The Black-Scholes option pricing model assumes, among other things, that stock prices followa lognormal distribution. Other writers have extended this assumption to currency options. However, the work in currency options has mainly assumed floating exchange rates. Options on currencies such as the Chinese yuan and Peruvian sol, which historically have followed a steadily increasing trend over considerable periods of time, would be priced incorrectly given this assumption. To address this lack in the literature, a closed-form version of a model with a trend-stationary, stochastic volatility exchange rate is derived, using both a linear and quadratic trend. The results show that the model more accurately prices currency options such as the ones on the yuan and creates lower percentage hedging errors from the computed prices compared with the Garman-Kohlhagen and Heston models. The model will help institutions to more accurately hedge their foreign exchange risk in a world in which the yuan's, and other similar currencies', value is increasingly important.
{"title":"Pricing Options on Trend-Stationary Currencies: Applications to the Chinese Yuan","authors":"Michael W. Mebane","doi":"10.21314/J0R.2016.329","DOIUrl":"https://doi.org/10.21314/J0R.2016.329","url":null,"abstract":"The Black-Scholes option pricing model assumes, among other things, that stock prices followa lognormal distribution. Other writers have extended this assumption to currency options. However, the work in currency options has mainly assumed floating exchange rates. Options on currencies such as the Chinese yuan and Peruvian sol, which historically have followed a steadily increasing trend over considerable periods of time, would be priced incorrectly given this assumption. To address this lack in the literature, a closed-form version of a model with a trend-stationary, stochastic volatility exchange rate is derived, using both a linear and quadratic trend. The results show that the model more accurately prices currency options such as the ones on the yuan and creates lower percentage hedging errors from the computed prices compared with the Garman-Kohlhagen and Heston models. The model will help institutions to more accurately hedge their foreign exchange risk in a world in which the yuan's, and other similar currencies', value is increasingly important.","PeriodicalId":46697,"journal":{"name":"Journal of Risk","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2016-04-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67693346","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}
Recent literature has demonstrated the existence of an unbounded risk premium if one combines the most important models for pricing and hedging derivatives with coherent risk measures. There may exist combinations of derivatives (good deals) whose pair (return risk) converges to the pair (+∞, −∞). This paper goes beyond existence properties and looks for optimal explicit constructions and empirical tests. It will be shown that the optimal good deal above may be a simple portfolio of options. This theoretical finding will enable us to implement empirical experiments involving three international stock index futures (Standard & Poor's 500, Eurostoxx 50 and DAX 30) and three commodity futures (gold, Brent and the Dow Jones-UBS Commodity Index). According to the empirical results, the good deal always outperforms the underlying index/commodity. The good deal is built in full compliance with the standard derivative pricing theory. Properties of classical pricing models totally inspire the good deal construction. This is a very interesting difference in our paper with respect to previous literature attempting to outperform a benchmark.
{"title":"Outperforming Benchmarks with Their Derivatives: Theory and Empirical Evidence","authors":"A. Balbás, B. Balbás, Raquel Balbás","doi":"10.21314/J0R.2016.328","DOIUrl":"https://doi.org/10.21314/J0R.2016.328","url":null,"abstract":"Recent literature has demonstrated the existence of an unbounded risk premium if one combines the most important models for pricing and hedging derivatives with coherent risk measures. There may exist combinations of derivatives (good deals) whose pair (return risk) converges to the pair (+∞, −∞). This paper goes beyond existence properties and looks for optimal explicit constructions and empirical tests. It will be shown that the optimal good deal above may be a simple portfolio of options. This theoretical finding will enable us to implement empirical experiments involving three international stock index futures (Standard & Poor's 500, Eurostoxx 50 and DAX 30) and three commodity futures (gold, Brent and the Dow Jones-UBS Commodity Index). According to the empirical results, the good deal always outperforms the underlying index/commodity. The good deal is built in full compliance with the standard derivative pricing theory. Properties of classical pricing models totally inspire the good deal construction. This is a very interesting difference in our paper with respect to previous literature attempting to outperform a benchmark.","PeriodicalId":46697,"journal":{"name":"Journal of Risk","volume":"36 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2016-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67693342","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}
In this paper, we consider the portfolio optimization problem, with conditional value-at-risk as the objective. We summarize commonly used methods of solution and note that the linear programming (LP) approximation is the most generally applicable and easiest to use (the LP uses a Monte Carlo sample from the true asset returns distribution). The suboptimality of the obtained approximate portfolios is then analyzed using a numerical example, with up to 101 assets and Student t - distributed returns, ranging from light to heavy tails. The results can be used as an estimate of the portfolio suboptimality for more general asset returns distributions, based on the number of assets, tail heaviness and fineness of the discretization. Computation times using the different techniques available in the literature are also analyzed.
