Let's assume that the stock price follows the following geometric Brownian motion dS(t) = (a q) SUE) dt + ch(t) dW(t) (1) whereuq represents the dividend yield. Let's denote r the continuously compounded interest rate and W(t) the Ito process that verifies dWU) = Odt + awn) (2) 047" where 6 = is the market price of risk. W(t) is a Brownian motion under the risk neutral ~ 0 measure P. 1. Show that under the risk-neutral measure IF'. the stock process's dynamic is described by the following equation dS(t) = (r (1)302) cit + 030:) elf/17m. (3) 2. Deduce from question 1 that 30:) = 3(0) exp {cf/17a) + (r 7 q , $02) if} (4) 3. The price v(t, as) of a European derivative contract which pays off h(S(T)) where 3(T) is the price of the stock above at time T verifies the following partial differential equation 1 vt(t, as) + (T q)zvx(t, 3:) + iJZIZUm, 3:) = rv(t, 3:). Use the discounted FeynmanKat"; representation theorem to deduce that the value of the derivative contract is given by v(t, x) = E [e\"Ti) h(S(T))| 8(t) = :3] 4. Deduce from questions 2 and 3 the price of a European binary option on a continuously paying dividend stock is given by the following formula 1:05, 3:) = 1006-r(T-t)[N(d2(T t, 33)) N(d1(T t, x))] , (5) where for j = 1, 2 1 K 1 dj(T, 3:) = [log "" i (T i q :E 502) T] . CT I The payoff of a European binary option is given by 0 if 8(T) 3 K1 mm) = 100 if K1