We present a new approach to identifying asset price bubbles based on options data. Given their forward-looking nature, options are ideal instruments with which to investigate market expectations about the future evolution of asset prices, which are key to understanding price bubbles. By exploiting the differential pricing between put and call options, we can detect and quantify bubbles in the prices of underlying asset. We apply our methodology to two stock market indexes, the S&P 500 and the Nasdaq-100, and two technology stocks, Amazon and Facebook, over the 2014-2018 sample period. We find that, while indexes exhibit rare and modest bubbles, Amazon and Facebook show more frequent and much larger bubbles. Since our approach can be implemented in real time, it is useful to both policy-makers and investors. As an illustration, our methodology applied to GameStop identifies a significant bubble between December 2020 and January 2021.
{"title":"Testing for Asset Price Bubbles using Options Data","authors":"Nicola Fusari, R. Jarrow, Sujan Lamichhane","doi":"10.2139/ssrn.3670999","DOIUrl":"https://doi.org/10.2139/ssrn.3670999","url":null,"abstract":"We present a new approach to identifying asset price bubbles based on options data. Given their forward-looking nature, options are ideal instruments with which to investigate market expectations about the future evolution of asset prices, which are key to understanding price bubbles. By exploiting the differential pricing between put and call options, we can detect and quantify bubbles in the prices of underlying asset. We apply our methodology to two stock market indexes, the S&P 500 and the Nasdaq-100, and two technology stocks, Amazon and Facebook, over the 2014-2018 sample period. We find that, while indexes exhibit rare and modest bubbles, Amazon and Facebook show more frequent and much larger bubbles. Since our approach can be implemented in real time, it is useful to both policy-makers and investors. As an illustration, our methodology applied to GameStop identifies a significant bubble between December 2020 and January 2021.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"8 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-08-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127398611","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}
Fear of disastrous tail events embedded in the short-term option contracts is reflected in the long-term equity risk premiums (ERP). A novel formula is proposed to identify the risk-neutral return quantiles from European option prices in a model-free manner. We use this formula to extract risk-neutral return quantiles of the S&P 500 index from January 1996 to June 2019. In the uni-variate predictive regressions, we find the difference between 5% and 95% risk-neutral quantiles (TD) significantly predicts equity risk premiums at horizons of more than one year, based on the standard error estimates of Hansen and Hodrick (1980) corrected for heteroskedasticity, Hodrick (1992) and Newey-West. We argue that TD captures the aversion to disastrous tail events of the market participants and, consistent with this, we find that TD is highly persistent. In the bi-variate predictive regressions that control for the well-known market predictors, TD is complementary to the variance risk premium of Bollerslev, Tauchen and Zhou (2009), which is a significant predictor of the ERP at horizons of less than one year. The correlation of TD with the dividend price ratio is 35% and highly significant, which is consistent with the finding of Campbell and Shiller (1989) that the variation of dividend price ratio is driven mainly by the variation of future discount rates.
