Research On The Pricing And Application Of Compound Option Under Fuzzy And Random Environments

Compound option is widely used in financial derivatives pricing and real options.At present,the research on pricing and application of 2-fold compound option under stochastic model has achieved considerable results,because multiple-fold compound option has complex structure,the studies on multiple-fold compound option pricing are mainly confined to the Black-Scholes model and the sequential compound call options.On the other hand,the uncertainty of financial markets includes randomness and fuzziness,and the two sides permeate each other and combine with each other.In recent years,some scholars have introduced fuzzy theory into the pricing of options and other financial derivatives,which is a useful and necessary supplement to the traditional pricing methods of financial derivatives based on the stochastic theory.In this dissertation,the pricing and application of compound option under fuzzy and random environments are studied deeply.The main research contents and conclusions are as follows.(1)The major challenge in deriving the closed-form pricing formula for n-fold compound options is to calculate the complex multivariate normal integrals.Firstly,in order to overcome this difficulty,this dissertation proves several mathematical expectation formulas that play an important role in the pricing of compound options and other derivatives.Secondly,a geometric Levy model is adopted for modeling the underlying asset price dynamics.The jump height of the underlying asset price is established as a constant,and the logarithmic price is the sum of a Brownian motion with drift and a Poisson process describing jumps in the price.Using Esscher transformed martingale measure as equivalent martingale measure,based on the mathematical expectation of multivariate normal random variables,the analytic pricing formula of n-fold compound call options is derived.Finally,this dissertation generalizes the sequential compound call options(SCCs)to sequential compound options(SCOs)which are defined as compound options on(compound)options,where the call/put property of each fold can be arbitrarily assigned,and derives the n-fold SCOs pricing formulas for the diffusion model and the log-normal jump-difTusion model,respectively.(2)Considering the uncertainty of financial market including randomness and fuzziness at least,the dissertation studies the problem of compound options pricing under fuzzy and random environment,which mainly includes four aspects.Firstly,the interest rate and volatility are regarded as fuzzy numbers,considering the subjectivity and uncertainty of the interest rate and the volatility in reality.Based on the pricing formula of 2-fold compound option in the diffusion model,the fuzzy price of 2-fold compound option is obtained by replacing the arithmetic operations with the corresponding fuzzy operations.According to the definition of fuzzy numbers operation and the crisp possibilistic mean value,the fuzzy price interval of any level set and the crisp possibilistic mean value of compound option price are given,respectively.In addition,the confidence problem of calculating any given option price is transformed into an optimization problem.Secondly,an asymmetric triangular fuzzy stochastic process of the underlying asset price is constructed based on the geometric Levy model.It is assumed that the left spread and the right spread of’ the triangular fuzzy number are equal to the product of the left fuzzy factor and the right fuzzy factor and the underlying asset price,respectively.An arbitrary level set of the n-fold compound call options fuzzy price is obtained by calculating the probability expectation and defuzzification of the fuzzy payoffs.The advantage of this method is to avoid the fuzzification of the parameter in the pricing formula one by one,and the arbitrary level set of the fuzzy price is easily obtained.Thirdly,taking into account decision maker’s subjective judgment,two estimation methods for fuzzy price are presented:First,the fuzzy goal is introduced to represent the decision maker’s satisfaction with the expected option price,the fuzzy goal means a kind of utility function for expected price,the range of prices such that the reliability degree of the expected price is greater than the degree of buyer’s satisfaction is presented.Second,this dissertation uses fuzzy measure to estimate the confidence that a fuzzy number takes values in an interval values,and presents the probabilistic mean values with pessimistic-optimistic index to represent the decision maker’s degree of pessimism.Fourthly,some numerical examples are given to illustrate the obtained theory,which show that the results of fuzzy and random models have a certain degree of reliability and validity,the compound option pricing model considering the decision maker’s subjective judgment under fuzzy and random environment increase the flexibility of investment decision,and has more practical significance.(3)Firstly,according to the investment process of new drug research and development project,it is divided into seven stages,and the real option characteristics of drug R&D projects are analyzed;Secondly,taking into account the technical risks throughout the development process,and the existing literature seldom consider the R&D project failure risk,this dissertation consider the risks in a jump diffusion model of project value,and that in a period of time interval,the jump means that the project technical failure,the value of the project is to jump to zero.On this basis,a 6-fold compound real option pricing model considering the risk of technical failure is constructed to describe the value of pharmaceutical R&D projects.Then,considering the drug R&D project has the characteristic with long period and high risk,the investment costs and the project’s cash flows are expressed as fuzzy numbers,the fuzzy compound real option model is used to evaluate the project value,and an arbitrary level set of R&D project value is given.Finally,a case of drug R&D is used to explain the obtained theory,and the sensitivity analysis of the project value with respect to cash flow,volatility,technical risk intensity and dividend without considering the technical risks and technical risks are presented,respectively.The investment costs and the cash flows of the project are regarded as trapezoidal fuzzy numbers,and the membership function of the fuzzy R&D project value,possibilistic mean value with different weighting function,the probabilistic mean values with pessimistic-optimistic index when considering technical risk are obtained.This dissertation studies the compound option pricing and application under fuzzy and random environments,makes up for the deficiency of the existing literature and obtaines some valuable conclusions,as well as provides a new idea for further research on the pricing and application of other financial derivatives under uncertainty.