Stock price volatility is considered the main matter of concern within the investment grounds.However,the diffusivity of these prices should as well be considered.As such,proper modelling should be done for investors ...Stock price volatility is considered the main matter of concern within the investment grounds.However,the diffusivity of these prices should as well be considered.As such,proper modelling should be done for investors to stay healthy-informed.This paper suggest to model stock price diffusions using the heat equation from physics.We hypothetically state that,our model captures and model the diffusion bubbles of stock prices with a better precision of reality.We compared our model with the standard geometric Brownian motion model which is the wide commonly used stochastic differential equation in asset valuation.Interestingly,the models proved to agree as evidenced by a bijective relation between the volatility coefficients of the Brownian motion model and the diffusion coefficients of our heat diffusion model as well as the corresponding drift components.Consequently,a short proof for the martingale of our model is done which happen to hold.展开更多
This paper presents two new versions of uncertain market models for valuing vulnerable European call option.The dynamics of underlying asset,counterparty asset,and corporate liability are formulated on the basis of un...This paper presents two new versions of uncertain market models for valuing vulnerable European call option.The dynamics of underlying asset,counterparty asset,and corporate liability are formulated on the basis of uncertain differential equations and uncertain fractional differential equations of Caputo type,respectively,and the solution to an uncertain fractional differential equation of Caputo type is presented by employing the Mittag-Leffler function andα-path.Then,the pricing formulas of vulnerable European call option based on the proposed models are investigated as well as some algorithms.Some numerical experiments are performed to verify the effectiveness of the results.展开更多
In this paper,a rough Heston model with variable volatility of volatility(vol-of-vol)is derived by modifying the generalized nonlinear Hawkes process and extending the scaling techniques.Then the nonlinear fractional ...In this paper,a rough Heston model with variable volatility of volatility(vol-of-vol)is derived by modifying the generalized nonlinear Hawkes process and extending the scaling techniques.Then the nonlinear fractional Ric-cati equation for the characteristic function of the asset log-price is derived.The existence,uniqueness and regularity of the solution to the nonlinear fractional Riccati equation are proved and the equation is solved by the Adams methods.Finally the Fourier-cosine methods are combined with the Adams methods to price the options.展开更多
A barrier option valuation model with stochastic barrier which was regarded as the main feature of the model was developed under the Hull-White interest rate model.The purpose of this study was to deal with the stocha...A barrier option valuation model with stochastic barrier which was regarded as the main feature of the model was developed under the Hull-White interest rate model.The purpose of this study was to deal with the stochastic barrier by means of partial differential equation methods and then derive the exact analytical solutions of the barrier options.Furthermore,a numerical example was given to show how to apply this model to pricing one structured product in realistic market.Therefore,this model can provide new insight for future research on structured products involving barrier options.展开更多
In this article, we derive a boundary element formulation for the pricing of barrier option. The price of a barrier option is modeled as the solution of Black-Scholes’ equation. Then the problem is transformed to a b...In this article, we derive a boundary element formulation for the pricing of barrier option. The price of a barrier option is modeled as the solution of Black-Scholes’ equation. Then the problem is transformed to a boundary value problem of heat equation with a moving boundary. The boundary integral representation and integral equation are derived. A boundary element method is designed to solve the integral equation. Special quadrature rules for the singular integral are used. A numerical example is also demonstrated. This boundary element formulation is correct.展开更多
Two kinds of mathematical expressions of stock price, one of which based on certain description is the solution of the simplest differential equation (S.D.E.) obtained by method similar to that used in solid mechanics...Two kinds of mathematical expressions of stock price, one of which based on certain description is the solution of the simplest differential equation (S.D.E.) obtained by method similar to that used in solid mechanics,the other based on uncertain description (i.e., the statistic theory)is the assumption of Black_Scholes's model (A.B_S.M.) in which the density function of stock price obeys logarithmic normal distribution, can be shown to be completely the same under certain equivalence relation of coefficients. The range of the solution of S.D.E. has been shown to be suited only for normal cases (no profit, or lost profit news, etc.) of stock market, so the same range is suited for A.B_ S.M. as well.展开更多
The Operator Splitting method is applied to differential equations occurring as mathematical models in financial models. This paper provides various operator splitting methods to obtain an effective and accurate solut...The Operator Splitting method is applied to differential equations occurring as mathematical models in financial models. This paper provides various operator splitting methods to obtain an effective and accurate solution to the Black-Scholes equation with appropriate boundary conditions for a European option pricing problem. Finally brief comparisons of option prices are given by different models.展开更多
