The authors propose a unified approach to local volatility modeling, encompassing all asset classes, with straightforward application to equity and interest rate underlyings. Specifically, they consider a local volatility model for asset-for-asset or Margrabe (1978) options under general conditions that underlying dynamics follow Itô processes and derive a closed-form non-parametric local volatility formula. They then show that many standard contracts—European equity, FX, and interest rate options—can be seen as particular examples of the Margrabe-type payoff, which allows them to analyze equity and interest rate instruments, for example, as special cases of the same general local volatility model, rather than two separate models. They then derive a Markovian projection for the general model, with an approximate local volatility diffusion for the Margrabe option underlying. Finally, they discuss a specific application of the model to swaptions qua asset-for-asset options, where they consider the Markovian projection with some frozen parameters as a minimal “poor man’s” model, featuring equity-like dynamics for the swap rate with its own “short rate” and the “dividend” implied from the term structure of interest rates. Using a number of numerical examples, they compare the minimal model to a fully fledged Cheyette local volatility model and the market benchmark Hull–White one-factor model (Hull and White 1990), demonstrating the adequacy of the “poor man’s” model for pricing European and Bermudan payoffs. TOPICS: Options, statistical methods
作者提出了一种统一的本地波动率建模方法，包括所有资产类别，并直接应用于股票和利率基础。具体来说，他们考虑了资产对资产或Margrabe(1978)期权的局部波动率模型，在基本动态遵循Itô过程的一般条件下，并推导出封闭形式的非参数局部波动率公式。然后，他们展示了许多标准合约——欧洲股票、外汇和利率期权——可以被视为马尔格雷布式收益的特殊例子，这使他们能够分析股票和利率工具，例如，作为相同的一般本地波动模型的特殊案例，而不是两个独立的模型。然后，他们为一般模型推导出一个马尔可夫投影，并为Margrabe期权提供近似的局部波动扩散。最后，他们讨论了该模型在资产换资产期权互换中的具体应用，其中他们将带有一些固定参数的马尔可夫预测视为最小的“穷人”模型，其特征是具有自己的“短期利率”和利率期限结构隐含的“股息”的掉期利率的类似股票的动态。他们使用一些数值例子，将最小模型与成熟的Cheyette本地波动模型和市场基准Hull - White单因素模型(Hull and White 1990)进行比较，证明了“穷人”模型在定价欧洲和百慕大收益方面的适当性。主题:选项，统计方法
{"title":"Towards a General Local Volatility Model for All Asset Classes","authors":"D. Gatarek, J. Jabłecki","doi":"10.3905/jod.2019.1.079","DOIUrl":"https://doi.org/10.3905/jod.2019.1.079","url":null,"abstract":"The authors propose a unified approach to local volatility modeling, encompassing all asset classes, with straightforward application to equity and interest rate underlyings. Specifically, they consider a local volatility model for asset-for-asset or Margrabe (1978) options under general conditions that underlying dynamics follow Itô processes and derive a closed-form non-parametric local volatility formula. They then show that many standard contracts—European equity, FX, and interest rate options—can be seen as particular examples of the Margrabe-type payoff, which allows them to analyze equity and interest rate instruments, for example, as special cases of the same general local volatility model, rather than two separate models. They then derive a Markovian projection for the general model, with an approximate local volatility diffusion for the Margrabe option underlying. Finally, they discuss a specific application of the model to swaptions qua asset-for-asset options, where they consider the Markovian projection with some frozen parameters as a minimal “poor man’s” model, featuring equity-like dynamics for the swap rate with its own “short rate” and the “dividend” implied from the term structure of interest rates. Using a number of numerical examples, they compare the minimal model to a fully fledged Cheyette local volatility model and the market benchmark Hull–White one-factor model (Hull and White 1990), demonstrating the adequacy of the “poor man’s” model for pricing European and Bermudan payoffs. TOPICS: Options, statistical methods","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"27 1","pages":"14 - 31"},"PeriodicalIF":0.0,"publicationDate":"2019-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47035201","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}
1. Menachem Brenner 2. Emanuel Derman 3. Robert Jarrow 4. Eric Reiner 1. To order reprints of this article, please contact David Rowe at d.rowe{at}pageantmedia.com or 646-891-2157. An icon of the derivatives world, Mark Rubinstein, passed away recently, but his immense
{"title":"Remembering Mark Rubinstein","authors":"M. Brenner, E. Derman, R. Jarrow, Eric S. Reiner","doi":"10.3905/jod.2019.1.082","DOIUrl":"https://doi.org/10.3905/jod.2019.1.082","url":null,"abstract":"1. Menachem Brenner\u00002. Emanuel Derman\u00003. Robert Jarrow\u00004. Eric Reiner\u0000\u0000\u0000 \u0000\u00001. To order reprints of this article, please contact David Rowe at d.rowe{at}pageantmedia.com or 646-891-2157. \u0000\u0000An icon of the derivatives world, Mark Rubinstein, passed away recently, but his immense","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"27 1","pages":"13 - 8"},"PeriodicalIF":0.0,"publicationDate":"2019-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44065671","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 article describes in detail the