# A Valuation Formula for Chained Options withn-Barriers

Authors: Won Choi; Doobae Jun; Hyejin Ku
Journal: Journal of Mathematics
 (2022)

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## Abstract

This study examines chained options that are connected in the sense that another barrier option becomes active continuously after the underlying asset price crosses a primary barrier. These barrier options have several advantages. First, they preserve the merit of regular barrier options, but demand far lower option premiums, which appeal to option traders. Second, they reduce the higher risk of loss of double barrier options, making option strategies more profitable in certain cases. Third, they have closed-form pricing formulas, unlike double-barrier options, and, thus, avoid the complexity of option pricing. Therefore, they help to enlarge the range of trader’s choice according to a variety of demand of buyers. The values of chained options are compared to those of similar single- and double-barrier options. This study extends the chained option with two barriers to a generalized chained option withn-barriers. In addition, this paper proves the closed formulas of generalized chained options with n-barriers using mathematical induction.


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## Body

## 1. Introduction

The rapid spread of the novel coronavirus (COVID-19) has dramatically impacted financial markets worldwide. It has created an unprecedented level of risk, causing investors to suffer significant losses in a very short period of time [1]. In a market facing sudden crisis, financial derivatives, such as barrier options, can be considered to reduce the risk of volatility and save money.Barrier options are a widely used class of path-dependent financial derivatives because they are flexible and less expensive than vanilla options. Merton [2], Rubinstein and Reiner [3], and Rich [4] provided a mathematical framework and derived closed-form pricing formulas for various types of single-barrier options. These studies assume that the underlying asset price is monitored with respect to a single constant barrier for the entire life of the option. Owing to their popularity in the market, more complicated structures of barrier options have been studied. Kunitomo and Ikeda [5], Geman and Yor [6], and Pelsser [7] provided double-barrier options with two barriers.Additional modifications of the barrier options include window barrier (partial barrier) options. In the case of a window barrier option, the trigger is valid only within a certain period. Heynen and Kat [8] studied partial barrier options, where the underlying price was monitored during only part of the option’s lifetime. By solving the Black–Scholes partial differential eqnarray under the appropriate boundary conditions, Hui [9] priced front-end and rear-end double barrier options, featuring early-ending and forward-starting monitoring, respectively. Guillaume [10] provided expressions of standard and partial window double-barrier option values as the infinite sums of three-dimensional and seven-dimensional normal distribution functions.All the papers described above are concerned with barrier options, where the monitoring of the barrier starts at a predetermined date. There is another class of barrier options with two barriers. The option is a chained option in which another barrier option is activated when a primary barrier is hit. This option has become popular in over-the-counter equity and foreign exchange derivative markets. Jun and Ku [11, 12] derived closed-form valuation formulas for chained barrier options of various types. Furthermore, Jun [13] explored the continuity correction approach for the approximate pricing of discretely monitored chained options.An international pattern of exotic derivatives, whose value is determined by the price or volatility of the underlying asset, appeared to have helped to transmit the financial crisis of 2008 from the United States and European Union to many different emerging market economies. The direct cost of these derivatives losses on nonfinancial firms is approximately$530 billion, based on the sum of national estimates [14]. The complexity of pricing exotic derivatives may put investors at risk if they cannot independently evaluate the risks and price the derivatives.In this study, the features of the chained option provide several desirable properties. The chained options can provide an initial protection period during which the option cannot be knocked out so that it reduces the risk of loss when a trader chooses to buy a double barrier option and is still cheaper than a single barrier option. In addition, chained options are the least expensive in certain cases among similar single and double barrier options.In Section2, the values of chained options are compared to those of similar single- and double-barrier options, and graphs are shown to illustrate that chained options are less expensive and more effective in various circumstances. Section 3 gives the valuation formula of the generalized chained option, which is an extended version of the chained option with two barriers. Finally, conclusions are presented in Section 4.


