We extend the scheme developed in B. During, A. We extend the scheme developed in B. During, A. Pitkin, "High-order compact finite difference scheme for option pricing in stochastic volatility jump models", 2019, to the so-called stochastic volatility with contemporaneous jumps (SVCJ) model, derived by Duffie, Pan and Singleton. The performance of the scheme is assessed through a number of numerical experiments, using comparisons against a standard second-order central difference scheme. We observe that the new high-order compact scheme achieves fourth order convergence and discuss the effects on efficiency and computation time.
我们将B. During, A. Pitkin，“随机波动率跳跃模型中期权定价的高阶紧凑有限差分格式”，2019中开发的方案推广到Duffie, Pan和Singleton推导的所谓的随机波动率与同期跳跃(SVCJ)模型。通过与标准二阶中心差分格式的比较，对该格式的性能进行了评估。我们观察到新的高阶紧凑格式达到了四阶收敛，并讨论了对效率和计算时间的影响。
{"title":"High-Order Compact Finite Difference Scheme for Option Pricing in Stochastic Volatility With Contemporaneous Jump Models","authors":"Bertram Düring, A. Pitkin","doi":"10.2139/ssrn.3275199","DOIUrl":"https://doi.org/10.2139/ssrn.3275199","url":null,"abstract":"We extend the scheme developed in B. During, A. We extend the scheme developed in B. During, A. Pitkin, \"High-order compact finite difference scheme for option pricing in stochastic volatility jump models\", 2019, to the so-called stochastic volatility with contemporaneous jumps (SVCJ) model, derived by Duffie, Pan and Singleton. The performance of the scheme is assessed through a number of numerical experiments, using comparisons against a standard second-order central difference scheme. We observe that the new high-order compact scheme achieves fourth order convergence and discuss the effects on efficiency and computation time.","PeriodicalId":129812,"journal":{"name":"Financial Engineering eJournal","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124816937","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 first part of this presentation gives an introduction to stochastic automatic differentiation and its application. The second part of the presentation introduces a simple "static hedge" approximation for an SIMM based MVA and compares it with an exact solution (where the exact solution was obtained by the stochastic automatic differentiation).
{"title":"Stochastic Automatic Differentiation - AAD for Monte-Carlo Simulations - MVA Approximation Methods (Presentation Slides from the 14th Quant Finance Conference, Nice)","authors":"Christian P. Fries","doi":"10.2139/ssrn.3263526","DOIUrl":"https://doi.org/10.2139/ssrn.3263526","url":null,"abstract":"This first part of this presentation gives an introduction to stochastic automatic differentiation and its application. The second part of the presentation introduces a simple \"static hedge\" approximation for an SIMM based MVA and compares it with an exact solution (where the exact solution was obtained by the stochastic automatic differentiation).","PeriodicalId":129812,"journal":{"name":"Financial Engineering eJournal","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121430588","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}
Sharpe ratio is widely used in asset management to compare and benchmark funds and asset managers. It computes the ratio of the excess return over the strategy standard deviation. However, the elements to compute the Sharpe ratio, namely, the expected returns and the volatilities are unknown numbers and need to be estimated statistically. This means that the Sharpe ratio used by funds is subject to be error prone because of statistical estimation error. Lo (2002), Mertens (2002) derive explicit expressions for the statistical distribution of the Sharpe ratio using standard asymptotic theory under several sets of assumptions (independent normally distributed - and identically distributed returns). In this paper, we provide the exact distribution of the Sharpe ratio for independent normally distributed return. In this case, the Sharpe ratio statistic is up to a rescaling factor a non centered Student distribution whose characteristics have been widely studied by statisticians. The asymptotic behavior of our distribution provide the result of Lo (2002). We also illustrate the fact that the empirical Sharpe ratio is asymptotically optimal in the sense that it achieves the Cramer Rao bound. We then study the empirical SR under AR(1) assumptions and investigate the effect of compounding period on the Sharpe (computing the annual Sharpe with monthly data for instance). We finally provide general formula in this case of heteroscedasticity and autocorrelation.
