Without considering the underlying risk dynamics and jumps, Wu and Zhu (2016) recently proposed an ingenious approach of hedging options statically with an option portfolio. We improve their scheme in three ways. First, we theoretically make the Wu-Zhu approach more accurate by utilizing the Black-Scholes-Merton dual equation. Second, we propose a better error measure, the so-called “true hedge error,” that takes the initial cost of the hedge into consideration. Finally, we suggest two measures of percentage hedge errors to assess hedge performance more precisely. With extensive simulations under both the Black-Scholes-Merton and Heston models, we show that our proposal significantly improves the hedge performance, especially for in-the-money and at-the-money options. Importantly, we extend Wu-Zhu to options with a payoff of homogeneous of degree one.
{"title":"Improving and Extending the Wu-Zhu Static Hedge","authors":"Shuxin Guo, Qiang Liu","doi":"10.3905/jod.2022.1.173","DOIUrl":"https://doi.org/10.3905/jod.2022.1.173","url":null,"abstract":"Without considering the underlying risk dynamics and jumps, Wu and Zhu (2016) recently proposed an ingenious approach of hedging options statically with an option portfolio. We improve their scheme in three ways. First, we theoretically make the Wu-Zhu approach more accurate by utilizing the Black-Scholes-Merton dual equation. Second, we propose a better error measure, the so-called “true hedge error,” that takes the initial cost of the hedge into consideration. Finally, we suggest two measures of percentage hedge errors to assess hedge performance more precisely. With extensive simulations under both the Black-Scholes-Merton and Heston models, we show that our proposal significantly improves the hedge performance, especially for in-the-money and at-the-money options. Importantly, we extend Wu-Zhu to options with a payoff of homogeneous of degree one.","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"30 1","pages":"26 - 41"},"PeriodicalIF":0.0,"publicationDate":"2022-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46643757","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We examine options listed on sector ETFs that constitute the S&P 500 and find evidence of predictability in implied volatilities associated with abnormally high or low implied correlations. We show that sector-implied volatilities evolve to maintain stable relations between sector correlation premiums and the correlation premium on the S&P 500, allowing the calculation of a sector-specific, idiosyncratic correlation premium. The sector-specific correlation premium is a more reliable signal of future changes in sector-implied volatility relative to simple level measures of the volatility or correlation premiums due to its focus on correlation rather than volatility, and its adjustment for aggregate levels. Moreover, we find that one-day reversals in sector-implied volatilities are related only to reversals in the sector-specific correlation premium, and that information extracted from stock-implied volatilities has little or no predictive ability for sector-implied volatility. The predictable variation in sector-implied volatilities associated with the sector-specific component of the correlation premium forms the basis for profitable trading signals that dominate strategies based directly on sector volatility premiums.
{"title":"Sector Option Correlation Premiums and Predictable Changes in Implied Volatility","authors":"Apoorva Koticha, Chen Li, Joseph M. Marks","doi":"10.3905/jod.2022.1.171","DOIUrl":"https://doi.org/10.3905/jod.2022.1.171","url":null,"abstract":"We examine options listed on sector ETFs that constitute the S&P 500 and find evidence of predictability in implied volatilities associated with abnormally high or low implied correlations. We show that sector-implied volatilities evolve to maintain stable relations between sector correlation premiums and the correlation premium on the S&P 500, allowing the calculation of a sector-specific, idiosyncratic correlation premium. The sector-specific correlation premium is a more reliable signal of future changes in sector-implied volatility relative to simple