# Valuation on an Outside-Reset Option with Multiple Resettable Levels and Dates

Authors: Guangming Xue; Bin Qin; Guohe Deng
Journal: Complexity
 (2018)

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## Abstract

This paper studies an outside-reset option with multiple strike resets and reset dates, in which the strike price is adjusted by an external process associated with the underlying risky asset. We obtain analytical pricing formula for this option and the hedging parameters Delta and Gamma. Furthermore, some numerical examples are provided to analyze some characteristics of the outside-reset option and to examine the impacts of the external parameters on option prices and Greeks. These results show that the external process can significantly affect option prices and Greeks.


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## Body

## 1. Introduction

Reset options, whose strike price will be adjusted to a new strike price only on each of a set of prespecified dates if the stock price is below one of the reset levels, have greatly evolved in the past two decades. This reset clause embedded in derivative products can protect the investors amidst stock price declines. This makes a reset option useful in portfolio insurance (see, e.g., Boyle et al. [1]).There are only a few articles studying the reset options in the academic literature. Gray and Whaley [2] were the first ones to examine the value of S&P 500 index bear market warrants with a periodic reset feature. In their other paper (see Gray and Whaley [3]), they provided an explicit formula for the reset option with a periodic reset date. Hsueh and Guo [4], on the other hand, analyzed the multiple reset feature that is included in most covered warrants traded in Taiwan. More recently, Cheng and Zhang [5] discussed the pricing and hedging of reset options and propose a closed-form pricing formula for this increasingly popular derivative instrument. Li et al. [6] derived a generalization of price formula for the reset call options with predetermined rates when the spot interest rate and volatility of stock are all time-dependent and deterministic. Liu et al. [7] evaluated the pricing of reset options when the underlying assets are autocorrelated. François-Heude and Yousfi [8] proposed a general valuation of reset option studied in Gray and Whaley [2] in which all options are replaced by ATM (At-The-Money) ones. Subsequent contributions include analytic extensions to multiple reset rights with shouting moment of Dai et al. [9, 10], Dai and Kwok [11], Yang et al. [12], and Goard [13], step (or snapshot)-reset design of Hsueh and Liu [14], and Yu and Shaw [15], average trigger reset clauses of Kao and Lyuu [16], Liao and Wang [17], Kim et al. [18], Chang et al. [19], Dai et al. [20], and Costabile et al. [21, 22], window reset option with continuous reset constraints of Hsiao [23], and reset rights embedded in the Quanto options of Chen and Jiang [24].In general, the reset call option withm predetermined reset dates 0<t1<t2<⋯<tm<T has a payoff at a fixed maturity T of (1)VT=max⁡ST-min⁡K0,St1,St2,…,Stm,0,where St denotes the underlying asset price at time t and K0 denotes the initial strike price of option. In a real application, the terminal payoff of reset call option is usually set as (2)VT=max⁡ST-K∗,0=ST-K∗+,where K∗ is defined by (3)K∗=K0,ifminSt1,St2,…,Stm≥D1,Kj,ifDj>minSt1,St2,…,Stm≥Dj+1,j=1,2,…,d-1,Kd,ifDd>minSt1,St2,…,Stm,where Kj, j=1,2,…,d, denote the strike price resets such that K0>K1>K2>⋯>Kd>0 and Dj, j=1,2,…,d, are the reset levels. In particular, there is only one reset price when d=1. For valuation on the general-reset options, Liao and Wang [25] provided an explicit pricing formula of this option and analyzed the phenomena of Delta jump and Gamma jump across reset dates. In fact, there is essentially no explicit pricing formula for the discrete reset options, except when they resort to multivariate cumulative normal distribution function.A common disadvantage of the reset option whose payoff is defined in (1), however, is that the reset trigger depends on the underlying asset price alone. This exposes the holder and the writer of the reset option to the risk that the counterparty may manipulate the underlying asset price such that the payoff of the reset option being benefits according to the counterparty favorable way. In other words, the strike price reset event is triggered by a price fluctuation intentionally caused by the counterparty. In order to prevent such price manipulation, reset options have been innovated where the trigger event does not depend on the underlying asset price but on an external variable Yt. For example, the underlying asset may be a foreign stock and the external variable may be an average of the underlying asset, the exchange rate or another asset correlating with the underlying asset. We will study different reset conditions imposed on a distinct but correlated underlying asset process. Such reset conditions are often called outside resets (see Heynen and Kat [26] and Kwok et al. [27]), and they are rather less studied than the regular type defined in (1). In this paper we propose an outside-reset option where the strike price K∗ is replaced by (4)K∗=K0,ifmin⁡Yt1,Yt2,…,Ytm≥D1,Kj,ifDj>min⁡Yt1,Yt2,…,Ytm≥Dj+1,j=1,2,…,d-1,Kd,ifDd>min⁡Yt1,Yt2,…,Ytm.This novel design of using the external variableYt as a reset trigger replacing the underlying asset St, as in the general-reset option specified in (2) and (3), offers three important advantages to both issuers and investors. First, the outside-reset option reduces the price manipulation around the reset level. Second, the outside-reset specification rules out jumps in Delta and thus makes the reset option more amenable to dynamic hedging. Finally, the outside-reset option provides a strike price correlated with an external variable fluctuation. The payoff that comes with these mentioned advantages above is complexity. The outside-reset option contains a simultaneous generalization of the reset option discussed by Liao and Wang [25] and a multivariate-based option of which both have a number of useful applications. For example, the arithmetic average reset options listed on the Taiwan Stock Exchange used an average of the stock prices as a trigger variable for reset. A barrier option on one or multiple assets with single external barrier also used other assets as resettable barrier variables for determining whether the options knock in or out. For valuation on these options, one can look to Cheng and Zhang [5], Liao and Wang [17], Kim et al. [18], Costabile et al. [22], Zhang [28], and so forth.This paper is to discuss the pricing problem of the outside-reset option, defined by (2) and (4), and has two contributions. The first is to derive the analytical pricing formula for the outside-reset option under the proposed model. Furthermore, we provide the analytical pricing formula for the outside-reset option with d strike resets and continuous reset dates, which is the limiting case of that with a set of discrete reset dates, and the hedging parameters Delta and Gamma. The second is to analyze the impacts of the external process on the option price and the values of Delta and Gamma and the phenomena of Delta and Gamma jumps across reset dates using numerical experiments.The paper is organized as follows: in Section2, a closed-form pricing formula is derived first for the outside-reset option under the assumption that the underlying asset price process and the external process follow two correlation geometric Brownian motions. In Section 3, we discuss some properties of the outside-reset option and present our numerical results. Finally, in Section 4, we give conclusion remarks and directions of future research.


