Implied volatility surface has been studied extensively for various option markets including equities, foreign currencies, and commodities. Previous studies report that option implied volatility varies across moneyness, maturity, and time, yet, once the level is controlled for, the shape of the volatility surface relative to the volatility implied for at-the-money (ATM) option is stable even over the period of the 1987 stock market crash. This study examines the dynamics of the implied volatility surface for euro-US dollar options, using a recently developed panel cointegration test that allows multiple structural breaks while accounting for cross-sectional dependence. In the model, the option implied volatility Is specified as a quadratic function of ATM volatility, spot and forward rates. The three factors together account for 98 percent of variations in the option implied volatilities across five moneyness, five maturities and over eight years of daily observations from Jan. 2006 to Dec. 2014. The estimated volatility surface however is not stable over time. Rather, its relationship with the three underlying factors exhibits substantial changes around the periods of the Global Financial Crisis and subsequent Euro-zone crisis. This finding is in a stark contrast to previous studies which report the shape of volatility surface is stable over time.
{"title":"A Panel Cointegration Analysis of the Dynamics of FX Option Implied Volatility Surface","authors":"Hiroaki Suenaga","doi":"10.2139/ssrn.2558902","DOIUrl":"https://doi.org/10.2139/ssrn.2558902","url":null,"abstract":"Implied volatility surface has been studied extensively for various option markets including equities, foreign currencies, and commodities. Previous studies report that option implied volatility varies across moneyness, maturity, and time, yet, once the level is controlled for, the shape of the volatility surface relative to the volatility implied for at-the-money (ATM) option is stable even over the period of the 1987 stock market crash. This study examines the dynamics of the implied volatility surface for euro-US dollar options, using a recently developed panel cointegration test that allows multiple structural breaks while accounting for cross-sectional dependence. In the model, the option implied volatility Is specified as a quadratic function of ATM volatility, spot and forward rates. The three factors together account for 98 percent of variations in the option implied volatilities across five moneyness, five maturities and over eight years of daily observations from Jan. 2006 to Dec. 2014. The estimated volatility surface however is not stable over time. Rather, its relationship with the three underlying factors exhibits substantial changes around the periods of the Global Financial Crisis and subsequent Euro-zone crisis. This finding is in a stark contrast to previous studies which report the shape of volatility surface is stable over time.","PeriodicalId":40006,"journal":{"name":"Journal of Derivatives","volume":"22 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2015-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87577868","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper analyzes credit risk transfer in banking. Specifically, we model loan sales and loan insurance (e.g. credit default swaps) as the two instruments of risk transfer. Recent empirical evidence suggests that the adverse selection problem is as relevant in loan insurance as it is in loan sales. Contrary to previous literature, this paper allows for informational asymmetries in both markets. We show how credit risk transfer can achieve optimal investment and minimize the social costs associated with excess risk taking by a bank. Furthermore, we find that no separation of loan types can occur in equilibrium. Our results show that a well capitalized bank will tend to use loan insurance regardless of loan quality in the presence of moral hazard and relationship banking costs of loan sales. Finally, we show that a poorly capitalized bank may be forced into the loan sales market, even in the presence of possibly significant relationship and moral hazard costs that can depress the selling price.
{"title":"Credit Risk Transfer: To Sell or to Insure","authors":"J. Thompson","doi":"10.2139/ssrn.1278087","DOIUrl":"https://doi.org/10.2139/ssrn.1278087","url":null,"abstract":"This paper analyzes credit risk transfer in banking. Specifically, we model loan sales and loan insurance (e.g. credit default swaps) as the two instruments of risk transfer. Recent empirical evidence suggests that the adverse selection problem is as relevant in loan insurance as it is in loan sales. Contrary to previous literature, this paper allows for informational asymmetries in both markets. We show how credit risk transfer can achieve optimal investment and minimize the social costs associated with excess risk taking by a bank. Furthermore, we find that no separation of loan types can occur in equilibrium. Our results show that a well capitalized bank will tend to use loan insurance regardless of loan quality in the presence of moral hazard and relationship banking costs of loan sales. Finally, we show that a poorly capitalized bank may be forced into the loan sales market, even in the presence of possibly significant relationship and moral hazard costs that can depress the selling price.","PeriodicalId":40006,"journal":{"name":"Journal of Derivatives","volume":"42 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2014-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74131214","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Dealing with default risk on sovereign debt became an urgent matter with the Financial Crisis of 2008. Depending on both exogenous and endogenous factors, the risk of eventual default may become significant over the lifetime of a loan. Creditors are then faced with a dilemma, since they cannot force the borrowing country into bankruptcy court. Should they ease repayment by reducing the principal of the debt (“forgiveness”), or should they offer an extra award if the borrower repays in full? The latter award could come in the form of lower borrowing costs in the future. In this article, Sun and Chen propose to model the possibility of loan modification for risky sovereign debt as a kind of down-and-in put option: There is a boundary value (the “default threshold”) for the country’s perceived ability to pay, measured as the ratio of debt principal to GDP, at which the terms of the debt are revised, either to reduce the principal or to offer a repayment award. Both approaches have been tried in practice. What the authors show is that forgiveness is better for the creditors than a repayment award. The sensitivity of bond value to default boundary, forgiveness amount, and repayment award within the model are explored through simulation. The authors then conduct an empirical analysis on the bonds of 17 of the G20 countries to estimate the values of these three model parameters, and show that their model performs well in practice.