短期期权合约对灾难性尾部事件的恐惧反映在长期股权风险溢价(ERP)中。提出了一种新的公式，以无模型的方式从欧式期权价格中识别风险中性收益分位数。我们使用该公式提取了1996年1月至2019年6月期间标准普尔500指数的风险中性收益分位数。在单变量预测回归中，根据Hansen和Hodrick(1980)、Hodrick(1992)和new - west对异方差校正后的标准误差估计，我们发现5%和95%风险中性分位数(TD)之间的差异显著地预测了一年以上的股票风险溢价。我们认为，TD抓住了市场参与者对灾难性尾部事件的厌恶，与此一致，我们发现TD具有高度的持久性。在控制知名市场预测因子的双变量预测回归中，TD与Bollerslev、Tauchen和Zhou(2009)的方差风险溢价(variance risk premium)是互补的，后者是小于一年的ERP的显著预测因子。TD与股利价格比的相关性为35%，且高度显著，这与Campbell and Shiller(1989)认为股利价格比的变化主要受未来贴现率的变化驱动的发现是一致的。
{"title":"Option-Implied Quantiles and Market Returns (Extended Abstract)","authors":"Yan Wang","doi":"10.2139/ssrn.3654110","DOIUrl":"https://doi.org/10.2139/ssrn.3654110","url":null,"abstract":"Fear of disastrous tail events embedded in the short-term option contracts is reflected in the long-term equity risk premiums (ERP). A novel formula is proposed to identify the risk-neutral return quantiles from European option prices in a model-free manner. We use this formula to extract risk-neutral return quantiles of the S&P 500 index from January 1996 to June 2019. In the uni-variate predictive regressions, we find the difference between 5% and 95% risk-neutral quantiles (TD) significantly predicts equity risk premiums at horizons of more than one year, based on the standard error estimates of Hansen and Hodrick (1980) corrected for heteroskedasticity, Hodrick (1992) and Newey-West. We argue that TD captures the aversion to disastrous tail events of the market participants and, consistent with this, we find that TD is highly persistent. In the bi-variate predictive regressions that control for the well-known market predictors, TD is complementary to the variance risk premium of Bollerslev, Tauchen and Zhou (2009), which is a significant predictor of the ERP at horizons of less than one year. The correlation of TD with the dividend price ratio is 35% and highly significant, which is consistent with the finding of Campbell and Shiller (1989) that the variation of dividend price ratio is driven mainly by the variation of future discount rates.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"78 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116069762","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}
Vasileios E. Kontosakos, Keegan Mendonca, A. Pantelous, K. Zuev
Abstract In this paper, we propose a new stochastic simulation-based methodology for pricing discretely-monitored double barrier options and estimating the corresponding probabilities of execution. We develop our framework by employing a versatile tool for the estimation of rare event probabilities known as subset simulation algorithm. In this regard, considering plausible dynamics for the price evolution of the underlying asset, we are able to compare and demonstrate clearly that our treatment always outperforms the standard Monte Carlo approach and becomes substantially more efficient (measured in terms of the sample coefficient of variation) when the underlying asset has high volatility and the barriers are set close to the spot price of the underlying asset. In addition, we test and report that our approach performs better when it is compared to the multilevel Monte Carlo method for special cases of barrier options and underlying assets that make the pricing problem a rare event estimation. These theoretical findings are confirmed by numerous simulation results.
{"title":"Pricing Discretely-Monitored Double Barrier Options with Small Probabilities of Execution","authors":"Vasileios E. Kontosakos, Keegan Mendonca, A. Pantelous, K. Zuev","doi":"10.2139/ssrn.3132336","DOIUrl":"https://doi.org/10.2139/ssrn.3132336","url":null,"abstract":"Abstract In this paper, we propose a new stochastic simulation-based methodology for pricing discretely-monitored double barrier options and estimating the corresponding probabilities of execution. We develop our framework by employing a versatile tool for the estimation of rare event probabilities known as subset simulation algorithm. In this regard, considering plausible dynamics for the price evolution of the underlying asset, we are able to compare and demonstrate clearly that our treatment always outperforms the standard Monte Carlo approach and becomes substantially more efficient (measured in terms of the sample coefficient of variation) when the underlying asset has high volatility and the barriers are set close to the spot price of the underlying asset. In addition, we test and report that our approach performs better when it is compared to the multilevel Monte Carlo method for special cases of barrier options and underlying assets that make the pricing problem a rare event estimation. These theoretical findings are confirmed by numerous simulation results.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"21 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126128400","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}
Abstract This study examines the value effects of financial and operational hedging in a managed floating exchange rate regime with strict limitations on the trading of Malaysian Ringgit for a sample of 109 Malaysian multinationals from 2004–2018. Using Tobin’s Q as a proxy for company value, the two-step system GMM estimation results show that, on average, derivatives hedging creates a value premium range of 7.88–8.21 % in the short-run, and 18.81–19.80 % in the long-run. This value premium emerged both after controlling for non-operational foreign exchange profits (losses), and its two components: transaction and translation profits (losses). In contrast, foreign debt hedging, on average, creates a value discount range of 8.19–8.54 % in the short-run and 12.70–13.12 % in the long-run. No evidence shows value effect for operational hedging though. The positive value effect of derivatives hedging should motivate managers of Malaysian multinationals to hedge foreign currency exposure through derivatives and encourage policymakers to take steps in developing derivatives market and products. However, the negative value effect of foreign debt hedging indicates that it destroys value. This negative effect might reflect two potential causes; higher company risk due to FC debt financing, and improper hedging practices including high costs of hedging in the underdeveloped derivatives market. These potential causes need further empirical evaluations.
