Option pricing has become one of the quite important parts of the financial market. As the market is always dynamic, it is really difficult to predict the option price accurately. For this reason, various machine lear...Option pricing has become one of the quite important parts of the financial market. As the market is always dynamic, it is really difficult to predict the option price accurately. For this reason, various machine learning techniques have been designed and developed to deal with the problem of predicting the future trend of option price. In this paper, we compare the effectiveness of Support Vector Machine (SVM) and Artificial Neural Network (ANN) models for the prediction of option price. Both models are tested with a benchmark publicly available dataset namely SPY option price-2015 in both testing and training phases. The converted data through Principal Component Analysis (PCA) is used in both models to achieve better prediction accuracy. On the other hand, the entire dataset is partitioned into two groups of training (70%) and test sets (30%) to avoid overfitting problem. The outcomes of the SVM model are compared with those of the ANN model based on the root mean square errors (RMSE). It is demonstrated by the experimental results that the ANN model performs better than the SVM model, and the predicted option prices are in good agreement with the corresponding actual option prices.展开更多
We develop two new pricing formulae for European options. The purpose of these formulae is to better understand the impact of each term of the model, as well as improve the speed of the calculations. We consider the S...We develop two new pricing formulae for European options. The purpose of these formulae is to better understand the impact of each term of the model, as well as improve the speed of the calculations. We consider the SABR model (with β=1) of stochastic volatility, which we analyze by tools from Malliavin Calculus. We follow the approach of Alòs et al. (2006) who showed that under stochastic volatility framework, the option prices can be written as the sum of the classic Hull-White (1987) term and a correction due to correlation. We derive the Hull-White term, by using the conditional density of the average volatility, and write it as a two-dimensional integral. For the correction part, we use two different approaches. Both approaches rely on the pairing of the exponential formula developed by Jin, Peng, and Schellhorn (2016) with analytical calculations. The first approach, which we call “Dyson series on the return’s idiosyncratic noise” yields a complete series expansion but necessitates the calculation of a 7-dimensional integral. Two of these dimensions come from the use of Yor’s (1992) formula for the joint density of a Brownian motion and the time-integral of geometric Brownian motion. The second approach, which we call “Dyson series on the common noise” necessitates the calculation of only a one-dimensional integral, but the formula is more complex. This research consisted of both analytical derivations and numerical calculations. The latter show that our formulae are in general more exact, yet more time-consuming to calculate, than the first order expansion of Hagan et al. (2002).展开更多
The SABR stochastic volatility model with β-volatility β ? (0,1) and an absorbing barrier in zero imposed to the forward prices/rates stochastic process is studied. The presence of (possibly) nonzero correlation bet...The SABR stochastic volatility model with β-volatility β ? (0,1) and an absorbing barrier in zero imposed to the forward prices/rates stochastic process is studied. The presence of (possibly) nonzero correlation between the stochastic differentials that appear on the right hand side of the model equations is considered. A series expansion of the transition probability density function of the model in powers of the correlation coefficient of these stochastic differentials is presented. Explicit formulae for the first three terms of this expansion are derived. These formulae are integrals of known integrands. The zero-th order term of the expansion is a new integral formula containing only elementary functions of the transition probability density function of the SABR model when the correlation coefficient is zero. The expansion is deduced from the final value problem for the backward Kolmogorov equation satisfied by the transition probability density function. Each term of the expansion is defined as the solution of a final value problem for a partial differential equation. The integral formulae that give the solutions of these final value problems are based on the Hankel and on the Kontorovich-Lebedev transforms. From the series expansion of the probability density function we deduce the corresponding expansions of the European call and put option prices. Moreover we deduce closed form formulae for the moments of the forward prices/rates variable. The moment formulae obtained do not involve integrals or series expansions and are expressed using only elementary functions. The option pricing formulae are used to study synthetic and real data. In particular we study a time series (of real data) of futures prices of the EUR/USD currency's exchange rate and of the corresponding option prices. The website: http://www.econ.univpm.it/recchioni/finance/w18 contains material including animations, an interactive application and an app that helps the understanding of the paper. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website:http://www.econ.univpm.it/recchioni/finance.展开更多