## 2. The Proposed Model and Pricing Outside-Reset Option

Following Heynen and Kat [26], we consider the two stochastic differential equations for the underlying risky asset price St and the external process Yt (as a reset trigger) under a risk-neutral measure Q: (5)dYtYt=rdt+σ1dWt1,dStSt=rdt+ρσ2dWt1+1-ρ2σ2dWt2,where σ1>0, σ2>0, and -1≤ρ≤1 are constants, and Wt=(Wt1,Wt2) is a bidimensional standard Brownian motion defined on a filtered probability space Ω,F,Ft0≤t≤T,Q, where Ft0≤t≤T is the Q-augmentation of the filtration generated by Wt and satisfies the usual conditions, r≥0 is the constant risk-free interest rate, and ρ presents the correlation coefficient between St and Yt. It should be noted that model (5) may reduce to the results discussed by Liao and Wang [25] for the case when σ1=σ2 and ρ=1.Using Itô formula to model (5), then, for t<s, (6)Ys=Ytexp⁡r-12σ12s-t+σ1Ws1-Wt1,Ss=Stexp⁡r-12σ22s-t+ρσ2Ws1-Wt1+1-ρ2σ2Ws2-Wt2.In view of (2) and (4), the terminal payoff of the outside-reset option with d strike resets and m predetermined reset dates can be written as (7)VT=ST-K0+IA1+ST-K1+IA2-IA1+⋯+ST-Kd-1+IAd-IAd-1+ST-Kd+1-IAd,where Aj=min⁡Yt1,Yt2,…,Ytm≥Dj, j∈1,2,…,d, and I(·) is indicator function. Following the pricing theory, under the risk-neutral measure Q, the value at time t of the outside-reset option is (8)ORCt=EQe-rT-tVT∣Ft=EtQe-rT-tVT=∑l=1dEtQe-rT-tST-Kl-1+IAl-∑l=1dEtQe-rT-tST-Kl+IAl+EtQe-rT-tST-Kd+,where 0≤t<t1<t2<⋯<tm<T.From (8), we know that the key of solution is to compute the following expression: (9)EtQe-rT-tST-Kh+IAl,where h=l-1 or l. Now we present the result in the following proposition.Proposition 1.
The value at timet of the expression, EtQ{e-r(T-t)ST-Kh+IAl}, assuming the risky asset price given in the model (5), is (10)∑g=1mStNm+1Dgl,h;Σg-Khe-rT-tNm+1D-gl,h;Σg,where Nm+1(·;Σ) is the (m+1)-dimensional cumulative normal distribution function with mean vector 0 and correlation matrix Σ, and Dgl,h stands for the transpose of the gth row of matrix Dl,h as follows: (11)Dl,h=al,1b1,2b1,3⋯b1,mdhb2,1al,2b2,3⋯b2,mdhb3,1b3,2al,3⋯b3,mdh⋮⋮⋱⋮⋮bm,1bm,2bm,3⋯al,mdhm×m+1.The components of the matrixDl,h are defined as follows: (12)al,g=ln⁡Yt/Dl+r-1/2σ12+ρσ1σ2tg-tσ1tg-t,bg,j=r-1/2σ12+ρσ1σ2tj-tgσ1tj-tg,dh=ln⁡St/Kh+r+1/2σ22T-tσ2T-t.The components of the matrix D-l,h are the same as those of the ones of the matrix Dl,h except the components al,g,bg,j, and dh are replaced by a-l,g,b-g,j, and d-h, respectively. Here (13)a-l,g=al,g-ρσ2tg-t,b-g,j=r-1/2σ12tj-tgσ1tj-tg,d-h=dh-σ2T-t.The correlation matrix Σg=(ρk,jg)(m+1)×(m+1), k, j=1,2,…,m+1, where ρk,jg is given by (14)ρk,jg=ρj,kg=1,k=j,tg-tjtg-tk,1≤k<j≤g-1,org+1≤k<j≤m,-tg-tktg-t,1≤k≤g-1,j=g,-ρtg-tkT-t,1≤k≤g-1,j=m+1,ρtg-tT-t,k=g,j=m+1,ρtk-tgT-t,g+1≤k≤m,j=m+1,0,otherwise.Proof.
See AppendixA.It is noticed that the 3rd term in the right hand of (8) is the value of the European vanilla call option given by Black-Scholes formula. In view of (10) and (8), the closed-form pricing formula at time t for the outside-reset option is gained by the following proposition.Proposition 2.
The price at timet of the outside-reset option with m-periodic reset and d strike resets, assuming the risky asset price given in model (5), is (15)ORCt=StNdd+∑l=1d∑g=1mNm+1Dgl,l-1;Σg-Nm+1Dgl,l;Σg-e-rT-tKdNd-d+∑l=1d∑g=1mKl-1Nm+1D-gl,l-1;Σg-KlNm+1D-gl,l;Σg,where N(·) is the cumulative probability function of a standard normal variable.The numerical valuation ofNm+1(·;Σ) appearing in the analytic formula (15) (where m may take value beyond 100) can be very computationally demanding. Monte Carlo and Quasi-Monte Carlo simulation methods seem to be the most promising for higher-order probabilities. Genz [29] proposed a quasi-randomized Monte Carlo procedure to evaluate the multivariate normal probabilities. Boyle et al. [30] and Lai [31] applied the Quasi-Monte Carlo simulation methods to finance. In this paper, we use Genz’s method to compute the cumulative multivariate probability function in (15).


