WEIPING LI (2) (25pts) (Quanto option) A quanto option pays off in one currency the price in another currency of an underlying asset without taking the currency conversion into account. Let the domestic and foreign interest rates to be constants r and respectively. Assume 01 > 0.02 > 0 and pe (-1.1). Under the risk-neutral measure P, ds(t) = S(t)(rdt +odWi(t)) and (MOQ(0) = M(OQOrdt + o2pd (0) +02V1-p20). (a) Show that under , S(t) = S(0) exp{o} W (t) + (b) Show that under P, Q(t) = Q(0) exp{C2pW (t) + 02V1 - p2Wz(t) + (r-pl (c) Show that B(t) = 02 (t) - V1-W2(t) is a Brownian motion, where 01= Va - 200102 +03. (d) Show that under P, S(0) S(0) Q(0) Qo exp{04B(t) + (r-a where a=r-/+ po 102 - oz. (e) Consider a quanto call pays off on-)+ units of domestic currency a time T, with r = 560. The price of the quanto call at time t is t, 3) = E-CTN ( t )- t-TTKN(d2), where du = calln +(-a )). (3) (20pts) Suppose the spot rate satisfies dr(t) = o(t) +a(t)(b - r(t))}dt+o(t)r(t)dw (t). The zero-coupon bond with fixed T is C{t,a) = Erit)=[exp(- r(s)ds)| Fo]. (i) Show that C(t, x) satisfies the following PDE acac + (t) +a(t)(b - 2) = xC(t, x). 20(t)28C WEIPING LI (2) (25pts) (Quanto option) A quanto option pays off in one currency the price in another currency of an underlying asset without taking the currency conversion into account. Let the domestic and foreign interest rates to be constants r and respectively. Assume 01 > 0.02 > 0 and pe (-1.1). Under the risk-neutral measure P, ds(t) = S(t)(rdt +odWi(t)) and (MOQ(0) = M(OQOrdt + o2pd (0) +02V1-p20). (a) Show that under , S(t) = S(0) exp{o} W (t) + (b) Show that under P, Q(t) = Q(0) exp{C2pW (t) + 02V1 - p2Wz(t) + (r-pl (c) Show that B(t) = 02 (t) - V1-W2(t) is a Brownian motion, where 01= Va - 200102 +03. (d) Show that under P, S(0) S(0) Q(0) Qo exp{04B(t) + (r-a where a=r-/+ po 102 - oz. (e) Consider a quanto call pays off on-)+ units of domestic currency a time T, with r = 560. The price of the quanto call at time t is t, 3) = E-CTN ( t )- t-TTKN(d2), where du = calln +(-a )). (3) (20pts) Suppose the spot rate satisfies dr(t) = o(t) +a(t)(b - r(t))}dt+o(t)r(t)dw (t). The zero-coupon bond with fixed T is C{t,a) = Erit)=[exp(- r(s)ds)| Fo]. (i) Show that C(t, x) satisfies the following PDE acac + (t) +a(t)(b - 2) = xC(t, x). 20(t)28C