This article proposes a semi-martingale approximation to a fractional Lévy process that is capable of capturing long and short memory in the stochastic process together with fat tails. The authors use the semi-martingale process in option pricing and empirically compare its performance to other option pricing models, including a stochastic volatility Lévy process. They contribute to the empirical literature by being the first to report the implied Hurst index computed from observed option prices using the Lévy process model. Calibrating the implied Hurst index of S&P 500 option prices in a period that covers the 2008 financial crisis, they find that the risk-neutral measure is characterized by a short memory in turbulent markets and a long memory in calm markets. TOPICS: Options, statistical methods, performance measurement
{"title":"Long and Short Memory in the Risk-Neutral Pricing Process","authors":"Y. S. Kim, Danling Jiang, Stoyan Stoyanov","doi":"10.3905/jod.2019.1.077","DOIUrl":"https://doi.org/10.3905/jod.2019.1.077","url":null,"abstract":"This article proposes a semi-martingale approximation to a fractional Lévy process that is capable of capturing long and short memory in the stochastic process together with fat tails. The authors use the semi-martingale process in option pricing and empirically compare its performance to other option pricing models, including a stochastic volatility Lévy process. They contribute to the empirical literature by being the first to report the implied Hurst index computed from observed option prices using the Lévy process model. Calibrating the implied Hurst index of S&P 500 option prices in a period that covers the 2008 financial crisis, they find that the risk-neutral measure is characterized by a short memory in turbulent markets and a long memory in calm markets. TOPICS: Options, statistical methods, performance measurement","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"26 1","pages":"71 - 88"},"PeriodicalIF":0.0,"publicationDate":"2019-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.3905/jod.2019.1.077","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44586469","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}
Volatility Index (VIX) futures are among the most actively traded contracts at the Chicago Board Options Exchange, in response to the growing need for protection against volatility risk. The authors develop a new class of discrete-time and closed-form VIX futures pricing models, in which the S&P 500 returns follow the time-varying infinite-activity Normal Inverse Gaussian (NIG) and finite-activity compound Poisson (CP) jump-GARCH models, and which are risk-neutralized by the variance-dependent pricing kernel used by Christoffersen et al. (2013). They estimate these models using several data sets, including the S&P 500 returns, VIX Index, and VIX futures. The empirical results indicate that the time-varying NIG and CP jump-GARCH models significantly outperform the Heston-Nandi (HN) GARCH model in asset returns fitting and VIX futures pricing. TOPICS: Futures and forward contracts, derivatives
{"title":"VIX Futures Pricing with Affine Jump-GARCH Dynamics and Variance-Dependent Pricing Kernels","authors":"Xinglin Yang, Pengguo Wang, Ji Chen","doi":"10.3905/jod.2019.1.075","DOIUrl":"https://doi.org/10.3905/jod.2019.1.075","url":null,"abstract":"Volatility Index (VIX) futures are among the most actively traded contracts at the Chicago Board Options Exchange, in response to the growing need for protection against volatility risk. The authors develop a new class of discrete-time and closed-form VIX futures pricing models, in which the S&P 500 returns follow the time-varying infinite-activity Normal Inverse Gaussian (NIG) and finite-activity compound Poisson (CP) jump-GARCH models, and which are risk-neutralized by the variance-dependent pricing kernel used by Christoffersen et al. (2013). They estimate these models using several data sets, including the S&P 500 returns, VIX Index, and VIX futures. The empirical results indicate that the time-varying NIG and CP jump-GARCH models significantly outperform the Heston-Nandi (HN) GARCH model in asset returns fitting and VIX futures pricing. TOPICS: Futures and forward contracts, derivatives","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"27 1","pages":"110 - 127"},"PeriodicalIF":0.0,"publicationDate":"2019-04-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.3905/jod.2019.1.075","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47809398","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, the author presents a model with jumps and stochastic volatility, based on two correlated variance-gamma processes combined with an Ornstein–Uhlenbeck process with gamma innovations. The objective is to analyze pricing methods for a European-style equity option to exchange one stock for another, as well as two important classes of its variants, which raise the stock prices and the standard option payoff, respectively, to certain powers. These option variants are particularly useful in adjusting the risk level of exchange options and can also be viewed as generalizations of traditional power-type options. The pricing formulas are obtained under risk neutrality in terms of characteristic functions and are thus independent from the model distribution. Numerical results are given for illustrating the efficiency of the presented formulas along with various advantages of the proposed stochastic-volatility model. TOPICS: Options, statistical methods, performance measurement
