Pub Date : 2017-07-04DOI: 10.1080/1350486X.2017.1384701
J. Kirkby
ABSTRACT We present an efficient method for robustly pricing discretely monitored barrier and occupation time derivatives under exponential Lévy models. This includes ordinary barrier options, as well as (resetting) Parisian options, delayed barrier options (also known as cumulative Parisian or Parasian options), fader options and step options (soft-barriers), all with single and double barriers, which have yet to be priced with more general Lévy processes, including KoBoL (CGMY), Merton’s jump diffusion and NIG. The method’s efficiency is derived in part from the use of frame-projected transition densities, which transform the problem into the Fourier domain and accelerate the convergence of intermediate expectations. Moreover, these expectations are approximated by Toeplitz matrix-vector multiplications, resulting in a fast implementation. We devise an augmentation approach that contributes to the method’s robustness, adding protection against mis-specifying a proper truncation support of the transition density. Theoretical convergence is verified by a series of numerical experiments which demonstrate the method’s efficiency and accuracy.
{"title":"Robust barrier option pricing by frame projection under exponential Lévy dynamics","authors":"J. Kirkby","doi":"10.1080/1350486X.2017.1384701","DOIUrl":"https://doi.org/10.1080/1350486X.2017.1384701","url":null,"abstract":"ABSTRACT We present an efficient method for robustly pricing discretely monitored barrier and occupation time derivatives under exponential Lévy models. This includes ordinary barrier options, as well as (resetting) Parisian options, delayed barrier options (also known as cumulative Parisian or Parasian options), fader options and step options (soft-barriers), all with single and double barriers, which have yet to be priced with more general Lévy processes, including KoBoL (CGMY), Merton’s jump diffusion and NIG. The method’s efficiency is derived in part from the use of frame-projected transition densities, which transform the problem into the Fourier domain and accelerate the convergence of intermediate expectations. Moreover, these expectations are approximated by Toeplitz matrix-vector multiplications, resulting in a fast implementation. We devise an augmentation approach that contributes to the method’s robustness, adding protection against mis-specifying a proper truncation support of the transition density. Theoretical convergence is verified by a series of numerical experiments which demonstrate the method’s efficiency and accuracy.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"3 1","pages":"337 - 386"},"PeriodicalIF":0.0,"publicationDate":"2017-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85591368","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}
Pub Date : 2017-07-04DOI: 10.1080/1350486X.2017.1378583
S. Singor, A. Boer, J. Alberts, C. Oosterlee
ABSTRACT We propose a modelling framework for risk-neutral stochastic processes nested in a real-world stochastic process. The framework is important for insurers that deal with the valuation of embedded options and in particular at future points in time. We make use of the class of State Space Hidden Markov models for modelling the joint behaviour of the parameters of a risk-neutral model and the dynamics of option market instruments. This modelling concept enables us to perform non-linear estimation, forecasting and robust calibration. The proposed method is applied to the Heston model for which we find highly satisfactory results. We use the estimated Heston model to compute the required capital of an insurance company under Solvency II and we find large differences compared to a basic calibration method.
{"title":"On the modelling of nested risk-neutral stochastic processes with applications in insurance","authors":"S. Singor, A. Boer, J. Alberts, C. Oosterlee","doi":"10.1080/1350486X.2017.1378583","DOIUrl":"https://doi.org/10.1080/1350486X.2017.1378583","url":null,"abstract":"ABSTRACT We propose a modelling framework for risk-neutral stochastic processes nested in a real-world stochastic process. The framework is important for insurers that deal with the valuation of embedded options and in particular at future points in time. We make use of the class of State Space Hidden Markov models for modelling the joint behaviour of the parameters of a risk-neutral model and the dynamics of option market instruments. This modelling concept enables us to perform non-linear estimation, forecasting and robust calibration. The proposed method is applied to the Heston model for which we find highly satisfactory results. We use the estimated Heston model to compute the required capital of an insurance company under Solvency II and we find large differences compared to a basic calibration method.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"94 1","pages":"302 - 336"},"PeriodicalIF":0.0,"publicationDate":"2017-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89898341","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}
Pub Date : 2017-06-01DOI: 10.1080/1350486X.2018.1438904
F. Benth, Anca Pircalabu
ABSTRACT The recent introduction of wind power futures written on the German wind power production index has brought with it new interesting challenges in terms of modelling and pricing. Some particularities of this product are the strong seasonal component embedded in the underlying, the fact that the wind index is bounded from both above and below and also that the futures are settled against a synthetically generated spot index. Here, we consider the non-Gaussian Ornstein–Uhlenbeck type processes proposed by Barndorff-Nielsen and Shephard in the context of modelling the wind power production index. We discuss the properties of the model and estimation of the model parameters. Further, the model allows for an analytical formula for pricing wind power futures. We provide an empirical study, where the model is calibrated to 37 years of German wind power production index that is synthetically generated assuming a constant level of installed capacity. Also, based on 1 year of observed prices for wind power futures with different delivery periods, we study the market price of risk. Generally, we find a negative risk premium whose magnitude decreases as the length of the delivery period increases. To further demonstrate the benefits of our proposed model, we address the pricing of European options written on wind power futures, which can be achieved through Fourier techniques.