{"title":"Forgive, or Award, Your Debtor? - A Barrier Option Approach","authors":"David S. Sun, D. Chow","doi":"10.2139/SSRN.2142458","DOIUrl":"https://doi.org/10.2139/SSRN.2142458","url":null,"abstract":"Dealing with default risk on sovereign debt became an urgent matter with the Financial Crisis of 2008. Depending on both exogenous and endogenous factors, the risk of eventual default may become significant over the lifetime of a loan. Creditors are then faced with a dilemma, since they cannot force the borrowing country into bankruptcy court. Should they ease repayment by reducing the principal of the debt (“forgiveness”), or should they offer an extra award if the borrower repays in full? The latter award could come in the form of lower borrowing costs in the future. In this article, Sun and Chen propose to model the possibility of loan modification for risky sovereign debt as a kind of down-and-in put option: There is a boundary value (the “default threshold”) for the country’s perceived ability to pay, measured as the ratio of debt principal to GDP, at which the terms of the debt are revised, either to reduce the principal or to offer a repayment award. Both approaches have been tried in practice. What the authors show is that forgiveness is better for the creditors than a repayment award. The sensitivity of bond value to default boundary, forgiveness amount, and repayment award within the model are explored through simulation. The authors then conduct an empirical analysis on the bonds of 17 of the G20 countries to estimate the values of these three model parameters, and show that their model performs well in practice.","PeriodicalId":40006,"journal":{"name":"Journal of Derivatives","volume":"24 1","pages":"67-95"},"PeriodicalIF":0.7,"publicationDate":"2013-11-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82290956","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the valuation of credit-contingent asset or options by modelling the correlation between asset price and credit default. We provide three ways of modelling such correlation: (1) asset value follows a diffusion process with a one-time jump (such as currency devaluation) at the time of credit default; (2) Default intensity and asset price are driven by correlated Brownian motions in addition to the jump; (3) Default time and future asset price are correlated through a copula. When both asset price and credit default are independent of interest rates, such contract can be valued on a two-dimensional lattice (or finite-difference grid) in the second approach. We show that for a large class of one-factor default rate models, the computation can be reduced to one-dimension, a property often reserved for the affine class of models. We also obtain analytical solutions if default hazard rate, asset price return, and the copula are all Gaussian. Our experience shows that valuation is much more sensitive to the first and third type of correlations. We apply the model to the valuation of extinguishable FX swaps that terminate upon a credit event and quanto credit default swaps where premium and protection legs are paid in different currencies.
{"title":"Valuation of Credit Contingent Options with Applications to Quanto CDS","authors":"Anlong Li","doi":"10.2139/ssrn.1153400","DOIUrl":"https://doi.org/10.2139/ssrn.1153400","url":null,"abstract":"We study the valuation of credit-contingent asset or options by modelling the correlation between asset price and credit default. We provide three ways of modelling such correlation: (1) asset value follows a diffusion process with a one-time jump (such as currency devaluation) at the time of credit default; (2) Default intensity and asset price are driven by correlated Brownian motions in addition to the jump; (3) Default time and future asset price are correlated through a copula. When both asset price and credit default are independent of interest rates, such contract can be valued on a two-dimensional lattice (or finite-difference grid) in the second approach. We show that for a large class of one-factor default rate models, the computation can be reduced to one-dimension, a property often reserved for the affine class of models. We also obtain analytical solutions if default hazard rate, asset price return, and the copula are all Gaussian. Our experience shows that valuation is much more sensitive to the first and third type of correlations. We apply the model to the valuation of extinguishable FX swaps that terminate upon a credit event and quanto credit default swaps where premium and protection legs are paid in different currencies.","PeriodicalId":40006,"journal":{"name":"Journal of Derivatives","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2013-11-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89950779","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
One of the many counterintuitive things students in a first course on options learn is that premature exercise of an American call option on a nondividend paying stock is a mistake, and that for a dividend-paying stock, early exercise is never rational except just before the stock goes ex-dividend. Efforts to incorporate counterparty credit risk in the calculation have suggested assuming no change in exercise policy and simply discount the projected future cash flows at a higher risky interest rate. Others have argued that counterparty default should never happen with American options because the holder will exercise just before the counterparty defaults and effectively step in front of the other creditors to be paid in full. Klein and Yang show that these ideas are incorrect. Early exercise gives up the option’s time value, so that even when imminent counterparty default is perfectly predictable, there is still a cost to exercise for credit reasons. Moreover, properly taking counterparty risk into account leads to optimal exercise behavior different from that in the nonvulnerable case, so simply discounting the same cash flows at a different rate undervalues the vulnerable option. The difference is particularly important when counterparty risk is wrong-way risk, such that the counterparty’s credit weakens under the same conditions that the option is in the money.