## 3. Characteristics of the Outside-Reset Option

In this section, we will analyze some characteristics of outside-reset option with the implementation of the pricing formula (15) given in Section 2. We then report some numerical results to illustrate the changes of option price with different parameters value in the model and the impacts of parameters of the external process on the option price. We also provide explicit closed-form formula for the outside-reset option with d strike resets and continuous reset dates and the hedging parameters Delta and Gamma. Finally, we analyze the phenomena of Delta jumps and Gamma jumps across reset dates.


### 3.1. Reset Features of the Outside-Reset Option

First, we discuss some properties of outside-reset option. In what follows, we let the current timet=0, the maturity T=1 (year), and initial strike price K0=100 of the outside-reset option. The strike price Kj, j=1,2,…,d, will be adjusted if the value of the external process Yt falls below the strike resets Dj. We will calculate the values of the outside-reset option with one strike reset and two strike resets under three reset dates, respectively. Three reset dates are assumed to be t1=1/12 (year), t2=2/12 (year), and t3=3/12 (year). These results are presented in Tables 1 and 2.Table 1
Values of the outside-reset option with single strike reset and three reset dates.
σ 2 S 0 K 1 r=0.05 r=0.07 m m 1 2 3 1 2 3 0.1 85 85 1.0983 1.3492 2.5126 1.4386 2.0270 3.0660 90 0.8019 0.8264 1.4968 1.1419 1.3303 1.9530 95 0.6658 0.6714 0.8900 0.9536 1.0532 1.2390 100 85 8.1683 8.4090 11.7214 9.4943 9.6839 13.0503 90 7.6661 7.6884 9.8844 9.0126 9.0186 11.2641 95 7.1837 7.5695 8.2075 8.5298 8.5466 9.5928  0.3 85 85 6.9739 7.5350 8.5148 7.5683 8.1354 9.1442 90 6.7524 7.0896 7.6832 7.3454 7.6879 8.3018 95 6.5682 6.7199 6.9890 7.1584 7.3132 7.5928 100 85 15.1206 16.0240 17.5469 16.0921 16.9846 18.5190 90 14.7867 15.3503 16.3057 15.7632 16.3221 17.2882 95 14.4905 14.7533 15.2013 15.4695 15.7311 16.1858 The parameters areσ1=0.3, ρ=0.25, Y0=100, and D1=90.Table 2
Values of the outside reset option with two strike resets and three reset dates.
σ 2 S 0 K 1 , K 2 r=0.05 r=0.07 m m 1 2 3 1 2 3 0.1 85 ( 85,75 ) 1.1504 1.2708 4.0516 1.4954 3.8198 4.6219 ( 90,80 ) 0.8540 0.8784 2.6921 1.1705 1.3878 3.2213 ( 95,85 ) 0.6836 0.6955 1.6860 0.9731 1.2277 2.1376 100 ( 85,75 ) 8.2040 8.6264 13.5817 9.4960 9.8386 14.8388 ( 90,80 ) 7.6975 8.0957 11.7032 9.0367 9.3292 13.0265 ( 95,85 ) 7.2358 7.6122 9.9175 8.6111 8.7650 11.2795  0.3 85 ( 85,75 ) 6.9973 7.7872 9.5545 7.5911 8.4549 10.1876 ( 90,80 ) 6.7721 7.2209 8.5729 7.3647 7.8165 9.2021 ( 95,85 ) 6.5847 6.8278 7.7387 7.1747 7.4199 8.3578 100 ( 85,75 ) 15.1534 17.9018 18.9530 16.1234 18.7775 19.9037 ( 90,80 ) 14.8161 15.9776 17.5850 15.7915 18.0760 18.5562 ( 95,85 ) 14.5166 15.6236 16.3492 15.4948 15.6759 17.3312 The parameters areσ1=0.3, ρ=0.25, Y0=100, T=1, D1=90, and D2=80.From Tables1 and 2, we can see that some characteristics of the outside-reset option are similar to the general-reset option studied by Liao and Wang [25]. For example, the values of the outside-reset call option are increasing functions of stock price, risk-free interest rate, and volatility of stock returns, and they are also increasing with the numbers of reset dates. In addition, under the same reset levels Dj, lower reset strike prices Kj will result in higher option values. Due to the higher protection to the option holder, the values of the outside-reset call option with two strike resets are always greater than that with one strike reset under the same model parameter values.


