The analytic recovery method (ARM) recovers arbitrage-free density functions from a given set of option prices with maximum accuracy and speed. For arbitrage-free option prices, ARM provides extremely fast convergence and arbitrary accuracy. In the presence of noise, the closest arbitrage-free approximation is identified. Option prices and densities, as well as their moments and other parameters, are easy-to-handle analytic functions defined for arbitrary strike prices. ARM reveals inconsistencies between quoted option prices, particularly for longer durations. ARM is essentially based on the no-arbitrage assumptions; it is not related to a specific model. It has been tested for a selection of S&P 500, EuroStoxx 50, and DAX data. Excellent no-arbitrage fit to call and put prices is obtained; extrapolations are in line with the market.
{"title":"ARM: The Analytic Recovery Method","authors":"E. Linden","doi":"10.3905/jod.2023.1.178","DOIUrl":"https://doi.org/10.3905/jod.2023.1.178","url":null,"abstract":"The analytic recovery method (ARM) recovers arbitrage-free density functions from a given set of option prices with maximum accuracy and speed. For arbitrage-free option prices, ARM provides extremely fast convergence and arbitrary accuracy. In the presence of noise, the closest arbitrage-free approximation is identified. Option prices and densities, as well as their moments and other parameters, are easy-to-handle analytic functions defined for arbitrary strike prices. ARM reveals inconsistencies between quoted option prices, particularly for longer durations. ARM is essentially based on the no-arbitrage assumptions; it is not related to a specific model. It has been tested for a selection of S&P 500, EuroStoxx 50, and DAX data. Excellent no-arbitrage fit to call and put prices is obtained; extrapolations are in line with the market.","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"30 1","pages":"8 - 27"},"PeriodicalIF":0.0,"publicationDate":"2023-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44621548","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}
Although there are many well-established models for valuing corporate debt and equity, option pricing literature rarely takes these models as their starting point. This happens in part because such models value equity as an option on the firm’s assets, and options on equity then become compound options that cannot generally be priced analytically. In this article, the authors present a consistent and unified framework for valuing equity and options on equity within the 1994 Leland model. The authors show that it is possible to value not only European call and put options but also exotic options such as barriers and lookbacks in closed form. Moreover, the authors show that the model produces an implied volatility skew that is typically observed in the equity options market.
{"title":"The Leland Model as a Consistent Framework for Analytic Valuation of Equity and Options on Equity","authors":"Oh Kang Kwon, Andrew R. Grant, S. Satchell","doi":"10.3905/jod.2023.1.176","DOIUrl":"https://doi.org/10.3905/jod.2023.1.176","url":null,"abstract":"Although there are many well-established models for valuing corporate debt and equity, option pricing literature rarely takes these models as their starting point. This happens in part because such models value equity as an option on the firm’s assets, and options on equity then become compound options that cannot generally be priced analytically. In this article, the authors present a consistent and unified framework for valuing equity and options on equity within the 1994 Leland model. The authors show that it is possible to value not only European call and put options but also exotic options such as barriers and lookbacks in closed form. Moreover, the authors show that the model produces an implied volatility skew that is typically observed in the equity options market.","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"30 1","pages":"28 - 45"},"PeriodicalIF":0.0,"publicationDate":"2023-02-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42062150","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 : 2022-11-30DOI: 10.3905/jod.2022.30.2.065
Tom P. Davis
The Malliavin calculus has been used successfully to derive efficient formulas for delta and gamma. This article extends these results to all higher-order spatial derivatives with respect to the underlying asset for arbitrary payoffs in both the Black-Scholes (Black and Scholes 1973) (lognormal) and Bachelier (normal) models. The former reproduces a well-known result from Peter Carr (2000), whereas the latter extends this work to the normal case.
Malliavin微积分已被成功地用于推导有效的delta和gamma公式。本文将这些结果推广到Black-Scholes（Black and Scholes 1973）（lognormal）和Bachelier（normal）模型中任意收益的所有关于基础资产的高阶空间导数。前者再现了彼得·卡尔（2000）的一个著名结果，而后者将这项工作扩展到了正常情况。
{"title":"Malliavin Derivatives of Derivative Securities","authors":"Tom P. Davis","doi":"10.3905/jod.2022.30.2.065","DOIUrl":"https://doi.org/10.3905/jod.2022.30.2.065","url":null,"abstract":"The Malliavin calculus has been used successfully to derive efficient formulas for delta and gamma. This article extends these results to all higher-order spatial derivatives with respect to the underlying asset for arbitrary payoffs in both the Black-Scholes (Black and Scholes 1973) (lognormal) and Bachelier (normal) models. The former reproduces a well-known result from Peter Carr (2000), whereas the latter extends this work to the normal case.","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"30 1","pages":"65 - 73"},"PeriodicalIF":0.0,"publicationDate":"2022-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41526560","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 : 2022-11-30DOI: 10.3905/jod.2022.30.2.032
R. Galeeva
In his pioneer paper traced back to 1993, “Deriving Derivatives of Derivatives Securities,” Peter Carr used the operator calculus to show that that all partial derivatives of path independent claims can be represented in terms of the spatial derivatives. We generalized these results for multiasset situations. Reversing the relationships and expressing the higher-order Greeks (as gamma or cross gamma) in terms of the first-order Greeks leads to better numerical stability and convergence properties. We apply the results to evaluation and risk of an important energy asset as storage. In addition, we consider Greeks for the CEV model and the stochastic volatility case. At the time of our discussions, dating back in 2010–2011, I was mostly interested in applications for commodity derivatives. Peter suggested including the exponential Lévy model, his favorite subject; the CEV models; and the stochastic volatility case. In preparing this article, I kept the original draft, dated December 2011, of the write-up we worked out together. I reworked the write-up, and added storage models, numerical examples, and derivations.