{"title":"A non-Gaussian Ornstein–Uhlenbeck model for pricing wind power futures","authors":"F. Benth, Anca Pircalabu","doi":"10.1080/1350486X.2018.1438904","DOIUrl":"https://doi.org/10.1080/1350486X.2018.1438904","url":null,"abstract":"ABSTRACT The recent introduction of wind power futures written on the German wind power production index has brought with it new interesting challenges in terms of modelling and pricing. Some particularities of this product are the strong seasonal component embedded in the underlying, the fact that the wind index is bounded from both above and below and also that the futures are settled against a synthetically generated spot index. Here, we consider the non-Gaussian Ornstein–Uhlenbeck type processes proposed by Barndorff-Nielsen and Shephard in the context of modelling the wind power production index. We discuss the properties of the model and estimation of the model parameters. Further, the model allows for an analytical formula for pricing wind power futures. We provide an empirical study, where the model is calibrated to 37 years of German wind power production index that is synthetically generated assuming a constant level of installed capacity. Also, based on 1 year of observed prices for wind power futures with different delivery periods, we study the market price of risk. Generally, we find a negative risk premium whose magnitude decreases as the length of the delivery period increases. To further demonstrate the benefits of our proposed model, we address the pricing of European options written on wind power futures, which can be achieved through Fourier techniques.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"11 1","pages":"36 - 65"},"PeriodicalIF":0.0,"publicationDate":"2017-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88668348","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}
Pub Date : 2017-05-29DOI: 10.1080/1350486X.2018.1507750
Tim Leung, Brian Ward
ABSTRACT We develop a methodology for index tracking and risk exposure control using financial derivatives. Under a continuous-time diffusion framework for price evolution, we present a pathwise approach to construct dynamic portfolios of derivatives in order to gain exposure to an index and/or market factors that may be not directly tradable. Among our results, we establish a general tracking condition that relates the portfolio drift to the desired exposure coefficients under any given model. We also derive a slippage process that reveals how the portfolio return deviates from the targeted return. In our multi-factor setting, the portfolio’s realized slippage depends not only on the realized variance of the index but also the realized covariance among the index and factors. We implement our trading strategies under a number of models, and compare the tracking strategies and performances when using different derivatives, such as futures and options.
{"title":"Dynamic Index Tracking and Risk Exposure Control Using Derivatives","authors":"Tim Leung, Brian Ward","doi":"10.1080/1350486X.2018.1507750","DOIUrl":"https://doi.org/10.1080/1350486X.2018.1507750","url":null,"abstract":"ABSTRACT We develop a methodology for index tracking and risk exposure control using financial derivatives. Under a continuous-time diffusion framework for price evolution, we present a pathwise approach to construct dynamic portfolios of derivatives in order to gain exposure to an index and/or market factors that may be not directly tradable. Among our results, we establish a general tracking condition that relates the portfolio drift to the desired exposure coefficients under any given model. We also derive a slippage process that reveals how the portfolio return deviates from the targeted return. In our multi-factor setting, the portfolio’s realized slippage depends not only on the realized variance of the index but also the realized covariance among the index and factors. We implement our trading strategies under a number of models, and compare the tracking strategies and performances when using different derivatives, such as futures and options.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"92 1","pages":"180 - 212"},"PeriodicalIF":0.0,"publicationDate":"2017-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74863114","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}
Pub Date : 2017-05-04DOI: 10.1080/1350486X.2017.1374870
S. Jaimungal, D. Kinzebulatov, D. Rubisov
ABSTRACT An accelerated share repurchase allows a firm to repurchase a significant portion of its shares immediately, while shifting the burden of reducing the impact and uncertainty in the trade to an intermediary. The intermediary must then purchase the shares from the market over several days, weeks or as much as several months. Some contracts allow the intermediary to specify when the repurchase ends, at which point the firm and the intermediary exchange the difference between the arrival price and the TWAP over the trading period plus a spread. Hence, the intermediary effectively has an American option embedded within an optimal execution problem. As a result, the firm receives a discounted spread relative to the no early exercise case. Here, we address the intermediary’s optimal execution and exit strategy taking into account the impact that trading has on the market. We demonstrate that it is optimal to exercise when the TWAP exceeds where is the midprice of the asset and is a deterministic function of time and inventory. Moreover, we develop a dimensional reduction of the stochastic control and stopping problem and implement an efficient numerical scheme to compute the optimal trading and exit strategies. We also provide bounds on the optimal strategy and characterize the convexity and monotonicity of the optimal strategies in addition to exploring its behaviour numerically and through simulation studies.