This paper studies the critical exercise price of American floating strike lookback options under the mixed jump-diffusion model. By using It formula and Wick-It-Skorohod integral, a new market pricing model estab...This paper studies the critical exercise price of American floating strike lookback options under the mixed jump-diffusion model. By using It formula and Wick-It-Skorohod integral, a new market pricing model established under the environment of mixed jumpdiffusion fractional Brownian motion. The fundamental solutions of stochastic parabolic partial differential equations are estimated under the condition of Merton assumptions. The explicit integral representation of early exercise premium and the critical exercise price are also given, then the American floating strike lookback options factorization formula is obtained, the results is generalized the classical Black-Scholes market pricing model.展开更多
This paper presents a method to solve the American option pricing problem in the Black Scholes framework that generalizes the Barone-Adesi, Whaley method [1]. An auxiliary parameter is introduced in the American optio...This paper presents a method to solve the American option pricing problem in the Black Scholes framework that generalizes the Barone-Adesi, Whaley method [1]. An auxiliary parameter is introduced in the American option pricing problem. Power series expansions in this parameter of the option price and of the corresponding free boundary are derived. These series expansions have the Baroni-Adesi, Whaley solution of the American option pricing problem as zero-th order term. The coefficients of the option price series are explicit formulae. The partial sums of the free boundary series are determined solving numerically nonlinear equations that depend from the time variable as a parameter. Numerical experiments suggest that the series expansions derived are convergent. The evaluation of the truncated series expansions on a grid of values of the independent variables is easily parallelizable. The cost of computing the n-th order truncated series expansions is approximately proportional to n as n goes to infinity. The results obtained on a set of test problems with the first and second order approximations deduced from the previous series expansions outperform in accuracy and/or in computational cost the results obtained with several alternative methods to solve the American option pricing problem [1]-[3]. For example when we consider options with maturity time between three and ten years and positive cost of carrying parameter (i.e. when the continuous dividend yield is smaller than the risk free interest rate) the second order approximation of the free boundary obtained truncating the series expansions improves substantially the Barone-Adesi, Whaley free boundary [1]. The website: http://www.econ.univpm.it/recchioni/finance/w20 contains material including animations, an interactive application and an app that helps the understanding of the paper. A general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.展开更多
This study investigates an option pricing method called g-pricing based on backward stochastic differential equations combined with deep learning.We adopted a datadriven approach to find a market-appropriate generator...This study investigates an option pricing method called g-pricing based on backward stochastic differential equations combined with deep learning.We adopted a datadriven approach to find a market-appropriate generator of the backward stochastic differential equation,which is achieved by leveraging the universal approximation capabilities of neural networks.Option pricing,which is the solution to the equation,is approximated using a recursive procedure.The empirical results for the S&P 500 index options show that the proposed deep learning g-pricing model has lower absolute errors than the classical Black–Scholes–Merton model for the same forward stochastic differential equations.The g-pricing mechanism has potential applications in option pricing.展开更多
Put options are known to be priced unusually high in the market,which we refer to as the overpriced put puzzle.This study proposes a quantum model(QM)that can explain such high put option prices as fair prices.Startin...Put options are known to be priced unusually high in the market,which we refer to as the overpriced put puzzle.This study proposes a quantum model(QM)that can explain such high put option prices as fair prices.Starting from a stochastic differential equation of stock returns,we convert the Fokker–Planck equation into the Schr鰀inger equation.To model the market force that always draws excess returns back to equilibrium,we specify a diffusion process corresponding to a QM with a delta potential.The results demonstrate that stock returns follow a Laplace distribution and exhibit power law in the tail.We then construct a closed-form solution for European put option pricing,determining that our model better explains the returns of the S&P 500 index and its corresponding put option prices than do geometric Brownian motion-based models.This study has significant implications for investors and risk managers,presenting a model that can potentially improve derivative pricing.Future studies can generalize the model assumptions by introducing asymmetric potential drawing back excess returns to equilibrium.展开更多
The primary goal of this paper is to price European options in the Merton＇s frame- work with underlying assets following jump-diffusion using fuzzy set theory. Owing to the vague fluctuation of the real financial mar...The primary goal of this paper is to price European options in the Merton＇s frame- work with underlying assets following jump-diffusion using fuzzy set theory. Owing to the vague fluctuation of the real financial market, the average jump rate and jump sizes cannot be recorded or collected accurately. So the main idea of this paper is to model the rate as a triangular fuzzy number and jump sizes as fuzzy random variables and use the property of fuzzy set to deduce two different jump-diffusion models underlying principle of rational expectations equilibrium price. Unlike many conventional models, the European option price will now turn into a fuzzy number. One of the major advantages of this model is that it allows investors to choose a reasonable European option price under an acceptable belief degree. The empirical results will serve as useful feedback information for improvements on the proposed model.展开更多