{"title":"Connecting Sharpe Ratio and Student T-Statistic, and Beyond","authors":"E. Benhamou","doi":"10.2139/ssrn.3223152","DOIUrl":"https://doi.org/10.2139/ssrn.3223152","url":null,"abstract":"Sharpe ratio is widely used in asset management to compare and benchmark funds and asset managers. It computes the ratio of the excess return over the strategy standard deviation. However, the elements to compute the Sharpe ratio, namely, the expected returns and the volatilities are unknown numbers and need to be estimated statistically. This means that the Sharpe ratio used by funds is subject to be error prone because of statistical estimation error. Lo (2002), Mertens (2002) derive explicit expressions for the statistical distribution of the Sharpe ratio using standard asymptotic theory under several sets of assumptions (independent normally distributed - and identically distributed returns). In this paper, we provide the exact distribution of the Sharpe ratio for independent normally distributed return. In this case, the Sharpe ratio statistic is up to a rescaling factor a non centered Student distribution whose characteristics have been widely studied by statisticians. The asymptotic behavior of our distribution provide the result of Lo (2002). We also illustrate the fact that the empirical Sharpe ratio is asymptotically optimal in the sense that it achieves the Cramer Rao bound. We then study the empirical SR under AR(1) assumptions and investigate the effect of compounding period on the Sharpe (computing the annual Sharpe with monthly data for instance). We finally provide general formula in this case of heteroscedasticity and autocorrelation.","PeriodicalId":129812,"journal":{"name":"Financial Engineering eJournal","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-07-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128915038","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 consider the Black and Scholes (1973) and Heston (1993) models and we generalize them to stochastic interest rates and maturity-dependent volatilities. In the Black-Scholes case we solve the extended model and provide a concrete form for the term structure of volatilities. In the Heston case we prove that, under some conditions, the generalized model is equivalent to a hybrid model and we find semi-closed-form solutions in the Hull and White (1990) and Cox et al. (1985) cases. We address the problem of the consistency of the Black-Scholes model with the volatility surface and we show that, under general conditions, the Black-Scholes formula cannot be generalized to account for the volatility smile.
我们考虑布莱克和斯科尔斯(1973)和赫斯顿(1993)模型，并将其推广到随机利率和到期依赖的波动率。在Black-Scholes案例中，我们求解了扩展模型，并提供了波动率期限结构的具体形式。在Heston案例中，我们证明了在某些条件下，广义模型等价于混合模型，并在Hull and White(1990)和Cox et al.(1985)案例中找到了半封闭形式的解。我们解决了Black-Scholes模型与波动面的一致性问题，并证明在一般条件下，Black-Scholes公式不能推广到波动面。
{"title":"Revisiting the Classical Models: Black-Scholes and Heston With Stochastic Interest Rates and Term Structure of Volatilities","authors":"Alberto Bueno-Guerrero","doi":"10.2139/ssrn.3192823","DOIUrl":"https://doi.org/10.2139/ssrn.3192823","url":null,"abstract":"We consider the Black and Scholes (1973) and Heston (1993) models and we generalize them to stochastic interest rates and maturity-dependent volatilities. In the Black-Scholes case we solve the extended model and provide a concrete form for the term structure of volatilities. In the Heston case we prove that, under some conditions, the generalized model is equivalent to a hybrid model and we find semi-closed-form solutions in the Hull and White (1990) and Cox et al. (1985) cases. \u0000 \u0000We address the problem of the consistency of the Black-Scholes model with the volatility surface and we show that, under general conditions, the Black-Scholes formula cannot be generalized to account for the volatility smile.","PeriodicalId":129812,"journal":{"name":"Financial Engineering eJournal","volume":"26 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123730599","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 new approach to obtaining market--directional information, based on a non-stationary solution to the dynamic equation "future price tends to the value that maximizes the number of shares traded per unit time" [1] is presented. In our previous work[2], we established that it is the share execution flow ($I=dV/dt$) and not the share trading volume ($V$) that is the driving force of the market, and that asset prices are much more sensitive to the execution flow $I$ (the dynamic impact) than to the traded volume $V$ (the regular impact). In this paper, an important advancement is achieved: we define the "scalp-price" ${cal P}$ as the sum of only those price moves that are relevant to market dynamics; the criterion of relevance is a high $I$. Thus, only "follow the market" (and not "little bounce") events are included in ${cal P}$. Changes in the scalp-price defined this way indicate a market trend change - not a bear market rally or a bull market sell-off; the approach can be further extended to non-local price change. The software calculating the scalp--price given market observations triples (time, execution price, shares traded) is available from the authors.