Mitigating the heat stress via a derivative policy is a vital financial option for agricultural producers and other business sectors to strategically adapt to the climate change scenario. This study has provided an ap...Mitigating the heat stress via a derivative policy is a vital financial option for agricultural producers and other business sectors to strategically adapt to the climate change scenario. This study has provided an approach to identifying heat stress events and pricing the heat stress weather derivative due to persistent days of high surface air temperature （SAT）. Cooling degree days （CDD） are used as the weather index for trade. In this study, a call-option model was used as an example for calculating the price of the index. Two heat stress indices were developed to describe the severity and physical impact of heat waves. The daily Global Historical Climatology Network （GHCN-D） SAT data from 1901 to 2007 from the southern California, USA, were used. A major California heat wave that occurred 20-25 October 1965 was studied. The derivative price was calculated based on the call-option model for both long-term station data and the interpolated grid point data at a regular 0.1~ x0.1~ latitude-longitude grid. The resulting comparison indicates that （a） the interpolated data can be used as reliable proxy to price the CDD and （b） a normal distribution model cannot always be used to reliably calculate the CDD price. In conclusion, the data, models, and procedures described in this study have potential application in hedging agricultural and other risks.展开更多
.Option pricing is a major problem in quantitative finance.The Black-Scholes model proves to be an effective model for the pricing of options.In this paper a com-putational method known as the modified differential tr....Option pricing is a major problem in quantitative finance.The Black-Scholes model proves to be an effective model for the pricing of options.In this paper a com-putational method known as the modified differential transform method has been em-ployed to obtain the series solution of Black-Scholes equation with boundary condi-tions for European call and put options paying continuous dividends.The proposed method does not need discretization to find out the solution and thus the computa-tional work is reduced considerably.The results are plotted graphically to establish the accuracy and efficacy of the proposed method.展开更多
In this paper,we construct and analyze a Crank-Nicolson fitted finite volume scheme for pricing European options under regime-switching Kou’s jumpdiffusion model which is governed by a system of partial integro-diffe...In this paper,we construct and analyze a Crank-Nicolson fitted finite volume scheme for pricing European options under regime-switching Kou’s jumpdiffusion model which is governed by a system of partial integro-differential equations(PIDEs).We show that this scheme is consistent,stable and monotone as the mesh sizes in space and time approach zero,hence it ensures the convergence to the solution of continuous problem.Finally,numerical experiments are performed to demonstrate the efficiency,accuracy and robustness of the proposed method.展开更多
There is a need for very fast option pricers when the financial objects are modeled by complex systems of stochastic differential equations.Here the authors investigate option pricers based on mixed Monte-Carlo partia...There is a need for very fast option pricers when the financial objects are modeled by complex systems of stochastic differential equations.Here the authors investigate option pricers based on mixed Monte-Carlo partial differential solvers for stochastic volatility models such as Heston's.It is found that orders of magnitude in speed are gained on full Monte-Carlo algorithms by solving all equations but one by a Monte-Carlo method,and pricing the underlying asset by a partial differential equation with random coefficients,derived by Ito calculus.This strategy is investigated for vanilla options,barrier options and American options with stochastic volatilities and jumps optionally.展开更多
文摘Stock price volatility is considered the main matter of concern within the investment grounds.However,the diffusivity of these prices should as well be considered.As such,proper modelling should be done for investors to stay healthy-informed.This paper suggest to model stock price diffusions using the heat equation from physics.We hypothetically state that,our model captures and model the diffusion bubbles of stock prices with a better precision of reality.We compared our model with the standard geometric Brownian motion model which is the wide commonly used stochastic differential equation in asset valuation.Interestingly,the models proved to agree as evidenced by a bijective relation between the volatility coefficients of the Brownian motion model and the diffusion coefficients of our heat diffusion model as well as the corresponding drift components.Consequently,a short proof for the martingale of our model is done which happen to hold.
基金supported by the National Natural Science Foundation of China under Grant Nos.11871010 and 11971040by the Fundamental Research Funds for the Central Universities under Grant No.2019XD-A11supported by the National Natural Science Foundation of China under Grant No.71073020.
文摘This paper presents two new versions of uncertain market models for valuing vulnerable European call option.The dynamics of underlying asset,counterparty asset,and corporate liability are formulated on the basis of uncertain differential equations and uncertain fractional differential equations of Caputo type,respectively,and the solution to an uncertain fractional differential equation of Caputo type is presented by employing the Mittag-Leffler function andα-path.Then,the pricing formulas of vulnerable European call option based on the proposed models are investigated as well as some algorithms.Some numerical experiments are performed to verify the effectiveness of the results.
基金supported by National Natural Science Foundation of China (No. 12171 122)Shenzhen Science and Technology Program (No. RCJC20210609103755110)+1 种基金Fundamental Research Project of Shenzhen (No. JCYJ20190806143201649)supported by National Natural Science Foundation of China (Grant No. 12071373).
文摘In this paper,a rough Heston model with variable volatility of volatility(vol-of-vol)is derived by modifying the generalized nonlinear Hawkes process and extending the scaling techniques.Then the nonlinear fractional Ric-cati equation for the characteristic function of the asset log-price is derived.The existence,uniqueness and regularity of the solution to the nonlinear fractional Riccati equation are proved and the equation is solved by the Adams methods.Finally the Fourier-cosine methods are combined with the Adams methods to price the options.