level measures of the volatility or correlation premiums due to its focus on correlation rather than volatility, and its adjustment for aggregate levels. Moreover, we find that one-day reversals in sector-implied volatilities are related only to reversals in the sector-specific correlation premium, and that information extracted from stock-implied volatilities has little or no predictive ability for sector-implied volatility. The predictable variation in sector-implied volatilities associated with the sector-specific component of the correlation premium forms the basis for profitable trading signals that dominate strategies based directly on sector volatility premiums.","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"30 1","pages":"84 - 115"},"PeriodicalIF":0.0,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49309790","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Before the global equity crash in October 1987, volatility could be reasonably approximated as a constant, consistent with Black-Scholes (1973) dynamics. Thereafter, a stylized feature of equity options markets is that both single-name and index options have exhibited consistent, regular deviations of volatility in both strike and maturity. The resulting volatility surface has been studied extensively (Rubinstein 1994, Jackwerth and Rubinstein 1996, Derman 1999, Cont and da Fonseca 2002, Gatheral 2006). Moreover, reduced-form representations of major equity indices’ volatility surfaces corresponding to “average” volatility (over strikes) accumulated through fixed maturities, for example, Cboe’s (formerly Chicago Board Options Exchange) Volatility Index (VIX), have been popularized as gauges of investor sentiment and risk-aversion. Likewise, there has been considerable interest in quantifying and interpreting the term structure of futures whose payoffs are tied to these indices (Zhu and Zhang 2007; Lu and Zhu 2009; Egloff et al. 2010). In the context of the risk-neutral distribution characterizing asset prices at contract maturity, these studies focus on futures’ expectations—their first moments; higher-order moments are less well-studied. Daigler et al. (2016) introduce implied convexity as a measure of variance, that is, the second moment. However, although many authors have studied the term structure of VIX futures’ expectations, to our knowledge, none has examined the term structure of their variances. This article extends the research of Daigler et al. in two important ways. First, it provides an alternative to their intermediate adjustments of the VIX near-term (VIN) and VIX far-term (VIF) component indices that is consistent with the assumptions underlying the calculation of all Cboe volatility indices. It is likewise consistent with volatility indices in foreign markets, for example, the Euro STOXX 50 Volatility (VSTOXX) index (Deutsche Börse Group 2022). Second, it characterizes the entire term structure of VIX futures’ second moments, rather than that of a single contract with a maturity of approximately one month. Additionally, we quantify the differences arising from various interpolation choices. We find that extrapolation based only on two maturities near thirty calendar days produces estimates of variance that differ considerably from interpolations based on all available expiries. Furthermore, the accuracy of extrapolation degrades as the absolute differences between a contract’s maturity and the maturities of the interpolants increase.
在1987年10月全球股市崩盘之前，波动性可以合理地近似为一个常数，与Black-Scholes(1973)动力学一致。此后，股票期权市场的一个风格化特征是，单名期权和指数期权在到期日和到期日都表现出一致的、有规律的波动偏离。由此产生的波动面已被广泛研究(Rubinstein 1994, Jackwerth和Rubinstein 1996, Derman 1999, Cont和da Fonseca 2002, Gatheral 2006)。此外，主要股指的波动率曲面的简化形式表示，对应于通过固定期限积累的“平均”波动率(超过罢工)，例如，Cboe(原芝加哥期权交易所)波动率指数(VIX)，已被普及为投资者情绪和风险厌恶的衡量标准。同样，人们对量化和解释收益与这些指数挂钩的期货的期限结构也有相当大的兴趣(Zhu and Zhang 2007;Lu and Zhu 2009;Egloff et al. 2010)。在合约到期时资产价格具有风险中性分布特征的背景下，这些研究侧重于期货的预期——它们的初始时刻;高阶矩的研究较少。Daigler等人(2016)引入隐含凸性作为方差的度量，即第二矩。然而，尽管许多作者研究了VIX期货预期的期限结构，但据我们所知，没有人研究过其方差的期限结构。本文在两个重要方面对Daigler等人的研究进行了扩展。首先，它为波动率指数短期(VIN)和长期(VIF)组成指数的中间调整提供了另一种选择，这与Cboe所有波动率指数计算的基本假设是一致的。它同样与国外市场的波动率指数一致，例如，欧洲斯托克50波动率指数(VSTOXX) (Deutsche Börse Group 2022)。其次，它描述了波动率指数期货第二时刻的整个期限结构，而不是期限约为一个月的单一合约。此外，我们量化了各种插值选择所产生的差异。我们发现，仅基于两个接近30个日历日的到期日的外推法产生的方差估计与基于所有可用到期日的内推法有很大不同。此外，外推的准确性随着合约期限与内插期限之间的绝对差值的增加而降低。
