This paper concerns an inverse problem of recovering implied volatility in short-term interest rate model from the market prices of zero-coupon bonds. Based on lineariza-tion, an analytic solution, which is given as a...This paper concerns an inverse problem of recovering implied volatility in short-term interest rate model from the market prices of zero-coupon bonds. Based on lineariza-tion, an analytic solution, which is given as a power series, is derived for the direct problem.By neglecting high order terms in the power series, an integral equation about the pertur-bation of volatility is formulated and the Tikhonov regularization method is applied to solvethe integral equation. Finally numerical experiments are given and the results show that the method is effective.展开更多
The recently developed Bitcoin futures and options contracts in cryptocurrency derivatives exchanges mark the beginning of a new era in Bitcoin price risk hedging.The need for these tools dates back to the market cras...The recently developed Bitcoin futures and options contracts in cryptocurrency derivatives exchanges mark the beginning of a new era in Bitcoin price risk hedging.The need for these tools dates back to the market crash of 1987,when investors needed better ways to protect their portfolios through option insurance.These tools provide greater flexibility to trade and hedge volatile swings in Bitcoin prices effectively.The violation of constant volatility and the log-normality assumption of the Black–Scholes option pricing model led to the discovery of the volatility smile,smirk,or skew in options markets.These stylized facts;that is,the volatility smile and implied volatilities implied by the option prices,are well documented in the option literature for almost all financial markets.These are expected to be true for Bitcoin options as well.The data sets for the study are based on short-dated Bitcoin options(14-day maturity)of two time periods traded on Deribit Bitcoin Futures and Options Exchange,a Netherlandsbased cryptocurrency derivative exchange.The estimated results are compared with benchmark Black–Scholes implied volatility values for accuracy and efficiency analysis.This study has two aims:(1)to provide insights into the volatility smile in Bitcoin options and(2)to estimate the implied volatility of Bitcoin options through numerical approximation techniques,specifically the Newton Raphson and Bisection methods.The experimental results show that Bitcoin options belong to the commodity class of assets based on the presence of a volatility forward skew in Bitcoin option data.Moreover,the Newton Raphson and Bisection methods are effective in estimating the implied volatility of Bitcoin options.However,the Newton Raphson forecasting technique converges faster than does the Bisection method.展开更多
Modeling implied volatility(IV)is important for option pricing,hedging,and risk management.Previous studies of deterministic implied volatility functions(DIVFs)propose two parameters,moneyness and time to maturity,to ...Modeling implied volatility(IV)is important for option pricing,hedging,and risk management.Previous studies of deterministic implied volatility functions(DIVFs)propose two parameters,moneyness and time to maturity,to estimate implied volatility.Recent DIVF models have included factors such as a moving average ratio and relative bid-ask spread but fail to enhance modeling accuracy.The current study offers a generalized DIVF model by including a momentum indicator for the underlying asset using a relative strength index(RSI)covering multiple time resolutions as a factor,as momentum is often used by investors and speculators in their trading decisions,and in contrast to volatility,RSI can distinguish between bull and bear markets.To the best of our knowledge,prior studies have not included RSI as a predictive factor in modeling IV.Instead of using a simple linear regression as in previous studies,we use a machine learning regression algorithm,namely random forest,to model a nonlinear IV.Previous studies apply DVIF modeling to options on traditional financial assets,such as stock and foreign exchange markets.Here,we study options on the largest cryptocurrency,Bitcoin,which poses greater modeling challenges due to its extreme volatility and the fact that it is not as well studied as traditional financial assets.Recent Bitcoin option chain data were collected from a leading cryptocurrency option exchange over a four-month period for model development and validation.Our dataset includes short-maturity options with expiry in less than six days,as well as a full range of moneyness,both of which are often excluded in existing studies as prices for options with these characteristics are often highly volatile and pose challenges to model building.Our in-sample and out-sample results indicate that including our proposed momentum indicator significantly enhances the model’s accuracy in pricing options.The nonlinear machine learning random forest algorithm also performed better than a simple linear regression.Compared to prevailing option pricing models that employ stochastic variables,our DIVF model does not include stochastic factors but exhibits reasonably good performance.It is also easy to compute due to the availability of real-time RSIs.Our findings indicate our enhanced DIVF model offers significant improvements and may be an excellent alternative to existing option pricing models that are primarily stochastic in nature.展开更多