### 3.2. Impacts of Parameters of External Process on Outside-Reset Option

Next, we investigate the impacts of three parameters including the initial inputsY0, volatility σ1, and correlation coefficient ρ in the external process on the option price. In Figures 1, 2, and 3, the left panel depicts the option value with one strike reset (d=1) and the right panel depicts the option value with two strike resets (d=2).Figure 1
Option values as a function ofY0.
(a)
d = 1 (b)
d = 2Figure 2
Option values as a function of volatilityσ1.
(a)
d = 1 (b)
d = 2Figure 3
Option values as a function of correlation coefficientρ.
(a)
d = 1 (b)
d = 2In Figure1, we compare option values as a function of initial inputs Y0 with d(=1,2) strike resets and m(=1,2,3) reset dates. Assume that the model parameters are S0=100, r=0.05, σ1=0.3, σ2=0.3, K1=95, K2=85, D1=90, D2=80, and ρ=1. From Figure 1, we see that the option values decrease as Y0 increases for any reset dates. Also, we see that the outside-reset option value decreases more rapidly when the value of Y0 is in the interval [75,110]. However, the change is very little for Y0<75, which reduces the price manipulation around the reset level.In Figure2, we plot option values of the outside-reset calls versus the volatility σ1 of the external process when the correlated coefficient is ρ=1. Assume that the model parameters are S0=100, r=0.05, Y0=100, σ2=0.3, K1=95, K2=85, D1=90, D2=80, and ρ=1. The effect on price of the outside-reset option of volatility is remarkable, and option values increase as volatility increases. This is the reason that fluctuation of external process brings the profit opportunity for the option.Finally, we compare option values as a function of correlation coefficientρ in Figure 3, in which the model parameters are assumed to be S0=100, r=0.05, Y0=100, σ1=0.3, σ2=0.3, K1=95, K2=85, D1=90, and D2=80. Figure 3 illustrates that the correlation coefficient influences option value very significantly, and lower values of correlation coefficient will result in higher option values. In addition, if σ1=σ2 and ρ=1, then the outside-reset option reduces to the general-reset option studied by Liao and Wang [25]. Therefore, Figure 3 indicates that the option value of outside-reset calls is greater than that of the general-reset calls studied by Liao and Wang [25].


### 3.3. Outside-Reset Option with Continuous Reset Dates

Whenm approaches infinity with a remaining time to T-t, the set of discrete reset dates become a continuous reset period. From (7), the terminal payoff function of the outside-reset option with continuous reset period is as follows: (16)VS,Y,T=ST-Kd++∑l=1dST-Kl-1+Imin0≤t≤T⁡Yt≥Dl-∑l=1dST-Kl+Imin0≤t≤T⁡Yt≥Dl.Based on the closed-form formula of European down-and-out outside-barrier call option studied by Heynen and Kat [26], we can present the explicit expression for the outside-reset option with continuous reset period, because (17)EtQe-rT-tST-Kh+Imin0≤t≤T⁡Yt≥Dl=StN2c1h,-e1l;ρ-DlYt2r-1/2σ12+ρσ1σ2/σ12N2c-1h,-e-1l;ρ-e-rT-tKhN2c2h,l,-e2l;ρ-DlYt2r/σ12N2c-2h,l,-e-2l;ρ,where (18)c1h=ln⁡St/Kh+r+1/2σ22T-tσ2T-t,c1h,l=c1h-σ2T-t,c-1h=c1h+2ρσ1T-tln⁡DlYt,c-2h,l=c-1h,l+2ρσ1T-tln⁡DlYt,e1l=ln⁡Dl/Yt-r-1/2σ12+ρσ1σ2T-tσ1T-t,e2l=e1l+ρσ2T-t,e-1l=e1l-2σ1T-tln⁡DlYt,e-2l=e-2l-2σ1T-tln⁡DlYt.Therefore, we present the value at time t of the outside-reset option with continuous reset period in the following proposition.Proposition 3.
The price at timet of the outside-reset option with continuous reset period, assuming the risky asset price given in (5), is given by (19)ORCCt=StNdd+∑l=1dN2c1l-1,-e1l;ρ-N2c1l,-e1l;ρ-DlYt2r+ρσ12/σ12N2c-1l-1,-e-1l;ρ-N2c-1l,-e-1l;ρ-e-rT-tKdNd-d+∑l=1dKl-1N2c2l-1,l,-e2l;ρ-KlN2c2l,l,-e2l;ρ-DlYt2r/σ12Kl-1N2c-2l-1,l,-e-2l;ρ-KlN2c-2l,l,-e-2l;ρ.