{"title":"On the Term Structure of VIX Futures’ Implied Convexity","authors":"D. Annis, D. Abasto","doi":"10.3905/jod.2022.1.170","DOIUrl":"https://doi.org/10.3905/jod.2022.1.170","url":null,"abstract":"Before the global equity crash in October 1987, volatility could be reasonably approximated as a constant, consistent with Black-Scholes (1973) dynamics. Thereafter, a stylized feature of equity options markets is that both single-name and index options have exhibited consistent, regular deviations of volatility in both strike and maturity. The resulting volatility surface has been studied extensively (Rubinstein 1994, Jackwerth and Rubinstein 1996, Derman 1999, Cont and da Fonseca 2002, Gatheral 2006). Moreover, reduced-form representations of major equity indices’ volatility surfaces corresponding to “average” volatility (over strikes) accumulated through fixed maturities, for example, Cboe’s (formerly Chicago Board Options Exchange) Volatility Index (VIX), have been popularized as gauges of investor sentiment and risk-aversion. Likewise, there has been considerable interest in quantifying and interpreting the term structure of futures whose payoffs are tied to these indices (Zhu and Zhang 2007; Lu and Zhu 2009; Egloff et al. 2010). In the context of the risk-neutral distribution characterizing asset prices at contract maturity, these studies focus on futures’ expectations—their first moments; higher-order moments are less well-studied. Daigler et al. (2016) introduce implied convexity as a measure of variance, that is, the second moment. However, although many authors have studied the term structure of VIX futures’ expectations, to our knowledge, none has examined the term structure of their variances. This article extends the research of Daigler et al. in two important ways. First, it provides an alternative to their intermediate adjustments of the VIX near-term (VIN) and VIX far-term (VIF) component indices that is consistent with the assumptions underlying the calculation of all Cboe volatility indices. It is likewise consistent with volatility indices in foreign markets, for example, the Euro STOXX 50 Volatility (VSTOXX) index (Deutsche Börse Group 2022). Second, it characterizes the entire term structure of VIX futures’ second moments, rather than that of a single contract with a maturity of approximately one month. Additionally, we quantify the differences arising from various interpolation choices. We find that extrapolation based only on two maturities near thirty calendar days produces estimates of variance that differ considerably from interpolations based on all available expiries. Furthermore, the accuracy of extrapolation degrades as the absolute differences between a contract’s maturity and the maturities of the interpolants increase.","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"30 1","pages":"10 - 25"},"PeriodicalIF":0.0,"publicationDate":"2022-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47364373","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Sara Wagner, Theo Vermaelen, Christian C. P. Wolff
This article explores the reasons why some banks issue Contingent Convertible (CoCo) bonds, but others do not. To this end, we use a binary logistic model and control for the determinants suggested by the literature. Our findings suggest that larger banks and those with higher Tier 1 capital, higher net loans, higher wholesale funding, lower levels of leverage, and lower risk-weighted assets have a higher tendency to issue CoCos and were the early adopters of this novel financing instrument. Our results also suggest that banks in countries with higher annual growth rate of GDP per capita and those listed as Globally Systematically Important Banks (G-SIBs) were more likely to issue CoCos. These results are difficult to explain by traditional capital structure theory, which assumes that companies voluntarily choose their optimal capital structures, but suggest that banks were more likely to be encouraged or nudged to issue CoCos by following regulators’ advice.
