Finite Element Method For Pricing Rainbow Option

As a derivative tool,options are a great advantage of versatility and can be applied to a variety of investment strategies.Options can diversify risks,help strengthen the overall anti-risk capabilities of the financial market,and enhance the soundness of the financial system.It is precisely because of the continuous innovation and development of the do-mestic option futures market that strengthening the study of the options market is particu-larly important for improving the capital market business.With the rapid development of financial markets and the increasing number of options,the option pricing theory is also constantly improving.In the 1970s,since the birth of the Black-Scholes option pricing formula,scholars have substantially increased their research results on option pricing for-mulas.These include European options,Asian options,American options and so on.In recent years,the international financial market has emerged a large number of new vari-eties,variant options derived from the changes,combinations,and derivatives of standard options.Multi-factory options are one of them.For example,rainbow options,basket options,etc.are all such options.We derive general analytic approximations for pricing European rainbow options on N assets.Option trading is a brand new derivative product and effective risk management tool based on futures trading.It has unique economic functions and high investment value.One is that options are more conducive to spot business operations and hedging.By pur-chasing options,they can avoid the risk of additional margin in futures trading.Second,options are conducive to the development of order agriculture and solve the”three rural issues”.The government guides and encourages farmers to enter the options market by providing farmland with financial subsidies for the option premiums and payment pro-cessing fees.Third,futures investors can use options to circumvent market risks.Options can be futures”reinsurance”.Different combinations of the two can constitute a variety of trading strategies with different risk preferences,providing investors with more trading options.Rainbow option,first proposed by Margrabe（1978）,is a kind of option which in-volves many kinds of risk assets,and the aim is to get the best return on a variety of assets.Margrabe derives an exact solution for pricing an exchange option,under the assumption that the price of two assets follow correlated log-normal processes.Since a linear combi-nation of log-normal processes is no longer log-normal,we can reduce the dimension of the correlated log-normal processes to one only when the strike of a two-asset option is zero.Johnson（1987）extends the two-asset rainbow option pricing formula provided by Stulz（1982）to the general case of N cases.Rubinstein derived the rainbow option pricing formula under the risk-neutral assumption,which depends on the lowest or highest price of the underlying asset.Hucki and kolokoltsov used game theory to study the pricing of rainbow options with fixed transaction costs.Meng et al.propose a rainbow option pricing method based on sine series expansion.Dockendorf et al.study the pricing of European rainbow options by stochastic co-integration model.Rainbow options differ from traditional American and European options in that they are excellent tools for hedging risks arising from holding multiple assets.Rainbow op-tions are most commonly used to assess natural resources because they depend on both the price and quantity of natural resources in stock.Rainbow options are divided into three categories:better-of options,outer performance options,maximum call options and minimum call options.In this paper,we try to use numerical method to value European rainbow options.Option pricing is an ancient issue.As early as 1900 years,Louis Bachelier published the dissertation”Speculative Trading Theory”.It was recognized as a milestone in modern finance.For the first time in his dissertation,a random walk model was used to give a random model of stock price operation.In this paper,he mentioned the pricing of options.In 1964,Paul Samuelson（Nobel Prize winner in economics）amended L.Bachelier’s model.Replace the stock price in the original model with the stock return.If S_trepresents the stock price,then_St^dStrepresents the return of the stock.The stochastic differential equa-tion proposed by P.Samuelson corrects the unreasonable situation in the original model that makes the stock price negative.Based on this model,P.Samuelson also studies the pricing of call options.Suppose V is the value of option,S is the stock price,K is the strike price,T is the maturity time,then the value of V is related toα_c,α_s.Hereα_c,α_sare the mathematical expected value of the return of the original asset S_tand the option price V_tat time t=T,respectively.These two quantities depend on the individual preferences of investors.Therefore,although this formula is beautiful,it can not be used in actual transactions.In 1973,Fishcher Black and Myron Sholes established the pricing formula of call options.Hereα_c,α_sdo not appear,but instead riskless interest rate r.The innovation of this formula is that it does not depend on the preferences of investors.It leads all investors to a risk-neutral world in which risk-free interest rates are returned.Black-Scholes model is based on the ideal market which is quite different from the real market.In the past twenty years,economists have tried to find a more realistic option pricing model under these