In order to price European contingent claim in a class of fractional Black-Scholes market, where the prices of assets follow a Wick-Ito stochastic differential equation driven by the fractional Brownian motion and mar...In order to price European contingent claim in a class of fractional Black-Scholes market, where the prices of assets follow a Wick-Ito stochastic differential equation driven by the fractional Brownian motion and market coefficients are deterministic functions, the pricing formula of European call option was explicitly derived by the method of the stochastic calculus of tile fractional Brownian motion. A result about fractional Clark derivative was also obtained.展开更多
Deepwater oil and gas projects embody high risks from geology and engineering aspects, which exert substantial influence on project valuation. But the uncer- tainties may be converted to additional value to the projec...Deepwater oil and gas projects embody high risks from geology and engineering aspects, which exert substantial influence on project valuation. But the uncer- tainties may be converted to additional value to the projects in the case of flexible management. Given the flexibility of project management, this paper extends the classical real options model to a multi-factor model which contains oil price, geology, and engineering uncertainties. It then gives an application example of the new model to evaluate deepwater oil and gas projects with a numerical analytical method. Compared with other methods and models, this multi-factor real options model contains more project information. It reflects the potential value deriving not only from oil price variation but also from geology and engi- neering uncertainties, which provides more accurate and reliable valuation information for decision makers.展开更多
This paper aims to develop a power penalty method for a linear parabolic variational inequality （VI） in two spatial dimensions governing the two-asset American option valuation. This method yields a two-dimensional ...This paper aims to develop a power penalty method for a linear parabolic variational inequality （VI） in two spatial dimensions governing the two-asset American option valuation. This method yields a two-dimensional nonlinear parabolic PDE containing a power penalty term with penalty constant λ〉 1 and a power parameter k 〉 0. We show that the nonlinear PDE is uniquely solvable and the solution of the PDE converges to that of the VI at the rate of order O（λ^-k/2）. A fitted finite volume method is designed to solve the nonlinear PDE, and some numerical experiments are performed to illustrate the usefulness of this method.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
This paper investigates several competing procedures for computing the prices of vanilla European options, such as puts, calls and binaries, in which the underlying model has a characteristic function that is known in...This paper investigates several competing procedures for computing the prices of vanilla European options, such as puts, calls and binaries, in which the underlying model has a characteristic function that is known in semi-closed form. The algorithms investigated here are the half-range Fourier cosine series, the half-range Fourier sine series and the full-range Fourier series. Their performance is assessed in simulation experiments in which an analytical solution is available and also for a simple affine model of stochastic volatility in which there is no closed-form solution. The results suggest that the half-range sine series approximation is the least effective of the three proposed algorithms. It is rather more difficult to distinguish between the performance of the half-range cosine series and the full-range Fourier series. However there are two clear differences. First, when the interval over which the density is approximated is relatively large, the full-range Fourier series is at least as good as the half-range Fourier cosine series, and outperforms the latter in pricing out-of-the-money call options, in particular with maturities of three months or less. Second, the computational time required by the half-range Fourier cosine series is uniformly longer than that required by the full-range Fourier series for an interval of fixed length. Taken together, these two conclusions make a case for pricing options using a full-range range Fourier series as opposed to a half-range Fourier cosine series if a large number of options are to be priced in as short a time as possible.展开更多
In this paper, the binomial tree method is introduced to price the European option under a class of jump-diffusion model. The purpose of the addressed problem is to find the parameters of the binomial tree and design ...In this paper, the binomial tree method is introduced to price the European option under a class of jump-diffusion model. The purpose of the addressed problem is to find the parameters of the binomial tree and design the pricing formula for European option. Compared with the continuous situation, the proposed value equation of option under the new binomial tree model converges to Merton’s accurate analytical solution, and the established binomial tree method can be proved to work better than the traditional binomial tree. Finally, a numerical example is presented to illustrate the effectiveness of the proposed pricing methods.展开更多