This paper deals with options on assets, such as stocks or indexes, which pay cash dividends. Pricing methods which consider discrete dividends are usually computationally expensive and become infeasible when one cons...This paper deals with options on assets, such as stocks or indexes, which pay cash dividends. Pricing methods which consider discrete dividends are usually computationally expensive and become infeasible when one considers multiple dividends paid during the option lifetime. This is the case of long-term options and options on indexes. The first purpose of this paper is to assess efficient and accurate numerical procedures which yield consistent prices for both European and American options when the underlying asset pays discrete dividends. The authors then analyze some methodologies to extract information on implied volatilities and dividends from quoted option prices. Implied dividends can also be computed using a modified version of the well-known put-call parity relationship. This technique is straightforward, nevertheless, its use is limited to European options, and when dealing with equities, most traded options are of American type. As an alternative, the numerical inversion of pricing methods, such as efficient interpolated binomial method, can be used. This paper applies different procedures to obtain implied volatilities and dividends of listed stocks of the Italian derivatives market （IDEM）.展开更多
Volatility is an important variable in the financial market. We propose a model-free implied volatility method to measure the volatility and test the volatility risk premium. The model-free implied volatility does not...Volatility is an important variable in the financial market. We propose a model-free implied volatility method to measure the volatility and test the volatility risk premium. The model-free implied volatility does not depend on the option pricing model, and extracts information from all the option contracts. We provide empirical evidence from the S & P 500 index option that model-free implied volatility is more accurate to forecast the future volatility and the volatility risk premium does not exist.展开更多
如何对期权的隐含波动率曲面建模,一直以来是金融衍生品领域的难点问题之一。已有文献针对某些特定结构的随机波动率模型,推导出期权的隐含波动率解析表达式(Analytical formula of implie dvolatility,AFIV),但通用性差、或精度不够好...如何对期权的隐含波动率曲面建模,一直以来是金融衍生品领域的难点问题之一。已有文献针对某些特定结构的随机波动率模型,推导出期权的隐含波动率解析表达式(Analytical formula of implie dvolatility,AFIV),但通用性差、或精度不够好。本文针对一般形式的随机波动率模型,给出获取高精度AFIV的通用方法:首先应用特征函数局部结构微扰法,推导出一般形式随机波动率模型的特征函数表达式;然后基于特征函数与期权价格的关系式,应用参数微扰法推导出隐含波动率的渐近表达式。数值实验和实证结果表明:(1)与同类最新研究(如Ait-Sahalia等,2021)[1-2]相比,本文方法在长到期期权上的定价精度有显著提升,在短到期期权上的定价精度也略优些;(2)与孙有发等(2022)[3]提出的基于特征函数局部结构微扰的期权定价方法相比,本文提出的AFIV方法不仅定价精度高,且拥有更优越的定价效率和参数校准的稳健性。本文提出的AFIV方法,不仅解决了众多性能良好的随机波动率模型在期权市场的运用问题,而且还为公平比较不同标的模型的市场刻画性能提供一种可行方法。展开更多
基金Supported by the National Natural Science Foundation of China(11171349)
文摘This paper concerns an inverse problem of recovering implied volatility in short-term interest rate model from the market prices of zero-coupon bonds. Based on lineariza-tion, an analytic solution, which is given as a power series, is derived for the direct problem.By neglecting high order terms in the power series, an integral equation about the pertur-bation of volatility is formulated and the Tikhonov regularization method is applied to solvethe integral equation. Finally numerical experiments are given and the results show that the method is effective.
文摘The recently developed Bitcoin futures and options contracts in cryptocurrency derivatives exchanges mark the beginning of a new era in Bitcoin price risk hedging.The need for these tools dates back to the market crash of 1987,when investors needed better ways to protect their portfolios through option insurance.These tools provide greater flexibility to trade and hedge volatile swings in Bitcoin prices effectively.The violation of constant volatility and the log-normality assumption of the Black–Scholes option pricing model led to the discovery of the volatility smile,smirk,or skew in options markets.These stylized facts;that is,the volatility smile and implied volatilities implied by the option prices,are well documented in the option literature for almost all financial markets.These are expected to be true for Bitcoin options as well.The data sets for the study are based on short-dated Bitcoin options(14-day maturity)of two time periods traded on Deribit Bitcoin Futures and Options Exchange,a Netherlandsbased cryptocurrency derivative exchange.The estimated results are compared with benchmark Black–Scholes implied volatility values for accuracy and efficiency analysis.This study has two aims:(1)to provide insights into the volatility smile in Bitcoin options and(2)to estimate the implied volatility of Bitcoin options through numerical approximation techniques,specifically the Newton Raphson and Bisection methods.The experimental results show that Bitcoin options belong to the commodity class of assets based on the presence of a volatility forward skew in Bitcoin option data.Moreover,the Newton Raphson and Bisection methods are effective in estimating the implied volatility of Bitcoin options.However,the Newton Raphson forecasting technique converges faster than does the Bisection method.