{"title":"A Stochastic-Volatility Model for Pricing Power Variants of Exchange Options","authors":"W. Xia","doi":"10.3905/jod.2019.1.074","DOIUrl":"https://doi.org/10.3905/jod.2019.1.074","url":null,"abstract":"In this article, the author presents a model with jumps and stochastic volatility, based on two correlated variance-gamma processes combined with an Ornstein–Uhlenbeck process with gamma innovations. The objective is to analyze pricing methods for a European-style equity option to exchange one stock for another, as well as two important classes of its variants, which raise the stock prices and the standard option payoff, respectively, to certain powers. These option variants are particularly useful in adjusting the risk level of exchange options and can also be viewed as generalizations of traditional power-type options. The pricing formulas are obtained under risk neutrality in terms of characteristic functions and are thus independent from the model distribution. Numerical results are given for illustrating the efficiency of the presented formulas along with various advantages of the proposed stochastic-volatility model. TOPICS: Options, statistical methods, performance measurement","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"26 1","pages":"113 - 127"},"PeriodicalIF":0.0,"publicationDate":"2019-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.3905/jod.2019.1.074","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48495377","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}
Several studies have investigated the magnitude, drivers, and even reasons for the existence of cross-currency basis swap spreads. However, studies examining the interrelations among these spreads have surprisingly been lacking. In this article, the author examines the long-run relationships and short-run dynamic linkages among nine major cross-currency swap spreads, emphasizing how crisis periods have impacted the long-run relationships and short-run dynamics. Results show that the long-run relationships were slightly weakened after crisis, while the short-run linkages were generally strengthened. The influence of euro and Swiss cross-currency swaps on other European cross-currency swaps generally increased after the crisis period, and the Swiss cross-currency swap became much more influential on all European cross-currency swaps. The findings are robust to alternative reordering of variables in the author’s nine-variable VaR system, computation of generalized impulse response functions, and consideration of rolling variance decompositions. TOPICS: Currency, interest-rate and currency swaps, developed markets, VAR and use of alternative risk measures of trading risk
{"title":"Interrelations among Cross-Currency Basis Swap Spreads: Pre- and Post-Crisis Analysis","authors":"O. Ibhagui","doi":"10.3905/jod.2019.1.073","DOIUrl":"https://doi.org/10.3905/jod.2019.1.073","url":null,"abstract":"Several studies have investigated the magnitude, drivers, and even reasons for the existence of cross-currency basis swap spreads. However, studies examining the interrelations among these spreads have surprisingly been lacking. In this article, the author examines the long-run relationships and short-run dynamic linkages among nine major cross-currency swap spreads, emphasizing how crisis periods have impacted the long-run relationships and short-run dynamics. Results show that the long-run relationships were slightly weakened after crisis, while the short-run linkages were generally strengthened. The influence of euro and Swiss cross-currency swaps on other European cross-currency swaps generally increased after the crisis period, and the Swiss cross-currency swap became much more influential on all European cross-currency swaps. The findings are robust to alternative reordering of variables in the author’s nine-variable VaR system, computation of generalized impulse response functions, and consideration of rolling variance decompositions. TOPICS: Currency, interest-rate and currency swaps, developed markets, VAR and use of alternative risk measures of trading risk","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"26 1","pages":"112 - 89"},"PeriodicalIF":0.0,"publicationDate":"2019-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.3905/jod.2019.1.073","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49226084","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}
The author studies T. Cover’s rebalancing option (Ordentlich and Cover 1998) under discrete hindsight optimization in continuous time. The payoff in question is equal to the final wealth that would have accrued to an initial deposit of 1 unit of the numéraire into the best of some finite set of (perhaps levered) rebalancing rules determined in hindsight. A rebalancing rule (or fixed-fraction betting scheme) amounts to fixing an asset allocation (i.e., 200% equities and −100% bonds) and then continuously executing rebalancing trades so as to counteract allocation drift. Restricting the hindsight optimization to a small number of rebalancing rules (i.e., 2) has some advantages over the pioneering approach taken by Cover & Company in their theory of universal portfolios (1986, 1991, 1996, 1998), wherein one’s trading performance is benchmarked relative to the final wealth of the best unlevered rebalancing rule (of any kind) in hindsight. Our approach lets practitioners express an a priori view that one of the favored asset allocations (“bets”) in the set {b1, …, bn} will turn out to have performed spectacularly well in hindsight. In limiting our robustness to some discrete set of asset allocations (rather than all possible asset allocations), we reduce the price of the rebalancing option and guarantee that we will achieve a correspondingly higher percentage of the hindsight-optimized wealth at the end of the planning period. A practitioner who lives to delta-hedge this variant of Cover’s rebalancing option through several decades is guaranteed to see the day that his realized compound-annual capital growth rate is very close to that of the best bi in hindsight, hence the point of the rock-bottom option price.