multiplicative quadrinomial tree numerical method with nonconstant volatility, based on a system of stochastic differential equations of the GARCH-diffusion type. The methodology allowed for the derivation of the first two moments of the proposed equations to estimate the respective recombination between discrete and continuous processes and, as a result, a numerical methodological proposal is formally presented to value, with relative ease, both real and financial options, when the volatility is stochastic. The main findings showed that in the proposed method, when volatility approaches zero, the multiplicative binomial traditional method is a particular case, and the results are comparable between these methodologies, as well as to the exact solution offered by the Black–Scholes model. Finally, the originality of the methodological proposal is that it allows for the emulation in a simple way of the presence of a nonconstant volatility in the price of the underlying asset, and it can be used to value all kinds of options both in the real world and in risk-neutral situations. TOPICS: Options, derivatives
{"title":"Quadrinomial Trees to Value Options in Stochastic Volatility Models","authors":"Julian A. Pareja-Vasseur, Freddy H. Marín-Sánchez","doi":"10.3905/jod.2019.1.076","DOIUrl":"https://doi.org/10.3905/jod.2019.1.076","url":null,"abstract":"This article describes in detail the multiplicative quadrinomial tree numerical method with nonconstant volatility, based on a system of stochastic differential equations of the GARCH-diffusion type. The methodology allowed for the derivation of the first two moments of the proposed equations to estimate the respective recombination between discrete and continuous processes and, as a result, a numerical methodological proposal is formally presented to value, with relative ease, both real and financial options, when the volatility is stochastic. The main findings showed that in the proposed method, when volatility approaches zero, the multiplicative binomial traditional method is a particular case, and the results are comparable between these methodologies, as well as to the exact solution offered by the Black–Scholes model. Finally, the originality of the methodological proposal is that it allows for the emulation in a simple way of the presence of a nonconstant volatility in the price of the underlying asset, and it can be used to value all kinds of options both in the real world and in risk-neutral situations. TOPICS: Options, derivatives","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"27 1","pages":"49 - 66"},"PeriodicalIF":0.0,"publicationDate":"2019-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.3905/jod.2019.1.076","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49563124","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 article proposes an improved estimation and calibration method to a family of GARCH models. The suggested method fixes one parameter such that the unconditional kurtosis of the model matches the sample kurtosis. An empirical analysis using Engle and Ng’s (1993) NGARCH(1,1) model shows that the method dominates previous estimation methods in multiple ways. The optimization problem is simplified and made less sensitive to initial values. The optimization time, both when estimating based on historical returns and calibrating to option prices, is reduced by roughly 50%. The in-sample fit is barely affected, while the option pricing, both in sample and out of sample, is improved. TOPICS: Statistical methods, quantitative methods, options, derivatives
{"title":"An Improved Estimation Method for a Family of GARCH Models","authors":"P. Létourneau","doi":"10.3905/jod.2019.1.081","DOIUrl":"https://doi.org/10.3905/jod.2019.1.081","url":null,"abstract":"This article proposes an improved estimation and calibration method to a family of GARCH models. The suggested method fixes one parameter such that the unconditional kurtosis of the model matches the sample kurtosis. An empirical analysis using Engle and Ng’s (1993) NGARCH(1,1) model shows that the method dominates previous estimation methods in multiple ways. The optimization problem is simplified and made less sensitive to initial values. The optimization time, both when estimating based on historical returns and calibrating to option prices, is reduced by roughly 50%. The in-sample fit is barely affected, while the option pricing, both in sample and out of sample, is improved. TOPICS: Statistical methods, quantitative methods, options, derivatives","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"27 1","pages":"67 - 91"},"PeriodicalIF":0.0,"publicationDate":"2019-07-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42225471","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 article considers the Black–Scholes and Heston models and generalize them to stochastic interest rates and maturity-dependent volatilities. In the Black–Scholes case, the author solves the extended model and provides a concrete form for the term structure of volatilities. In the Heston case, he proves that, under some conditions, the generalized model is equivalent to a hybrid model and finds semi-closed-form solutions in the Hull and White and CIR cases. TOPICS: Options, statistical methods, fixed income and structured finance
