| Since the seventies of last century, there happened far-reaching changes in western financial system, which fastened the development of finance subject towards commercialization and engineering-orientation. And new demands, new products, new organizations and new subdivided markets sprang up in derivative financial instrumental markets. Investors hold various opinions toward inner value and future price fluctuation of option due to option's uncertainty of fluctuation ratio in objective capitalism market. Meanwhile the profit and loss of option demonstrates nonlinear-ity which makes option an indispensable instrument to satisfy various investment portfolios. All these factors turn option into one of favorite financial instruments, and innovation in option field emerges in an endless stream.In the eighties of last century, option trading on an organized exchange won investors' favor and exchanges kept on increasing. So financial institutes began to study new types of options to meet special demands as well as expand the scopes of business. Since the original option had limitation and lacking of flexible systems, which failed to meet the new demands in financial markets, there emerged a variety of options such as Asian options , barrier options , lookback options ,basket options and so on. All these burgeoning options broke the limitations of the original types with regard to path dependence, multiple capitalism objectives and date ofoption execution.Cliquet, a new type of option, appeared in this century. It reduced the investors' risks because it restricted the upper and lower limit of profit not only in the sectional but only in the global. Therefore, at the beginning of this century, which could be characterized by economic depression and sharp falling of stock market, new types of options enjoyed investors' favors and became to prevalent.The first cliquet options to be traded on a public exchange were S&P 500 bear market warrant with a periodic reset.Gray & Whaley (1999) (GW) proposed a closed-form solution for the European reset (cliquet) put option described above.as do Haug & Haug (2001) (HH). Another closed-form solution can be found for a different type of cliquet (ratchet) option with n periodic reset.Let T be a future point in time, and divide the interval[0,T]into N subintervals called reset periods of lengthAi = Tn - Tn_i,where{Tn}^=0 , To = 0 , TN -Tare called the reset days. The return of an asset with price processStover a reset period[Tn_i,Tn) isthen denned asRn = max(min(Rn, C),F) are returns truncated at some floor and cap levels F and C . A general cliquet option has a payoff Y at time T ofNY = B x min(max(^2%i,Fg),Cg). n=iwhere the global floor Fg and global cap Cg are minimum and maximum returns respectively and B is a notional amount which is set to one for the remainder of this thesis.This very general form of cliquet option is not very common on the market but removing the global cap Cg gives the popular cliquet option with global floor, which pays the holderNTherefore ,cliquet option with global floor is a kind of path-dependent option. Denote J% as path-dependent variable:/?? CA F\ i P \T 1 T \ n — 1 ? ? ? N__ rpThe payoff of cliquet at time T depend on R4, in each reset period. Denote a new stochastic variable-RS is the value of the underlying instrument of the reset days before St. as,￡ € [ra_i,ro) , S = Sra.lta = 1,2,-?? ,iV,and tn -> T^Rj,- = ^-. Jt can be rewritedj =iT,iZ{?*x(rnin(RT--l,C),F), t e [Ta.uTa),a = 1, ? ? ? ,JV. \￡A = T.The price of the cliquet option depend on R,J and t,V=V(R, J, t). Suppose the change of St suite one period and two-state model,aslo R suite one period and two-state model,and periodical changed.When t = Ta — taM,a = 1, ? ? ? ,N — 1,begin a new period , R in two different reset periods is independent.When tn E [TQ_i,Ta), J"fcn is stable, kn e /? is the index of path-dependent variable Rf, In is the muster of the index,because Jffc is stable, denote it as J"^1N tN _ VJQ,kNM - Z￡Y = max(JT, Fg) = max(j￡kNM, Fg).max(mm(Rf4 - 1, C), F) = 70ak u.when t = TaThe change process of ^and Jn have been established , use A-hedging and the risk-neutral probability measure, put forward the binomial tree method for valuing cliquet option.R?\ A,,? ^yn+lm+lthatVi+V,kn+1 =when t = Ta{a = 1,2, ? ? ? , N - 1),whenT =In each reset period tn 6 [Ta,Ta+i) give portfolion = v - as,choose the suitble A , make II is risk-free,we can getq = ^3 , 1 — q = jj^d , p — 1 + rjfjtf , r is risk-free rate. Now, the binomial tree method for valuing cliquet option is gived.In section 3.3 of this thesis give the concrete algorithm,and in section 3.4 give a simple example to explain how the algorithm practice.In chapter 4 we derive a partial differential equation for the cliquet with global floor. Change of variables x = log(|),the number of dimensions can be reduced from four to three.Theoreml.3 Let x = log(|) , the price V = V(x, J, i)satisfies the partial differential equationV(x,J,T-) = V(O,J + max(min(ex-l,C),F),Tn)Discretization:At ^2 (Az)2+ (r — ^-)—fc+i' ' ' i+l2Ax— ' '1+1----rV{x%, J", ij) = 0 , ^n-i 0.Theorem2.3 (Lax Theorem)For the initied-boundary value problem of linear partial differential equation, if the difference scheme is compatibal, the convergenceand the stability of this difference scheme have equivalence relation this difference scheme.Theorem 2.4then when At ->? 0lim VA(R,J,t) =UVa(-R, J, t) is the linear extend of V(R, J,t)) ?... |
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