Question 3 [20 marks] Consider two risky assets $(1) and (2) with dynamics under Q dsc") = $!" ((-+** ) at +01dw, .). T + 2 7. a TT ds/2) = ) (rdt +02dW+), where 0 1 such that Z4 := e-rt(s),- (s?))1-a is a martingale. (b) For 1 > 1, define the stopping time 71 := inf{t > 0:5"> 15). Prove that Ele (597) $2))+) = (1 - 1)E [e-752) and B[e="+(54) 5(2)+] = " 1}. Prove that 1"-1 sup E [e-(1- $2))+) = (5%) (52)- (L) where l* = (d) Assume the Black & Scholes model with r = Deduce from part (c) the price of the perpetual American put option. 1 - 1 TET Question 3 [20 marks] Consider two risky assets $(1) and (2) with dynamics under Q dsc") = $!" ((-+** ) at +01dw, .). T + 2 7. a TT ds/2) = ) (rdt +02dW+), where 0 1 such that Z4 := e-rt(s),- (s?))1-a is a martingale. (b) For 1 > 1, define the stopping time 71 := inf{t > 0:5"> 15). Prove that Ele (597) $2))+) = (1 - 1)E [e-752) and B[e="+(54) 5(2)+] = " 1}. Prove that 1"-1 sup E [e-(1- $2))+) = (5%) (52)- (L) where l* = (d) Assume the Black & Scholes model with r = Deduce from part (c) the price of the perpetual American put option. 1 - 1 TET