{"title":"Counterparty Credit Risk and American Options","authors":"Peter G. Klein, Jun Yang","doi":"10.2139/SSRN.1600911","DOIUrl":"https://doi.org/10.2139/SSRN.1600911","url":null,"abstract":"One of the many counterintuitive things students in a first course on options learn is that premature exercise of an American call option on a nondividend paying stock is a mistake, and that for a dividend-paying stock, early exercise is never rational except just before the stock goes ex-dividend. Efforts to incorporate counterparty credit risk in the calculation have suggested assuming no change in exercise policy and simply discount the projected future cash flows at a higher risky interest rate. Others have argued that counterparty default should never happen with American options because the holder will exercise just before the counterparty defaults and effectively step in front of the other creditors to be paid in full. Klein and Yang show that these ideas are incorrect. Early exercise gives up the option’s time value, so that even when imminent counterparty default is perfectly predictable, there is still a cost to exercise for credit reasons. Moreover, properly taking counterparty risk into account leads to optimal exercise behavior different from that in the nonvulnerable case, so simply discounting the same cash flows at a different rate undervalues the vulnerable option. The difference is particularly important when counterparty risk is wrong-way risk, such that the counterparty’s credit weakens under the same conditions that the option is in the money.","PeriodicalId":40006,"journal":{"name":"Journal of Derivatives","volume":"18 1","pages":"7-21"},"PeriodicalIF":0.7,"publicationDate":"2013-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78769169","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Equity and credit are options on firm assets. As options, actual returns to equity and credit are functions of two distinct sources of value; 1) expected firm asset returns and 2) the difference between option price implied and subsequent realized firm asset volatility. Equity and credit are subject to option value arbitrage pricing and therefore no compensating returns accrue to equity and credit for stochastic firm asset volatility. Portfolio construction that eliminates the capital structure option exposure will increase realized risk adjusted returns relative to investments with capital structure option exposure.
{"title":"The Realized Sharpe Ratio and Uncertainty of Firm Earnings Distribution (Starting with Modigliani Miller)","authors":"William A. Barr","doi":"10.2139/ssrn.1079662","DOIUrl":"https://doi.org/10.2139/ssrn.1079662","url":null,"abstract":"Equity and credit are options on firm assets. As options, actual returns to equity and credit are functions of two distinct sources of value; 1) expected firm asset returns and 2) the difference between option price implied and subsequent realized firm asset volatility. Equity and credit are subject to option value arbitrage pricing and therefore no compensating returns accrue to equity and credit for stochastic firm asset volatility. Portfolio construction that eliminates the capital structure option exposure will increase realized risk adjusted returns relative to investments with capital structure option exposure.","PeriodicalId":40006,"journal":{"name":"Journal of Derivatives","volume":"141 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2013-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86741746","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The concept of equivalent martingale measures is of key importance for pricing of nancial derivative contracts. The goal of the paper is to apply in nitesimals in the non-standard analysis set-up to provide an elementary construction of the equivalent martingale measure built on hyper nite binomial trees with in nitesimal time steps.
{"title":"Construction of Equivalent Martingale Measures with Infinitesimals","authors":"J. Witzany","doi":"10.2139/ssrn.1274962","DOIUrl":"https://doi.org/10.2139/ssrn.1274962","url":null,"abstract":"The concept of equivalent martingale measures is of key importance for pricing of nancial derivative contracts. The goal of the paper is to apply in nitesimals in the non-standard analysis set-up to provide an elementary construction of the equivalent martingale measure built on hyper nite binomial trees with in nitesimal time steps.","PeriodicalId":40006,"journal":{"name":"Journal of Derivatives","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2013-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86765023","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Binomial model and similar lattice methods are workhorses of practical derivatives valuation. But returns processes more realistic than lognormal diffusions with constant parameters easily create difficulties for them. One of the most important extensions of the Black-Scholes paradigm is to allow stochastic volatility, but even nonstochastic timevarying volatility destroys the important property that the tree recombines, which limits the growth in the number of nodes as time advances. Stochastic volatility introduces a second random variable, which then requires adding another dimension to the tree, under the constraint that the return and volatility changes must maintain the same degree of correlation as in the data. The Heston model features correlation in return and volatility shocks, but building it into a lattice is tricky. In this article, Ruckdeschel, Sayer, and Szimayer develop a lattice method that begins with a binomial tree for the volatility and a trinomial tree for stock price, and then connects them in such a way that the empirical degree of correlation between return and volatility is maintained. Efficiency relative to existing methods is increased, and in some cases it is possible to improve performance further by matching higher moments as well.