## 2. Comparison of Chained Options with Other Barrier Options

The valuation formulas for chained barrier options of various types are given in [11] for the constant barrier cases and [12] for general curved barriers. In chained options, a regular barrier option is activated when a primary barrier is hit. For example, a down-and-in chained call DICu is a down-and-in call option activated when the underlying asset price hits an upper barrier level. This option gives the option holder the payoff of a call if the price of the underlying asset rises above an upper barrier and then falls below a lower barrier before expiry, and it pays off zero otherwise. An up-and-in doubly chained call UICud is an up-and-in call option which is activated at a time when the asset price crosses two different barrier levels (an up barrier followed by a down barrier).Chained options differ from double-barrier options in the sense that the other barrier does not exist until one barrier is hit according to their predetermined order. In other words, crossing a barrier leads to the creation or cancelation of a barrier option rather than a call or put.Letr be the risk-free interest rate and σ be a constant. Assume that the price of the underlying asset S follows a geometric Brownian motion:(1)St=S0eμ¯t+σWt,where μ¯=r−σ2/2 and Wt is a standard Brownian motion under a filtered probability space Ω,F,Ft,P¯. Here, P¯ is the risk-neutral probability measure.Let us denote withτu and τd the first time that the underlying asset price St at time t hits the up barrier U and down barrier D, respectively, and with τud the first time after τu that the underlying asset price touches the down barrier D:(2)τu≔mint:St=U,U>S0,τd≔mint:St=D,D<S0,τud≔mint>τu:St=D.LetXt=1/σlnSt/S0,k=1/σlnK/S0,u=1/σlnU/S0 and d=1/σlnD/S0. The minimum and maximum for Xt are(3)mab=inft∈a,bXt,Mab=supt∈a,bXt,and are denoted by EP¯ the expectation operator under the P¯-measure.The chained optionsDOCu,DICu and double-barrier options UIDOC,UIDIC can be defined as follows:(4)DOCu=e−rTEP¯ST−K+1mτuT>d,τu≤T,DICu=e−rTEP¯ST−K+1mτuT≤d,τu≤T,UIDOC=e−rTEP¯ST−K+1τu≤T<τd,UIDIC=e−rTEP¯ST−K+1τu≤T,τd≤T,where 1 is an indicator function, K is the strike price, and T is the maturity of the options.The down-and-out chained call optionDOCu is similar to a double-barrier option with upper “in” and lower “out” barriers UIDOC in the sense that they pay off a call if the up barrier is touched and the down barrier is not before expiry. However, they differ in the case of τd<τu<T<τud; the payoff of UIDOC is zero because the down barrier has been touched, while DOCu pays off a call ST−K+. In addition, the down-and-in call option DICu and up-and-in and down-and-in double-barrier option UIDIC are similar, but have different payoffs when τd<τu<T<τud. Because DICu ignores the movement of the underlying asset price before reaching the up barrier, even if the down barrier has been hit before τu, it pays off 0 unless the down barrier is breached again after τu.If one expects the underlying stock to rally strongly, an up-and-in call option can result in a higher profitability than the vanilla call option, and it is cheaper. If the trader wants to maximize their profits, they may bet on a double barrier option with an upper knock-in and a lower knock-out barrierUIDOC at an even cheaper price. However, the trader faces a higher risk of loss if the underlying asset moves downward and crosses the lower barrier at any time. What if the trader holds a down-and-out chained option DOCu? In this case, a lower knock-out barrier does not appear until the upper barrier is touched. Once the underlying asset price reaches the upper-barrier level, the probability of hitting the down barrier afterward is much smaller than before. The chained option has a much lower risk of loss than a double-barrier option UIDOC and is still cheaper than an up-and-in call option UIC, which results in higher profits.Figure1 shows the graphs of the option values UIC, UIDOC, and DOCu at time t<τu and tτd<τu<t<τud. The parameter values in the left figure are U=105,D=90,K=100,r=0.05,σ=0.3, and T=1. The closer the current asset price is to the up barrier, the greater the likelihood of the option being knocked in and thus the higher the option values.Figure 1
Comparison ofUIC,DOCu,UIDOC when (a) t<τu and when (b) τd<τu<t<τud.
(a)(b)The parameter values that we used in the right figure areU=110,D=95,K=100,r=0.05,σ=0.3, and T=1. Once the down barrier was breached, the UIDOC was 0. If the underlying asset price then rises to the up barrier, UIC equals the vanilla call price, while DOCu is equivalent to the regular down-and-out call price. From Figure 1, it is observed that the chained down-and-out call option DOCu is always in between UIC and UIDOC. Furthermore, when the volatility is higher, DOCu is farther from the UIC because the likelihood of the option being knocked out is greater.As the down barrier decreases, the values of the three options become close. However, if the down barrier approaches the initial spotS0, the difference between DOCu and UIDOC becomes larger. Because they pay off a call option, barrier criteria are satisfied, and the values naturally increase as the strike price K increases. When S0 approaches the up barrier, the value UIDOC approaches DOCu.Next, we consider chained options with knock-in barriers. A down-and-in chained call optionDICu is a down-and-in call option activated at a time when the underlying asset price hits an upper-barrier level. DICu demands a far lower premium than both the single up-and-in call option UIC and double touch knock-in option UIDIC. Figure 2 shows that DICu is lower than UIC and UIDIC at different times. The parameter values are U=110,D=90,K=100,r=0.05,σ=0.3, and T=1. UIDIC is the same as UIC if the current spot price is below the down barrier 90 and decreases as the asset price St increases otherwise. When τd<τu<t<τud, both UIC and UIDIC are equal to the vanilla call price. However, DICu is the same as the vanilla call option if St is below the down barrier, while it is equal to the regular down-and-in barrier option price otherwise.Figure 2
Comparison ofUIC,UIDIC,DICu when (a) t<τu and (b) τd<τu<t<τud.
(a)(b)