文摘Modeling implied volatility(IV)is important for option pricing,hedging,and risk management.Previous studies of deterministic implied volatility functions(DIVFs)propose two parameters,moneyness and time to maturity,to estimate implied volatility.Recent DIVF models have included factors such as a moving average ratio and relative bid-ask spread but fail to enhance modeling accuracy.The current study offers a generalized DIVF model by including a momentum indicator for the underlying asset using a relative strength index(RSI)covering multiple time resolutions as a factor,as momentum is often used by investors and speculators in their trading decisions,and in contrast to volatility,RSI can distinguish between bull and bear markets.To the best of our knowledge,prior studies have not included RSI as a predictive factor in modeling IV.Instead of using a simple linear regression as in previous studies,we use a machine learning regression algorithm,namely random forest,to model a nonlinear IV.Previous studies apply DVIF modeling to options on traditional financial assets,such as stock and foreign exchange markets.Here,we study options on the largest cryptocurrency,Bitcoin,which poses greater modeling challenges due to its extreme volatility and the fact that it is not as well studied as traditional financial assets.Recent Bitcoin option chain data were collected from a leading cryptocurrency option exchange over a four-month period for model development and validation.Our dataset includes short-maturity options with expiry in less than six days,as well as a full range of moneyness,both of which are often excluded in existing studies as prices for options with these characteristics are often highly volatile and pose challenges to model building.Our in-sample and out-sample results indicate that including our proposed momentum indicator significantly enhances the model’s accuracy in pricing options.The nonlinear machine learning random forest algorithm also performed better than a simple linear regression.Compared to prevailing option pricing models that employ stochastic variables,our DIVF model does not include stochastic factors but exhibits reasonably good performance.It is also easy to compute due to the availability of real-time RSIs.Our findings indicate our enhanced DIVF model offers significant improvements and may be an excellent alternative to existing option pricing models that are primarily stochastic in nature.
文摘This paper deals with options on assets, such as stocks or indexes, which pay cash dividends. Pricing methods which consider discrete dividends are usually computationally expensive and become infeasible when one considers multiple dividends paid during the option lifetime. This is the case of long-term options and options on indexes. The first purpose of this paper is to assess efficient and accurate numerical procedures which yield consistent prices for both European and American options when the underlying asset pays discrete dividends. The authors then analyze some methodologies to extract information on implied volatilities and dividends from quoted option prices. Implied dividends can also be computed using a modified version of the well-known put-call parity relationship. This technique is straightforward, nevertheless, its use is limited to European options, and when dealing with equities, most traded options are of American type. As an alternative, the numerical inversion of pricing methods, such as efficient interpolated binomial method, can be used. This paper applies different procedures to obtain implied volatilities and dividends of listed stocks of the Italian derivatives market （IDEM）.
文摘Volatility is an important variable in the financial market. We propose a model-free implied volatility method to measure the volatility and test the volatility risk premium. The model-free implied volatility does not depend on the option pricing model, and extracts information from all the option contracts. We provide empirical evidence from the S & P 500 index option that model-free implied volatility is more accurate to forecast the future volatility and the volatility risk premium does not exist.
文摘如何对期权的隐含波动率曲面建模,一直以来是金融衍生品领域的难点问题之一。已有文献针对某些特定结构的随机波动率模型,推导出期权的隐含波动率解析表达式(Analytical formula of implie dvolatility,AFIV),但通用性差、或精度不够好。本文针对一般形式的随机波动率模型,给出获取高精度AFIV的通用方法:首先应用特征函数局部结构微扰法,推导出一般形式随机波动率模型的特征函数表达式;然后基于特征函数与期权价格的关系式,应用参数微扰法推导出隐含波动率的渐近表达式。数值实验和实证结果表明:(1)与同类最新研究(如Ait-Sahalia等,2021)[1-2]相比,本文方法在长到期期权上的定价精度有显著提升,在短到期期权上的定价精度也略优些;(2)与孙有发等(2022)[3]提出的基于特征函数局部结构微扰的期权定价方法相比,本文提出的AFIV方法不仅定价精度高,且拥有更优越的定价效率和参数校准的稳健性。本文提出的AFIV方法,不仅解决了众多性能良好的随机波动率模型在期权市场的运用问题,而且还为公平比较不同标的模型的市场刻画性能提供一种可行方法。
