| According to public data,since 1998,the average annual turnover of options market is about 90 tril ion US dollars abroad,of which the average proportion of OTC options in the option market is 63%,which is always higher than that of exchange options.In the domestic market,in recent years,with the support of the regulatory policy,the institutions mainly securities firms are also vigorously developing FICC OTC business.According to the statistics of China Securities Association,in March2019 alone,the nominal principal of OTC options transactions carried out by securities companies amounted to 138.241 billion yuan,and the unsettled nominal principal of 408.129 billion yuan.The OTC market deals with contracts tailored to meet the individual needs of customers,in which is different from the standardized contracts in the exchange market.Shout options studied in this paper are one of them.At present,a large number of mutual fund products embedded shout options have been issued in European and American markets.Therefore,whether for individual customers,institutional investors or even regulators,the demand for reasonable pricing of shout options is more urgent.The study of shout options in this paper can provide reference for pricing research of other exotic options,and it also has great significance for domestic institutions to design shout options products and realize price discovery function.In the financial market,price jump is a common phenomenon.At present,about the research of shout options,the description of underlying asset price path is still at the level of Brownian motion,and there is no description of this price jump phenomenon.In this paper,jump-diffusion model is introduced to describe this phenomenon under risk-neutral measure.This paper constructs the shout options value model under the assumption that the underlying asset price path obeys jump-diffusion model.Compared with the partial differential equation(PDE)derived from the pricing of standardized options,this paper first derives the PDE for the value of shout options by using It?-Doeblin formula when the jump size is discrete random variable.Aiming at the partial differential inequalities(PDIs)in which the value of shout options satisfies the discrete part of the compound Poisson process through variable substitution.Then,similarly,we obtain partial integro-differential inequality(PIDI)satisfied by the shout options value when the jump amplitude is a continuous random variable.In view of the unique product characteristics of shout options,it is proposed that the value of new contracts received by investors after exercise of their rights is equal to the value of the original contract and the dividend of the underlying assets caused by the exercise of call rights by the holder of shout options.At the same time,because the holder exercises the shout power,the shout options holder receives fewer shouts right for new contracts,so the value of new contracts is less than that of the original contract.Finally,the boundary conditions of shout options are studied.Based on the above research,the value model of shout option is constructed.Then,under the assumption that jump size is a continuous random variable and obeys exponential distribution and logarithmic normal distribution respectively,a simplified form of partial integrodifferential inequality for shout options value is obtained.Finally,the penalty function is introduced and the equation is discretized to obtain the discretization equation of the value of the insurance product embedded in shout options,thus realizing the reasonable pricing of this kind of insurance products.In the part of algorithm design,the calculation efficiency and valuation accuracy are proposed and used.The existing options pricing algorithms are summarized and compared including binary tree algorithm,trigeminal tree algorithm,implicit difference method,explicit difference method,quadratic approximation algorithm and other numerical algorithms,as well as binding method,analog random number method and least square method.The results show that under the same parameters,the binary tree algorithm is the fastest in the numerical algorithm.It only takes 0.328 seconds to calculate the value of a shout call option,which is faster than 1.047 seconds of the trigeminal tree,0.541 seconds of the explicit difference and 0.572 seconds of the implicit difference.For the estimation error,the quadratic approximation method has the largest error,which is 2.9E02.The estimation errors of binary tree and trigeminal tree are smal er,4.5E04 and 3.3E04,respectively.Comparing comprehensively,the numerical algorithm performs best in the shout call option valuation.For the simulation algorithm,under the same parameters,the calculation error of the least squares algorithm can be controlled at an astonishing 0.001,and the estimation accuracy is much higher than that of the bundling method 0.008 and the simulation tree algorithm 0.012.Similar conclusions can be drawn for shout put options.On the basis of summing up the existing algorithms and combining the unique characteristics of shout options,Howard high-order algorithm is designed to solve the value model of shout options.Firstly,the integral region in the model is divided into two parts.The differential operator is introduced into the differential term part,and the fourth-order discretization form is obtained by central difference approximation.The cubic spline approximation is used for the nodes in a given time interval.Then,the original model is transformed into a minimization problem.Given the value of a node,the value of the shout options of the next node is