### 3.4. Hedging the Outside-Reset Option

It is well known that hedge parameters (or Greeks) of options are very important in risk management of reset option. Of particular importance are the Delta and the Gamma, respectively, the rate of change of the option value with respect to the stock price, and the rate of change of the option Delta with respect to the stock price. Here, we provide the Delta and Gamma of the outside-reset option in Proposition4. Other Greeks can be similarly derived.Proposition 4.
The values at timet of Δ and Γ for the outside-reset call option with m predecided reset dates and d strike resets, assuming the risky asset price given in (5), are given as follows: (20)Δ=Ndd+∑l=1d∑g=1mNm+1Dgl,l-1;Σg-Nm+1Dgl,l;Σg+1σ2T-t∑l=1d∑g=1m∂Nm+1Dgl,l-1;Σg∂dl-1-∂Nm+1Dgl,l;Σg∂dl-e-rT-tSKl-1∂Nm+1D-gl,l-1;Σg∂d-l-1-Kl∂Nm+1D-gl,l;Σg∂d-l,(21)Γ=1Sσ2T-t12πe-dd2/2+∑l=1d∑g=1m∂Nm+1Dgl,l-1;Σg∂dl-1-∂Nm+1Dgl,l;Σg∂dl+1σ2T-t∂2Nm+1Dgl,l-1;Σg∂dl-12-∂2Nm+1Dgl,l;Σg∂dl2+e-rT-tSKl-1∂Nm+1D-gl,l-1;Σg∂d-l-1-Kl∂Nm+1D-gl,l;Σg∂d-l-e-rT-tσ2T-tKl-1∂2Nm+1D-gl,l-1;Σg∂d-l-12-Kl∂2Nm+1D-gl,l;Σg∂d-l2.Proof.
See AppendixB.In order to describe clearly the phenomena of Delta and Gamma for the outside-reset option, in the following, we limit ourselves to considering the simple case for this option with one reset date. Then, we have(22)N2a,b;ϱ=12π1-ϱ2∫-∞a∫-∞be-x2-2ϱxy+y2/21-ϱ2dxdy,∂N2a,b;ϱ∂a=12πe-a2/2Nb-ϱa1-ϱ2,∂2N2a,b;ϱ∂a2=12πe-a2/2-aNb-ϱa1-ϱ2-ϱ2π1-ϱ2e-b-ϱa2/21-ϱ2and the following corollary.Corollary 5.
The values at timet of Δ and Γ for the outside-reset call option with one reset date are, respectively, given by (23)Δ=Ndd+∑l=1dN2al,1,dl-1;ϱ-N2al,1,dl;ϱ+1σ22πT-te-dl-12/2Nal,1-ϱdl-11-ϱ2-e-dl2/2Nal,1-ϱdl1-ϱ2-e-rT-tSKl-1e-d-l-12/2Na-l,1-ϱd-l-11-ϱ2-Kle-d-l2/2Na-l,1-ϱd-l1-ϱ2,Γ=1Sσ22πT-te-dd2/2+∑l=1de-dl-12/2Nal,1-ϱdl-11-ϱ2-e-dl2/2Nal,1-ϱdl1-ϱ2+1σ2T-t-dl-1e-dl-12/2Nal,1-ϱdl-11-ϱ2-ϱe-al,12-2ϱal,1dl-1+dl-12/21-ϱ22π1-ϱ2+dle-dl2/2Nal,1-ϱdl1-ϱ2-ϱe-al,12-2ϱal,1dl+dl2/21-ϱ22π1-ϱ2+e-rT-tSKl-1e-d-l-12/2Na-l,1-ϱd-l-11-ϱ2-Kle-d-l2/2Na-l,1-ϱd-l1-ϱ2-e-rT-tσ2T-t-d-l-1Kl-1e-d-l-12/2Na-l,1-ϱd-l-11-ϱ2+d-lKle-d-l2/2Na-l,1-ϱd-l1-ϱ2-ϱKl-1e-a-l,12-2ϱa-l,1d-l-1+d-l-12/21-ϱ22π1-ϱ2-ϱKle-a-l,12-2ϱa-l,1d-l+d-l2/21-ϱ22π1-ϱ2,where ϱ=ρt1-t/T-t and t1 is the reset date.Whent→t1-, then (24)N2al,1,dl-1;ϱ⟶Ndl-1,12πe-dl2/2Nal,1-ϱdl1-ϱ2⟶12πe-dl2/2t=t1,N2al,1,dl;ϱ⟶0,12πe-d-l2/2Nal,1-ϱd-l1-ϱ2⟶12πe-d-l2/2t=t1.ConsequentlyΔ and Γ at time t1 become as follows: (25)Δ=Nd0+1σ22πT-t-1e-dd2/2-e-d02/2t=t1,Γ=1Stσ22πT-te-d02/2+1σ2T-t-2π·dde-dd2/2-d0e-d02/2t=t1.However, Δ and Γ at time t>t1 are given as follows: (26)Δ=Nd0IYt1≥D1+∑l=1d-1NdlIDl≥Yt1≥Dl+1+NddIYt1<Dd,Γ=1Stσ22πT-te-d02/2IYt1≥D1+∑l=1d-1e-dl2/2IDl≥Yt1≥Dl+1+e-dd2/2IYt1<Dd.From (25)-(26), we find that Δ and Γ at time t1 are continuous as a function of the stock price. Therefore, Delta and Gamma do not have significant jumps when the stock price goes across the reset dates because of using the external process as a reset trigger, and it makes outside-reset options more amenable to dynamic hedging. Figure 4 illustrates the phenomena at time t1 of Delta and Gamma for the outside-reset options at various underlying stock prices and contrasts those with the general-reset option specified in (2) and (3) which is studied by Liao and Wang [25]. Assume K0=100, σ1=σ2=0.3, ρ=1, Y0=100, r=0.05, T=1 (year), and t1=3/12 (year) and the current stock price is 100. We consider the strike price of the outside-reset option will be adjusted to 80,70,60,50, and 40 if the external variable Yt falls below 80,70,60,50, and 40. As shown in Figure 4, the Delta and Gamma of the general-reset options will fluctuate dramatically and oscillate if the stock prices near strike resets. This shows that the general-reset option has the phenomena of Delta jump and Gamma jump across reset date. With the outside-reset design, however, the oscillations in the delta are replaced by a much smoother double U-shaped curve. This added continuity of delta around reset level greatly facilitates option holders to hedge outside-reset options dynamically by trading the underlying stock. Therefore, it is evident that the outside-reset option does not have Delta and Gamma jumps as the general-reset option.Figure 4
Comparison of optionΔ and Γ between the general-reset option and outside-reset option.Figure5 plots the option Delta and Gamma of the outside-reset call option at time t1 for different values of Y0 shown in Figure 5, and other parameter values are similar to that in Figure 4. We can see that the option Delta and Gamma of the outside-reset calls are sensitive to the changes in S and Y.Figure 5
OptionΔ and Γ at time t1 for different Y0.