## 3. Pricing Formulas for General Chained Option

The chained option pricing formula with only two barriers is provided in the work of Jun and Ku [11]. In this section, the valuation formula of the chained option with n≥2-barriers is described.Consider European options expiring atT with strike price K, up barriers Uii=1,…,nn≥2, and down barriers Dii=1,…,nn≥2. These options are chained options, where a barrier option is given when the underlying asset price hits up and down barriers in the following four orders:(5)A1:U1D1⋯Un−1Dn−1UnDi<Ui,Di<Ui+1i=1,…,n−1,A2:D1U1⋯DnUnDi<Uii=1,…,n,Ui>Di+1i=1,…,n−1,A3:U1D1⋯UnDnDi<Uii=1,…,n,Di<Ui+1i=1,…,n−1,A4:D1U1⋯Dn−1Un−1DnDi<Ui,Ui>Di+1i=1,…,n−1.The following theorem presents the valuation formula of the general chained option, where the down-and-in call option is given when the underlying asset price hits the barriers in the orderA1 or A2 with down barrier D<Un.Theorem 1.
The valuation formula for the down-and-in call option commencing at a time the underlying asset hits the barriers in the orderA1 or A2 and across the last up barrier Un when the strike price K is greater than the down barrier D is(6)IDICU1D1⋯Un−1Dn−1Un=S0DUn∏i=1n−1DiUi2μ˜/σ2Nz1−e−rTKDUn∏i=1n−1DiUi2μ¯/σ2Nz1−σT,IIDICD1U1⋯DnUn=S0DS0∏i=1nDiUi2μ˜/σ2Nz2−e−rTKDS0∏i=1nDiUi2μ¯/σ2Nz2−σT,where(7)z1=1σTlnD2S0Un2K∏i=1n−1DiUi2+μ˜σT,μ˜=r+σ22,z2=1σTlnD2KS0∏i=1nDiUi2+μ˜σT,and N⋅ is the cumulative standard normal distribution function.Proof.
Mathematical induction can be used to prove the closed formula of chained optionDICU1D1⋯Un−1Dn−1Un.
Whenn=1 in I, then(8)DICU1=S0DU12μ˜/σ2Nz1−e−rTKDU12μ˜/σ2Nz1−σTwhere(9)z1=1σTlnD2S0U12K+μ˜σT,μ˜=r+σ22
This formula is the same as Theorem 2.1 in [11]. Thus, I holds for n=1.
Suppose formulaI holds for n=k:(10)DICU1D1⋯Uk−1Dk−1Uk=S0DUk∏i=1k−1DiUi2μ˜/σ2Nz1−e−rTKDUk∏i=1k−1DiUi2μ˜/σ2Nz1−σT,where(11)z1=1σTlnD2S0Un2K∏i=1k−1DiUi2+μ˜σT,μ˜=r+σ22.
Then, we show thatI holds for n=k+1.
Under the risk-neutral probabilityP¯, the chained option is(12)DICU1D1⋯UkDkUk+1=e−rTEP¯ST−K+1mτ2k+1T<d,τ1<τ2<⋯<τ2k+1≤T,Sτ1=U1,Sτ2=D1,…,Sτ2k+1=Uk+1=e−rTEP¯ST−K1mτ2k+1T<d,ST>K,τ1<τ2<⋯<τ2k+1≤T,Sτ1=U1,Sτ2=D1,…,Sτ2k+1=Uk+1,where mτ2k+1T=inft∈τ2k+1,TSt. Let us define a new measure P˜ such that(13)dP˜dP¯=e−1/2σ2T+σWT.