In this paper, we study the price of catastrophe Options with counterparty credit risk in a reduced form model. We assume that the loss process is generated by a doubly stochastic Poisson process, the share price proc...In this paper, we study the price of catastrophe Options with counterparty credit risk in a reduced form model. We assume that the loss process is generated by a doubly stochastic Poisson process, the share price process is modeled through a jump-diffusion process which is correlated to the loss process, the interest rate process and the default intensity process are modeled through the Vasicek model： We derive the closed form formulae for pricing catastrophe options in a reduced form model. Furthermore, we make some numerical analysis on the explicit formulae.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
Using physical probability measure of price process and the principle of fair premium, the results of Mogens Bladt and Hina Hviid Rydberg are generalized. In two cases of paying intermediate divisends and no intermedi...Using physical probability measure of price process and the principle of fair premium, the results of Mogens Bladt and Hina Hviid Rydberg are generalized. In two cases of paying intermediate divisends and no intermediate dividends, the Black_Scholes model is generalized to the case where the risk_less asset (bond or bank account) earns a time_dependent interest rate and risk asset (stock) has time_dependent the continuously compounding expected rate of return, volatility. In these cases the accurate pricing formula and put_call parity of European option are obtained. The general approach of option pricing is given for the general Black_Scholes of the risk asset (stock) has the continuously compounding expected rate of return, volatility. The accurate pricing formula and put_call parity of European option on a stock whose price process is driven by general Ornstein_Uhlenback (O_U) process are given by actuarial approach.展开更多
A numerical method for American options pricing on assets under the Heston stochastic volatility model is developed.A preliminary transformation is applied to remove the mixed derivative term avoiding known numerical ...A numerical method for American options pricing on assets under the Heston stochastic volatility model is developed.A preliminary transformation is applied to remove the mixed derivative term avoiding known numerical drawbacks and reducing computational costs.Free boundary is treated by the penalty method.Transformed nonlinear partial differential equation is solved numerically by using the method of lines.For full discretization the exponential time differencing method is used.Numerical analysis establishes the stability and positivity of the proposed method.The numerical convergence behaviour and effectiveness are investigated in extensive numerical experiments.展开更多
文摘Option pricing has become one of the quite important parts of the financial market. As the market is always dynamic, it is really difficult to predict the option price accurately. For this reason, various machine learning techniques have been designed and developed to deal with the problem of predicting the future trend of option price. In this paper, we compare the effectiveness of Support Vector Machine (SVM) and Artificial Neural Network (ANN) models for the prediction of option price. Both models are tested with a benchmark publicly available dataset namely SPY option price-2015 in both testing and training phases. The converted data through Principal Component Analysis (PCA) is used in both models to achieve better prediction accuracy. On the other hand, the entire dataset is partitioned into two groups of training (70%) and test sets (30%) to avoid overfitting problem. The outcomes of the SVM model are compared with those of the ANN model based on the root mean square errors (RMSE). It is demonstrated by the experimental results that the ANN model performs better than the SVM model, and the predicted option prices are in good agreement with the corresponding actual option prices.
文摘We develop two new pricing formulae for European options. The purpose of these formulae is to better understand the impact of each term of the model, as well as improve the speed of the calculations. We consider the SABR model (with β=1) of stochastic volatility, which we analyze by tools from Malliavin Calculus. We follow the approach of Alòs et al. (2006) who showed that under stochastic volatility framework, the option prices can be written as the sum of the classic Hull-White (1987) term and a correction due to correlation. We derive the Hull-White term, by using the conditional density of the average volatility, and write it as a two-dimensional integral. For the correction part, we use two different approaches. Both approaches rely on the pairing of the exponential formula developed by Jin, Peng, and Schellhorn (2016) with analytical calculations. The first approach, which we call “Dyson series on the return’s idiosyncratic noise” yields a complete series expansion but necessitates the calculation of a 7-dimensional integral. Two of these dimensions come from the use of Yor’s (1992) formula for the joint density of a Brownian motion and the time-integral of geometric Brownian motion. The second approach, which we call “Dyson series on the common noise” necessitates the calculation of only a one-dimensional integral, but the formula is more complex. This research consisted of both analytical derivations and numerical calculations. The latter show that our formulae are in general more exact, yet more time-consuming to calculate, than the first order expansion of Hagan et al. (2002).
