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

Xue, Guangming; Qin, Bin; Deng, Guohe

doi:https://doi.org/10.1155/2018/2825483

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Research Article | Open Access

Volume 2018 | Article ID 2825483 | https://doi.org/10.1155/2018/2825483

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

Guangming Xue,¹Bin Qin,¹and Guohe Deng²

Academic Editor: Michele Scarpiniti

Received14 Oct 2017

Accepted22 Feb 2018

Published08 Apr 2018

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.

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 with predetermined reset dates has a payoff at a fixed maturity of where denotes the underlying asset price at time and denotes the initial strike price of option. In a real application, the terminal payoff of reset call option is usually set as where is defined by where , , denote the strike price resets such that and , , are the reset levels. In particular, there is only one reset price when . 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 . 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 is replaced by

This novel design of using the external variable as a reset trigger replacing the underlying asset , 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 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 Section 2, 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 and the external process (as a reset trigger) under a risk-neutral measure : where , , and are constants, and is a bidimensional standard Brownian motion defined on a filtered probability space , where is the -augmentation of the filtration generated by and satisfies the usual conditions, is the constant risk-free interest rate, and presents the correlation coefficient between and . It should be noted that model (5) may reduce to the results discussed by Liao and Wang [25] for the case when and .

Using Itô formula to model (5), then, for ,

In view of (2) and (4), the terminal payoff of the outside-reset option with strike resets and predetermined reset dates can be written as where , , and is indicator function. Following the pricing theory, under the risk-neutral measure , the value at time of the outside-reset option is where .

From (8), we know that the key of solution is to compute the following expression: where or . Now we present the result in the following proposition.

Proposition 1. The value at time of the expression, , assuming the risky asset price given in the model (5), is where is the -dimensional cumulative normal distribution function with mean vector 0 and correlation matrix , and stands for the transpose of the th row of matrix as follows:

The components of the matrix are defined as follows: The components of the matrix are the same as those of the ones of the matrix except the components , and are replaced by , and , respectively. Here The correlation matrix , , , where is given by

Proof. See Appendix A.

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 for the outside-reset option is gained by the following proposition.

Proposition 2. The price at time of the outside-reset option with -periodic reset and strike resets, assuming the risky asset price given in model (5), is where is the cumulative probability function of a standard normal variable.

The numerical valuation of appearing in the analytic formula (15) (where 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 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 time , the maturity (year), and initial strike price of the outside-reset option. The strike price , , will be adjusted if the value of the external process falls below the strike resets . 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 (year), (year), and (year). These results are presented in Tables 1 and 2.

From Tables 1 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 , lower reset strike prices 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 inputs , volatility , and correlation coefficient 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 () and the right panel depicts the option value with two strike resets ().

(a)

(b)

(a)

(b)

(a)

(b)

In Figure 1, we compare option values as a function of initial inputs with strike resets and reset dates. Assume that the model parameters are , , , , , , , , and . From Figure 1, we see that the option values decrease as increases for any reset dates. Also, we see that the outside-reset option value decreases more rapidly when the value of is in the interval . However, the change is very little for , which reduces the price manipulation around the reset level.

In Figure 2, we plot option values of the outside-reset calls versus the volatility of the external process when the correlated coefficient is . Assume that the model parameters are , , , , , , , , and . 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 coefficient in Figure 3, in which the model parameters are assumed to be , , , , , , , , and . 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 and , 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

When approaches infinity with a remaining time to , 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: 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 where Therefore, we present the value at time of the outside-reset option with continuous reset period in the following proposition.

Proposition 3. The price at time of the outside-reset option with continuous reset period, assuming the risky asset price given in (5), is given by

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 Proposition 4. Other Greeks can be similarly derived.

Proposition 4. The values at time of and for the outside-reset call option with predecided reset dates and strike resets, assuming the risky asset price given in (5), are given as follows:

Proof. See Appendix B.

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 and the following corollary.

Corollary 5. The values at time of and for the outside-reset call option with one reset date are, respectively, given by where and is the reset date.

When , then

Consequently and at time become as follows: However, and at time are given as follows:

From (25)-(26), we find that and at time 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 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 , , , , , (year), and (year) and the current stock price is 100. We consider the strike price of the outside-reset option will be adjusted to , and 40 if the external variable falls below , 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 5 plots the option Delta and Gamma of the outside-reset call option at time for different values of 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 and .