{"title":"Optimal accelerated share repurchases","authors":"S. Jaimungal, D. Kinzebulatov, D. Rubisov","doi":"10.1080/1350486X.2017.1374870","DOIUrl":"https://doi.org/10.1080/1350486X.2017.1374870","url":null,"abstract":"ABSTRACT An accelerated share repurchase allows a firm to repurchase a significant portion of its shares immediately, while shifting the burden of reducing the impact and uncertainty in the trade to an intermediary. The intermediary must then purchase the shares from the market over several days, weeks or as much as several months. Some contracts allow the intermediary to specify when the repurchase ends, at which point the firm and the intermediary exchange the difference between the arrival price and the TWAP over the trading period plus a spread. Hence, the intermediary effectively has an American option embedded within an optimal execution problem. As a result, the firm receives a discounted spread relative to the no early exercise case. Here, we address the intermediary’s optimal execution and exit strategy taking into account the impact that trading has on the market. We demonstrate that it is optimal to exercise when the TWAP exceeds where is the midprice of the asset and is a deterministic function of time and inventory. Moreover, we develop a dimensional reduction of the stochastic control and stopping problem and implement an efficient numerical scheme to compute the optimal trading and exit strategies. We also provide bounds on the optimal strategy and characterize the convexity and monotonicity of the optimal strategies in addition to exploring its behaviour numerically and through simulation studies.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"56 1","pages":"216 - 245"},"PeriodicalIF":0.0,"publicationDate":"2017-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86458902","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}
Pub Date : 2017-03-04DOI: 10.1080/1350486X.2017.1353917
D. Madan
ABSTRACT Learning the pre-limited liability value process of equity claims and its relationship to the stock price is an answer to the financial jeopardy question when observed option prices are the answer being given by the market. Constant dollar equity holder values, prior to the imposition of limited liability, are the signed conditional expectations of the integral of discounted net residual equity claims through all time. The stock is modelled as a limited liability claim imputing positive dividend flows to shareholders in certain circumstances coupled with a call option written on the integral of all discounted net residual equity claims. The underlying signed value has a known characteristic function when revenues and expenses are modelled as independent gamma processes. The stock price is a positive function of this signed underlying value, given by the solution of a partial integro differential equation. Options on the stock are then options on this function of the signed underlying value and are solved for using its density obtained by Fourier inversion of the characteristic function. The calibration of model parameters, the imputed dividend function and the terminal call strike is conducted on option prices at a single maturity for four underliers, and In all these cases it is observed that risk neutrally up moves arrive more frequently and are generally smaller while down moves are less frequent and are larger. The terminal option strikes were in the money for and , and out of the money for and
{"title":"Financial jeopardy","authors":"D. Madan","doi":"10.1080/1350486X.2017.1353917","DOIUrl":"https://doi.org/10.1080/1350486X.2017.1353917","url":null,"abstract":"ABSTRACT Learning the pre-limited liability value process of equity claims and its relationship to the stock price is an answer to the financial jeopardy question when observed option prices are the answer being given by the market. Constant dollar equity holder values, prior to the imposition of limited liability, are the signed conditional expectations of the integral of discounted net residual equity claims through all time. The stock is modelled as a limited liability claim imputing positive dividend flows to shareholders in certain circumstances coupled with a call option written on the integral of all discounted net residual equity claims. The underlying signed value has a known characteristic function when revenues and expenses are modelled as independent gamma processes. The stock price is a positive function of this signed underlying value, given by the solution of a partial integro differential equation. Options on the stock are then options on this function of the signed underlying value and are solved for using its density obtained by Fourier inversion of the characteristic function. The calibration of model parameters, the imputed dividend function and the terminal call strike is conducted on option prices at a single maturity for four underliers, and In all these cases it is observed that risk neutrally up moves arrive more frequently and are generally smaller while down moves are less frequent and are larger. The terminal option strikes were in the money for and , and out of the money for and","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"36 1","pages":"155 - 173"},"PeriodicalIF":0.0,"publicationDate":"2017-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77564466","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}
Pub Date : 2017-03-04DOI: 10.1080/1350486X.2017.1342551
Gary Quek, C. Atkinson
ABSTRACT In this article, we study a multi-period portfolio selection model in which a generic class of probability distributions is assumed for the returns of the risky asset. An investor with a power utility function rebalances a portfolio comprising a risk-free and risky asset at the beginning of each time period in order to maximize expected utility of terminal wealth. Trading the risky asset incurs a cost that is proportional to the value of the transaction. At each time period, the optimal investment strategy involves buying or selling the risky asset to reach the boundaries of a certain no-transaction region. In the limit of small transaction costs, dynamic programming and perturbation analysis are applied to obtain explicit approximations to the optimal boundaries and optimal value function of the portfolio at each stage of a multi-period investment process of any length.