{"title":"Market Dynamics: On Directional Information Derived from (Time, Execution Price, Shares Traded) Transaction Sequences","authors":"V. Malyshkin","doi":"10.2139/ssrn.3361478","DOIUrl":"https://doi.org/10.2139/ssrn.3361478","url":null,"abstract":"A new approach to obtaining market--directional information, based on a non-stationary solution to the dynamic equation \"future price tends to the value that maximizes the number of shares traded per unit time\" [1] is presented. In our previous work[2], we established that it is the share execution flow ($I=dV/dt$) and not the share trading volume ($V$) that is the driving force of the market, and that asset prices are much more sensitive to the execution flow $I$ (the dynamic impact) than to the traded volume $V$ (the regular impact). In this paper, an important advancement is achieved: we define the \"scalp-price\" ${cal P}$ as the sum of only those price moves that are relevant to market dynamics; the criterion of relevance is a high $I$. Thus, only \"follow the market\" (and not \"little bounce\") events are included in ${cal P}$. Changes in the scalp-price defined this way indicate a market trend change - not a bear market rally or a bull market sell-off; the approach can be further extended to non-local price change. The software calculating the scalp--price given market observations triples (time, execution price, shares traded) is available from the authors.","PeriodicalId":129812,"journal":{"name":"Financial Engineering eJournal","volume":"22 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126896480","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}
In this paper, time-inconsistent model was established under stochastic differential game framework. The investment portfolio includes multi-risky assets, whose returns are assumed to be correlated in a time-varying manner and change cyclically. The claim losses of insurance companies and investment are also assumed to be correlated with each other. The Solution to extended HJBI equations results in the portion of retention and an optimal portfolio with equally weighted allocations of risky assets. An optimal control bound is proposed for monitoring and predicting the optimal wealth level. The proposed model is expected to be effective in making decision for investment and reinsurance strategies, controlling and predicting optimal wealth under uncertain environment. Especially, it can be applied easily in the situation of very high dimensional investment portfolio.
{"title":"Time Inconsistent Stochastic Differential Game: Theory and an Example in Insurance","authors":"Hong Mao, Zhongkai Wen","doi":"10.2139/ssrn.3041861","DOIUrl":"https://doi.org/10.2139/ssrn.3041861","url":null,"abstract":"In this paper, time-inconsistent model was established under stochastic differential game framework. The investment portfolio includes multi-risky assets, whose returns are assumed to be correlated in a time-varying manner and change cyclically. The claim losses of insurance companies and investment are also assumed to be correlated with each other. The Solution to extended HJBI equations results in the portion of retention and an optimal portfolio with equally weighted allocations of risky assets. An optimal control bound is proposed for monitoring and predicting the optimal wealth level. The proposed model is expected to be effective in making decision for investment and reinsurance strategies, controlling and predicting optimal wealth under uncertain environment. Especially, it can be applied easily in the situation of very high dimensional investment portfolio.","PeriodicalId":129812,"journal":{"name":"Financial Engineering eJournal","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122388425","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 written statement accompanied Professor Omarova’s oral testimony given on June 15, 2017, in a hearing held by the U.S. Senate Banking Committee on the necessity of relaxing certain aspects of post-crisis financial regulation applicable to midsized, regional and large banks, as a means of fostering America’s economic growth. In her written statement, Professor Omarova systematically lays out the reasons why massive deregulation urged by the banking industry will hinder, rather than foster, sustainable long-term growth in the real (i.e., non-financial) sector of the American economy.
{"title":"Testimony before the U.S. Senate Banking Committee: ‘Fostering Economic Growth: Midsized, Regional and Large Institution Perspective’","authors":"S. Omarova","doi":"10.31228/osf.io/8mxsg","DOIUrl":"https://doi.org/10.31228/osf.io/8mxsg","url":null,"abstract":"This written statement accompanied Professor Omarova’s oral testimony given on June 15, 2017, in a hearing held by the U.S. Senate Banking Committee on the necessity of relaxing certain aspects of post-crisis financial regulation applicable to midsized, regional and large banks, as a means of fostering America’s economic growth. In her written statement, Professor Omarova systematically lays out the reasons why massive deregulation urged by the banking industry will hinder, rather than foster, sustainable long-term growth in the real (i.e., non-financial) sector of the American economy.","PeriodicalId":129812,"journal":{"name":"Financial Engineering eJournal","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"130146424","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 research involved developing an optimal stock investment decision strategy which offers minimum risk to the potential investor for counters listed on the Zimbabwe Stock Exchange. Three counters were compared namely Econet Wireless, Delta and Old Mutual. The first phase of the project involved understanding how stock returns behaved by examining the transition matrices of the counters. The second phase of the project involved determining how much time it took for a counter to be in a positive return state. The final phase involved determining the long run probability of an investor being able to get any positive returns from a specific counter. From the three counters, Old Mutual had the highest long run probability 96.1% of being in the bull market, followed by Delta which had a 49.6% chance and lastly Econet which had a 47.1% probability of being in the bull market in the long run. Hence, Old Mutual was found to be the most preferred counter since it had a greater chance of providing the potential investor with positive returns in the future.