{"title":"The Effects of Financial and Operational Hedging on Company Value: The Case of Malaysian Multinationals","authors":"Azadeh Hadian, Cahit Adaoglu","doi":"10.2139/ssrn.3724984","DOIUrl":"https://doi.org/10.2139/ssrn.3724984","url":null,"abstract":"Abstract This study examines the value effects of financial and operational hedging in a managed floating exchange rate regime with strict limitations on the trading of Malaysian Ringgit for a sample of 109 Malaysian multinationals from 2004–2018. Using Tobin’s Q as a proxy for company value, the two-step system GMM estimation results show that, on average, derivatives hedging creates a value premium range of 7.88–8.21 % in the short-run, and 18.81–19.80 % in the long-run. This value premium emerged both after controlling for non-operational foreign exchange profits (losses), and its two components: transaction and translation profits (losses). In contrast, foreign debt hedging, on average, creates a value discount range of 8.19–8.54 % in the short-run and 12.70–13.12 % in the long-run. No evidence shows value effect for operational hedging though. The positive value effect of derivatives hedging should motivate managers of Malaysian multinationals to hedge foreign currency exposure through derivatives and encourage policymakers to take steps in developing derivatives market and products. However, the negative value effect of foreign debt hedging indicates that it destroys value. This negative effect might reflect two potential causes; higher company risk due to FC debt financing, and improper hedging practices including high costs of hedging in the underdeveloped derivatives market. These potential causes need further empirical evaluations.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122002078","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}
Cryptocurrencies provide a unique opportunity to identify how derivatives impact cash markets. They are fully fungible across multiple trading venues and futures contracts were selectively introduced on bitcoin (BTC) exchange rates against the USD in December 2017. Following the futures introduction, we find a significantly greater increase in cross-exchange price synchronicity for BTC-USD relative to other exchange rate pairs, as demonstrated by an increase in price correlations and a reduction in arbitrage opportunities. We also find support for an increase in price efficiency, market quality, and liquidity. Overall, our analysis supports the view that the introduction of BTC-USD futures was beneficial to the bitcoin cash market by making the underlying prices more informative.
{"title":"The Impact of Derivatives on Cash Markets: Evidence From the Introduction of Bitcoin Futures Contracts","authors":"Patrick Augustin, Alexey Rubtsov, Donghwa Shin","doi":"10.2139/ssrn.3648406","DOIUrl":"https://doi.org/10.2139/ssrn.3648406","url":null,"abstract":"Cryptocurrencies provide a unique opportunity to identify how derivatives impact cash markets. They are fully fungible across multiple trading venues and futures contracts were selectively introduced on bitcoin (BTC) exchange rates against the USD in December 2017. Following the futures introduction, we find a significantly greater increase in cross-exchange price synchronicity for BTC-USD relative to other exchange rate pairs, as demonstrated by an increase in price correlations and a reduction in arbitrage opportunities. We also find support for an increase in price efficiency, market quality, and liquidity. Overall, our analysis supports the view that the introduction of BTC-USD futures was beneficial to the bitcoin cash market by making the underlying prices more informative.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"173 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122066957","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}
Patryk Gierjatowicz, Marc Sabate Vidales, D. Šiška, L. Szpruch, Zan Zuric
Mathematical modelling is ubiquitous in the financial industry and drives key decision processes. Any given model provides only a crude approximation to reality and the risk of using an inadequate model is hard to detect and quantify. By contrast, modern data science techniques are opening the door to more robust and data-driven model selection mechanisms. However, most machine learning models are "black-boxes" as individual parameters do not have meaningful interpretation. The aim of this paper is to combine the above approaches achieving the best of both worlds. Combining neural networks with risk models based on classical stochastic differential equations (SDEs), we find robust bounds for prices of derivatives and the corresponding hedging strategies while incorporating relevant market data. The resulting model called neural SDE is an instantiation of generative models and is closely linked with the theory of causal optimal transport. Neural SDEs allow consistent calibration under both the risk-neutral and the real-world measures. Thus the model can be used to simulate market scenarios needed for assessing risk profiles and hedging strategies. We develop and analyse novel algorithms needed for efficient use of neural SDEs. We validate our approach with numerical experiments using both local and stochastic volatility models.