{"title":"Portfolio selection in discrete time with transaction costs and power utility function: a perturbation analysis","authors":"Gary Quek, C. Atkinson","doi":"10.1080/1350486X.2017.1342551","DOIUrl":"https://doi.org/10.1080/1350486X.2017.1342551","url":null,"abstract":"ABSTRACT In this article, we study a multi-period portfolio selection model in which a generic class of probability distributions is assumed for the returns of the risky asset. An investor with a power utility function rebalances a portfolio comprising a risk-free and risky asset at the beginning of each time period in order to maximize expected utility of terminal wealth. Trading the risky asset incurs a cost that is proportional to the value of the transaction. At each time period, the optimal investment strategy involves buying or selling the risky asset to reach the boundaries of a certain no-transaction region. In the limit of small transaction costs, dynamic programming and perturbation analysis are applied to obtain explicit approximations to the optimal boundaries and optimal value function of the portfolio at each stage of a multi-period investment process of any length.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"22 1","pages":"111 - 77"},"PeriodicalIF":0.0,"publicationDate":"2017-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76140096","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}
Pub Date : 2017-02-04DOI: 10.1080/1350486X.2018.1506257
Ryan Francis Donnelly, Luhui Gan
ABSTRACT We show how the position of a limit order (LO) in the queue influences the decision of whether to cancel the order or let it rest. Using ultra-high-frequency data from the Nasdaq exchange, we perform empirical analysis on various LO book events and propose novel ways for modelling some of these events, including cancellation of LOs in various positions and size of market orders. Based on our empirical findings, we develop a queuing model that captures stylized facts on the data. This model includes a distinct feature which allows for a potentially random effect due to the agent’s impulse control. We apply the queuing model in an algorithmic trading setting by considering an agent maximizing her expected utility through placing and cancelling of LOs. The agent’s optimal strategy is presented after calibrating the model to real data. A simulation study shows that for the same level of standard deviation of terminal wealth, the optimal strategy has a 2.5% higher mean compared to a strategy which ignores the effect of position, or an 8.8% lower standard deviation for the same level of mean. This extra gain stems from posting an LO during adverse conditions and obtaining a good queue position before conditions become favourable.
{"title":"Optimal Decisions in a Time Priority Queue","authors":"Ryan Francis Donnelly, Luhui Gan","doi":"10.1080/1350486X.2018.1506257","DOIUrl":"https://doi.org/10.1080/1350486X.2018.1506257","url":null,"abstract":"ABSTRACT We show how the position of a limit order (LO) in the queue influences the decision of whether to cancel the order or let it rest. Using ultra-high-frequency data from the Nasdaq exchange, we perform empirical analysis on various LO book events and propose novel ways for modelling some of these events, including cancellation of LOs in various positions and size of market orders. Based on our empirical findings, we develop a queuing model that captures stylized facts on the data. This model includes a distinct feature which allows for a potentially random effect due to the agent’s impulse control. We apply the queuing model in an algorithmic trading setting by considering an agent maximizing her expected utility through placing and cancelling of LOs. The agent’s optimal strategy is presented after calibrating the model to real data. A simulation study shows that for the same level of standard deviation of terminal wealth, the optimal strategy has a 2.5% higher mean compared to a strategy which ignores the effect of position, or an 8.8% lower standard deviation for the same level of mean. This extra gain stems from posting an LO during adverse conditions and obtaining a good queue position before conditions become favourable.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"5 1","pages":"107 - 147"},"PeriodicalIF":0.0,"publicationDate":"2017-02-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"91354405","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}
Pub Date : 2017-01-30DOI: 10.1080/1350486X.2018.1507751
A. Al-Aradi, S. Jaimungal
ABSTRACT Portfolio management problems are often divided into two types: active and passive, where the objective is to outperform and track a preselected benchmark, respectively. Here, we formulate and solve a dynamic asset allocation problem that combines these two objectives in a unified framework. We look to maximize the expected growth rate differential between the wealth of the investor’s portfolio and that of a performance benchmark while penalizing risk-weighted deviations from a given tracking portfolio. Using stochastic control techniques, we provide explicit closed-form expressions for the optimal allocation and we show how the optimal strategy can be related to the growth optimal portfolio. The admissible benchmarks encompass the class of functionally generated portfolios (FGPs), which include the market portfolio, as the only requirement is that they depend only on the prevailing asset values. Finally, some numerical experiments are presented to illustrate the risk–reward profile of the optimal allocation.