基金National Natural Science Foundations of China(Nos.11471175,11171221)
文摘A barrier option valuation model with stochastic barrier which was regarded as the main feature of the model was developed under the Hull-White interest rate model.The purpose of this study was to deal with the stochastic barrier by means of partial differential equation methods and then derive the exact analytical solutions of the barrier options.Furthermore,a numerical example was given to show how to apply this model to pricing one structured product in realistic market.Therefore,this model can provide new insight for future research on structured products involving barrier options.
文摘In this article, we derive a boundary element formulation for the pricing of barrier option. The price of a barrier option is modeled as the solution of Black-Scholes’ equation. Then the problem is transformed to a boundary value problem of heat equation with a moving boundary. The boundary integral representation and integral equation are derived. A boundary element method is designed to solve the integral equation. Special quadrature rules for the singular integral are used. A numerical example is also demonstrated. This boundary element formulation is correct.
文摘Two kinds of mathematical expressions of stock price, one of which based on certain description is the solution of the simplest differential equation (S.D.E.) obtained by method similar to that used in solid mechanics,the other based on uncertain description (i.e., the statistic theory)is the assumption of Black_Scholes's model (A.B_S.M.) in which the density function of stock price obeys logarithmic normal distribution, can be shown to be completely the same under certain equivalence relation of coefficients. The range of the solution of S.D.E. has been shown to be suited only for normal cases (no profit, or lost profit news, etc.) of stock market, so the same range is suited for A.B_ S.M. as well.
文摘The Operator Splitting method is applied to differential equations occurring as mathematical models in financial models. This paper provides various operator splitting methods to obtain an effective and accurate solution to the Black-Scholes equation with appropriate boundary conditions for a European option pricing problem. Finally brief comparisons of option prices are given by different models.
基金supportedin part by the US National Science Foundation (GrantNos. AGS-1015926 and AGS-1015957)supported in part by a U.S. National Oceanographic and Atmospheric Administration (NOAAGrantNo. EL133E09SE4048)
文摘Mitigating the heat stress via a derivative policy is a vital financial option for agricultural producers and other business sectors to strategically adapt to the climate change scenario. This study has provided an approach to identifying heat stress events and pricing the heat stress weather derivative due to persistent days of high surface air temperature （SAT）. Cooling degree days （CDD） are used as the weather index for trade. In this study, a call-option model was used as an example for calculating the price of the index. Two heat stress indices were developed to describe the severity and physical impact of heat waves. The daily Global Historical Climatology Network （GHCN-D） SAT data from 1901 to 2007 from the southern California, USA, were used. A major California heat wave that occurred 20-25 October 1965 was studied. The derivative price was calculated based on the call-option model for both long-term station data and the interpolated grid point data at a regular 0.1~ x0.1~ latitude-longitude grid. The resulting comparison indicates that （a） the interpolated data can be used as reliable proxy to price the CDD and （b） a normal distribution model cannot always be used to reliably calculate the CDD price. In conclusion, the data, models, and procedures described in this study have potential application in hedging agricultural and other risks.
文摘.Option pricing is a major problem in quantitative finance.The Black-Scholes model proves to be an effective model for the pricing of options.In this paper a com-putational method known as the modified differential transform method has been em-ployed to obtain the series solution of Black-Scholes equation with boundary condi-tions for European call and put options paying continuous dividends.The proposed method does not need discretization to find out the solution and thus the computa-tional work is reduced considerably.The results are plotted graphically to establish the accuracy and efficacy of the proposed method.
基金supported by the National Natural Science Foundation of China(Nos.11971354,and 11701221)the Special Basic Cooperative Research Programs of Yunnan Provincial Undergraduate Universities’Association(No.2019FH001-079)the Fundamental Research Funds for the Central Universities(No.22120210555).
文摘In this paper,we construct and analyze a Crank-Nicolson fitted finite volume scheme for pricing European options under regime-switching Kou’s jumpdiffusion model which is governed by a system of partial integro-differential equations(PIDEs).We show that this scheme is consistent,stable and monotone as the mesh sizes in space and time approach zero,hence it ensures the convergence to the solution of continuous problem.Finally,numerical experiments are performed to demonstrate the efficiency,accuracy and robustness of the proposed method.
文摘There is a need for very fast option pricers when the financial objects are modeled by complex systems of stochastic differential equations.Here the authors investigate option pricers based on mixed Monte-Carlo partial differential solvers for stochastic volatility models such as Heston's.It is found that orders of magnitude in speed are gained on full Monte-Carlo algorithms by solving all equations but one by a Monte-Carlo method,and pricing the underlying asset by a partial differential equation with random coefficients,derived by Ito calculus.This strategy is investigated for vanilla options,barrier options and American options with stochastic volatilities and jumps optionally.
