Consider the standard Black-Scholes model with bond price B =ert,and shareprice S=Soent+ow for t∈O,T,where So>,o>.0 and a are constants,and W=(W)t>o is a Wiener process under a probability measure P.(1)We want to compute Vo,the price at t=0 of the European type optionwith payoffh:=Lif mintep.]S≤K0 otherwiseat expiry date T,where L>0 and K 0 are some constants such thatS>K.(a)Using the main theorem on pricing European type options,show that6=le-P((+a)≤)where a:=-,b:=o-1 In(K/So).(b)Using Girsanov’s theorem show that[10 marks]V%=Le-rTE(1.n W≤qe-where a and b are the constants defined in (a).(c)Calculate the probability density function p=p(z,)[10 marks]dom vector (Wr,.7 W),knowing that for F=for the ran-distribution function of (Wr,min,ep.W.),we haveF(z,y),theF(r,y)=P(Wr≤x,minW,≤y)E0.TP(W≤x)P(min elo.r W,≤y)-P(Wr≤2y-x)if r
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