Sparse Grid Method For Pricing RMB Yield Enhancement Product

RMB yield enhancement product is a kind of financial product based on thecommon deposit of bank’s customer, and combined with the structure of derivatives.It also links up with paraments of international or domestic financial market suchas the exchange rate, interest rate, and makes the customers to take some risks butcan achieve more interests than the common deposits.In this thesis, we shall discuss the first pure RMB financial products in thedomestic financial market of MinSheng bank, which is the third phase of MinShengwealth RMB Feifan financial product and links up with the american dow Jonesstock index.Assume the american the dow Jones stock index (denoted by St) conform tothe Geometric Brownian motion, the riskless interst rate r>0is a constant, thereare no arbitrage, transaction costs and tex in the market, and both the bank andthe customer can’t breach the contract. Under above assumptios, the problem canbe treated as a price model of lookback option with discrete observation intervals.Firstly, we assume that there is only one observation interval, then the termi-nal payof isand V=V (S, M, t) can be regarded as a lookback option with the underlying assetSt. By Making the portfolio Π=V S, using-hedging principle, choosingthe appropriate, letting Π be riskless in (t, t+dt) and through Ito formula, wecan obtain the price model for the one-observation interval. For more observatonintervals, by similar arguments and calculating from the last time interval to thefirst interval, we can also acheive the corresponding mathenmatical model. In chapter2, we have derived the formula of n observation interval option:When N=24, it means the maturity date of the product is two years, which impliesa48-dimension integration. Noting the traditional monte carlo method work wellwhen dimension is no more than four, so in chapter3and chapter4, we shallpresent importance sampling Monte Carlo method and sparse grid method to solvethe numerical solution respectively.Importance sampling reduce the variance by the change of the probabilitymeasure. Consider the following problemwhere ρ is the probability density and h is a multivariate function. The classicalMonte Carlo method usenQto estimate the problem with points x1,, xnindependent draws from ρ. Here,we change the probability measure and let g be another probability density onsatisfies ρ(x)>0g(x)>0for all x∈d. So we can represent I(h) asI(h)=h(x)ρ(x)g(x)g(x)dx. If points x1,, xnare also independent draws from g,importance sampling method with probability measure g is given byNow, we should compare the variances of the importance sampling MonteCarlo method and the Monte Carlo method. Since these two methods’ expection are the same, we only need to compare second moments. The second moments ofimportance sampling method isand the second moments of Monte Carlo method is E[h(x)2]. We can observe thatwhether the variance can be smaller depends on the choice of g, and the choice ofg is the key to importance sampling Monte Carlo method. Importance samplingfunction g(x) should satisfies the following properties1. When h(x)=0, g(x)>0.2. g(x) should be close to the proportion of|h(x)ρ(x)|.3. It needs to be easy to simulate from g(x).4. It should be easy to cumpute the density g(x) for any points in the interval.5. It should be that g(x) gets small much slower than ρ(x) in the tails.Sinceas the probability density funcion of variables (X(1), Y(1), X(2), Y(2),, X(N), Y(N)).Then we can get the numerical solution by use importance sampling Monte Carlomethod.In chapter4, we introduce the algorithm of tensor product problem of multi-variate funtion and sparse grid method.Univariate quadrature formulas can be extend to multivariate problem withthe product rule. Now assume that each univariate quadrature formula has dif-ferent occuracy levels in each dimension, can be denoted by multi-index[i1,, id], and the notes Xi1,, Xid, weights ωi1,, ωid, are decided by the respecting one-dimensional quadrature rules Vi1,, Vid.The tensor product method needs to evaluate the function f at all grid pointsXm Xm, if the univariate quadrature rules is defined on m nodes, tensorproduct method for d dimension problem requires mdevaluations of function f,the The computational costs is growed with dimension in exponential speed,this iscalled “curse of dimensionality” of multi-dimension problem in many bibliographies,it’s computational cost is too much.Sparse grid method costs much smaller number of function evaluations thanthe tensor product method in multi-dimension problem. It is combined by univariatequadrature rules, this idea is first put forward by Smolyak in1963.The accuracy of the sparse grid method: It can be assumed that a univariate quadra-ture rule sequence V={Vi: i∈N}is existed, where Viis exact for, ω and allunivariate polynomials of order no more than2i1. It can be infered that theSmolyak rule using V is exact for d-variate polynomials of total order no more thanIf the sparse grid method above is based on Gaussian quadrature rules in eachdimension, the logarithm of nodes Xd,kis of order O(log(d)). The multi-dimensionnumerical integration of sparse grid method is: where Xikis the univariate point set of accuracy level ik, it includes more than ikpoints. the weight of xkin Iikis ωik, and the term in the braces above is the weightof grid point in d dimension., it depends on Ndq. If one point appears many times,it’s weight equals the sum of these weights on all the combinations of points.Chapter5is devoted to the numerical experiment, and we have shown hathsparse grid method is better than importance sampling Monte Carlo method in thisoption problem.