外文翻譯--違約風(fēng)險的歐式期權(quán)定價模型（英文）

1、第 3 8卷 第 6期 上海師范大學(xué)學(xué)報 (自然科學(xué)版 ) Vol. 38, No. 62 0 0 9年 1 2月 Journal of Shanghai Nor mal University(Natural Sciences) Dec . , 2 0 0 9Prici ng models for default - risky european optionsFU Yi , ZHAN

2、G Ji2 zhou, WANG Yang(Mathematics and Science College, Shanghai Nor mal University, Shanghai 200234, China)Abstract: W e establish the models for optionswith constant and variable rate parameter . An explicit p ricing fo

3、r mu2la for the models is obtained by the method of PDE . W e also set up the model for the op tion with stochastic rate pa2rameter, and the Monte Carlo method is used for the model .Key words: credit risk; European opti

4、on; PDE; Monte CarloCLC num ber: O23 Docum ent code: A Article I D : 10002 5137 (2009) 062 05732 07Received da te: 20092 062 10Foundation item : National Basic Research Program of China (2007CB814903) ; the Science and

5、 Technology Comm issionof ShanghaiMunicipality grant (075105118) ; Leading Academ ic D iscip line Project of Shanghai Nor mal University (DZ L707) ;Scientific Research Project of Shanghai Nor mal University ( SK200933, S

6、K200812).Biography: FU Yi(1980 - ) , male, lecturer, Mathematics and Science College, Shanghai Nor mal University .1 IntroductionGenerally, options have been considered the financial derivative without credit risk becaus

7、e the marginsystem plays an i mportant role in avoiding the risk . However, there is no margin system in the over - the -counter markets . The holders have to face with the potential credit risk that the option writers d

8、o not dischargetheir contractual obligations at maturity . Thus the credit risk should be taken into account when we decide theprice of this kind of options .There are generally t wo approaches to model the credit risk .

9、 The first approach is the structuralmodels inwhich default is deter m ined based on the evolution of the assets and the liabilities of the fir m. Examples of thisapproach begin with B lack and Scholes[ 1 ] and Merton[ 1

10、0 ]. In their model, they posit a single point defaultboundary, and default can only happen at maturity . To i mprove thismodel, B lack and Cox[ 2 ] allow default tooccur at any ti me . Longstaff and Schwartz[ 8 ] extend

11、 the risky debt model of B lack and Cox to allow stochasticinterest rates to follow the O rnstein– Uhlenbeck process . The second approach is the reduced - for m modelsin which the default process is regarded as the exog

12、enous Poisson process . The reduced for m model was firstintroduced by Jarrow and Turnbull in 1992[ 5 ]. In this model, the first ti me of bankruptcy which is exponen2tially distributed with an intensity parameter result

13、s in a jump process . Longstaff and Schwartz[ 9 ] regarded de2fault spreads as mean reverting process . The model in Jarrow et al[ 6 ] is the first model that incorporates thecredit rating infor mation into valuation met

14、hodology . Duffie and Lando[ 3 ] considered the hazard - rate processa function of the value of the fir m, conditional on accounting data . The defaultable clai m is discounted at a de2fault adjusted short - ter m intere

15、st rate for the risk - neutral in [ 4 ].In this paper, our work is based on reduced - for m models . The main advantage of this app roach is itscomputational tractability because it is restricted within observable variab

16、les in contrast to structural models .第 6期 傅 毅 ,張寄洲 ,王 楊 :違約風(fēng)險的歐式期權(quán)定價模型d1 - σ T - t ( see [ 1 ]).2. 2 The m odel with var i able rate param eterλ ( t)2. 2. 1 Basic assumptionsThe default event follows an inhomogeneou

17、s Poisson process . In general, the rate parameter may changeover ti me . In this case, the generalized rate function is given as λ ( t) . Now the expected number of events in[ t, t + dt] isλ t, t + dt = ∫ tt + dtλ ( t)

18、dt. Thus, the number of events in the ti me interval [ t, t + dt] , given asN ( t + dt) - N ( t) , follows a Poisson distribution with associated parameterλ t, t+dtP [ (N ( t + dt) - N ( t) ) = k ] = e- λt, t + dt ( λ t,

19、 t + dt )kk! , k = 0, 1, 2 .2. 2. 2 Establish and solve equationSi m ilarly, we establish the equation by the hedge of the portfolio. Becauseλ ( t) is ti me dependent, wecan getλ t, t + dt = ∫ tt + dtλ ( t) dtλ ( t) dt,P

20、 [ (N ( t + dt) - N ( t) ) = 1 ] = e- λt, t + dt ( λ t, t + dt )1 λ ( t) dt.Then,(1 - λ ( t) dt) [ ( 5 V5 t + 12σ2 S2 52 V 5 S2 ) dt + 5 V5 S dS - Δ dS ] +λ( t) dt( - V ) = r(V - Δ S) dt.Therefore, the call option functi

21、on satisfies5 V 5 t + 12σ2 S2 52 V 5 S2 + rS 5 V5 S - ( r +λ( t) ) V = 0 (0 ≤ S < + ∞ , 0 ≤ t < T)V (S, T) = (S - K)+. (2)To s olve the equation, let u =Veβ ( t) and y = Seα ( t) , whereα ( t) = r(T - t) andβ( t) =

22、 r(T - t) + ∫ tT λ ( t) dt .Then the equation (2) becomes5 u 5 t +σ22 y2 52 u 5 y2 = 0 (0 ≤ y < + ∞ , 0 ≤ t < T)u t = T = ( y - K)+. (3)Let x = ln y. Then we obtain by (3) that5 u 5τ - σ2252 u 5 x2 +σ225 u 5 x = 0u

23、 τ = 0 = ( ex - K)+, (4)Let u = w e- σ28τ + 12 x . Then the above equation (4) is rewritten as5 w 5τ - σ2252 w 5 x2 = 0w τ = 0 = e- x 2 ( ex - K)+ . (5)Consequently, the solution of (5) isV = Se- ∫Tt λ ( t) dt N ( d1 ) +