{"title":"Pricing American Options in the Heston Model: A Close Look on Incorporating Correlation","authors":"P. Ruckdeschel, Tilman Sayer, Alexander Szimayer","doi":"10.2139/SSRN.1797962","DOIUrl":"https://doi.org/10.2139/SSRN.1797962","url":null,"abstract":"The Binomial model and similar lattice methods are workhorses of practical derivatives valuation. But returns processes more realistic than lognormal diffusions with constant parameters easily create difficulties for them. One of the most important extensions of the Black-Scholes paradigm is to allow stochastic volatility, but even nonstochastic timevarying volatility destroys the important property that the tree recombines, which limits the growth in the number of nodes as time advances. Stochastic volatility introduces a second random variable, which then requires adding another dimension to the tree, under the constraint that the return and volatility changes must maintain the same degree of correlation as in the data. The Heston model features correlation in return and volatility shocks, but building it into a lattice is tricky. In this article, Ruckdeschel, Sayer, and Szimayer develop a lattice method that begins with a binomial tree for the volatility and a trinomial tree for stock price, and then connects them in such a way that the empirical degree of correlation between return and volatility is maintained. Efficiency relative to existing methods is increased, and in some cases it is possible to improve performance further by matching higher moments as well.","PeriodicalId":40006,"journal":{"name":"Journal of Derivatives","volume":"27 1","pages":"9-29"},"PeriodicalIF":0.7,"publicationDate":"2013-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81318998","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
M. Szymanowska, Frans de Roon, T. Nijman, Rob van den Goorbergh
We identify two types of risk premia in commodity futures returns: spot premia related to the risk in the underlying commodity, and term premia related to changes in the basis. Sorting on forecasting variables such as the futures basis, return momentum, volatility, inflation, hedging pressure, and liquidity results in sizable spot premia between 5% and 14% per annum and term premia between 1% and 3% per annum. We show that a single factor, the high-minus-low portfolio from basis sorts, explains the cross-section of spot premia. Two additional basis factors are needed to explain the term premia.
{"title":"An Anatomy of Commodity Futures Risk Premia","authors":"M. Szymanowska, Frans de Roon, T. Nijman, Rob van den Goorbergh","doi":"10.2139/ssrn.1343809","DOIUrl":"https://doi.org/10.2139/ssrn.1343809","url":null,"abstract":"We identify two types of risk premia in commodity futures returns: spot premia related to the risk in the underlying commodity, and term premia related to changes in the basis. Sorting on forecasting variables such as the futures basis, return momentum, volatility, inflation, hedging pressure, and liquidity results in sizable spot premia between 5% and 14% per annum and term premia between 1% and 3% per annum. We show that a single factor, the high-minus-low portfolio from basis sorts, explains the cross-section of spot premia. Two additional basis factors are needed to explain the term premia.","PeriodicalId":40006,"journal":{"name":"Journal of Derivatives","volume":"18 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2013-01-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82510474","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
These days, both exchange-traded and most OTC derivatives transactions are collateralized: The upfront payment, if any, and regular mark-to-market variation margin are held in escrow, possibly earning interest, until the contract is exercised. In theoretical modeling, interest paid on the collateral is typically ignored, under the assumption that the mark-tomarket cash flows can go either way and, in any case, the interest amounts are small. Kan and Pedersen demonstrate that this intuition is not necessarily true. For credit default swaps (CDS), the impact of margin interest can be significant; in fact, the margin interest rate should be used for discounting in calculating the fair value of CDS.
{"title":"The Impact of Margin Interest on the Valuation of Credit Default Swaps","authors":"Y. Kan, Claus M. Pedersen","doi":"10.2139/SSRN.1777196","DOIUrl":"https://doi.org/10.2139/SSRN.1777196","url":null,"abstract":"These days, both exchange-traded and most OTC derivatives transactions are collateralized: The upfront payment, if any, and regular mark-to-market variation margin are held in escrow, possibly earning interest, until the contract is exercised. In theoretical modeling, interest paid on the collateral is typically ignored, under the assumption that the mark-tomarket cash flows can go either way and, in any case, the interest amounts are small. Kan and Pedersen demonstrate that this intuition is not necessarily true. For credit default swaps (CDS), the impact of margin interest can be significant; in fact, the margin interest rate should be used for discounting in calculating the fair value of CDS.","PeriodicalId":40006,"journal":{"name":"Journal of Derivatives","volume":"16 1","pages":"60-79"},"PeriodicalIF":0.7,"publicationDate":"2012-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81717394","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"经济学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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