文摘The SABR stochastic volatility model with β-volatility β ? (0,1) and an absorbing barrier in zero imposed to the forward prices/rates stochastic process is studied. The presence of (possibly) nonzero correlation between the stochastic differentials that appear on the right hand side of the model equations is considered. A series expansion of the transition probability density function of the model in powers of the correlation coefficient of these stochastic differentials is presented. Explicit formulae for the first three terms of this expansion are derived. These formulae are integrals of known integrands. The zero-th order term of the expansion is a new integral formula containing only elementary functions of the transition probability density function of the SABR model when the correlation coefficient is zero. The expansion is deduced from the final value problem for the backward Kolmogorov equation satisfied by the transition probability density function. Each term of the expansion is defined as the solution of a final value problem for a partial differential equation. The integral formulae that give the solutions of these final value problems are based on the Hankel and on the Kontorovich-Lebedev transforms. From the series expansion of the probability density function we deduce the corresponding expansions of the European call and put option prices. Moreover we deduce closed form formulae for the moments of the forward prices/rates variable. The moment formulae obtained do not involve integrals or series expansions and are expressed using only elementary functions. The option pricing formulae are used to study synthetic and real data. In particular we study a time series (of real data) of futures prices of the EUR/USD currency's exchange rate and of the corresponding option prices. The website: http://www.econ.univpm.it/recchioni/finance/w18 contains material including animations, an interactive application and an app that helps the understanding of the paper. A more general reference to the work of the authors and of their coauthors in mathematical finance is the website:http://www.econ.univpm.it/recchioni/finance.
基金Supported by the Fundamental Research Funds of Lanzhou University of Finance and Economics(Lzufe2017C-09)
文摘This paper studies the critical exercise price of American floating strike lookback options under the mixed jump-diffusion model. By using It formula and Wick-It-Skorohod integral, a new market pricing model established under the environment of mixed jumpdiffusion fractional Brownian motion. The fundamental solutions of stochastic parabolic partial differential equations are estimated under the condition of Merton assumptions. The explicit integral representation of early exercise premium and the critical exercise price are also given, then the American floating strike lookback options factorization formula is obtained, the results is generalized the classical Black-Scholes market pricing model.
文摘This paper presents a method to solve the American option pricing problem in the Black Scholes framework that generalizes the Barone-Adesi, Whaley method [1]. An auxiliary parameter is introduced in the American option pricing problem. Power series expansions in this parameter of the option price and of the corresponding free boundary are derived. These series expansions have the Baroni-Adesi, Whaley solution of the American option pricing problem as zero-th order term. The coefficients of the option price series are explicit formulae. The partial sums of the free boundary series are determined solving numerically nonlinear equations that depend from the time variable as a parameter. Numerical experiments suggest that the series expansions derived are convergent. The evaluation of the truncated series expansions on a grid of values of the independent variables is easily parallelizable. The cost of computing the n-th order truncated series expansions is approximately proportional to n as n goes to infinity. The results obtained on a set of test problems with the first and second order approximations deduced from the previous series expansions outperform in accuracy and/or in computational cost the results obtained with several alternative methods to solve the American option pricing problem [1]-[3]. For example when we consider options with maturity time between three and ten years and positive cost of carrying parameter (i.e. when the continuous dividend yield is smaller than the risk free interest rate) the second order approximation of the free boundary obtained truncating the series expansions improves substantially the Barone-Adesi, Whaley free boundary [1]. The website: http://www.econ.univpm.it/recchioni/finance/w20 contains material including animations, an interactive application and an app that helps the understanding of the paper. A general reference to the work of the authors and of their coauthors in mathematical finance is the website: http://www.econ.univpm.it/recchioni/finance.
基金supported by Taishan Scholar Project of Shandong Province of China(Grant tstp20240803)the National Key R&D Program of China(Grant No.2023YFA1008903)the Major Fundamental Research Project of Shandong Province of China(Grant No.ZR2023ZD33).