{"title":"Cover’s Rebalancing Option with Discrete Hindsight Optimization","authors":"Alex Garivaltis","doi":"10.2139/ssrn.3346107","DOIUrl":"https://doi.org/10.2139/ssrn.3346107","url":null,"abstract":"The author studies T. Cover’s rebalancing option (Ordentlich and Cover 1998) under discrete hindsight optimization in continuous time. The payoff in question is equal to the final wealth that would have accrued to an initial deposit of 1 unit of the numéraire into the best of some finite set of (perhaps levered) rebalancing rules determined in hindsight. A rebalancing rule (or fixed-fraction betting scheme) amounts to fixing an asset allocation (i.e., 200% equities and −100% bonds) and then continuously executing rebalancing trades so as to counteract allocation drift. Restricting the hindsight optimization to a small number of rebalancing rules (i.e., 2) has some advantages over the pioneering approach taken by Cover & Company in their theory of universal portfolios (1986, 1991, 1996, 1998), wherein one’s trading performance is benchmarked relative to the final wealth of the best unlevered rebalancing rule (of any kind) in hindsight. Our approach lets practitioners express an a priori view that one of the favored asset allocations (“bets”) in the set {b1, …, bn} will turn out to have performed spectacularly well in hindsight. In limiting our robustness to some discrete set of asset allocations (rather than all possible asset allocations), we reduce the price of the rebalancing option and guarantee that we will achieve a correspondingly higher percentage of the hindsight-optimized wealth at the end of the planning period. A practitioner who lives to delta-hedge this variant of Cover’s rebalancing option through several decades is guaranteed to see the day that his realized compound-annual capital growth rate is very close to that of the best bi in hindsight, hence the point of the rock-bottom option price.","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"29 1","pages":"8 - 29"},"PeriodicalIF":0.0,"publicationDate":"2019-03-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44222947","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 : 2019-02-28DOI: 10.3905/jod.2019.26.3.007
F. Fabozzi, R. Shiller, R. Tunaru
Real estate derivatives have the potential to stabilize one of the most influential risks present in economies worldwide—real estate risk. Commercial and residential real estate represent a very large proportion of wealth in developed economies. In this article, the authors revisit the evolution of these instruments and describe the state of the art in modeling how they should be priced. The property derivatives market is still underdeveloped by comparison with its corresponding cash market, one main reason commonly cited being the lack of flexible and robust theoretical approaches that can be easily applied in practice. In recent years, several models have been proposed for pricing real estate derivatives, and this article reviews the most important ones. In addition, the authors highlight a discrete-time model that can be easily set up and applied for pricing real estate derivatives employing Monte Carlo simulation. It is reasonable to expect that the expanding literature on real estate derivatives valuation will provide the framework needed for this market to grow.
