Yi X j X j XY i X ij Yi Yi Yi Yi

Q Q Q based on the Girsanov theorem, where ε Yi ,ε Yi , and ε Yi follow the standard normal Xi X j Yi , distribution, we can express Xi X j , and Yi under the measure QYi as, respectively, Q ( ) = μ + ρ σ σ + σ ε Yi , Xi T exp Xi XY,i Yi Xi Xi X i Q ( ) = μ + ρ ρ σ σ + σ ε Yi , X j T exp X j XY,i X,ij Yi X j X j X j Q Y (T ) = exp μ + σ 2 + σ ε Yi . i Yi Yi Yi Yi

Consequently, I2 can be evaluated as μ + 1 σ 2 Q Q Q Q Yi 2 Y QY Yi Yi Yi Yi I2 = e i P i Z , ≤ d , ,Z , ≤ d , X j Xi X j Xi 1≤ j=i≤m Yi Xi Yi Xi μ + 1 σ 2 Q Q Yi 2 Y Yi Yi QY = e i Nm d , ,d , ; R i , X j Xi 1≤ j=i≤m Yi Xi 123 Author's personal copy

132 J.-Y. Wang et al. where

Q Q σ ε Yi − σ ε Yi Q X j X Xi X Yi ≡ j i Z X ,X j i Q Q var σ ε Yi − σ ε Yi X j X j Xi Xi Q Q σ ε Yi − σ ε Yi X j X Xi X = j i , for 1 ≤ j = i ≤ m, σX,ij

QYi QYi Q σY ε − σX ε Yi ≡ i Yi i Xi ZY ,X i i Q Q var σ ε Yi − σ ε Yi Yi Yi Xi Xi

QYi QYi σY ε − σX ε = i Yi i Xi , σ XY,i Q μX + ρXY,i σY σX − μX + ρX,ijρXY,i σY σX Yi = i i i j i j , ≤ = ≤ , dX ,X for 1 j i m j i σX,ij μ + ρ , σ σ − μ + σ 2 − Q Xi XY i Yi Xi Yi Y ln K Yi = i , dY ,X i i σXY,i and ( ) ( ) Q I (m−1)×(m−1) II (m−1)×1 R Yi = , (II) (III) × 1×(m−1) 1 1 m×m with QYi QYi (I )(m−1)×(m−1) = corr Z , ,Z , Xl Xi X j Xi − × − (m 1) (m 1) Q Xi Q Xi = corr Z , , Z , , Xl Xi X j Xi (m−1)×(m−1) for 1 ≤ l = i ≤ m and 1 ≤ j = i ≤ m,

QYi QYi (II)(m−1)×1 = corr Z , , Z , X j Xi Yi Xi − × (m 1) 1 Q Xi Q Xi = corr Z , , Z , , X j Xi Yi Xi (m−1)×1 ≤ = ≤ , for 1 j i m Q Q ( ) = Yi , Yi III 1×1 corr ZY ,X ZY ,X i i i i 1×1 Q Xi Q Xi = corr Z , , Z , = 1. Yi Xi Yi Xi 1×1 123 Author's personal copy

Rainbow trend options: valuation and applications 133

QY Q X Note that the correlation matrix R i in I2 is identical to the correlation matrix R i in I1. Finally, we combine everything to derive the general pricing formula m μ + 1 σ 2 Q Q −rτa Xi 2 X Xi Xi Q X V (ta) = e e i Nm d , , d , ; R i i=1 X j Xi 1≤ j=i≤m Yi Xi m μ + 1 σ 2 Q Q −rτa Yi 2 Y Yi Yi QY − Ke e i Nm d , , d , ; R i . i=1 X j Xi 1≤ j=i≤m Yi Xi

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