A put option is a financial contract that gives the holder the right to sell an asset at a specific price by (or at) a specific date. A put option can therefore provide its holder insurance against a large drop in the stock price. This makes the prices of put options an ideal source of information for a market-based measure of the probability of a firm’s default.
{"title":"A Simple Method for Extracting the Probability of Default from American Put Option Prices","authors":"B. Chang, Greg Orosi","doi":"10.2139/ssrn.3524525","DOIUrl":"https://doi.org/10.2139/ssrn.3524525","url":null,"abstract":"A put option is a financial contract that gives the holder the right to sell an asset at a specific price by (or at) a specific date. A put option can therefore provide its holder insurance against a large drop in the stock price. This makes the prices of put options an ideal source of information for a market-based measure of the probability of a firm’s default.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121968461","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}
Financial statements and an accompanying NPV calculation are embedded into a binomial tree. This generalization of traditional static NPV analysis allows the financial statements to both evolve through time and, at any given time, to vary with states of the world (similar to a Monte Carlo analysis). Modelling the component cash flows in a tree reveals dynamic detailed structure, leading to a more useful NPV analysis than if only the final cash flow value was modelled in a tree or if component cash flows were modelled without a tree. This dynamic detail provides credible cash flow forecasts that can improve hedging of adverse events and allow for leveraging of beneficial circumstances. The financial statements take the form of pro forma after-tax operating cash flows in this treatment. However, any cash flow model driven by the random variable in the tree and allowing for separate treatment of fixed costs, can be used. The benefits of this technique are illustrated via a real options example.
{"title":"Embedding an NPV Analysis into a Binomial Tree with a Real Options Application","authors":"Tom Arnold, T. Crack, Adam Schwartz","doi":"10.2139/ssrn.3526901","DOIUrl":"https://doi.org/10.2139/ssrn.3526901","url":null,"abstract":"Financial statements and an accompanying NPV calculation are embedded into a binomial tree. This generalization of traditional static NPV analysis allows the financial statements to both evolve through time and, at any given time, to vary with states of the world (similar to a Monte Carlo analysis). Modelling the component cash flows in a tree reveals dynamic detailed structure, leading to a more useful NPV analysis than if only the final cash flow value was modelled in a tree or if component cash flows were modelled without a tree. This dynamic detail provides credible cash flow forecasts that can improve hedging of adverse events and allow for leveraging of beneficial circumstances. The financial statements take the form of pro forma after-tax operating cash flows in this treatment. However, any cash flow model driven by the random variable in the tree and allowing for separate treatment of fixed costs, can be used. The benefits of this technique are illustrated via a real options example.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"24 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114763545","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 paper develops an option pricing model that admits probability measure ambiguity. It formulates a piecewise risk-ambiguity-neutral probability density function and derives analytical pricing formula. Options and their underlying assets are exposed to different scopes of ambiguity that cannot be hedged, implying that options are generically non-redundant assets and have different Sharpe ratios than the underlying assets. Introduction of probability measure ambiguity reduces the in-sample and 1-day (5-day) out-of-sample pricing errors of the Black-Scholes-Merton model by 80% and 66% (61%) in pricing S&P 500 index options, and remarkably alleviates volatility smile. Option-implied market ambiguity premium is counter-cyclical and contains distinct information compared to VIX.
{"title":"An Option Pricing Model with Probability Measure Ambiguity","authors":"Yu Liu, Hao Wang, Lihong Zhang","doi":"10.2139/ssrn.3521604","DOIUrl":"https://doi.org/10.2139/ssrn.3521604","url":null,"abstract":"This paper develops an option pricing model that admits probability measure ambiguity. It formulates a piecewise risk-ambiguity-neutral probability density function and derives analytical pricing formula. Options and their underlying assets are exposed to different scopes of ambiguity that cannot be hedged, implying that options are generically non-redundant assets and have different Sharpe ratios than the underlying assets. Introduction of probability measure ambiguity reduces the in-sample and 1-day (5-day) out-of-sample pricing errors of the Black-Scholes-Merton model by 80% and 66% (61%) in pricing S&P 500 index options, and remarkably alleviates volatility smile. Option-implied market ambiguity premium is counter-cyclical and contains distinct information compared to VIX.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"257 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116572198","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}
Fitting simultaneously SPX and VIX smiles is known to be one of the most challenging problems in volatility modeling. A long-standing conjecture due to Julien Guyon is that it may not be possible to calibrate jointly these two quantities with a model with continuous sample-paths. We present the quadratic rough Heston model as a counterexample to this conjecture. The key idea is the combination of rough volatility together with a price-feedback (Zumbach) effect.