彼得·卡尔（Peter Carr）在其1993年的先驱论文《衍生证券的衍生工具》（Deriving Derivatives of Derivatives-Securities）中，使用算子演算表明，路径独立债权的所有偏导数都可以用空间导数表示。我们将这些结果推广到多资产情况。颠倒关系并用一阶希腊语表示高阶希腊语（如伽马或交叉伽马），可以获得更好的数值稳定性和收敛性。我们将结果应用于存储等重要能源资产的评估和风险。此外，我们考虑了希腊的CEV模型和随机波动率的情况。在我们讨论的时候，可以追溯到2010-2011年，我主要对大宗商品衍生品的应用感兴趣。彼得建议包括指数莱维模型，这是他最喜欢的主题；CEV模型；以及随机波动情况。在准备这篇文章时，我保留了我们共同撰写的2011年12月的原始草稿。我重新编写了这篇文章，并添加了存储模型、数值示例和推导。
{"title":"Deriving Better Second-Order Derivatives","authors":"R. Galeeva","doi":"10.3905/jod.2022.30.2.032","DOIUrl":"https://doi.org/10.3905/jod.2022.30.2.032","url":null,"abstract":"In his pioneer paper traced back to 1993, “Deriving Derivatives of Derivatives Securities,” Peter Carr used the operator calculus to show that that all partial derivatives of path independent claims can be represented in terms of the spatial derivatives. We generalized these results for multiasset situations. Reversing the relationships and expressing the higher-order Greeks (as gamma or cross gamma) in terms of the first-order Greeks leads to better numerical stability and convergence properties. We apply the results to evaluation and risk of an important energy asset as storage. In addition, we consider Greeks for the CEV model and the stochastic volatility case. At the time of our discussions, dating back in 2010–2011, I was mostly interested in applications for commodity derivatives. Peter suggested including the exponential Lévy model, his favorite subject; the CEV models; and the stochastic volatility case. In preparing this article, I kept the original draft, dated December 2011, of the write-up we worked out together. I reworked the write-up, and added storage models, numerical examples, and derivations.","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"30 1","pages":"32 - 48"},"PeriodicalIF":0.0,"publicationDate":"2022-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44774858","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 : 2022-11-30DOI: 10.3905/jod.2022.30.2.016
K. Atteson, P. Carr
Drawdown is defined as the amount a portfolio has decreased from its running maximum. Drawdown has become ensconced in finance practice with some hedge funds shutting down portfolio managers who reach a certain drawdown limit. In this article, we show that, for every continuous local martingale that hits a given point m with probability 1, the running maximum of drawdown at the time of hitting m has the same inverse exponential distribution. We then derive prices and hedge ratios for binary calls on maximum absolute and relative drawdown maturing at the hitting time for m. We also derive prices for call spreads on maximum drawdown at the hitting time for m. These prices and hedge ratios are model independent across all continuous arbitrage-free stochastic processes that, with probability 1, either hit m or reach a drawdown equal to the strike price. This includes stochastic volatility models whose volatility is bounded away from 0 before hitting m or the strike. These results are both simpler and more general than prior work, which, while allowing for a fixed maturity, require infinite series representations, the use of complex derivatives to hedge and greater restrictions on the stochastic process. The key fact that facilitates our form of model independence is that the values of the derivatives at maturity are invariant to time changes.
{"title":"Carr Memorial: Maximum Drawdown Derivatives to a Hitting Time","authors":"K. Atteson, P. Carr","doi":"10.3905/jod.2022.30.2.016","DOIUrl":"https://doi.org/10.3905/jod.2022.30.2.016","url":null,"abstract":"Drawdown is defined as the amount a portfolio has decreased from its running maximum. Drawdown has become ensconced in finance practice with some hedge funds shutting down portfolio managers who reach a certain drawdown limit. In this article, we show that, for every continuous local martingale that hits a given point m with probability 1, the running maximum of drawdown at the time of hitting m has the same inverse exponential distribution. We then derive prices and hedge ratios for binary calls on maximum absolute and relative drawdown maturing at the hitting time for m. We also derive prices for call spreads on maximum drawdown at the hitting time for m. These prices and hedge ratios are model independent across all continuous arbitrage-free stochastic processes that, with probability 1, either hit m or reach a drawdown equal to the strike price. This includes stochastic volatility models whose volatility is bounded away from 0 before hitting m or the strike. These results are both simpler and more general than prior work, which, while allowing for a fixed maturity, require infinite series representations, the use of complex derivatives to hedge and greater restrictions on the stochastic process. The key fact that facilitates our form of model independence is that the values of the derivatives at maturity are invariant to time changes.","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"30 1","pages":"16 - 31"},"PeriodicalIF":0.0,"publicationDate":"2022-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48611074","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 : 2022-11-30DOI: 10.3905/jod.2022.30.2.049
P. Carr, C. Tebaldi
The infinitesimal generator of a time-homogeneous univariate diffusion process is a second-order linear ordinary differential operator. Feller (1952) famously factorized this generator into successive differentiations with respect to scale and speed measure. Later, Feller (1957) also factored an extended generator that loads also on the identity operator in a particular way. We provide a novel financial interpretation of these factorization results and show that they produce an operator representation of a conditionally linear risk-return tradeoff when the conditioning variable evolves like a one-dimensional diffusion process.