{"title":"Suboptimality in Portfolio Conditional Value-at-Risk Optimization","authors":"E. Jakobsons","doi":"10.21314/J0R.2016.330","DOIUrl":"https://doi.org/10.21314/J0R.2016.330","url":null,"abstract":"In this paper, we consider the portfolio optimization problem, with conditional value-at-risk as the objective. We summarize commonly used methods of solution and note that the linear programming (LP) approximation is the most generally applicable and easiest to use (the LP uses a Monte Carlo sample from the true asset returns distribution). The suboptimality of the obtained approximate portfolios is then analyzed using a numerical example, with up to 101 assets and Student t - distributed returns, ranging from light to heavy tails. The results can be used as an estimate of the portfolio suboptimality for more general asset returns distributions, based on the number of assets, tail heaviness and fineness of the discretization. Computation times using the different techniques available in the literature are also analyzed.","PeriodicalId":46697,"journal":{"name":"Journal of Risk","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2016-04-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67693390","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 develops a new financial product that allows the profit-and-loss sharing (PLS) principle to be enforced recursively in practice. A new equity-like financial product is proposed through a three-tier partnership to which a new contracting party (the risk moderator) is added to absorb the underlying risk of premature default and adjust the annual revenue to a predetermined annual cost. The financing mechanism pioneers a new type of option, dubbed the PLS option, to manage the underlying risk of revenue sharing. A dynamic capital structure methodology is developed for the valuation of the PLS option that allows for an annual adjustment of the project's revenue and recalculates the entitlements pertaining to contracting parties. Monte Carlo simulation is conducted to evaluate the project when the construction cost is deterministic and the streams of expected cashflows are stochastic. The simulation results show that the dynamic adjustment of the capital structure simultaneously endorses a recursive profit-and-loss sharing and a dynamic risk-hedging approach. Sheer evidence shows the immunization against premature default through the involvement of the risk moderator to absorb any potential loss, which is indicative of an incentive factor for the project's survival and business continuity.
{"title":"Recursive Profit-and-Loss Sharing","authors":"Walid Mansour, M. Abdelhamid, A. Heshmati","doi":"10.21314/JOR.2015.309","DOIUrl":"https://doi.org/10.21314/JOR.2015.309","url":null,"abstract":"This paper develops a new financial product that allows the profit-and-loss sharing (PLS) principle to be enforced recursively in practice. A new equity-like financial product is proposed through a three-tier partnership to which a new contracting party (the risk moderator) is added to absorb the underlying risk of premature default and adjust the annual revenue to a predetermined annual cost. The financing mechanism pioneers a new type of option, dubbed the PLS option, to manage the underlying risk of revenue sharing. A dynamic capital structure methodology is developed for the valuation of the PLS option that allows for an annual adjustment of the project's revenue and recalculates the entitlements pertaining to contracting parties. Monte Carlo simulation is conducted to evaluate the project when the construction cost is deterministic and the streams of expected cashflows are stochastic. The simulation results show that the dynamic adjustment of the capital structure simultaneously endorses a recursive profit-and-loss sharing and a dynamic risk-hedging approach. Sheer evidence shows the immunization against premature default through the involvement of the risk moderator to absorb any potential loss, which is indicative of an incentive factor for the project's survival and business continuity.","PeriodicalId":46697,"journal":{"name":"Journal of Risk","volume":"17 1","pages":"21-50"},"PeriodicalIF":0.7,"publicationDate":"2015-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67718188","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}
Islamic banks have access to only short-dated funding sources resulting in asset liability mismatches when financing assets with longer maturities. Maturity mismatches give rise to a risk that an unexpected increase in the cost of refinancing liabilities as they mature will not be offset by corresponding asset returns. Exposure to refinancing risk is exacerbated by paying returns to providers of off balance sheet funds which do not covary with the returns of corresponding assets as they would from a stricter application of sharia principles underlying these funding structures. The active hedging of refinancing risk by Islamic banks is also challenged due to a lack of suitable hedging instruments as well as differing sharia opinions concerning their permissibility. As an alternative to risk transference through hedging, this paper develops a framework to quantify a reserve to instead absorb refinancing risk which is distinguished from reserves already in use by Islamic banks, namely the investment risk and profit equalization reserves.
