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

Hindawi Complexity Volume 2018, Article ID 2825483, 13 pages https://doi.org/10.1155/2018/2825483 Research Article Valuation on an Outside-Reset Option with Multiple Resettable Levels and Dates Guangming Xue,1 Bin Qin,1 and Guohe Deng 1 2 School of Information and Statistics, Guangxi University of Finance and Economics, Nanning 530003, China School of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, China 2 Correspondence should be addressed to Guohe Deng; Received 14 October 2017; Accepted 22 February 2018; Published 8 April 2018 Academic Editor: Michele Scarpiniti 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. 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 outsidereset 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 closedform 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 𝑚 predetermined reset dates 0 < 𝑡1 < 𝑡2 < ⋅ ⋅ ⋅ < 𝑡𝑚 < 𝑇 has a payoff at a fixed maturity 𝑇 of 𝑉 (𝑇) = max {𝑆𝑇 − min [𝐾0 , 𝑆𝑡1 , 𝑆𝑡2 , . . . , 𝑆𝑡𝑚 ] , 0} , (1) 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 + 𝑉 (𝑇) = max {𝑆𝑇 − 𝐾∗ , 0} = (𝑆𝑇 − 𝐾∗ ) , (2) 2 Complexity where 𝐾∗ is defined by 𝐾0 , { { { { 𝐾∗ = {𝐾𝑗 , { { { {𝐾𝑑 , if min [𝑆𝑡1 , 𝑆𝑡2 , . . . , 𝑆𝑡𝑚 ] ≥ 𝐷1 , if 𝐷𝑗 > min [𝑆𝑡1 , 𝑆𝑡2 , . . . , 𝑆𝑡𝑚 ] ≥ 𝐷𝑗+1 , 𝑗 = 1, 2, . . . , 𝑑 − 1, if 𝐷𝑑 > min [𝑆𝑡1 , 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 𝐾0 , { { { { 𝐾∗ = {𝐾𝑗 , { { { {𝐾𝑑 , (3) 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 if min [𝑌𝑡1 , 𝑌𝑡2 , . . . , 𝑌𝑡𝑚 ] ≥ 𝐷1 , if 𝐷𝑗 > min [𝑌𝑡1 , 𝑌𝑡2 , . . . , 𝑌𝑡𝑚 ] ≥ 𝐷𝑗+1 , 𝑗 = 1, 2, . . . , 𝑑 − 1, (4) if 𝐷𝑑 > min [𝑌𝑡1 , 𝑌𝑡2 , . . . , 𝑌𝑡𝑚 ] . This novel design of using the external variable 𝑌𝑡 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 (...truncated)