## 3.1. Reset Features of the Outside-Reset Option

First, we discuss some properties of outside-reset option. In what follows, we let the current timet=0, the maturity T=1 (year), and initial strike price K0=100 of the outside-reset option. The strike price Kj, j=1,2,…,d, will be adjusted if the value of the external process Yt falls below the strike resets Dj. We will calculate the values of the outside-reset option with one strike reset and two strike resets under three reset dates, respectively. Three reset dates are assumed to be t1=1/12 (year), t2=2/12 (year), and t3=3/12 (year). These results are presented in Tables 1 and 2.Table 1
Values of the outside-reset option with single strike reset and three reset dates.
σ 2 S 0 K 1 r=0.05 r=0.07 m m 1 2 3 1 2 3 0.1 85 85 1.0983 1.3492 2.5126 1.4386 2.0270 3.0660 90 0.8019 0.8264 1.4968 1.1419 1.3303 1.9530 95 0.6658 0.6714 0.8900 0.9536 1.0532 1.2390 100 85 8.1683 8.4090 11.7214 9.4943 9.6839 13.0503 90 7.6661 7.6884 9.8844 9.0126 9.0186 11.2641 95 7.1837 7.5695 8.2075 8.5298 8.5466 9.5928  0.3 85 85 6.9739 7.5350 8.5148 7.5683 8.1354 9.1442 90 6.7524 7.0896 7.6832 7.3454 7.6879 8.3018 95 6.5682 6.7199 6.9890 7.1584 7.3132 7.5928 100 85 15.1206 16.0240 17.5469 16.0921 16.9846 18.5190 90 14.7867 15.3503 16.3057 15.7632 16.3221 17.2882 95 14.4905 14.7533 15.2013 15.4695 15.7311 16.1858 The parameters areσ1=0.3, ρ=0.25, Y0=100, and D1=90.Table 2
Values of the outside reset option with two strike resets and three reset dates.
σ 2 S 0 K 1 , K 2 r=0.05 r=0.07 m m 1 2 3 1 2 3 0.1 85 ( 85,75 ) 1.1504 1.2708 4.0516 1.4954 3.8198 4.6219 ( 90,80 ) 0.8540 0.8784 2.6921 1.1705 1.3878 3.2213 ( 95,85 ) 0.6836 0.6955 1.6860 0.9731 1.2277 2.1376 100 ( 85,75 ) 8.2040 8.6264 13.5817 9.4960 9.8386 14.8388 ( 90,80 ) 7.6975 8.0957 11.7032 9.0367 9.3292 13.0265 ( 95,85 ) 7.2358 7.6122 9.9175 8.6111 8.7650 11.2795  0.3 85 ( 85,75 ) 6.9973 7.7872 9.5545 7.5911 8.4549 10.1876 ( 90,80 ) 6.7721 7.2209 8.5729 7.3647 7.8165 9.2021 ( 95,85 ) 6.5847 6.8278 7.7387 7.1747 7.4199 8.3578 100 ( 85,75 ) 15.1534 17.9018 18.9530 16.1234 18.7775 19.9037 ( 90,80 ) 14.8161 15.9776 17.5850 15.7915 18.0760 18.5562 ( 95,85 ) 14.5166 15.6236 16.3492 15.4948 15.6759 17.3312 The parameters areσ1=0.3, ρ=0.25, Y0=100, T=1, D1=90, and D2=80.From Tables1 and 2, we can see that some characteristics of the outside-reset option are similar to the general-reset option studied by Liao and Wang [25]. For example, the values of the outside-reset call option are increasing functions of stock price, risk-free interest rate, and volatility of stock returns, and they are also increasing with the numbers of reset dates. In addition, under the same reset levels Dj, lower reset strike prices Kj will result in higher option values. Due to the higher protection to the option holder, the values of the outside-reset call option with two strike resets are always greater than that with one strike reset under the same model parameter values.