{"title":"The Management of Refinancing Risk in Islamic Banks","authors":"K. Baldwin","doi":"10.21314/jor.2015.308","DOIUrl":"https://doi.org/10.21314/jor.2015.308","url":null,"abstract":"Islamic banks have access to only short-dated funding sources resulting in asset liability mismatches when financing assets with longer maturities. Maturity mismatches give rise to a risk that an unexpected increase in the cost of refinancing liabilities as they mature will not be offset by corresponding asset returns. Exposure to refinancing risk is exacerbated by paying returns to providers of off balance sheet funds which do not covary with the returns of corresponding assets as they would from a stricter application of sharia principles underlying these funding structures. The active hedging of refinancing risk by Islamic banks is also challenged due to a lack of suitable hedging instruments as well as differing sharia opinions concerning their permissibility. As an alternative to risk transference through hedging, this paper develops a framework to quantify a reserve to instead absorb refinancing risk which is distinguished from reserves already in use by Islamic banks, namely the investment risk and profit equalization reserves.","PeriodicalId":46697,"journal":{"name":"Journal of Risk","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2015-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67718116","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 studies the predictive power of the time-varying shape of the credit default swap (CDS) term structure to explain changes in future implied and excess implied volatility (implied volatility of the company over and above the market volatility) and therefore provide a leading sign of potential financial distress in a company. The shape of the CDS curve is captured by fitting the Nelson-Siegel model to the term structure and creating a new binary indicator (shape indicator) to distinguish between "good" and "bad" CDS curves. Applying the methodology to twenty US-traded companies from the financial and non-financial sectors, we find that the credit market is generally a leading indicator for movements in the volatility market during the subprime crisis. After confirming the strong link between CDSs and implied volatility markets (the average R2 per sector between 36% and 63% is obtained using the Nelson-Siegel parameter and shape indicator), a partial F -test is executed to see if additional information is contained within the CDS markets over and above the volatility markets. For the period studied, this is the case for the majority of names, and it is particularly significant for Lehman Brothers.
{"title":"The Signalling Properties of the Shape of the Credit Default Swap Term Structure","authors":"J. Castellanos, N. Constantinou, W. Ng","doi":"10.21314/JOR.2015.298","DOIUrl":"https://doi.org/10.21314/JOR.2015.298","url":null,"abstract":"This paper studies the predictive power of the time-varying shape of the credit default swap (CDS) term structure to explain changes in future implied and excess implied volatility (implied volatility of the company over and above the market volatility) and therefore provide a leading sign of potential financial distress in a company. The shape of the CDS curve is captured by fitting the Nelson-Siegel model to the term structure and creating a new binary indicator (shape indicator) to distinguish between \"good\" and \"bad\" CDS curves. Applying the methodology to twenty US-traded companies from the financial and non-financial sectors, we find that the credit market is generally a leading indicator for movements in the volatility market during the subprime crisis. After confirming the strong link between CDSs and implied volatility markets (the average R2 per sector between 36% and 63% is obtained using the Nelson-Siegel parameter and shape indicator), a partial F -test is executed to see if additional information is contained within the CDS markets over and above the volatility markets. For the period studied, this is the case for the majority of names, and it is particularly significant for Lehman Brothers.","PeriodicalId":46697,"journal":{"name":"Journal of Risk","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2015-04-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67718108","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}
Alexios Theiakos, Jurgen M.C Tas, Han van der Lem, D. Kandhai
Stochastic scenario analysis of mortgage hedging strategies using single-CPU core machines is often too time consuming. In order to achieve a large practical speedup, we present two methods implemented on a many-core system consisting of graphical processing units (GPUs). The first method is based on Monte Carlo simulations, which are widely used in risk management. The second method relies on a parallel implicit finite difference (FD) discretization of a forward Kolmogorov equation. To estimate the speedup that can be achieved in practice, we compared the performance of both methods with an existing serial trinomial tree implementation on a single CPU core currently in use in our department. For both methods, a large speedup of roughly two orders of magnitude is achieved for realistic workloads. We show that the FD method is approximately four times faster than the Monte Carlo method when implemented on GPUs. On the other hand we argue that the Monte Carlo method is more adaptable to accommodate generic models, while the FD method is typically suitable to low dimensional models, such as single-factor interest rate models. To our knowledge, the application of GPUs for mortgage hedge calculations is new, as is the implementation of the FD method on GPUs.