文摘This study investigates an option pricing method called g-pricing based on backward stochastic differential equations combined with deep learning.We adopted a datadriven approach to find a market-appropriate generator of the backward stochastic differential equation,which is achieved by leveraging the universal approximation capabilities of neural networks.Option pricing,which is the solution to the equation,is approximated using a recursive procedure.The empirical results for the S&P 500 index options show that the proposed deep learning g-pricing model has lower absolute errors than the classical Black–Scholes–Merton model for the same forward stochastic differential equations.The g-pricing mechanism has potential applications in option pricing.
基金supported by the International Joint Research Grant by Yonsei Graduate School(Kwangwon Ahn)the National Research Foundation of Korea(NRF)grant funded by the Korea government(MSIT)(RS-2025-16067531:Kwangwon Ahn)+2 种基金the NRF of Korea grant funded by the Korea government(MSIT)(2020R1A2C1A01005949,RS-2023-00217705:Taeyoung Park)the MSIT(Ministry of Science and ICT)Korea,under the ICAN(ICT Challenge and Advanced Network of HRD)support program(RS-2023-00259934:Taeyoung Park)supervised by the IITP(Institute for Information&Communications Technology Planning&Evaluation)the Son Jiho Research Grant of Yonsei University(2023-22-0006:Taeyoung Park).
文摘Put options are known to be priced unusually high in the market,which we refer to as the overpriced put puzzle.This study proposes a quantum model(QM)that can explain such high put option prices as fair prices.Starting from a stochastic differential equation of stock returns,we convert the Fokker–Planck equation into the Schr鰀inger equation.To model the market force that always draws excess returns back to equilibrium,we specify a diffusion process corresponding to a QM with a delta potential.The results demonstrate that stock returns follow a Laplace distribution and exhibit power law in the tail.We then construct a closed-form solution for European put option pricing,determining that our model better explains the returns of the S&P 500 index and its corresponding put option prices than do geometric Brownian motion-based models.This study has significant implications for investors and risk managers,presenting a model that can potentially improve derivative pricing.Future studies can generalize the model assumptions by introducing asymmetric potential drawing back excess returns to equilibrium.
基金Supported by the Key Grant Project of Chinese Ministry of Education(309018)National Natural Science Foundation of China(70973104 and 11171304)the Zhejiang Natural Science Foundation of China(Y6110023)
文摘The primary goal of this paper is to price European options in the Merton＇s frame- work with underlying assets following jump-diffusion using fuzzy set theory. Owing to the vague fluctuation of the real financial market, the average jump rate and jump sizes cannot be recorded or collected accurately. So the main idea of this paper is to model the rate as a triangular fuzzy number and jump sizes as fuzzy random variables and use the property of fuzzy set to deduce two different jump-diffusion models underlying principle of rational expectations equilibrium price. Unlike many conventional models, the European option price will now turn into a fuzzy number. One of the major advantages of this model is that it allows investors to choose a reasonable European option price under an acceptable belief degree. The empirical results will serve as useful feedback information for improvements on the proposed model.
基金National Natural Science Foundation of China(No.10826098)Natural Science Foundation of Anhui Province,China(No.090416225)Anhui Natural Science Foundation of Universities,China(No.KJ2010A037)
文摘In order to price European contingent claim in a class of fractional Black-Scholes market, where the prices of assets follow a Wick-Ito stochastic differential equation driven by the fractional Brownian motion and market coefficients are deterministic functions, the pricing formula of European call option was explicitly derived by the method of the stochastic calculus of tile fractional Brownian motion. A result about fractional Clark derivative was also obtained.
基金supported from the National Science and Technology Major Project under Grant No.2011ZX05030
文摘Deepwater oil and gas projects embody high risks from geology and engineering aspects, which exert substantial influence on project valuation. But the uncer- tainties may be converted to additional value to the projects in the case of flexible management. Given the flexibility of project management, this paper extends the classical real options model to a multi-factor model which contains oil price, geology, and engineering uncertainties. It then gives an application example of the new model to evaluate deepwater oil and gas projects with a numerical analytical method. Compared with other methods and models, this multi-factor real options model contains more project information. It reflects the potential value deriving not only from oil price variation but also from geology and engi- neering uncertainties, which provides more accurate and reliable valuation information for decision makers.