{"title":"The Quadratic Rough Heston Model and the Joint S&P 500/VIX Smile Calibration Problem","authors":"Jim Gatheral, Paul Jusselin, M. Rosenbaum","doi":"10.2139/ssrn.3514894","DOIUrl":"https://doi.org/10.2139/ssrn.3514894","url":null,"abstract":"Fitting simultaneously SPX and VIX smiles is known to be one of the most challenging problems in volatility modeling. A long-standing conjecture due to Julien Guyon is that it may not be possible to calibrate jointly these two quantities with a model with continuous sample-paths. We present the quadratic rough Heston model as a counterexample to this conjecture. The key idea is the combination of rough volatility together with a price-feedback (Zumbach) effect.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"10 7","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121000090","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 paper offers an option value-based rationale for the consideration of non- compliance record in punishment. We study compliance decisions of a population of individuals who live for two periods, where each individual's non-compliance benefits are random and independent over time. Because increasing or decreasing sanction schemes produce different option values to current-period compliance and non-compliance, an optimal sanction scheme involves a trade-o§ between present and future compliance. This trade-o§ increases incentives for present compliance while facilitating a more efficient allocation of sanctions across periods. The optimal sanction scheme accordingly depends on the overall sanction and the distribution of non-compliance benefits.
{"title":"The Option Value of Record-Based Sanctions","authors":"Shmuel Leshem, Avraham Tabbach","doi":"10.2139/ssrn.3514094","DOIUrl":"https://doi.org/10.2139/ssrn.3514094","url":null,"abstract":"This paper offers an option value-based rationale for the consideration of non- compliance record in punishment. We study compliance decisions of a population of individuals who live for two periods, where each individual's non-compliance benefits are random and independent over time. Because increasing or decreasing sanction schemes produce different option values to current-period compliance and non-compliance, an optimal sanction scheme involves a trade-o§ between present and future compliance. This trade-o§ increases incentives for present compliance while facilitating a more efficient allocation of sanctions across periods. The optimal sanction scheme accordingly depends on the overall sanction and the distribution of non-compliance benefits.<br>","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124448686","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 paper proposes an innovative algorithm that significantly improves on the approximation of the optimal early exercise boundary obtained with simulation based methods for American option pricing. The method works by exploiting and leveraging the information in multiple cross-sectional regressions to the fullest by averaging the individually obtained estimates at each early exercise step, starting from just before maturity, in the backwards induction algorithm. With this method, less errors are accumulated, and as a result of this, the price estimate is essentially unbiased even for long maturity options. Numerical results demonstrate the improvements from our method and show that these are robust to the choice of simulation setup, the characteristics of the option, and the dimensionality of the problem. Finally, because our method naturally disassociates the estimation of the optimal early exercise boundary from the pricing of the option, significant efficiency gains can be obtained by using less simulated paths and repetitions to estimate the optimal early exercise boundary than with the regular method.