obtained by solving the discrete problem.The Howard algorithm generates a series of approximations for the shout option value on the node.If the coefficient matrix is M-matrix,it has a finite terminal.Because the shout option’s return function is not smooth,in order to achieve the fourth-order convergence rate,the grid refinement technique is used near the execution price.After constructing the Howard higher order algorithm for the shout option value model,some examples are given to il ustrate the convergence speed and the accuracy of the evaluation of the Howard method in solving the shout option value model.The results show that Howard cubic spline method has a fourth-order convergence rate because the logarithmic-logarithmic graph is parallel to the fourth-order convergence slope,and Crank-Nicolson time-step finite element method has a second-order convergence rate,so the convergence rate of Howard algorithm is faster than that of traditional finite element method in dealing with shout option valuation.In terms of estimation accuracy,the higher-order discretization of space and time means that the higher-order method has higher accuracy than the second-order discretization method under the same time step and number of spatial nodes.In the example,when 480 spatial steps are used,the error of Howard cubic spline algorithm is only 10E-7,which is much smal er than that of finite element method 10E-3.The same result is obtained when other spatial steps are used.When different parameters are used for comparison,the same conclusion is drawn from the results of the example.In view of the unique product characteristics of shout options which are different from other options,the index Kappa reflecting the sensitivity of its value to the change of strike price and the index Script reflecting the sensitivity to shout times are proposed.This paper studies and elaborates the Greek values of shout options: Delta,Theta,Gamma,Rho,Vega,Kappa and Script,their concepts,mathematical formulas and economic meanings,and explores the relationship between these Greek values.In this paper,Delta dynamic hedging in practice is studied.Aiming at the two major drawbacks of complete hedging strategy: the high cost caused by frequent position adjustment and the inability to meet the needs of customer asset value preservation and increment,a solution to optimize parameters and algorithms is proposed.Taking the standard deviation of hedging costs divided by the market value of the portfolio as a measure of the hedging effect,the hedging effect is 0.53 when adjusting positions every day,which is higher than that corresponding to other adjusting frequencies;from the perspective of cost,the annual transaction cost is only 3% of the initial market value of the portfolio,which is within acceptable range.General y speaking,daily adjustment is the best adjustment cycle.In the application part of shout options,two scenarios are proposed.One is to address the shortcomings of the traditional CPPI and TIPP portfolio insurance strategy,that is,when the risky asset price loses substantially at the early stage of the portfolio establishment,the buffer zone established by riskless assets will soon be broken down.Even if the subsequent risky asset price rises sharply,the CPPI and TIPP strategies will not have the opportunity to invest again.The portfolio insurance strategy based on shout put options(PIBOSO)is proposed.The core idea of this strategy is to give asset managers the right to lock in the minimum return in advance,so they can execute the option in time when the market is favorable to the portfolio,so as to lock in the return in advance and reduce the loss in time when the market is unfavorable to the portfolio.Empirical research regards HS300 index as risk asset and considers the portfolio performance after embedded shout put option.Assuming that the risk-free interest rate is 1.5% and remains unchanged during the inspection period,the HS300 index market since 2016 is divided into three stages: unilateral rising market is selected as2016/2/1-2018/1/24;unilateral fal ing market is selected as 2018/1/26-2019/1/7;volatile market is selected as 2019/5/6-2019/8/30,transaction cost is set as 0.3%,Risk Multiplier m is set as 2,3 and 4,insurance ratio parameter lambda was set at 80%,90% and 95% respectively.The Sharp ratio is used as a measure to compare the performance of PIBOSO portfolio insurance strategy and traditional CPPI and TIPP strategy.Empirical results show that,under the same parameters,the Sharp ratio of PIBOSO can reach 4.56 when the market is rising unilateral y,while that of CPPI and TIPP is only 1.84 and 1.14 respectively.In the case of unilateral downturn,although all three strategies have achieved the effect of preserving value,the yield of PIBOSO is negative,the highest yield of PIBOSO is-3.85%,and that of CPPI and TIPP is-4.37% and-4.58% respectively.When the market is turbulent,the Sharp Ratio of PIBOSO strategy can reach 6.80,while that of CPPI is negative,and that of TIPP is0.12.From the above analysis,we can see that PIBOSO portfolio insurance strategy achieves the goal of hedging,and its performance has been greatly improved compared with traditional CPPI and TIPP strategies.The second scenario is to study the insurance products with embedded call options,which are widely distributed in Canada and the United States.Considering the impact of insurance payment on product value,a valuation model for the value satisfaction of such insurance products is proposed.The effects of call number and volatility of underlying asset price on product value are analyzed in the form of numerical experiments. |