{"title":"Which Factors Play a Role in CoCo Issuance? Evidence from European Banks","authors":"Sara Wagner, Theo Vermaelen, Christian C. P. Wolff","doi":"10.3905/jod.2022.1.169","DOIUrl":"https://doi.org/10.3905/jod.2022.1.169","url":null,"abstract":"This article explores the reasons why some banks issue Contingent Convertible (CoCo) bonds, but others do not. To this end, we use a binary logistic model and control for the determinants suggested by the literature. Our findings suggest that larger banks and those with higher Tier 1 capital, higher net loans, higher wholesale funding, lower levels of leverage, and lower risk-weighted assets have a higher tendency to issue CoCos and were the early adopters of this novel financing instrument. Our results also suggest that banks in countries with higher annual growth rate of GDP per capita and those listed as Globally Systematically Important Banks (G-SIBs) were more likely to issue CoCos. These results are difficult to explain by traditional capital structure theory, which assumes that companies voluntarily choose their optimal capital structures, but suggest that banks were more likely to be encouraged or nudged to issue CoCos by following regulators’ advice.","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"30 1","pages":"58 - 73"},"PeriodicalIF":0.0,"publicationDate":"2022-08-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46311669","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Total return swap (TRS) involves a pricing dilemma: LIBOR discounting of its premium leg forces upfront payment of future funding premium, and yet replacing LIBOR with a firm’s own funding rate falls into the well-known FVA debate trap. We consider TRS hedge financing from a repo market perspective and apply postcrisis derivatives valuation with collateralization and funding to TRS. We find that the financing cost of the TRS hedge should be reflected on the security leg, and the funding premium can only be discounted in conjunction with the TRS as a whole, depending on margining schemes. An easy to implement, recursive tree model is developed to value TRS with repo-style margining or defaultable underlying, together with any value adjustments.
{"title":"Pricing Total Return Swaps","authors":"W. Lou","doi":"10.3905/jod.2022.1.167","DOIUrl":"https://doi.org/10.3905/jod.2022.1.167","url":null,"abstract":"Total return swap (TRS) involves a pricing dilemma: LIBOR discounting of its premium leg forces upfront payment of future funding premium, and yet replacing LIBOR with a firm’s own funding rate falls into the well-known FVA debate trap. We consider TRS hedge financing from a repo market perspective and apply postcrisis derivatives valuation with collateralization and funding to TRS. We find that the financing cost of the TRS hedge should be reflected on the security leg, and the funding premium can only be discounted in conjunction with the TRS as a whole, depending on margining schemes. An easy to implement, recursive tree model is developed to value TRS with repo-style margining or defaultable underlying, together with any value adjustments.","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"30 1","pages":"66 - 83"},"PeriodicalIF":0.0,"publicationDate":"2022-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48502229","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider semi-analytical pricing of barrier options for the time-dependent SABR stochastic volatility model (with drift in the instantaneous volatility) with zero correlation between spot and stochastic volatility. In doing so, we modify the general integral transform method (see Itkin et al. 2021) and deliver solution of this problem in the form of Fourier-Bessel series. The weights of this series solve a linear mixed Volterra-Fredholm equation (LMVF) of the second kind also derived in the article. Numerical examples illustrate the speed and accuracy of our method, which are comparable with those of the finite-difference approach at small maturities and outperform them at high maturities even by using a simplistic implementation of the RBF method for solving the LMVF.