{"title":"Black–Scholes and Heston Models with Stochastic Interest Rates and Term Structure of Volatilities","authors":"Alberto Bueno-Guerrero","doi":"10.3905/jod.2019.1.078","DOIUrl":"https://doi.org/10.3905/jod.2019.1.078","url":null,"abstract":"This article considers the Black–Scholes and Heston models and generalize them to stochastic interest rates and maturity-dependent volatilities. In the Black–Scholes case, the author solves the extended model and provides a concrete form for the term structure of volatilities. In the Heston case, he proves that, under some conditions, the generalized model is equivalent to a hybrid model and finds semi-closed-form solutions in the Hull and White and CIR cases. TOPICS: Options, statistical methods, fixed income and structured finance","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"27 1","pages":"32 - 48"},"PeriodicalIF":0.0,"publicationDate":"2019-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46038653","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 authors introduce to the literature an intelligent lattice search algorithm to efficiently locate the optimal exercise boundary for American options. Lattice models can be accelerated by incorporating intelligent lattice search, truncation, and dynamic memory. We reduce computational runtime from over 18 minutes down to less than 3 seconds to estimate a 15,000-step binomial tree where the results obtained are consistent with a widely acclaimed literature. Delta and implied volatility can also be accelerated relative to standard models. Lattice estimation, in general, is considered to be slow and not practical for valuing large books of options or for promptly rebalancing a risk-neutral portfolio. Our technique transforms standard binomial trees and renders them to be at least on par with commonly used analytical formulae. More importantly, intelligent lattice search can be tweaked to reach varying levels of accuracy with different step size, while conventional analytical formulae are less flexible. TOPICS: Options, derivatives
{"title":"American Option Pricing: An Accelerated Lattice Model with Intelligent Lattice Search","authors":"Qianru Shang, Brian Byrne","doi":"10.3905/jod.2019.1.080","DOIUrl":"https://doi.org/10.3905/jod.2019.1.080","url":null,"abstract":"The authors introduce to the literature an intelligent lattice search algorithm to efficiently locate the optimal exercise boundary for American options. Lattice models can be accelerated by incorporating intelligent lattice search, truncation, and dynamic memory. We reduce computational runtime from over 18 minutes down to less than 3 seconds to estimate a 15,000-step binomial tree where the results obtained are consistent with a widely acclaimed literature. Delta and implied volatility can also be accelerated relative to standard models. Lattice estimation, in general, is considered to be slow and not practical for valuing large books of options or for promptly rebalancing a risk-neutral portfolio. Our technique transforms standard binomial trees and renders them to be at least on par with commonly used analytical formulae. More importantly, intelligent lattice search can be tweaked to reach varying levels of accuracy with different step size, while conventional analytical formulae are less flexible. TOPICS: Options, derivatives","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"27 1","pages":"108 - 92"},"PeriodicalIF":0.0,"publicationDate":"2019-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43764587","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 authors formulate a risk-based swaption portfolio management framework for a profit-and-loss (P&L) explanation. They analyze the implication of using the right volatility backbone in the pricing model from a hedging perspective and demonstrate the importance of incorporating a stability and robustness measure as part of the calibration process for optimal model selection. They also derive a displaced-diffusion stochastic volatility model with a closed-form analytical expression to handle negative interest rates. Finally, they show that their framework is able to identify the optimal pricing model, which leads to a superior P&L explanation and hedging performance. TOPICS: Risk management, portfolio management/multi-asset allocation, derivatives, factor-based models Key Findings • A holistic, risk-based calibration framework allows one to select the optimal pricing model with superior P&L explanation performance. • A displaced-diffusion stochastic volatility model with closed-form expression provides a means to price swaptions efficiently under both positive and negative interest rate regimes while capturing the volatility backbone. • Using the Herfindahl index to measure the concentration of hedging performance, we show that the optimal model exhibits stability and robustness of model parameters, along with the economy of the explanatory power of daily P&L movement.