{"title":"Bootstrapping the Early Exercise Boundary in the Least-Squares Monte Carlo Method","authors":"P. Létourneau, Lars Stentoft","doi":"10.2139/ssrn.3503049","DOIUrl":"https://doi.org/10.2139/ssrn.3503049","url":null,"abstract":"This paper proposes an innovative algorithm that significantly improves on the approximation of the optimal early exercise boundary obtained with simulation based methods for American option pricing. The method works by exploiting and leveraging the information in multiple cross-sectional regressions to the fullest by averaging the individually obtained estimates at each early exercise step, starting from just before maturity, in the backwards induction algorithm. With this method, less errors are accumulated, and as a result of this, the price estimate is essentially unbiased even for long maturity options. Numerical results demonstrate the improvements from our method and show that these are robust to the choice of simulation setup, the characteristics of the option, and the dimensionality of the problem. Finally, because our method naturally disassociates the estimation of the optimal early exercise boundary from the pricing of the option, significant efficiency gains can be obtained by using less simulated paths and repetitions to estimate the optimal early exercise boundary than with the regular method.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123005130","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 document a higher bond return volatility around the time of default for bonds included in CDS auctions (especially cheapest-to-deliver bonds) versus those that are not, while controlling for firm fundamentals and bond illiquidity. This finding does not extend to time periods far ahead of default, and there is no significant difference between the idiosyncratic stock return volatility of CDS firms and non-CDS firms around the time of default. These results are more consistent with CDS buyers and sellers manipulating bond prices to achieve favorable CDS auction outcomes, rather than a spillover of price discovery by CDS traders into the stock and bond markets.
{"title":"Bond Volatility and CDS Auctions","authors":"Jennifer Mace, F. Yu, Ran Zhao","doi":"10.2139/ssrn.3497729","DOIUrl":"https://doi.org/10.2139/ssrn.3497729","url":null,"abstract":"We document a higher bond return volatility around the time of default for bonds included in CDS auctions (especially cheapest-to-deliver bonds) versus those that are not, while controlling for firm fundamentals and bond illiquidity. This finding does not extend to time periods far ahead of default, and there is no significant difference between the idiosyncratic stock return volatility of CDS firms and non-CDS firms around the time of default. These results are more consistent with CDS buyers and sellers manipulating bond prices to achieve favorable CDS auction outcomes, rather than a spillover of price discovery by CDS traders into the stock and bond markets.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133863204","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 lower and upper bounds on the conditional market autocorrelation index at various investment horizons without using the precise form of the utility function. The bounds are derived in terms of option prices and can be computed at daily frequency for any given horizon. The bounds incorporate all the information contained in the entire distribution of returns. We use options on the S&P 500 index to quantify the bounds and document that asset prices imply a negative upper bound on the market conditional autocorrelation index. The upper bound on the market conditional autocorrelation index is highly volatile, skewed, and exhibits fat tails. It varies from -28% to -3% and takes extremely negative values during crisis or recession periods while being close to zero during normal times. On average, the upper bound on the market conditional autocorrelation index is -14%. We also document that periods of extremely negative market conditional autocorrelation index coincide with periods of a high Sharpe ratio, and we show that leading asset pricing models cannot reproduce both the negative market conditional autocorrelation index and the negative average market conditional autocorrelation index implied by asset prices.
{"title":"What Is the Conditional Autocorrelation on the Stock Market?","authors":"Fousseni Chabi-Yo","doi":"10.2139/ssrn.3490938","DOIUrl":"https://doi.org/10.2139/ssrn.3490938","url":null,"abstract":"We derive lower and upper bounds on the conditional market autocorrelation index at various investment horizons without using the precise form of the utility function. The bounds are derived in terms of option prices and can be computed at daily frequency for any given horizon. The bounds incorporate all the information contained in the entire distribution of returns. We use options on the S&P 500 index to quantify the bounds and document that asset prices imply a negative upper bound on the market conditional autocorrelation index. The upper bound on the market conditional autocorrelation index is highly volatile, skewed, and exhibits fat tails. It varies from -28% to -3% and takes extremely negative values during crisis or recession periods while being close to zero during normal times. On average, the upper bound on the market conditional autocorrelation index is -14%. We also document that periods of extremely negative market conditional autocorrelation index coincide with periods of a high Sharpe ratio, and we show that leading asset pricing models cannot reproduce both the negative market conditional autocorrelation index and the negative average market conditional autocorrelation index implied by asset prices.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"144 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116372170","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}
A practical implementation of a CVA calculation on a portfolio of different types of derivatives with one counterpart can be complicated and needs to be validated in various ways. One very simple possibility is provided by the present paper: When the portfolio is made up only of CCY swaps (which differ only by the notional amounts and are otherwise identical) and the driving model is the one used here then one can use a closed form solution to predict the CVA with very high accuracy. This can be used for tests that an implementation should pass. For that purpose, the simplest possible model that includes vols and correlations for interest rates in both currencies, the FX rate and the default intensity is chosen. It allows for a closed form solution for the PV of a contingent credit default swap (CCDS) that pays in default the outstanding mark to market price of a cross currency swap provided the latter is positive. The paper also provides conditions which determine the directions of the sensitivities of this PV with respect to changes in the correlations.
