This paper develops a closed-form solution to an extended Black-Scholes (EBS) pricing formula which admits an implied drift parameter alongside the standard implied volatility. The market volatility smiles for vanilla...This paper develops a closed-form solution to an extended Black-Scholes (EBS) pricing formula which admits an implied drift parameter alongside the standard implied volatility. The market volatility smiles for vanilla call options on the S&P 500 index are recreated fitting the best volatility-drift combination in this new EBS. Using a likelihood ratio test, the implied drift parameter is seen to be quite significant in explaining volatility smiles. The implied drift parameter is sufficiently small to be undetectable via historical pricing analysis, suggesting that drift is best considered as an implied parameter rather than a historically-fit one. An overview of option-pricing models is provided as background.展开更多
We consider an economic model with a deterministic money market account and a finite set of basic economic risks. The real-world prices of the risks are represented by continuous time stochastic processes satisfying a...We consider an economic model with a deterministic money market account and a finite set of basic economic risks. The real-world prices of the risks are represented by continuous time stochastic processes satisfying a stochastic differential equation of diffusion type. For the simple class of log-normally distributed instantaneous rates of return, we construct an explicit state-price deflator. Since this includes the Black-Scholes and the Vasicek (Ornstein-Uhlenbeck) return models, the considered deflator is called Black-Scholes- Vasicek deflator. Besides a new elementary proof of the Black-Scholes and Margrabe option pricing formulas a validation of these in a multiple risk economy is achieved.展开更多
In this paper,the pricing formulae of the geometric average Asian call option with the fixed and floating strike price under the fractional Brownian motion(FBM)are given out by the method of partial differential equat...In this paper,the pricing formulae of the geometric average Asian call option with the fixed and floating strike price under the fractional Brownian motion(FBM)are given out by the method of partial differential equation(PDE).The call-put parity for the geometric average Asian options is given.The results are generalization of option pricing under standard Brownian motion.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
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
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
Discuss the no-arbitrage principle in a fuzzy market and present a model for pricing an option. Get a fuzzy price for the contingent claim in a market involving fuzzy elements, whose level set can be seen as the possi...Discuss the no-arbitrage principle in a fuzzy market and present a model for pricing an option. Get a fuzzy price for the contingent claim in a market involving fuzzy elements, whose level set can be seen as the possible price level interval with given belief degree. Use fuzzy densit） function and fuzzy mean as evidence for such model. Also give an example for comparing the result of the model in this article and that of another pricing method.展开更多
The method for pricing the option in a market with interval number factors is proposed. The no-arbitrage principle in the interval number valued market and the rule to judge the reasonability of a price interval are g...The method for pricing the option in a market with interval number factors is proposed. The no-arbitrage principle in the interval number valued market and the rule to judge the reasonability of a price interval are given. Using the method, the price interval where the riskless interest and the volatility under B-S setting is given. The price interval from binomial tree model when the key factors u, d, R are all interval numbers is also discussed.展开更多
Under the assumption of the underlying asset is driven by the mixed fractional Brownian motion, we obtain the mixed fractionalBlack-Scholes partial differential equation by fractional Ito formula, and the pricing form...Under the assumption of the underlying asset is driven by the mixed fractional Brownian motion, we obtain the mixed fractionalBlack-Scholes partial differential equation by fractional Ito formula, and the pricing formula of perpetual American put option bythis partial differential equation theory.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
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
Binomial no-arbitrage price have a method is the traditional approach for derivative pricing,which is,the complete model,which makes possible the perfect replication in the market.Risk neutral pricing is an appropriat...Binomial no-arbitrage price have a method is the traditional approach for derivative pricing,which is,the complete model,which makes possible the perfect replication in the market.Risk neutral pricing is an appropriate method of asset pricing in a complete market.We have discussed an incomplete market,a non-transaction asset that produces incompleteness of the market.An effective method of asset pricing in incomplete markets is the undifferentiated pricing method.This technique was firstly introduced by Bernoulli in(1738)the sense of gambling,lottery and their expected return.It is used to command investors'preferences and better returns the results they expect.In addition,we also discuss the utility function,which is the core element of the undifferentiated pricing.We also studied some important behavior preferences of agents,and injected exponential effect of risk aversion in the model,so that the model was nonlinear in the process of claim settlement.展开更多
文摘This paper develops a closed-form solution to an extended Black-Scholes (EBS) pricing formula which admits an implied drift parameter alongside the standard implied volatility. The market volatility smiles for vanilla call options on the S&P 500 index are recreated fitting the best volatility-drift combination in this new EBS. Using a likelihood ratio test, the implied drift parameter is seen to be quite significant in explaining volatility smiles. The implied drift parameter is sufficiently small to be undetectable via historical pricing analysis, suggesting that drift is best considered as an implied parameter rather than a historically-fit one. An overview of option-pricing models is provided as background.
文摘We consider an economic model with a deterministic money market account and a finite set of basic economic risks. The real-world prices of the risks are represented by continuous time stochastic processes satisfying a stochastic differential equation of diffusion type. For the simple class of log-normally distributed instantaneous rates of return, we construct an explicit state-price deflator. Since this includes the Black-Scholes and the Vasicek (Ornstein-Uhlenbeck) return models, the considered deflator is called Black-Scholes- Vasicek deflator. Besides a new elementary proof of the Black-Scholes and Margrabe option pricing formulas a validation of these in a multiple risk economy is achieved.
基金Shanghai Leading Academic Discipline Project,China(No.S30405)Special Funds for Major Specialties of Shanghai Education Committee,China
文摘In this paper,the pricing formulae of the geometric average Asian call option with the fixed and floating strike price under the fractional Brownian motion(FBM)are given out by the method of partial differential equation(PDE).The call-put parity for the geometric average Asian options is given.The results are generalization of option pricing under standard Brownian motion.
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.
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.
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Discuss the no-arbitrage principle in a fuzzy market and present a model for pricing an option. Get a fuzzy price for the contingent claim in a market involving fuzzy elements, whose level set can be seen as the possible price level interval with given belief degree. Use fuzzy densit） function and fuzzy mean as evidence for such model. Also give an example for comparing the result of the model in this article and that of another pricing method.
文摘The method for pricing the option in a market with interval number factors is proposed. The no-arbitrage principle in the interval number valued market and the rule to judge the reasonability of a price interval are given. Using the method, the price interval where the riskless interest and the volatility under B-S setting is given. The price interval from binomial tree model when the key factors u, d, R are all interval numbers is also discussed.
文摘Under the assumption of the underlying asset is driven by the mixed fractional Brownian motion, we obtain the mixed fractionalBlack-Scholes partial differential equation by fractional Ito formula, and the pricing formula of perpetual American put option bythis partial differential equation theory.
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.
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.
文摘Binomial no-arbitrage price have a method is the traditional approach for derivative pricing,which is,the complete model,which makes possible the perfect replication in the market.Risk neutral pricing is an appropriate method of asset pricing in a complete market.We have discussed an incomplete market,a non-transaction asset that produces incompleteness of the market.An effective method of asset pricing in incomplete markets is the undifferentiated pricing method.This technique was firstly introduced by Bernoulli in(1738)the sense of gambling,lottery and their expected return.It is used to command investors'preferences and better returns the results they expect.In addition,we also discuss the utility function,which is the core element of the undifferentiated pricing.We also studied some important behavior preferences of agents,and injected exponential effect of risk aversion in the model,so that the model was nonlinear in the process of claim settlement.