{"title":"Swaption Portfolio Risk Management: Optimal Model Selection in Different Interest Rate Regimes","authors":"Poh Ling Neo, C. W. Tee","doi":"10.3905/jod.2019.1.083","DOIUrl":"https://doi.org/10.3905/jod.2019.1.083","url":null,"abstract":"The authors formulate a risk-based swaption portfolio management framework for a profit-and-loss (P&L) explanation. They analyze the implication of using the right volatility backbone in the pricing model from a hedging perspective and demonstrate the importance of incorporating a stability and robustness measure as part of the calibration process for optimal model selection. They also derive a displaced-diffusion stochastic volatility model with a closed-form analytical expression to handle negative interest rates. Finally, they show that their framework is able to identify the optimal pricing model, which leads to a superior P&L explanation and hedging performance. TOPICS: Risk management, portfolio management/multi-asset allocation, derivatives, factor-based models Key Findings • A holistic, risk-based calibration framework allows one to select the optimal pricing model with superior P&L explanation performance. • A displaced-diffusion stochastic volatility model with closed-form expression provides a means to price swaptions efficiently under both positive and negative interest rate regimes while capturing the volatility backbone. • Using the Herfindahl index to measure the concentration of hedging performance, we show that the optimal model exhibits stability and robustness of model parameters, along with the economy of the explanatory power of daily P&L movement.","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"27 1","pages":"107 - 81"},"PeriodicalIF":0.0,"publicationDate":"2019-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46948243","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 : 2019-05-31DOI: 10.3905/jod.2019.26.4.054
Shuxin Guo, Qiang Liu
Known discrete dollar dividends lead to non-recombining binomial trees (NR-BT) with an explosion of nodes, and make the pricing of options much more complex. This article proposes a novel method for constructing a recombining binomial tree via balanced dividend adjustments (BDA). BDA is proved to converge to NR-BT for European options. Furthermore, BDA is shown heuristically to approximate NR-BT superbly for American options; for American calls, an error formula for BDA is derived and can be used to reduce further the pricing error. In a numerical illustration for American options, BDA turns out to be quite accurate, outperforming several existing approaches. A new insight emerges that BDA can be a competitive, yet simple, alternative to the industry practice of interpolating for dividends under binomial-tree or finite difference. TOPICS: Options, statistical methods, performance measurement
{"title":"A Simple Accurate Binomial Tree for Pricing Options on Stocks with Known Dollar Dividends","authors":"Shuxin Guo, Qiang Liu","doi":"10.3905/jod.2019.26.4.054","DOIUrl":"https://doi.org/10.3905/jod.2019.26.4.054","url":null,"abstract":"Known discrete dollar dividends lead to non-recombining binomial trees (NR-BT) with an explosion of nodes, and make the pricing of options much more complex. This article proposes a novel method for constructing a recombining binomial tree via balanced dividend adjustments (BDA). BDA is proved to converge to NR-BT for European options. Furthermore, BDA is shown heuristically to approximate NR-BT superbly for American options; for American calls, an error formula for BDA is derived and can be used to reduce further the pricing error. In a numerical illustration for American options, BDA turns out to be quite accurate, outperforming several existing approaches. A new insight emerges that BDA can be a competitive, yet simple, alternative to the industry practice of interpolating for dividends under binomial-tree or finite difference. TOPICS: Options, statistical methods, performance measurement","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"26 1","pages":"54 - 70"},"PeriodicalIF":0.0,"publicationDate":"2019-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.3905/jod.2019.26.4.054","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42661758","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 : 2019-05-31DOI: 10.3905/jod.2019.26.4.009
Deeveya Thakoor, M. Bhuruth
Lattice methods are often employed to price contingent claims with discrete dividends under the lognormal diffusion, but they are inclined to suffer from large decreases in execution speed as the number of dividends increases. Heteroskedastic assumptions for the stock price dynamics in between ex-dividend dates exacerbate these difficulties, and the option pricing problem with discrete dividends has thus been limited to the lognormal framework. This article proposes strategies to speed up lattice-based approximations under these general diffusions. A range-curtailing technique that bypasses superfluous computations of numerous subtrees at unrealistic stock prices is considered for European, American, and barrier options. The effect of discrete dividends on the premature exercise of American options is also studied. A benchmark method based on numerical integration is described to validate results obtained in the heteroskedastic framework. TOPICS: Options, statistical methods
{"title":"Range-Curtailing for Options with Discrete Dividend Payments under General Diffusions","authors":"Deeveya Thakoor, M. Bhuruth","doi":"10.3905/jod.2019.26.4.009","DOIUrl":"https://doi.org/10.3905/jod.2019.26.4.009","url":null,"abstract":"Lattice methods are often employed to price contingent claims with discrete dividends under the lognormal diffusion, but they are inclined to suffer from large decreases in execution speed as the number of dividends increases. Heteroskedastic assumptions for the stock price dynamics in between ex-dividend dates exacerbate these difficulties, and the option pricing problem with discrete dividends has thus been limited to the lognormal framework. This article proposes strategies to speed up lattice-based approximations under these general diffusions. A range-curtailing technique that bypasses superfluous computations of numerous subtrees at unrealistic stock prices is considered for European, American, and barrier options. The effect of discrete dividends on the premature exercise of American options is also studied. A benchmark method based on numerical integration is described to validate results obtained in the heteroskedastic framework. TOPICS: Options, statistical methods","PeriodicalId":34223,"journal":{"name":"Jurnal Derivat","volume":"26 1","pages":"34 - 9"},"PeriodicalIF":0.0,"publicationDate":"2019-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.3905/jod.2019.26.4.009","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45238468","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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