| Exotic options are more complex and flexible than European options.When pricing exotic options,more of boundary conditions should be considered.Under Black Scholes model proposed by Black and Scholes,the underlying asset price follows geometric Brownian motion,which means the mathematical distribution of asset price is lognormal distribution.Under this model the pricing formula of exotic options can be obtained by changing the boundary conditions of the Black-Scholes partial differential equation of the European optionsHowever,the Black-Scholes model has some defects.For instance,the logarithm of asset return is often leptokurtic and skewed to left.Also the real asset price volatility has the Smile Property.Due to the defects of Black Scholes model,this paper tries to introduce an exotic options’ pricing model which is more ordinary and more universally-used than Black-Scholes model.Based on a new assumption projected by Nowak[24],the paper induced Miyahara’s the idea[7]of measure-transformFirstly,we use an assumption that asset price distribution follows a geometric levy process instead of geometric Brownian motion,the same with Nowak’s assumption[24].In order to make the geometric levy process more general,the logarithmic distribution of asset price in this paper is the sum of a Brownian motion with drift term and a linear homogeneous Poisson process,which is St=SeLt where Lt=μt+σWt+k1Ntλ1+k2Ntλ2+···+kDNtλDt.Secondly,due to the fact that the risk neutral pricing formula is only valid under Black-Scholes model,in order to apply the risk neutral pricing formula,this paper will carry on a measure transformation process.After comparing the advantages and drawbacks of some equivalent martingale measure,we choose the least q-moment equivalent martingale measure(the minimal Lq equivalent martingale measure)which is more convenient to be applied in the field of quantitative finance.According to the definition of the minimal Lq equivalent martingale measure,the paper gives a proof of the minimal Lq equivalent martingale measure’s existence condition.Then the paper gives the proof to the expression theorem of the minimal Lq equivalent martingale measure.It is concluded under such equivalent measure when q∈R\{0,1},the Levy triplet(σ2,b,v(dx))satisfiesγq*σ2+∫((1+(q-1)γq*(ex-1))1/q-1(ex-1)-xI|x|≤<1(x))v(dx)=β.where β=r-(b+σ2/2).Suppose γq*=μq*/q(q-1),then fq*=γq*σ,esq*(x)=(1+(q-1)γq*(ex-1))1/q-1,vMLHEMM(dx)=(1+(q-1)γq*(ex-1))1/q-1 v(dx).Under new measure Pq,the logarithm of asset price distribution can be represented by Lt=μ’t+σWtq+k1Ntλ1q+k2Ntλ2q+···+kDNtλDq.With the theorems and formulas of the minimal Lq equivalent martingale,the paper deduces the pricing formulas of five kinds of exotic options:Gap option:the paper derives the pricing formula of a gap option with strike price K1 and hurdle price K2: Then the paper derives the price formula of a chooser option: where Then the paper derives the price formula of an all-or-nothing option:ANCtq=D·Φ(dt,(2)q,m(s)),where Then the paper derives a descriptive price formula of a barrier option: Finally the paper derives a descriptive price formula of a forward start option whose strike price at time T is cSt: ·[e(μ1q-r)T+σ2/2(T-t)+km)Φ(d1q,m)-ce-rTΦ(d2q,m)],where... |
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