conditions.Many excellent results have been achieved,which greatly enrich the option pricing theory.Since the 1990s,especially in recent years,many economists have made extensive research on Option Pricing in incomplete markets,abnormal price jumps of underlying assets,or the variance of underlying assets return is not constant,and many important research results have been achieved.Among them,it is worth mentioning that it greatly enriches the relevant theory of option pricing model.On the basis of classical models,many new models are put forward.Starting from the Black-Scholes model,the option pricing theory has a history of nearly 40 years.With the development of modern financial markets,the market has cre-ated a lot of complex options products,along with stocks,bonds,foreign exchange and futures.Rainbow option is an important new financial derivative product.In the devel-opment process of market,the rate of return and price of the option products based on the underlying assets have also changed,and the pricing problem have always been one of the core problems of financial mathematics.The numerical methods of option pricing are divided into five categories:analytical solutions,tree methods,numerical methods for partial differential equations,Monte Carlo methods,and Fourier transform methods.（1）Analytical solution methodActually,according to the known stochastic differential equation（SDE）model,we then solve the process of the expression of this random process function.If we can find the analytical solution of this SDE,then the price of an European pathless-dependent option is The discounted expected value at the time of final value.This is an analytical solution to the option pricing.The advantage of this kind of method is obvious.Once the analytical solution exists,then the option price formula of the calculation speed is very fast,no matter to do fitting and optimization will be efficient on quality,and shortcomings of this kind of method is obvious.That is,for most of the model and the most exotic options,analytical solution may not exist.（2）Tree methodInform you of the volatility of a target asset,then you can construct a Binary tree up and down fluctuations of N segments.Then use the inverse to get the option price for the initial time.The tree model has the advantage that any continuous-time model cannot replace it.That is every pricing,in the tree model,regardless of American,European,path-dependent,singular,through the Backward Induction Principle The price is always ac-companied by an explicit hedging strategy.In the continuous-time model,the problem of the continuous-time hedging strategy is a Backward Stochastic Differential Equation（BSDE）problemOn the other hand,the disadvantages of the tree model are also obvious.The high-dimensional problem tree model cannot be solved.Therefore,for the problems of multiple target assets,especially assets with a correlation coefficient,we can only appeal to PDE model.（3）PDE methodIn fact,different random models correspond to different PDE.BS PDE is just a PDE expression for a single asset that conforms to the geometric Brownian motion stochastic model.As for options,we often know the payoff of their final maturity date,so we use payoff function as the final value of PDE.If there are analytic solutions to PDE,the optimal solution is also the analytic solution.However,if the analytic solution does not exist,we must resort to numerical methods.The disadvantages of PDE methods are two main problems:path dependence prob-lem and high dimension problem.Many of the PDE forms of path-dependent problems are troublesome or even unspeakable.If the numerical method of PDE rises from the plane grid to the spatial grid,it is not only complicated in complexity,but also more difficult to control in edge value conditions.（4）Monte Carlo methodThe Monte Carlo method is the most widely used method at present.Because there is no option price with advanced exercise attribute is actually an expectation,we can simulate the many routes and use the average number to estimate the real expectation.For American-style or Bermuda-type options with advanced exercise attributes,its option price is actually a random optimization problem.However,the shortcomings of Monte Carlo are also obvious:because we need to simulate millions of paths,and we need to do path calculations for exotic options,Americans need to do more regression.The Monte Carlo method has become synonymous with long computing time.（5）Fourier methodThe Fourier method is also called the eigenfunction method.For many models,their eigenfunctions are often expressed explicitly.We can use the inverse Fourier transform to obtain the density of the original random variables,and thus achieve the purpose of solving the option price.In this paper,we derive efficient numerical methods for pricing European rainbow options on two assets.We can also express the option as a sum of risk assets and risk-assets exchange options.The underlying asset prices are assumed to follow log-normal process-es.Based on the principle of no arbitrage,stochastic differential equation and Black-Scholes model,we obtain the partial differential equation for pricing two-asset rainbow options.The continuous and discontinuous Galerkin discrete Finite Element method was applied to outer-performance options,better-of options and maximum call options respec-tively.The error analysis of the finite element method of pricing model was carried out,and the rationality and effectiveness of the method were verified by numerical examples.