{"title":"Robust Pricing and Hedging via Neural SDEs","authors":"Patryk Gierjatowicz, Marc Sabate Vidales, D. Šiška, L. Szpruch, Zan Zuric","doi":"10.2139/ssrn.3646241","DOIUrl":"https://doi.org/10.2139/ssrn.3646241","url":null,"abstract":"Mathematical modelling is ubiquitous in the financial industry and drives key decision processes. Any given model provides only a crude approximation to reality and the risk of using an inadequate model is hard to detect and quantify. By contrast, modern data science techniques are opening the door to more robust and data-driven model selection mechanisms. However, most machine learning models are \"black-boxes\" as individual parameters do not have meaningful interpretation. The aim of this paper is to combine the above approaches achieving the best of both worlds. Combining neural networks with risk models based on classical stochastic differential equations (SDEs), we find robust bounds for prices of derivatives and the corresponding hedging strategies while incorporating relevant market data. The resulting model called neural SDE is an instantiation of generative models and is closely linked with the theory of causal optimal transport. Neural SDEs allow consistent calibration under both the risk-neutral and the real-world measures. Thus the model can be used to simulate market scenarios needed for assessing risk profiles and hedging strategies. We develop and analyse novel algorithms needed for efficient use of neural SDEs. We validate our approach with numerical experiments using both local and stochastic volatility models.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121861347","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}
From the option prices with strike K, with K > S for calls and K < S for puts, a parameter can be estimated to calculate the prices of the entire options chain using a heuristic based on exponential decay, which has very well known applications in several natural and social phenomena. With the support of arbitrage restrictions such as the put-call parity, reasoning is validated and we can consider alternatives to evaluate forward conditions such as volatility and option prices in financial markets.
{"title":"Option Pricing: A Heuristic Based on Exponential Decay","authors":"Rogério Pereira","doi":"10.2139/ssrn.3643931","DOIUrl":"https://doi.org/10.2139/ssrn.3643931","url":null,"abstract":"From the option prices with strike K, with K > S for calls and K < S for puts, a parameter can be estimated to calculate the prices of the entire options chain using a heuristic based on exponential decay, which has very well known applications in several natural and social phenomena. With the support of arbitrage restrictions such as the put-call parity, reasoning is validated and we can consider alternatives to evaluate forward conditions such as volatility and option prices in financial markets.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"275 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130773070","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 introduce stochastic volatility models, in which the volatility is described by a time-dependent nonnegative function of a reflecting diffusion. The idea to use reflecting diffusions as building blocks of the volatility came into being because of a certain volatility misspecification in the classical Stein and Stein model. A version of this model that uses the reflecting Ornstein-Uhlenbeck process as the volatility process is a special example of a stochastic volatility model with reflection. The main results obtained in the present paper are sample path and small-noise large deviation principles for the log-price process in a stochastic volatility model with reflection under rather mild restrictions. We use these results to study the asymptotic behavior of binary barrier options and call prices in the small-noise regime.