{"title":"Evolution of Real Estate Derivatives and Their Pricing","authors":"F. Fabozzi, R. Shiller, R. Tunaru","doi":"10.3905/jod.2019.26.3.007","DOIUrl":"https://doi.org/10.3905/jod.2019.26.3.007","url":null,"abstract":"Real estate derivatives have the potential to stabilize one of the most influential risks present in economies worldwide—real estate risk. Commercial and residential real estate represent a very large proportion of wealth in developed economies. In this article, the authors revisit the evolution of these instruments and describe the state of the art in modeling how they should be priced. The property derivatives market is still underdeveloped by comparison with its corresponding cash market, one main reason commonly cited being the lack of flexible and robust theoretical approaches that can be easily applied in practice. In recent years, several models have been proposed for pricing real estate derivatives, and this article reviews the most important ones. In addition, the authors highlight a discrete-time model that can be easily set up and applied for pricing real estate derivatives employing Monte Carlo simulation. It is reasonable to expect that the expanding literature on real estate derivatives valuation will provide the framework needed for this market to grow.","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"26 1","pages":"21 - 7"},"PeriodicalIF":0.0,"publicationDate":"2019-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.3905/jod.2019.26.3.007","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43524423","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 : 2019-02-28DOI: 10.3905/jod.2019.26.3.022
Honglei Zhao, Rupak Chatterjee, T. Lonon, I. Florescu
In a recent study, Zhao et al. (2017) presented a tree methodology to evaluate the expected generalized realized variance in a general stochastic volatility model; it provided an efficient way of calculating the fair value of the strike for variance swaps. In this article, the authors expand the methodology to price nonlinear derivatives written on realized variance. They introduce a new option contract, a Bermudan variance swaption, defined as an option on variance swap with early exercise dates. Within the same framework they also show how to value forward-start variance swaps, VIX futures, and VIX options. Numerical tests show that the methodology is efficient and accurate.
{"title":"Pricing Bermudan Variance Swaptions Using Multinomial Trees","authors":"Honglei Zhao, Rupak Chatterjee, T. Lonon, I. Florescu","doi":"10.3905/jod.2019.26.3.022","DOIUrl":"https://doi.org/10.3905/jod.2019.26.3.022","url":null,"abstract":"In a recent study, Zhao et al. (2017) presented a tree methodology to evaluate the expected generalized realized variance in a general stochastic volatility model; it provided an efficient way of calculating the fair value of the strike for variance swaps. In this article, the authors expand the methodology to price nonlinear derivatives written on realized variance. They introduce a new option contract, a Bermudan variance swaption, defined as an option on variance swap with early exercise dates. Within the same framework they also show how to value forward-start variance swaps, VIX futures, and VIX options. Numerical tests show that the methodology is efficient and accurate.","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"26 1","pages":"22 - 34"},"PeriodicalIF":0.0,"publicationDate":"2019-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.3905/jod.2019.26.3.022","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45039544","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}
This study obtains a closed-form solution for the discrete-time global quadratic hedging problem of Schweizer (1995) applied to vanilla European options under the geometric Gaussian random walk model for the underlying asset. This extends the work of Rémillard and Rubenthaler (2013), who obtained closed-form formulas for some components of the hedging problem solution. Coefficients embedded in the closed-form expression can be computed either directly or through a recursive algorithm. The author also presents a brief sensitivity analysis to determine the impact of the underlying asset drift and the hedging portfolio rebalancing frequency on the optimal hedging capital and the initial hedge ratio.
{"title":"A Closed-Form Solution for the Global Quadratic Hedging of Options under Geometric Gaussian Random Walks","authors":"Frédéric Godin","doi":"10.3905/jod.2019.1.071","DOIUrl":"https://doi.org/10.3905/jod.2019.1.071","url":null,"abstract":"This study obtains a closed-form solution for the discrete-time global quadratic hedging problem of Schweizer (1995) applied to vanilla European options under the geometric Gaussian random walk model for the underlying asset. This extends the work of Rémillard and Rubenthaler (2013), who obtained closed-form formulas for some components of the hedging problem solution. Coefficients embedded in the closed-form expression can be computed either directly or through a recursive algorithm. The author also presents a brief sensitivity analysis to determine the impact of the underlying asset drift and the hedging portfolio rebalancing frequency on the optimal hedging capital and the initial hedge ratio.","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"26 1","pages":"107 - 97"},"PeriodicalIF":0.0,"publicationDate":"2019-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.3905/jod.2019.1.071","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47595350","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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