{"title":"Closed Form Solutions for Contingent CDS on Cross Currency Swaps","authors":"Rainer Hoehnle","doi":"10.2139/ssrn.3482443","DOIUrl":"https://doi.org/10.2139/ssrn.3482443","url":null,"abstract":"A practical implementation of a CVA calculation on a portfolio of different types of derivatives with one counterpart can be complicated and needs to be validated in various ways. One very simple possibility is provided by the present paper: When the portfolio is made up only of CCY swaps (which differ only by the notional amounts and are otherwise identical) and the driving model is the one used here then one can use a closed form solution to predict the CVA with very high accuracy. This can be used for tests that an implementation should pass. For that purpose, the simplest possible model that includes vols and correlations for interest rates in both currencies, the <br>FX rate and the default intensity is chosen. It allows for a closed form solution for the PV of a contingent credit default swap (CCDS) that pays in default the outstanding mark to market price of a cross currency swap provided the latter is positive. The paper also provides conditions which determine the directions of the sensitivities of this PV with respect to changes in the correlations.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-11-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130600577","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 present a perturbation theory of the market impact based on an extension of the framework proposed by [Loeper, 2018] – originally based on [Liu and Yong, 2005] – in which we consider only local linear market impact. We study the execution process of hedging derivatives and show how these hedging metaorders can explain some stylized facts observed in the empirical market impact literature. As we are interested in the execution process of hedging we will establish that the arbitrage opportunities that exist in the discrete time setting vanish when the trading frequency goes to infinity letting us to derive a pricing equation. Furthermore our approach retrieves several results already established in the option pricing literature such that the spot dynamics modified by the market impact. We also study the relaxation of our hedging metaorders based on the fair pricing hypothesis and establish a relation between the immediate impact and the permanent impact which is in agreement with recent empirical studies on the subject.
我们基于[Loeper, 2018]提出的框架的扩展(最初基于[Liu and Yong, 2005])提出了市场影响的扰动理论，其中我们只考虑局部线性市场影响。我们研究了套期保值衍生品的执行过程，并展示了这些套期保值元指令如何解释在实证市场影响文献中观察到的一些程式化事实。由于我们对对冲的执行过程感兴趣，我们将建立在离散时间设置中存在的套利机会，当交易频率趋于无穷大时消失，让我们推导出定价方程。此外，我们的方法检索了几个已经在期权定价文献中建立的结果，使现货动态受到市场影响。我们还研究了基于公平定价假设的套期保值元订单的放松，并建立了直接影响与永久影响之间的关系，这与最近关于该主题的实证研究一致。
{"title":"How Option Hedging Shapes Market Impact","authors":"Emilio Said","doi":"10.2139/ssrn.3470915","DOIUrl":"https://doi.org/10.2139/ssrn.3470915","url":null,"abstract":"We present a perturbation theory of the market impact based on an extension of the framework proposed by [Loeper, 2018] – originally based on [Liu and Yong, 2005] – in which we consider only local linear market impact. We study the execution process of hedging derivatives and show how these hedging metaorders can explain some stylized facts observed in the empirical market impact literature. As we are interested in the execution process of hedging we will establish that the arbitrage opportunities that exist in the discrete time setting vanish when the trading frequency goes to infinity letting us to derive a pricing equation. Furthermore our approach retrieves several results already established in the option pricing literature such that the spot dynamics modified by the market impact. We also study the relaxation of our hedging metaorders based on the fair pricing hypothesis and establish a relation between the immediate impact and the permanent impact which is in agreement with recent empirical studies on the subject.","PeriodicalId":293888,"journal":{"name":"Econometric Modeling: Derivatives eJournal","volume":"58 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131519393","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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