equations

Question 1

put call parity:

Answer 1

St+Pt-Ct=Ee^(-r(T-t))

Question 2

put and call bounds:

Answer 2

St-Ee^(-r(T-t))<=Ct<=St
for Pt switch all St and Ee^(-r(T-t)) w/ each other

Question 3

C0:

Answer 3

e^(-rT)(pCu-(1-p)Cd)
p=(e^(rT)-d)/(u-d)
for trees with more than 1 stage, replace all T with the change in t for the step, so if T=1 and there’s two stages replace T with 0.5 rather than 1 and work out the middle stage from the last and use those numbers for the original price, using the same replacement for T the whole time
is same for puts

Question 4

C0 but american:

Answer 4

max of same calc for european and the payoff of exercising the option at that time

Question 5

eliminate risk:

Answer 5

find the dW thing usually from ito’s lemma, make it equal 0 and that eliminates risk

Question 6

standard N(x) results:

Answer 6

N(infinity)=1
N(-infinity)=0
N(0)=1/2
N(-x)=1-N(x)

Question 7

Y(t,T):

Answer 7

-(ln(V(t,T))-ln(V(T,T)))/(T-t)

Question 8

V(t,T) and Y(t,T):

Answer 8

V(t,T)=Fe^(-Y(t,T)(T-t))

Question 9

asian option:

Answer 9

exercise price is based on the average share price over a given interval

Question 10

payoff profit and expected profit:

Answer 10

using call option as an example but goes same for puts
payoff -> max(S-E,0)
profit -> payoff - C0e^(rT)
expected profit -> expected payoff - C0e^(rT), expected payoff is NOT the payoff of the expected share price, it’s actually the funky things that involve p, the expected payoff of the Option

Question 11

value of B (a bond):

Answer 11

B0e^(-r(T-t))

Question 12

payoff of B (a bond):

Answer 12

B0e^(rT)

Question 13

no arbitrage means:

Answer 13

ΠT>=0 -> Πt>=0 for all t<=T
VT>=UT -> Vt>=Ut for all t<=T
dΠ/Π=dB/B=r dt, all risk free portfolios have the same rate of return