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.

Appendix

A. The Proof of Proposition 1

Since according to Geman et al. [32], in order to simplify calculations, we change numeraire. For the first term we choose the stock price as numeraire and switch measure to . The Radon-Nikodym derivative is then given by where and is again a martingale measure. By Girsanov’s theorem, defined by two independent Brownian motions are under . Under the new measure , the option price (A.1) can be restated as

Now we need to calculate two probabilities of (A.4), respectively. First we have In terms of (A.3) and (A.4), then where , , are defined by and with . Similar to the argument technique above, we get This completes the proof of Proposition 1.

B. The Proof of Proposition 4

Because the derivatives where , , therefore, the Delta of the outside-reset option is given by Notice that . Rearranging (B.2) then yields formula (16). Similar to (B.2), we also have the Gamma of the outside-reset option as follows:

By observing (B.2) and (B.3), we see the key elements for computing Delta and Gamma are , , , and .

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

The authors would like to thank participants at the 15th International Conference on Financial Systems Engineering and Risk Management (FSERM’2017) for valuable comments on an earlier version of this work. This work was supported by the NSF of China (Grant no. 11461008).

References

P. P. Boyle, A. W. Kolkiewicz, and K. S. Tan, “Valuation of the reset options embedded in some equity-linked insurance products,” North American Actuarial Journal, vol. 5, no. 3, pp. 1–18, 2001.
View at: Publisher Site | Google Scholar | MathSciNet
S. F. Gray and R. E. Whaley, “Valuing S&P 500 bear market warrants with a periodic reset,” The Journal of Derivatives, vol. 5, no. 1, pp. 99–106, 1997.
View at: Publisher Site | Google Scholar
S. F. Gray and R. E. Whaley, “Reset put options: Valuation, risk characteristics, and an application,” Australian Journal of Management, vol. 24, no. 1, pp. 1–20, 1999.
View at: Publisher Site | Google Scholar
L. P. Hsueh and B. H. Guo, “Reset warrants: Design and valuation,” Journal of Financial Studies, vol. 6, no. 2, pp. 1–18, 1998.
View at: Google Scholar
W.-Y. Cheng and S. Zhang, “The analytics of reset options,” The Journal of Derivatives, vol. 8, no. 1, pp. 59–71, 2000.
View at: Publisher Site | Google Scholar
S. J. Li, S. H. Li, and C. Sun, “A generalization of reset options pricing formulae with stochastic interest rates,” Research in International Business and Finance, vol. 21, no. 2, pp. 119–133, 2007.
View at: Publisher Site | Google Scholar
Y. H. Liu, I. M. Jiang, S. H. Lee, and Y. T. Cheng, “The valuation of reset options when underlying assets are autocorrelated,” International Journal of Business and Finance Research, vol. 5, no. 2, pp. 95–114, 2011.
View at: Google Scholar
A. François-Heude and O. Yousfi, “A generalization of Gray and Whaley's reset option,” Journal of Asset Management, vol. 16, no. 4, pp. 223–235, 2015.
View at: Publisher Site | Google Scholar
M. Dai, Y. K. Kwok, and L. X. Wu, “Options with multiple reset rights,” International Journal of Theoretical and Applied Finance, vol. 6, no. 6, pp. 637–653, 2003.
View at: Publisher Site | Google Scholar
M. Dai, Y. K. Kwok, and L. Wu, “Optimal shouting policies of options with strike reset right,” Mathematical Finance, vol. 14, no. 3, pp. 383–401, 2004.
View at: Publisher Site | Google Scholar
M. Dai and Y. K. Kwok, “Options with combined reset rights on strike and maturity,” Journal of Economic Dynamics & Control, vol. 29, no. 9, pp. 1495–1515, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
Z. Yang, F. Yi, and M. Dai, “A parabolic variational inequality arising from the valuation of strike reset options,” Journal of Differential Equations, vol. 230, no. 2, pp. 481–501, 2006.
View at: Publisher Site | Google Scholar | MathSciNet
J. Goard, “Exact solutions for a strike reset put option and a shout call option,” Mathematical and Computer Modelling, vol. 55, no. 5-6, pp. 1787–1797, 2012.