Then, we have(14)DICU1D1⋯UkDkUk+1=S0P˜mτ2k+1T<d,ST>K,τ1<τ2<⋯<τ2k+1≤T,Sτ1=U1,Sτ2=D1,…,Sτ2k+1=Uk+1−e−rTKP¯mτ2k+1T<d,ST>K,τ1<τ2<⋯<τ2k+1≤T,Sτ1=U1,Sτ2=D1,…,Sτ2k+1=Uk+1.
We calculate the required probability only under theP¯-measure. Note that(15)P¯mτ2k+1T<d,ST>K,τ1<τ2<⋯<τ2k+1≤T,Sτ1=U1,Sτ2=D1,…,Sτ2k+1=Uk+1=P¯mτ2k+1T<d,XT>k,τ1<τ2<⋯<τ2k+1≤T,Xτ1=u1,Xτ2=d1,…,Xτ2k+1=uk+1,where Xt=Wt+μ¯/σt is a standard Brownian motion under the equivalent probability measure Q, defined by(16)dQdP¯=exp−μ¯σWT−12μ¯σ2T.
Then,(17)P¯mτ2k+1T<d,XT>k,τ1<τ2<⋯<τ2k+1≤T,Xτ1=u1,Xτ2=d1,…,Xτ2k+1=uk+1=EQdP¯dQ1mτ2k+1T<d,XT>k,τ1<τ2<⋯<τ2k+1≤T,Xτ1=u1,Xτ2=d1,…,Xτ2k+1=uk+1=EQeμ¯/σXT−1/2μ¯2/σ2T1mτ2k+1T<d,XT>k,τ1<τ2<⋯<τ2k+1≤T,Xτ1=u1,Xτ2=d1,…,Xτ2k+1=uk+1.
Let us introduce a processX˜t,t∈0,T, defined by the formula(18)X˜t=Xt,t≤τ2k,2dk−Xt,t>τ2k.
By virtue of the reflection principle, processX˜t follows a standard Brownian motion under Q, and(19)EQeμ¯/σXT−1/2μ¯2/σ2T1B1=EQeμ¯/σ2dk−X˜T−1/2μ¯2/σ2T1B2where(20)B1=mτ2k+1T<d,XT>k,τ1<τ2<⋯<τ2k+1≤T,Xτ1=u1,Xτ2=d1,…,Xτ2k+1=uk+1,B2=M˜τ2k+1T≥2dk−d,X˜T<2dk−k,τ1<τ2<⋯<τ2k+1≤T,Xτ1=u1,Xτ2=d1,…,Xτ2k−1=uk−1,Xτ2k+1=2dk−uk+1.
Here,M˜τ2k+1T=supt∈τ2k+1,TX˜t. We apply the reflection principle again. Let us introduce a process X^t,t∈0,T defined by the formula(21)X^t=X˜t,t≤τ2k+1,22dk−uk+1−X˜t,t>τ2k+1.
Then, the processX^t also follows a standard Brownian motion under Q, and(22)EQeμ¯/σ2dk−X˜T−1/2μ¯2/σ2T1B2=EQeμ¯/σ−2dk+2uk+1+X^T−1/2μ¯2/σ2T1B3where(23)B3=m^τ2k+1T≤2dk−2uk+1+d,X^T>2dk−2uk+1+k,τ1<τ2<⋯<τ2k+1≤T,Xτ1=u1,Xτ2=d1,…,Xτ2k−1=uk−1,and m^τ2k+1T=inft∈τ2k+1,TX^t.
Sinceuk+1>dk, 2dk−2uk+1+d=d−2uk+1−dk<d, and(24)m^τ2k+1T≤2dk−2uk+1+d,τ2k+1≤T=m^τ2k−1T≤2dk−2uk+1+dthus,(25)EQeμ¯/σ−2dk+2uk+1+X^T−1/2μ¯2/σ2T1B3=eμ¯/σ−2dk+2uk+1EQeμ¯/σX^T−1/2μ¯2/σ2T1B4=Uk+1Dk2μ˜/σ2EQeμ¯/σX^T−1/2μ¯2/σ2T1B4,where(26)B4=m^τ2k−1T≤2dk−2uk+1+d,X^T>2dk−2uk+1+k,τ1<τ2<⋯<τ2k−1≤T,Xτ1=u1,Xτ2=d1,…,Xτ2k−1=uk−1.
Then,(27)P¯mτ2k+1T<d,XT>k,τ1<τ2<⋯<τ2k+1≤T,Xτ1=u1,Xτ2=d1,…,Xτ2k+1=uk+1=Uk+1Dk2μ˜/σ2P¯mτ2k−1T≤2dk−2uk+1+d,XT>2dk−2uk+1+k,τ1<τ2<⋯<τ2k−1≤T,Xτ1=u1,Xτ2=d1,…,Xτ2k−1=uk−1.
By combining the chained option formula forn=k, we complete the proof of I. The proof of II is similar to the proof process of I.