{"title":"A Comparative Analysis of Stock Return Behavior Using a Markov Switching Model (Case Study: Zimbabwe Stock Exchange)","authors":"Tawanda Dakwa, I. Moyo","doi":"10.2139/ssrn.2984852","DOIUrl":"https://doi.org/10.2139/ssrn.2984852","url":null,"abstract":"This research involved developing an optimal stock investment decision strategy which offers minimum risk to the potential investor for counters listed on the Zimbabwe Stock Exchange. Three counters were compared namely Econet Wireless, Delta and Old Mutual. The first phase of the project involved understanding how stock returns behaved by examining the transition matrices of the counters. The second phase of the project involved determining how much time it took for a counter to be in a positive return state. The final phase involved determining the long run probability of an investor being able to get any positive returns from a specific counter. From the three counters, Old Mutual had the highest long run probability 96.1% of being in the bull market, followed by Delta which had a 49.6% chance and lastly Econet which had a 47.1% probability of being in the bull market in the long run. Hence, Old Mutual was found to be the most preferred counter since it had a greater chance of providing the potential investor with positive returns in the future.","PeriodicalId":129812,"journal":{"name":"Financial Engineering eJournal","volume":"48 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2017-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121455624","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The paper presents a generalized algorithm of determining the payback period for either a conventional or a non-conventional cash flow of an investment project. A non-conventional cash flow may have more than one payback periods, if an investor makes additional investments during the operating phase of the project. I give numeric examples and explain in detail the calculation of the payback period with Excel formulas, as well as with Excel user-defined function written in VBA. In conclusion, I give some thoughts on why the payback period can be a useful performance measure in capital budgeting in spite of the criticisms against it in academic literature on the ground that it is not compatible with the NPV criterion.
{"title":"Generalized Method of Determining the Payback Period for both Conventional and Non-conventional Cash Flows: Ready-to-Use Excel Formulas and UDF","authors":"S. V. Cheremushkin","doi":"10.2139/ssrn.1982827","DOIUrl":"https://doi.org/10.2139/ssrn.1982827","url":null,"abstract":"The paper presents a generalized algorithm of determining the payback period for either a conventional or a non-conventional cash flow of an investment project. A non-conventional cash flow may have more than one payback periods, if an investor makes additional investments during the operating phase of the project. I give numeric examples and explain in detail the calculation of the payback period with Excel formulas, as well as with Excel user-defined function written in VBA. In conclusion, I give some thoughts on why the payback period can be a useful performance measure in capital budgeting in spite of the criticisms against it in academic literature on the ground that it is not compatible with the NPV criterion.","PeriodicalId":129812,"journal":{"name":"Financial Engineering eJournal","volume":"108 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-12-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121374953","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 considers pricing European options in a large class of one-dimensional Markovian jump processes known as subordinate diffusions, which are obtained by time changing a diffusion process with an independent Levy or additive random clock. These jump processes are non-Levy in general, and they can be viewed as a natural generalization of many popular Levy processes used in finance. Subordinate diffusions offer richer jump behavior than Levy processes and they have found a variety of applications in financial modeling. The pricing problem for these processes presents unique challenges, as existing numerical PIDE schemes fail to be efficient and the applicability of transform methods to many subordinate diffusions is unclear. We develop a novel method based on a finite difference approximation of spatial derivatives and matrix eigendecomposition, and it can deal with diffusions that exhibit various types of boundary behavior. Since financial payoffs are typically not smooth, we apply a smoothing tech...
{"title":"Option Pricing in Some Non-Levy Jump Models","authors":"Lingfei Li, Gongqiu Zhang","doi":"10.1137/15M1048926","DOIUrl":"https://doi.org/10.1137/15M1048926","url":null,"abstract":"This paper considers pricing European options in a large class of one-dimensional Markovian jump processes known as subordinate diffusions, which are obtained by time changing a diffusion process with an independent Levy or additive random clock. These jump processes are non-Levy in general, and they can be viewed as a natural generalization of many popular Levy processes used in finance. Subordinate diffusions offer richer jump behavior than Levy processes and they have found a variety of applications in financial modeling. The pricing problem for these processes presents unique challenges, as existing numerical PIDE schemes fail to be efficient and the applicability of transform methods to many subordinate diffusions is unclear. We develop a novel method based on a finite difference approximation of spatial derivatives and matrix eigendecomposition, and it can deal with diffusions that exhibit various types of boundary behavior. Since financial payoffs are typically not smooth, we apply a smoothing tech...","PeriodicalId":129812,"journal":{"name":"Financial Engineering eJournal","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2016-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"127679346","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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