View at: Publisher Site | Google Scholar | MathSciNet
L. P. Hsueh and Y. A. Liu, “Step-Reset Options: Design and Valuation,” Journal of Futures Markets, vol. 22, no. 2, pp. 155–171, 2002.
View at: Publisher Site | Google Scholar
E. C. Yu and W. T. Shaw, “On the valuation of derivatives with snapshot reset features,” International Journal of Theoretical and Applied Finance, vol. 11, no. 8, pp. 905–941, 2008.
View at: Publisher Site | Google Scholar | MathSciNet
C.-H. Kao and Y.-D. Lyuu, “Pricing of moving-average-type options with applications,” Journal of Futures Markets, vol. 23, no. 5, pp. 415–440, 2003.
View at: Publisher Site | Google Scholar
S.-L. Liao and C.-W. Wang, “Pricing arithmetic average reset options with control variates,” The Journal of Derivatives, vol. 10, no. 2, pp. 59–74, 2002.
View at: Publisher Site | Google Scholar
I. J. Kim, G. H. Chang, and S. J. Byun, “Valuation of arithmetic average reset options,” The Journal of Derivatives, vol. 11, no. 1, pp. 70–80, 2003.
View at: Publisher Site | Google Scholar
C.-C. Chang, S.-L. Chung, and M. B. Shackleton, “Pricing options with American-style average reset features,” Quantitative Finance, vol. 4, no. 3, pp. 292–300, 2004.
View at: Publisher Site | Google Scholar | MathSciNet
T.-S. Dai, Y.-Y. Fang, and Y.-D. Lyuu, “Analytics for geometric average trigger reset options,” Applied Economics Letters, vol. 12, no. 13, pp. 835–840, 2005.
View at: Publisher Site | Google Scholar
M. Costabile, I. Massabo, and E. Russo, “A binomial model for pricing US-style average options with reset features,” International Journal of Financial Markets and Derivatives, vol. 1, no. 3, p. 258, 2010.
View at: Publisher Site | Google Scholar
M. Costabile, I. Massabó, and E. Russo, “On pricing arithmetic average reset options with multiple reset dates in a lattice framework,” Journal of Computational and Applied Mathematics, vol. 235, no. 17, pp. 5307–5325, 2011.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Hsiao, “A Simple Method to Price Window Reset Options,” Journal of Mathematical Finance, vol. 3, no. 1, pp. 96–102, 2013.
View at: Publisher Site | Google Scholar
S. N. Chen and I. M. Jiang, “Quanto reset put options,” Journal of Accounting Finance & Management Strategy, vol. 11, no. 1, pp. 19–28, 2016.
View at: Google Scholar
S.-L. Liao and C.-W. Wang, “The valuation of reset options with multiple strike resets and reset dates,” Journal of Futures Markets, vol. 23, no. 1, pp. 87–107, 2003.
View at: Publisher Site | Google Scholar
R. Heynen and H. Kat, “Crossing barrier,” Risk, vol. 7, pp. 46–51, 1994.
View at: Google Scholar
Y. Kwok, L. Wu, and H. Yu, “Pricing Multi-Asset Options with an External Barrier,” International Journal of Theoretical and Applied Finance, vol. 1, no. 4, pp. 523–541, 1998.
View at: Publisher Site | Google Scholar
P. G. Zhang, “A unified formula for outside barrier options,” Journal of Fiancial Engineering, vol. 4, pp. 335–347, 1995.
View at: Google Scholar
A. Genz, “Numerical computation of multivariate normal probabilities,” Journal of Computational and Graphical Statistics, vol. 1, no. 2, pp. 141–149, 1992.
View at: Publisher Site | Google Scholar
P. P. Boyle, Y. Lai, and K. S. Tan, “Pricing options using lattice rules,” North American Actuarial Journal, vol. 9, no. 3, pp. 50–76, 2005.
View at: Publisher Site | Google Scholar | MathSciNet
Y. Lai, “Intermediate rank lattice rules and applications to finance,” Applied Numerical Mathematics, vol. 59, no. 1, pp. 1–20, 2009.
View at: Publisher Site | Google Scholar | MathSciNet
H. Geman, N. E. Karoui, and J. Rochet, “Changes of numeraire, change of probability measure and option pricing,” Journal of Applied Probability, vol. 32, no. 2, pp. 443–458, 1995.
View at: Publisher Site | Google Scholar | MathSciNet

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Copyright © 2018 Guangming Xue et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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