{"title":"Large Deviation Principles for Stochastic Volatility Models with Reflection and Three Faces of the Stein and Stein Model","authors":"Archil Gulisashvili","doi":"10.2139/ssrn.3757783","DOIUrl":"https://doi.org/10.2139/ssrn.3757783","url":null,"abstract":"We introduce stochastic volatility models, in which the volatility is described by a time-dependent nonnegative function of a reflecting diffusion. The idea to use reflecting diffusions as building blocks of the volatility came into being because of a certain volatility misspecification in the classical Stein and Stein model. A version of this model that uses the reflecting Ornstein-Uhlenbeck process as the volatility process is a special example of a stochastic volatility model with reflection. The main results obtained in the present paper are sample path and small-noise large deviation principles for the log-price process in a stochastic volatility model with reflection under rather mild restrictions. We use these results to study the asymptotic behavior of binary barrier options and call prices in the small-noise regime.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125634035","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}
Dynamic equilibrium models based on present value computation imply that returns are predictable but also generate particular patterns of predictability in asset returns. I take advantage of this to construct a set of tests of Equilibrium Generated Predictability (EGP). I apply the tests to document two puzzles: First, option implied or realized measures of volatility ought to predict returns but do not. Second, the Variance Risk Premium (VRP) predicts returns but only at long horizons. VRP fails the tests of EGP as the term structure of predictable variation is inconsistent with an equilibrium interpretation.
{"title":"Predictability Puzzles","authors":"Bjørn Eraker","doi":"10.2139/ssrn.3625709","DOIUrl":"https://doi.org/10.2139/ssrn.3625709","url":null,"abstract":"Dynamic equilibrium models based on present value computation imply that returns are predictable but also generate particular patterns of predictability in asset returns. I take advantage of this to construct a set of tests of Equilibrium Generated Predictability (EGP). I apply the tests to document two puzzles: First, option implied or realized measures of volatility ought to predict returns but do not. Second, the Variance Risk Premium (VRP) predicts returns but only at long horizons. VRP fails the tests of EGP as the term structure of predictable variation is inconsistent with an equilibrium interpretation.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"2 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128262649","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 derive generalizations of Dupire formula to the cases of general stochastic drift and/or stochastic local volatility. First, we handle a case in which the drift is given as difference of two stochastic short rates. Such a setting is natural in foreign exchange context where the short rates correspond to the short rates of the two currencies, equity single-currency context with stochastic dividend yield, or commodity context with stochastic convenience yield. We present the formula both in a call surface formulation as well as total implied variance formulation where the latter avoids calendar spread arbitrage by construction. We provide derivations for the case where both short rates are given as single factor processes and present the limits for a single stochastic rate or all deterministic short rates. The limits agree with published results. Then we derive a formulation that allows a more general stochastic drift and diffusion including one or more stochastic local volatility terms. In the general setting, our derivation allows the computation and calibration of the leverage function for stochastic local volatility models.
{"title":"Extensions of Dupire Formula: Stochastic Interest Rates and Stochastic Local Volatility","authors":"O. Ogetbil","doi":"10.2139/ssrn.3598736","DOIUrl":"https://doi.org/10.2139/ssrn.3598736","url":null,"abstract":"We derive generalizations of Dupire formula to the cases of general stochastic drift and/or stochastic local volatility. First, we handle a case in which the drift is given as difference of two stochastic short rates. Such a setting is natural in foreign exchange context where the short rates correspond to the short rates of the two currencies, equity single-currency context with stochastic dividend yield, or commodity context with stochastic convenience yield. We present the formula both in a call surface formulation as well as total implied variance formulation where the latter avoids calendar spread arbitrage by construction. We provide derivations for the case where both short rates are given as single factor processes and present the limits for a single stochastic rate or all deterministic short rates. The limits agree with published results. Then we derive a formulation that allows a more general stochastic drift and diffusion including one or more stochastic local volatility terms. In the general setting, our derivation allows the computation and calibration of the leverage function for stochastic local volatility models.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"32 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-05-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116677970","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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