## 3.2. Impacts of Parameters of External Process on Outside-Reset Option

Next, we investigate the impacts of three parameters including the initial inputsY0, volatility σ1, and correlation coefficient ρ in the external process on the option price. In Figures 1, 2, and 3, the left panel depicts the option value with one strike reset (d=1) and the right panel depicts the option value with two strike resets (d=2).Figure 1
Option values as a function ofY0.
(a)
d = 1 (b)
d = 2Figure 2
Option values as a function of volatilityσ1.
(a)
d = 1 (b)
d = 2Figure 3
Option values as a function of correlation coefficientρ.
(a)
d = 1 (b)
d = 2In Figure1, we compare option values as a function of initial inputs Y0 with d(=1,2) strike resets and m(=1,2,3) reset dates. Assume that the model parameters are S0=100, r=0.05, σ1=0.3, σ2=0.3, K1=95, K2=85, D1=90, D2=80, and ρ=1. From Figure 1, we see that the option values decrease as Y0 increases for any reset dates. Also, we see that the outside-reset option value decreases more rapidly when the value of Y0 is in the interval [75,110]. However, the change is very little for Y0<75, which reduces the price manipulation around the reset level.In Figure2, we plot option values of the outside-reset calls versus the volatility σ1 of the external process when the correlated coefficient is ρ=1. Assume that the model parameters are S0=100, r=0.05, Y0=100, σ2=0.3, K1=95, K2=85, D1=90, D2=80, and ρ=1. The effect on price of the outside-reset option of volatility is remarkable, and option values increase as volatility increases. This is the reason that fluctuation of external process brings the profit opportunity for the option.Finally, we compare option values as a function of correlation coefficientρ in Figure 3, in which the model parameters are assumed to be S0=100, r=0.05, Y0=100, σ1=0.3, σ2=0.3, K1=95, K2=85, D1=90, and D2=80. Figure 3 illustrates that the correlation coefficient influences option value very significantly, and lower values of correlation coefficient will result in higher option values. In addition, if σ1=σ2 and ρ=1, then the outside-reset option reduces to the general-reset option studied by Liao and Wang [25]. Therefore, Figure 3 indicates that the option value of outside-reset calls is greater than that of the general-reset calls studied by Liao and Wang [25].


## 3.3. Outside-Reset Option with Continuous Reset Dates

Whenm approaches infinity with a remaining time to T-t, the set of discrete reset dates become a continuous reset period. From (7), the terminal payoff function of the outside-reset option with continuous reset period is as follows: (16)VS,Y,T=ST-Kd++∑l=1dST-Kl-1+Imin0≤t≤T⁡Yt≥Dl-∑l=1dST-Kl+Imin0≤t≤T⁡Yt≥Dl.Based on the closed-form formula of European down-and-out outside-barrier call option studied by Heynen and Kat [26], we can present the explicit expression for the outside-reset option with continuous reset period, because (17)EtQe-rT-tST-Kh+Imin0≤t≤T⁡Yt≥Dl=StN2c1h,-e1l;ρ-DlYt2r-1/2σ12+ρσ1σ2/σ12N2c-1h,-e-1l;ρ-e-rT-tKhN2c2h,l,-e2l;ρ-DlYt2r/σ12N2c-2h,l,-e-2l;ρ,where (18)c1h=ln⁡St/Kh+r+1/2σ22T-tσ2T-t,c1h,l=c1h-σ2T-t,c-1h=c1h+2ρσ1T-tln⁡DlYt,c-2h,l=c-1h,l+2ρσ1T-tln⁡DlYt,e1l=ln⁡Dl/Yt-r-1/2σ12+ρσ1σ2T-tσ1T-t,e2l=e1l+ρσ2T-t,e-1l=e1l-2σ1T-tln⁡DlYt,e-2l=e-2l-2σ1T-tln⁡DlYt.Therefore, we present the value at time t of the outside-reset option with continuous reset period in the following proposition.Proposition 3.
The price at timet of the outside-reset option with continuous reset period, assuming the risky asset price given in (5), is given by (19)ORCCt=StNdd+∑l=1dN2c1l-1,-e1l;ρ-N2c1l,-e1l;ρ-DlYt2r+ρσ12/σ12N2c-1l-1,-e-1l;ρ-N2c-1l,-e-1l;ρ-e-rT-tKdNd-d+∑l=1dKl-1N2c2l-1,l,-e2l;ρ-KlN2c2l,l,-e2l;ρ-DlYt2r/σ12Kl-1N2c-2l-1,l,-e-2l;ρ-KlN2c-2l,l,-e-2l;ρ.