我们考虑了现货与随机波动率零相关的随时间变化的SABR随机波动率模型(瞬时波动率有漂移)的障碍期权的半解析定价。在此过程中，我们修改了一般的积分变换方法(见Itkin et al. 2021)，并以傅里叶-贝塞尔级数的形式给出了该问题的解。该级数的权值解出了第二类线性混合Volterra-Fredholm方程(LMVF)。数值例子说明了我们的方法的速度和准确性，在小期限时与有限差分方法相当，在高期限时甚至通过使用RBF方法的简化实现来求解LMVF也优于它们。
{"title":"Semi-Analytical Pricing of Barrier Options in the Time-Dependent λ-SABR Model: Uncorrelated Case","authors":"A. Itkin, D. Muravey","doi":"10.3905/jod.2022.1.166","DOIUrl":"https://doi.org/10.3905/jod.2022.1.166","url":null,"abstract":"We consider semi-analytical pricing of barrier options for the time-dependent SABR stochastic volatility model (with drift in the instantaneous volatility) with zero correlation between spot and stochastic volatility. In doing so, we modify the general integral transform method (see Itkin et al. 2021) and deliver solution of this problem in the form of Fourier-Bessel series. The weights of this series solve a linear mixed Volterra-Fredholm equation (LMVF) of the second kind also derived in the article. Numerical examples illustrate the speed and accuracy of our method, which are comparable with those of the finite-difference approach at small maturities and outperform them at high maturities even by using a simplistic implementation of the RBF method for solving the LMVF.","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"30 1","pages":"74 - 101"},"PeriodicalIF":0.0,"publicationDate":"2022-07-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42979858","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we continue the research of our recent interest rate tree model, called the Zero Black-Derman-Toy (ZBDT) model, which includes the possibility of a jump at each step to a practically zero interest rate. This approach allows a better match with the risk of financial slowdown caused by catastrophic events. We present how to valuate a wide range of financial derivatives using such a model. The classical Black-Derman-Toy (BDT) model and a novel ZBDT model are described, and analogies in their calibration methodology are established. Finally, two cases of applications of the novel ZBDT model are introduced. The first is the hypothetical case of an S-shaped term structure and decreasing volatility of yields. The second case is an application in the structure of US sovereign bonds in the 2020 economic slowdown caused by the coronavirus pandemic. The objective of this study is to understand the differences presented by the valuation in both models for exotic derivatives.
在这篇文章中，我们继续研究我们最近的利率树模型，称为Zero Black Derman Toy（ZBDT）模型，该模型包括每一步跳到实际零利率的可能性。这种方法可以更好地应对灾难性事件导致的金融放缓风险。我们介绍了如何使用这样一个模型来评估各种金融衍生品。介绍了经典的Black Derman Toy（BDT）模型和一种新的ZBDT模型，并在它们的校准方法上进行了类比。最后介绍了新型ZBDT模型的两个应用实例。第一种是S型期限结构和收益率波动性下降的假设情况。第二个案例是美国主权债券结构在冠状病毒大流行导致的2020年经济放缓中的应用。本研究的目的是了解奇异衍生品两种模型中估值所呈现的差异。
{"title":"Zero Black-Derman-Toy Model in Catastrophic Events: COVID-19 Performance Analysis","authors":"G. Krzyzanowski, Andr'es Sosa","doi":"10.3905/jod.2022.1.164","DOIUrl":"https://doi.org/10.3905/jod.2022.1.164","url":null,"abstract":"In this article, we continue the research of our recent interest rate tree model, called the Zero Black-Derman-Toy (ZBDT) model, which includes the possibility of a jump at each step to a practically zero interest rate. This approach allows a better match with the risk of financial slowdown caused by catastrophic events. We present how to valuate a wide range of financial derivatives using such a model. The classical Black-Derman-Toy (BDT) model and a novel ZBDT model are described, and analogies in their calibration methodology are established. Finally, two cases of applications of the novel ZBDT model are introduced. The first is the hypothetical case of an S-shaped term structure and decreasing volatility of yields. The second case is an application in the structure of US sovereign bonds in the 2020 economic slowdown caused by the coronavirus pandemic. The objective of this study is to understand the differences presented by the valuation in both models for exotic derivatives.","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"30 1","pages":"103 - 118"},"PeriodicalIF":0.0,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48829137","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-05-18DOI: 10.3905/jod.2022.29.4.001
Joseph M. Pimbley, Frank J. Fabozzi
{"title":"Editor’s Letter","authors":"Joseph M. Pimbley, Frank J. Fabozzi","doi":"10.3905/jod.2022.29.4.001","DOIUrl":"https://doi.org/10.3905/jod.2022.29.4.001","url":null,"abstract":"","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"29 1","pages":"1 - 5"},"PeriodicalIF":0.0,"publicationDate":"2022-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46477558","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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