MATH 5010, SPRING SEMESTER, 2008
CONTINUOUS-TIME FINANCE
INSTRUCTOR: WEIPING LI
FINAL EXAM
1. Strategy with consumption
A strategy with consumption is dened by three stochastic processes: Ht0 , Ht , ct , for 0 t T ,
and Ht0 , Ht are the quantities of risk
MATH 5010 CONTINUOUS-TIME FINANCE
ANSWER KEYS FOR HOMEWORK ASSIGNMENT 1
(1) The discounted price Sn is a martingale under P if and only if E (Sn+1 |Fn ) = Sn (This is
equivalent to E (Sn+1 /Sn |Fn ) = 1 by the conditional expectation property). Note that
MATH 5010 CONTINUOUS-TIME FINANCE
ANSWER KEYS FOR HOMEWORK ASSIGNMENT 2
(1) Show that W 2 (t) t is a martingale.
E [W (t)2 t|F (s)] = E [W 2 (t)|F (s)] t
= E [(W (t) W (s)2 + 2W (t)W (s) W (s)2 |F (s)] t
= E [(W (t) W (s)2 |F (s)] + 2E [W (t)W (s)|F (s)]
MATH 5010 CONTINUOUS-TIME FINANCE
HOMEWORK ASSIGNMENT 4
(1) Let Xt be a stochastic process satisfying
dXt = cXt dt + dWt ,
X0 = x.
(i) Use Yt = Xt ect , show that dYt = ect dWt and
t
Xt = xect + ect
ecs dWs .
0
(ii) Compute E (Xt ) and V ar(Xt ).
(iii) Pr