The following theorem presents the valuation formula of the general chained option, where the up-and-in call option is given when the underlying asset price hits the barriers in the orderA3 or A4 with down barrier U>Dn.Theorem 2.
The valuation formula for the up-and-in call option commencing at a time the underlying asset hits the barriers in the orderA3 or A4 and across the last up-barrier Dn when the strike price K is lower than the up barrier U>Dn is(28)IUICU1D1⋯UnDn=S0∏i=1nDiUi2μ˜/σ2Nz3+US0∏i=1nUiDi2μ˜/σ2Nz4−Nz5−e−rTK∏i=1nDiUi2μ¯/σ2Nz3−σT+US0∏i=1nUiDi2μ˜/σ2Nz4−σT−Nz5−σT,IIUICD1U1⋯Dn−1Un−1Dn=S0DnS0∏i=1n−1DiUi2μ˜/σ2Nz6+DnU∏i=1n−1DiUi2μ˜/σ2Nz7−Nz8−e−rTKDnS0∏i=1n−1DiUi2μ¯/σ2Nz6−σT+DnU∏i=1n−1DiUi2μ˜/σ2Nz7−σT−Nz8−σT,where(29)z3=1σTlnS0U∏i=1nDiUi2+μ˜σT,z4=1σTlnU2KS0∏i=1nUiDi2+μ˜σT,z5=1σTlnUS0∏i=1nUiDi2+μ˜σT,z6=1σTlnDn2US0∏i=1n−1DiUi2+μ˜σT,z7=1σTlnU2S0Dn2K∏i=1n−1UiDi2+μ˜σT,z8=1σTlnUS0Dn2∏i=1n−1UiDi2+μ˜σT.Remark 1.
Suppose thatn=1,U1=U, and D1=D in Theorem 2I. Then, UICU1D1 is equal to UICud in Theorem 3.1 of [11].
The following theorem presents the valuation formula of the general chained option, where the down-and-in put option is given when the underlying asset price hits the barriers in the orderA1 or A2 with down barrier D<Un.Theorem 3.
The valuation formula for the down-and-in put option commencing at a time the underlying asset hits the barriers in the orderA1 or A2 and across the last up barrier Un when the strike price K is greater than the down barrier D<Un is(30)IDIPU1D1⋯Un−1Dn−1Un=S0UnS0∏i=1n−1UiDi2μ˜/σ2Nz9+DUn∏i=1n−1DiUi2μ˜/σ2Nz10−Nz1−e−rTKUnS0∏i=1n−1UiDi2μ¯/σ2Nz9+σT+DUn∏i=1n−1DiUi2μ˜σ2Nz10−σT−Nz1−σT,IIDIPD1U1⋯DnUn=S0∏i=1nUiDi2μ˜/σ2Nz11+DS0∏i=1nDiUi2μ˜/σ2Nz12−Nz2−e−rTK∏i=1nUiDi2μ¯/σ2Nz11+σT+DS0∏i=1nDiUi2μ˜/σ2Nz12−σT−Nz2−σT,where(31)z9=1σTlnDS0Un2∏i=1n−1DiUi2−μ˜σT,z10=1σTlnDS0Un2∏i=1n−1DiUi2+μ˜σT,z11=1σTlnDS0∏i=1nDiUi2−μ˜σT,z12=1σTlnDS0∏i=1nDiUi2+μ˜σT.
The following theorem presents the valuation formula of the general chained option, where the up-and-in put option is given when the underlying asset price hits the barriers in the orderA3 or A4 with down barrier U>Dn.Theorem 4.