文摘This paper aims to develop a power penalty method for a linear parabolic variational inequality （VI） in two spatial dimensions governing the two-asset American option valuation. This method yields a two-dimensional nonlinear parabolic PDE containing a power penalty term with penalty constant λ〉 1 and a power parameter k 〉 0. We show that the nonlinear PDE is uniquely solvable and the solution of the PDE converges to that of the VI at the rate of order O（λ^-k/2）. A fitted finite volume method is designed to solve the nonlinear PDE, and some numerical experiments are performed to illustrate the usefulness of this method.
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This paper investigates several competing procedures for computing the prices of vanilla European options, such as puts, calls and binaries, in which the underlying model has a characteristic function that is known in semi-closed form. The algorithms investigated here are the half-range Fourier cosine series, the half-range Fourier sine series and the full-range Fourier series. Their performance is assessed in simulation experiments in which an analytical solution is available and also for a simple affine model of stochastic volatility in which there is no closed-form solution. The results suggest that the half-range sine series approximation is the least effective of the three proposed algorithms. It is rather more difficult to distinguish between the performance of the half-range cosine series and the full-range Fourier series. However there are two clear differences. First, when the interval over which the density is approximated is relatively large, the full-range Fourier series is at least as good as the half-range Fourier cosine series, and outperforms the latter in pricing out-of-the-money call options, in particular with maturities of three months or less. Second, the computational time required by the half-range Fourier cosine series is uniformly longer than that required by the full-range Fourier series for an interval of fixed length. Taken together, these two conclusions make a case for pricing options using a full-range range Fourier series as opposed to a half-range Fourier cosine series if a large number of options are to be priced in as short a time as possible.
文摘In this paper, the binomial tree method is introduced to price the European option under a class of jump-diffusion model. The purpose of the addressed problem is to find the parameters of the binomial tree and design the pricing formula for European option. Compared with the continuous situation, the proposed value equation of option under the new binomial tree model converges to Merton’s accurate analytical solution, and the established binomial tree method can be proved to work better than the traditional binomial tree. Finally, a numerical example is presented to illustrate the effectiveness of the proposed pricing methods.
基金supported by the National Natural Science Foundation of China(11371274)
文摘In this paper, we study the price of catastrophe Options with counterparty credit risk in a reduced form model. We assume that the loss process is generated by a doubly stochastic Poisson process, the share price process is modeled through a jump-diffusion process which is correlated to the loss process, the interest rate process and the default intensity process are modeled through the Vasicek model： We derive the closed form formulae for pricing catastrophe options in a reduced form model. Furthermore, we make some numerical analysis on the explicit formulae.
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Using physical probability measure of price process and the principle of fair premium, the results of Mogens Bladt and Hina Hviid Rydberg are generalized. In two cases of paying intermediate divisends and no intermediate dividends, the Black_Scholes model is generalized to the case where the risk_less asset (bond or bank account) earns a time_dependent interest rate and risk asset (stock) has time_dependent the continuously compounding expected rate of return, volatility. In these cases the accurate pricing formula and put_call parity of European option are obtained. The general approach of option pricing is given for the general Black_Scholes of the risk asset (stock) has the continuously compounding expected rate of return, volatility. The accurate pricing formula and put_call parity of European option on a stock whose price process is driven by general Ornstein_Uhlenback (O_U) process are given by actuarial approach.
基金This work has been supported by the Spanish Ministerio de Economía,Industria y Competitividad(MINECO),the Agencia Estatal de Investigación(AEI)and Fondo Europeo de Desarrollo Regional(FEDER UE)grant MTM2017-89664-P.
文摘A numerical method for American options pricing on assets under the Heston stochastic volatility model is developed.A preliminary transformation is applied to remove the mixed derivative term avoiding known numerical drawbacks and reducing computational costs.Free boundary is treated by the penalty method.Transformed nonlinear partial differential equation is solved numerically by using the method of lines.For full discretization the exponential time differencing method is used.Numerical analysis establishes the stability and positivity of the proposed method.The numerical convergence behaviour and effectiveness are investigated in extensive numerical experiments.