## 3.4. Hedging the Outside-Reset Option

It is well known that hedge parameters (or Greeks) of options are very important in risk management of reset option. Of particular importance are the Delta and the Gamma, respectively, the rate of change of the option value with respect to the stock price, and the rate of change of the option Delta with respect to the stock price. Here, we provide the Delta and Gamma of the outside-reset option in Proposition4. Other Greeks can be similarly derived.Proposition 4.
The values at timet of Δ and Γ for the outside-reset call option with m predecided reset dates and d strike resets, assuming the risky asset price given in (5), are given as follows: (20)Δ=Ndd+∑l=1d∑g=1mNm+1Dgl,l-1;Σg-Nm+1Dgl,l;Σg+1σ2T-t∑l=1d∑g=1m∂Nm+1Dgl,l-1;Σg∂dl-1-∂Nm+1Dgl,l;Σg∂dl-e-rT-tSKl-1∂Nm+1D-gl,l-1;Σg∂d-l-1-Kl∂Nm+1D-gl,l;Σg∂d-l,(21)Γ=1Sσ2T-t12πe-dd2/2+∑l=1d∑g=1m∂Nm+1Dgl,l-1;Σg∂dl-1-∂Nm+1Dgl,l;Σg∂dl+1σ2T-t∂2Nm+1Dgl,l-1;Σg∂dl-12-∂2Nm+1Dgl,l;Σg∂dl2+e-rT-tSKl-1∂Nm+1D-gl,l-1;Σg∂d-l-1-Kl∂Nm+1D-gl,l;Σg∂d-l-e-rT-tσ2T-tKl-1∂2Nm+1D-gl,l-1;Σg∂d-l-12-Kl∂2Nm+1D-gl,l;Σg∂d-l2.Proof.
See AppendixB.In order to describe clearly the phenomena of Delta and Gamma for the outside-reset option, in the following, we limit ourselves to considering the simple case for this option with one reset date. Then, we have(22)N2a,b;ϱ=12π1-ϱ2∫-∞a∫-∞be-x2-2ϱxy+y2/21-ϱ2dxdy,∂N2a,b;ϱ∂a=12πe-a2/2Nb-ϱa1-ϱ2,∂2N2a,b;ϱ∂a2=12πe-a2/2-aNb-ϱa1-ϱ2-ϱ2π1-ϱ2e-b-ϱa2/21-ϱ2and the following corollary.Corollary 5.
The values at timet of Δ and Γ for the outside-reset call option with one reset date are, respectively, given by (23)Δ=Ndd+∑l=1dN2al,1,dl-1;ϱ-N2al,1,dl;ϱ+1σ22πT-te-dl-12/2Nal,1-ϱdl-11-ϱ2-e-dl2/2Nal,1-ϱdl1-ϱ2-e-rT-tSKl-1e-d-l-12/2Na-l,1-ϱd-l-11-ϱ2-Kle-d-l2/2Na-l,1-ϱd-l1-ϱ2,Γ=1Sσ22πT-te-dd2/2+∑l=1de-dl-12/2Nal,1-ϱdl-11-ϱ2-e-dl2/2Nal,1-ϱdl1-ϱ2+1σ2T-t-dl-1e-dl-12/2Nal,1-ϱdl-11-ϱ2-ϱe-al,12-2ϱal,1dl-1+dl-12/21-ϱ22π1-ϱ2+dle-dl2/2Nal,1-ϱdl1-ϱ2-ϱe-al,12-2ϱal,1dl+dl2/21-ϱ22π1-ϱ2+e-rT-tSKl-1e-d-l-12/2Na-l,1-ϱd-l-11-ϱ2-Kle-d-l2/2Na-l,1-ϱd-l1-ϱ2-e-rT-tσ2T-t-d-l-1Kl-1e-d-l-12/2Na-l,1-ϱd-l-11-ϱ2+d-lKle-d-l2/2Na-l,1-ϱd-l1-ϱ2-ϱKl-1e-a-l,12-2ϱa-l,1d-l-1+d-l-12/21-ϱ22π1-ϱ2-ϱKle-a-l,12-2ϱa-l,1d-l+d-l2/21-ϱ22π1-ϱ2,where ϱ=ρt1-t/T-t and t1 is the reset date.Whent→t1-, then (24)N2al,1,dl-1;ϱ⟶Ndl-1,12πe-dl2/2Nal,1-ϱdl1-ϱ2⟶12πe-dl2/2t=t1,N2al,1,dl;ϱ⟶0,12πe-d-l2/2Nal,1-ϱd-l1-ϱ2⟶12πe-d-l2/2t=t1.ConsequentlyΔ and Γ at time t1 become as follows: (25)Δ=Nd0+1σ22πT-t-1e-dd2/2-e-d02/2t=t1,Γ=1Stσ22πT-te-d02/2+1σ2T-t-2π·dde-dd2/2-d0e-d02/2t=t1.However, Δ and Γ at time t>t1 are given as follows: (26)Δ=Nd0IYt1≥D1+∑l=1d-1NdlIDl≥Yt1≥Dl+1+NddIYt1<Dd,Γ=1Stσ22πT-te-d02/2IYt1≥D1+∑l=1d-1e-dl2/2IDl≥Yt1≥Dl+1+e-dd2/2IYt1<Dd.From (25)-(26), we find that Δ and Γ at time t1 are continuous as a function of the stock price. Therefore, Delta and Gamma do not have significant jumps when the stock price goes across the reset dates because of using the external process as a reset trigger, and it makes outside-reset options more amenable to dynamic hedging. Figure 4 illustrates the phenomena at time t1 of Delta and Gamma for the outside-reset options at various underlying stock prices and contrasts those with the general-reset option specified in (2) and (3) which is studied by Liao and Wang [25]. Assume K0=100, σ1=σ2=0.3, ρ=1, Y0=100, r=0.05, T=1 (year), and t1=3/12 (year) and the current stock price is 100. We consider the strike price of the outside-reset option will be adjusted to 80,70,60,50, and 40 if the external variable Yt falls below 80,70,60,50, and 40. As shown in Figure 4, the Delta and Gamma of the general-reset options will fluctuate dramatically and oscillate if the stock prices near strike resets. This shows that the general-reset option has the phenomena of Delta jump and Gamma jump across reset date. With the outside-reset design, however, the oscillations in the delta are replaced by a much smoother double U-shaped curve. This added continuity of delta around reset level greatly facilitates option holders to hedge outside-reset options dynamically by trading the underlying stock. Therefore, it is evident that the outside-reset option does not have Delta and Gamma jumps as the general-reset option.Figure 4
Comparison of optionΔ and Γ between the general-reset option and outside-reset option.Figure5 plots the option Delta and Gamma of the outside-reset call option at time t1 for different values of Y0 shown in Figure 5, and other parameter values are similar to that in Figure 4. We can see that the option Delta and Gamma of the outside-reset calls are sensitive to the changes in S and Y.Figure 5
OptionΔ and Γ at time t1 for different Y0.


## 4. Conclusion

In this paper, we proposed an outside-reset option with multiple strike reset and reset dates determined by an external process which correlates to the underlying asset and provided analytical pricing formula for this option. We analyzed some characteristics of this option and the impacts of the external process on the option through numerical experiments. Besides, it is also shown that the outside-reset option does not have Delta and Gamma jumps as the general-reset option. This useful property reduces the hedging risk dramatically. The model may be seen as an extension of Liao and Wang [25] to include further realism by introducing the external process. Also the model could be extended to value a large class of path-dependent options with discrete monitoring. This extension still remains as future work.



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