{"title":"Outperformance and Tracking: Dynamic Asset Allocation for Active and Passive Portfolio Management","authors":"A. Al-Aradi, S. Jaimungal","doi":"10.1080/1350486X.2018.1507751","DOIUrl":"https://doi.org/10.1080/1350486X.2018.1507751","url":null,"abstract":"ABSTRACT Portfolio management problems are often divided into two types: active and passive, where the objective is to outperform and track a preselected benchmark, respectively. Here, we formulate and solve a dynamic asset allocation problem that combines these two objectives in a unified framework. We look to maximize the expected growth rate differential between the wealth of the investor’s portfolio and that of a performance benchmark while penalizing risk-weighted deviations from a given tracking portfolio. Using stochastic control techniques, we provide explicit closed-form expressions for the optimal allocation and we show how the optimal strategy can be related to the growth optimal portfolio. The admissible benchmarks encompass the class of functionally generated portfolios (FGPs), which include the market portfolio, as the only requirement is that they depend only on the prevailing asset values. Finally, some numerical experiments are presented to illustrate the risk–reward profile of the optimal allocation.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"38 1","pages":"268 - 294"},"PeriodicalIF":0.0,"publicationDate":"2017-01-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72850024","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}
Pub Date : 2017-01-10DOI: 10.1080/1350486X.2017.1409641
A. Itkin
ABSTRACT It is known that the implied volatility skew of Forex (FX) options demonstrates a stochastic behaviour which is called stochastic skew. In this paper, we create stochastic skew by assuming the spot/instantaneous variance (InV) correlation to be stochastic. Accordingly, we consider a class of Stochastic Local Volatility (SLV) models with stochastic correlation where all drivers – the spot, InV and their correlation – are modelled by processes. We assume all diffusion components to be fully correlated, as well as all jump components. A new fully implicit splitting finite-difference scheme is proposed for solving forward PIDE which is used when calibrating the model to market prices of the FX options with different strikes and maturities. The scheme is unconditionally stable, of second order of approximation in time and space, and achieves a linear complexity in each spatial direction. The results of simulation obtained by using this model demonstrate the capacity of the presented approach in modelling stochastic skew.
{"title":"Modelling stochastic skew of FX options using SLV models with stochastic spot/vol correlation and correlated jumps","authors":"A. Itkin","doi":"10.1080/1350486X.2017.1409641","DOIUrl":"https://doi.org/10.1080/1350486X.2017.1409641","url":null,"abstract":"ABSTRACT It is known that the implied volatility skew of Forex (FX) options demonstrates a stochastic behaviour which is called stochastic skew. In this paper, we create stochastic skew by assuming the spot/instantaneous variance (InV) correlation to be stochastic. Accordingly, we consider a class of Stochastic Local Volatility (SLV) models with stochastic correlation where all drivers – the spot, InV and their correlation – are modelled by processes. We assume all diffusion components to be fully correlated, as well as all jump components. A new fully implicit splitting finite-difference scheme is proposed for solving forward PIDE which is used when calibrating the model to market prices of the FX options with different strikes and maturities. The scheme is unconditionally stable, of second order of approximation in time and space, and achieves a linear complexity in each spatial direction. The results of simulation obtained by using this model demonstrate the capacity of the presented approach in modelling stochastic skew.","PeriodicalId":35818,"journal":{"name":"Applied Mathematical Finance","volume":"28 2 1","pages":"485 - 519"},"PeriodicalIF":0.0,"publicationDate":"2017-01-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83457164","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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