{"title":"Ultra-Fast Scenario Analysis of Mortgage Prepayment Risk","authors":"Alexios Theiakos, Jurgen M.C Tas, Han van der Lem, D. Kandhai","doi":"10.21314/JOR.2015.323","DOIUrl":"https://doi.org/10.21314/JOR.2015.323","url":null,"abstract":"Stochastic scenario analysis of mortgage hedging strategies using single-CPU core machines is often too time consuming. In order to achieve a large practical speedup, we present two methods implemented on a many-core system consisting of graphical processing units (GPUs). The first method is based on Monte Carlo simulations, which are widely used in risk management. The second method relies on a parallel implicit finite difference (FD) discretization of a forward Kolmogorov equation. To estimate the speedup that can be achieved in practice, we compared the performance of both methods with an existing serial trinomial tree implementation on a single CPU core currently in use in our department. For both methods, a large speedup of roughly two orders of magnitude is achieved for realistic workloads. We show that the FD method is approximately four times faster than the Monte Carlo method when implemented on GPUs. On the other hand we argue that the Monte Carlo method is more adaptable to accommodate generic models, while the FD method is typically suitable to low dimensional models, such as single-factor interest rate models. To our knowledge, the application of GPUs for mortgage hedge calculations is new, as is the implementation of the FD method on GPUs.","PeriodicalId":46697,"journal":{"name":"Journal of Risk","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2015-02-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67718247","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}
Using classical Taylor series techniques, we develop a unified approach to pricing and implied volatility for European-style options in a general local–stochastic volatility setting. Our price approximations require only a normal cumulative distribution function and our implied volatility approximations are fully explicit (ie, they require no special functions, no infinite series and no numerical integration). As such, approximate prices can be computed as efficiently as Black– Scholes prices, and approximate implied volatilities can be computed nearly instantaneously.
{"title":"A Taylor Series Approach to Pricing and Implied Volatility for Local–Stochastic Volatility Models","authors":"Matthew J. Lorig, S. Pagliarani, A. Pascucci","doi":"10.21314/JOR.2014.297","DOIUrl":"https://doi.org/10.21314/JOR.2014.297","url":null,"abstract":"Using classical Taylor series techniques, we develop a unified approach to pricing and implied volatility for European-style options in a general local–stochastic volatility setting. Our price approximations require only a normal cumulative distribution function and our implied volatility approximations are fully explicit (ie, they require no special functions, no infinite series and no numerical integration). As such, approximate prices can be computed as efficiently as Black– Scholes prices, and approximate implied volatilities can be computed nearly instantaneously.","PeriodicalId":46697,"journal":{"name":"Journal of Risk","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2014-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67718094","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 consider the class of risk measures associated with optimized certainty equivalents. This class includes several popular examples, such as conditional value-at-risk (CVaR) and monotone mean–variance. Numerical schemes are developed for the computation of these risk measures using Fourier transform methods. This leads, in particular, to a very competitive method for the calculation of CVaR, which is comparable in computational time to the calculation of VaR. We also develop methods for the efficient computation of risk contributions.
{"title":"A Fourier Approach to the Computation of Conditional Value-at-Risk and Optimized Certainty Equivalents","authors":"Samuel Drapeau, M. Kupper, A. Papapantoleon","doi":"10.21314/JOR.2014.281","DOIUrl":"https://doi.org/10.21314/JOR.2014.281","url":null,"abstract":"We consider the class of risk measures associated with optimized certainty equivalents. This class includes several popular examples, such as conditional value-at-risk (CVaR) and monotone mean–variance. Numerical schemes are developed for the computation of these risk measures using Fourier transform methods. This leads, in particular, to a very competitive method for the calculation of CVaR, which is comparable in computational time to the calculation of VaR. We also develop methods for the efficient computation of risk contributions.","PeriodicalId":46697,"journal":{"name":"Journal of Risk","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2014-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67717925","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 evaluate the relative performance of logistic credit risk models that were selected by means of standard stepwise model selection methods and “average” models obtained by Bayesian model averaging (BMA). Our bootstrap analysis shows that BMA should be considered as an alternative to the stepwise model selection procedures that are currently often used in practice.
{"title":"Selection Versus Averaging of Logistic Credit Risk Models","authors":"E. Hayden, A. Stomper, Arne Westerkamp","doi":"10.21314/jor.2014.285","DOIUrl":"https://doi.org/10.21314/jor.2014.285","url":null,"abstract":"We evaluate the relative performance of logistic credit risk models that were selected by means of standard stepwise model selection methods and “average” models obtained by Bayesian model averaging (BMA). Our bootstrap analysis shows that BMA should be considered as an alternative to the stepwise model selection procedures that are currently often used in practice.","PeriodicalId":46697,"journal":{"name":"Journal of Risk","volume":"1 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2014-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67717929","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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