Final

Question 1

There is no arbitrage if and only if

Answer 1

d<1+r<u></u>

Question 2

Derivative security

Answer 2

A security that pays some amount V1 at time 1

Question 3

European Call Option

Answer 3

Contract that allows owner to buy 1 share of stock at time 1 for price k

Question 4

Arbitrage

Answer 4

1. X0=0 (no initial wealth)
2. P(X1>0)>0 (positive probability of gaining money)
3. P(X1<0)=0 (no chance of losing money)

Question 5

Wealth at time n+1 (Xn+1)

Answer 5

Xn+1 = 🔺n Sn+1 + (1+r)(Xn - 🔺n Sn)

Question 6

European Call Option

Answer 6

V1 = (S1 - K)^+

Question 7

European put option

Answer 7

V1 = (K - S1)^+

Question 8

Risk neutral probabilities

Answer 8

~
p = (1+r-d)/(u-d)
~
q = (u-1-r)/(u-d)

Question 9

Shares held at time n (🔺n)

Answer 9

🔺n = (Vn+1(…,H) - Vn+1(…,T))/(Sn+1(…,H) - Sn+1(…,T))

Question 10

Martingale

Answer 10

If M is adapted and for all n En [Mn+1] = Mn (expected value conditional on n of M at time n+1 equals M at time n) then M is a martingale (under P)

Question 11

Submartingale

Answer 11

En [Mn+1] >= Mn

Question 12

Adapted

Answer 12

A stochastic process Zn is adapted if Zn only depends on the first n coin flips

Question 13

Markov Process

Answer 13

X0,X1,…,XN is adapted and for all n and for all f there exists g such that En [f(Xn+1)] = g(Xn) (markov property)

Question 14

Radon-Nikodym derivative of ~P with respect to P

Answer 14

Z(w) = (~P(w)) / (P(w))

Question 15

State price

Answer 15

((Z(w)) / ((1+r)^N)) P(w)

State price density) (P(w)