#Nathan Fox
#Homework 12
#I give permission for this work to be posted online

#Read packages
with(`combinat`):

#Read procedures from class
read(`C12.txt`):

#Help procedure
Help:=proc():
    print(` OP2scr(S1, S2, D11, D12, D13, D21, D22, D23, r, T, C1, C2, C3) `):
    print(` OP2nice(S1, S2, D11, D12, D13, D21, D22, D23, r, T, C1, C2, C3) `):
    print(` ExpBuyer2(S1, S2, D11, D12, D13, D21, D22, D23, r, T, C1, C2, C3, q1, q2) `):
    print(` ExpSeller2(S1, S2, D11, D12, D13, D21, D22, D23, r, T, C1, C2, C3, q1, q2) `):
end:

##PROBLEM 2##

#OP2scr(S1, S2, D11, D12, D13, D21, D22, D23, r, T, C1, C2, C3)
#There are two stocks, 1 and 2, whose current prices are S1 and S2.
#Their values in time T in the future depends whether the economy
#will be really bad, OK, or really good (there are only three
#scenarios).
#-If the economy will be really bad, the prices of stock 1 and stock
#2 will be D11 and D21 respectively
#-If the economy will be OK, the prices of stock 1 and stock 2 will
#be D12 and D22 respectively
#-If the economy will be great, the prices of stock 1 and stock 2
#will be D13 and D23 respectively
#
#This procedure calculates the FAIR price (no arbitrage!) of an
#option that pays C1 dollars if the economy is bad, C2 dollars if
#the economy is OK, and C3 dollars if the economy is great,
#assuming inerest rate r.
OP2scr:=proc(S1, S2, D11, D12, D13, D21, D22, D23, r, T, C1, C2, C3)
local x1, x2, x3, eq, var, P:
    eq:={x1*exp(r*T)+x2*D11+x3*D21=C1,
        x1*exp(r*T)+x2*D12+x3*D22=C2,
        x1*exp(r*T)+x2*D13+x3*D23=C3}:
    var:={x1, x2, x3}:
    
    var:=solve(eq, var):
    x1:=subs(var, x1):
    x2:=subs(var, x2):
    x3:=subs(var, x3):
    
    P:=x1+x2*S1+x3*S2:
    
    return normal(P), x2, x3:
end:

##PROBLEM 3##

#OP2nice(S1, S2, D11, D12, D13, D21, D22, D23, r, T, C1, C2, C3):
#outputs the same thing as OP2scr, but does it with martingales
OP2nice:=proc(S1, S2, D11, D12, D13, D21, D22, D23, r, T, C1, C2, C3)
local q1, q2, eq, var:
    eq:={q1*D11+q2*D12+(1-q1-q2)*D13=exp(r*T)*S1,
        q1*D21+q2*D22+(1-q1-q2)*D23=exp(r*T)*S2}:
    var:={q1, q2}:
    var:=solve(eq, var):
    q1:=subs(var, q1):
    q2:=subs(var, q2):
    return normal(q1*(exp(-r*T)*C1) + q2*(exp(-r*T)*C2) + (1-q1-q2)*(exp(-r*T)*C3)):
end:

#This gives something equivalent to OP2scr

##PROBLEM 4##

#The expected gain to the buyer of the option
#if she believes that the probability of the economy going down is
#q1, the probability of ok is q1, and the probability of going up
#is 1-q1-q2
ExpBuyer2:=proc(S1, S2, D11, D12, D13, D21, D22, D23, r, T, C1, C2, C3, q1, q2) local P:
    P:=OP2scr(S1, S2, D11, D12, D13, D21, D22, D23, r, T, C1, C2, C3)[1]:
    return q1*(C1*exp(-r*T)-P)+q2*(C2*exp(-r*T)-P)+(1-q1-q2)*(C3*exp(-r*T)-P):
end:

#The expected gain to the seller of the option
#if he believes that the probability of the stock going up is q
#following the do nothing portfolio
ExpBuyer2:=proc(S1, S2, D11, D12, D13, D21, D22, D23, r, T, C1, C2, C3, q1, q2) local P:
    P:=OP2scr(S1, S2, D11, D12, D13, D21, D22, D23, r, T, C1, C2, C3)[1]:
    return q1*(P-C1*exp(-r*T))+q2*(P-C2*exp(-r*T))+(1-q1-q2)*(P-C3*exp(-r*T)):
end:

#Yes, they add up to zero