The valuation formula for the up-and-in put option commencing at a time the underlying asset hits the barriers in the orderA3 or A4 and across the last down barrier Dn when the strike price K is lower than the up barrier U>Dn is(32)IUIPU1D1⋯UnDn=S0US0∏i=1nUiDi2μ˜/σ2N−z4−e−rTKUS0∏i=1nUiDi2μ¯/σ2N−z4+σT,IIUIPD1U1⋯Dn−1Un−1Dn=S0UDn∏i=1n−1UiDi2μ˜/σ2N−z7−e−rTKUDn∏i=1n−1UiDi2μ¯/σ2N−z7+σT.
The closed formulas of the generalized chained option of the knock-out type can be obtained as follows. The down-and-out option can be represented by the difference between the up-and-in option and the down-and-in option. The up-and-out option can be expressed as the difference between the down-and-in option and the up-and-in option:(33)1DOCU1D1⋯Un−1Dn−1Un=UICU1D1⋯Un−1Dn−1U=Un−DICU1D1⋯Un−1Dn−1Un,2DOCD1U1⋯DnUn=UICD1U1⋯Un−1DnU=Un−DICD1U1⋯DnUn,3UOCU1D1⋯UnDn=DICU1D1⋯Dn−1UnD=Dn−UICU1D1⋯UnDn,4UOCD1U1⋯Dn−1Un−1Dn=DICD1U1⋯Dn−1Un−1D=Dn−UICD1U1⋯Dn−1Un−1Dn,5DOPU1D1⋯Un−1Dn−1Un=UIPU1D1⋯Un−1Dn−1U=Un−DIPU1D1⋯Un−1Dn−1Un,6DOPD1U1⋯DnUn=UIPD1U1⋯Un−1DnU=Un−DIPD1U1⋯DnUn,7UOPU1D1⋯UnDn=DIPU1D1⋯Dn−1UnD=Dn−UIPU1D1⋯UnDn,8UOPD1U1⋯Dn−1Un−1Dn=DIPD1U1⋯Dn−1Un−1D=Dn−UIPD1U1⋯Dn−1Un−1Dn.Remark 2.
The generalized chained option of the knock-out type is obtained from the knock-in and knock-out properties of the chained option. The property is that the sum of the knock-in chained option and knock-out chained option is equal to the opposite side knock-in chained option of the previous step.


## 4. Conclusion

This paper reviews chained options and compares chained options with similar single- and double-barrier options. A chained option is viewed as a barrier option that is chained together, each with a payoff contingent on a specified barrier. When the underlying asset price hits a primary barrier, a secondary barrier option is given to the primary barrier option holder. If the asset price hits another barrier, a third barrier option is given, and so on. This paper proves the closed formulas of generalized chained options withn-barriers using mathematical induction. These formulas include valuations of down-and-in call options when the underlying asset price hits the barriers in the order A1 or A2 and the up-and-in call option commencing at a time the underlying asset hits the barriers in the order A3 or A4.



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