{"title":"Financial Interpretation of Feller’s Factorization","authors":"P. Carr, C. Tebaldi","doi":"10.3905/jod.2022.30.2.049","DOIUrl":"https://doi.org/10.3905/jod.2022.30.2.049","url":null,"abstract":"The infinitesimal generator of a time-homogeneous univariate diffusion process is a second-order linear ordinary differential operator. Feller (1952) famously factorized this generator into successive differentiations with respect to scale and speed measure. Later, Feller (1957) also factored an extended generator that loads also on the identity operator in a particular way. We provide a novel financial interpretation of these factorization results and show that they produce an operator representation of a conditionally linear risk-return tradeoff when the conditioning variable evolves like a one-dimensional diffusion process.","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"30 1","pages":"49 - 63"},"PeriodicalIF":0.0,"publicationDate":"2022-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46592268","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}
Realized semivariance, computed from intraday positive/negative squared returns, provides an accurate measure of the upside/downside variations of stock returns. This article investigates the role of realized semivariance in pricing the CBOE VIX and VIX futures, using a realized semivariance-based model. We obtain the closed-form pricing formula for the VIX index and VIX futures prices, and show that the new model provides superior pricing performance, both in-sample and out-of-sample. We further analytically derive the pricing formulas for the upside/downside components of the VIX (risk-neutral semivariance). Such a decomposition shows that the information gains from the conventional unsigned realized variance are concentrated on pricing the downside part of the VIX; the new realized semivariance-based model provides a larger and more balanced improvement for both the upside and downside components of the VIX. Our results provide strong evidence that the spread between upside/downside variance is the main driver of the asymmetry in return distributions.
{"title":"Good Volatility, Bad Volatility, and VIX Futures Pricing: Evidence from the Decomposition of VIX","authors":"Chen Tong, Zhuo Huang","doi":"10.3905/jod.2022.1.174","DOIUrl":"https://doi.org/10.3905/jod.2022.1.174","url":null,"abstract":"Realized semivariance, computed from intraday positive/negative squared returns, provides an accurate measure of the upside/downside variations of stock returns. This article investigates the role of realized semivariance in pricing the CBOE VIX and VIX futures, using a realized semivariance-based model. We obtain the closed-form pricing formula for the VIX index and VIX futures prices, and show that the new model provides superior pricing performance, both in-sample and out-of-sample. We further analytically derive the pricing formulas for the upside/downside components of the VIX (risk-neutral semivariance). Such a decomposition shows that the information gains from the conventional unsigned realized variance are concentrated on pricing the downside part of the VIX; the new realized semivariance-based model provides a larger and more balanced improvement for both the upside and downside components of the VIX. Our results provide strong evidence that the spread between upside/downside variance is the main driver of the asymmetry in return distributions.","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"30 1","pages":"117 - 143"},"PeriodicalIF":0.0,"publicationDate":"2022-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49157529","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 explore the pricing of compound derivatives under the newly introduced conjugate-power Dagum distribution. Assuming a discrete-time multiplicative conjugate-power Dagum random walk, we first provide an alternative derivation of the price of a married put based on a change of measure, which is helpful for the pricing of compound options. Then, we apply these results to obtain the equivalent of the Roll-Geske-Whaley formula for the pricing of American options in presence of one known discrete dividend under this alternative distribution.
{"title":"Compound Option Pricing and the Roll-Geske-Whaley Formula under the Conjugate-Power Dagum Distribution","authors":"P. Carr, Federico Maglione","doi":"10.3905/jod.2022.1.172","DOIUrl":"https://doi.org/10.3905/jod.2022.1.172","url":null,"abstract":"We explore the pricing of compound derivatives under the newly introduced conjugate-power Dagum distribution. Assuming a discrete-time multiplicative conjugate-power Dagum random walk, we first provide an alternative derivation of the price of a married put based on a change of measure, which is helpful for the pricing of compound options. Then, we apply these results to obtain the equivalent of the Roll-Geske-Whaley formula for the pricing of American options in presence of one known discrete dividend under this alternative distribution.","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"30 1","pages":"94 - 125"},"PeriodicalIF":0.0,"publicationDate":"2022-10-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41460382","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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