:: Elementary Introduction to Stochastic Finance in Discrete Time
:: by Peter Jaeger
::
:: Received March 22, 2011
:: Copyright (c) 2011-2021 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies FINANCE1, NUMBERS, XBOOLE_0, PROB_1, SUBSET_1, FUNCT_1, TARSKI,
RELAT_1, CARD_1, ARYTM_1, CARD_3, PROB_3, NAT_1, ARYTM_3, XXREAL_0,
SERIES_1, EQREL_1, SEQ_1, XXREAL_1, MESFUNC1, RANDOM_1, RANDOM_2,
FUNCOP_1, VALUED_1, FUNCT_7, REAL_1, RANDOM_3;
notations TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, XCMPLX_0, XXREAL_0, NAT_1,
XREAL_0, NUMBERS, FUNCT_1, RELSET_1, FUNCT_2, PROB_3, SERIES_1, PROB_1,
MEASURE6, SEQ_1, MESFUNC1, MESFUNC6, RANDOM_1, XXREAL_1, RCOMP_1,
FUNCOP_1, VALUED_1, RANDOM_2;
constructors REAL_1, PROB_3, SERIES_1, BINOP_2, RELSET_1, MEASURE6, RCOMP_1,
MESFUNC1, MESFUNC6, RANDOM_2;
registrations XBOOLE_0, SUBSET_1, ORDINAL1, RELSET_1, NUMBERS, XREAL_0,
MEMBERED, PROB_1, VALUED_0, XXREAL_0, NAT_1, XCMPLX_0, VALUED_1, FUNCT_2;
requirements SUBSET, NUMERALS, BOOLE, ARITHM, REAL;
begin
reserve Omega, Omega2 for non empty set;
reserve Sigma, F for SigmaField of Omega;
reserve Sigma2, F2 for SigmaField of Omega2;
notation
let a,r be Real;
synonym halfline_fin(a,r) for [.a,r.[;
end;
definition
let a,r be Real;
redefine func halfline_fin(a,r) -> Subset of REAL;
end;
theorem :: FINANCE1:1
for k being Real holds REAL \ [.k,+infty.[ = ].-infty,k.[;
theorem :: FINANCE1:2
for k being Real holds REAL \ ].-infty,k.[ = [.k,+infty.[;
definition
let a,b be Real;
func half_open_sets(a,b) -> SetSequence of REAL means
:: FINANCE1:def 1
it.0 = halfline_fin(a,b+1) &
for n being Nat holds it.(n+1) = halfline_fin(a,b+1/(n+1));
end;
definition
mode pricefunction -> Real_Sequence means
:: FINANCE1:def 2
it.0 = 1 & for n being Element of NAT holds it.n >= 0;
end;
notation
let phi,jpi be Real_Sequence;
synonym ElementsOfBuyPortfolio(phi,jpi) for phi (#) jpi;
end;
definition
let phi,jpi be Real_Sequence;
redefine func ElementsOfBuyPortfolio(phi,jpi) -> Real_Sequence;
end;
definition
let d be Nat;
let phi,jpi be Real_Sequence;
func BuyPortfolioExt(phi,jpi,d) -> Real equals
:: FINANCE1:def 3
Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)).d;
func BuyPortfolio(phi,jpi,d) -> Real equals
:: FINANCE1:def 4
Partial_Sums(ElementsOfBuyPortfolio(phi,jpi)^\1).(d-1);
end;
definition
let Omega, Omega2 be set;
let Sigma be SigmaField of Omega;
let Sigma2 be SigmaField of Omega2;
let X be Function of Omega, Omega2;
attr X is (Sigma,Sigma2)-random_variable-like means
:: FINANCE1:def 5
for x being set st x in Sigma2 holds X"x in Sigma;
end;
registration
let Omega1, Omega2 be non empty set;
let S1 be SigmaField of Omega1;
let S2 be SigmaField of Omega2;
cluster (S1,S2)-random_variable-like for Function of Omega1, Omega2;
end;
definition
let Omega, Omega2 be non empty set;
let F be SigmaField of Omega;
let F2 be SigmaField of Omega2;
mode random_variable of F, F2 is
(F,F2)-random_variable-like Function of Omega, Omega2;
end;
definition
let Omega, Omega2 be set;
let F be SigmaField of Omega;
let F2 be SigmaField of Omega2;
func set_of_random_variables_on (F,F2) -> set equals
:: FINANCE1:def 6
{ M where M is Function of Omega, Omega2:
M is (F,F2)-random_variable-like };
end;
registration
let Omega,Omega2,F,F2;
cluster set_of_random_variables_on(F,F2) -> non empty;
end;
registration
let Omega,Omega2,F,F2;
cluster set_of_random_variables_on(F,F2) -> functional;
end;
definition
let Omega, Omega2 be non empty set;
let F be SigmaField of Omega;
let F2 be SigmaField of Omega2;
let X be set such that
X = set_of_random_variables_on(F,F2);
let k be Element of X;
func Change_Element_to_Func(F,F2,k) -> Function of Omega,Omega2 equals
:: FINANCE1:def 7
k;
end;
definition
let Omega be non empty set;
let F be SigmaField of Omega;
let X be non empty set;
let k be Element of X;
func ElementsOfPortfolioValueProb_fut(F,k) -> Function of Omega,REAL means
:: FINANCE1:def 8
for w being Element of Omega holds
it.w = Change_Element_to_Func(F,Borel_Sets,k).w;
end;
definition
let p be Nat;
let Omega, Omega2 be non empty set;
let F be SigmaField of Omega;
let F2 be SigmaField of Omega2;
let X be set such that
X = set_of_random_variables_on(F,F2);
let G be sequence of X;
func Element_Of(F,F2,G,p) -> Function of Omega,Omega2 equals
:: FINANCE1:def 9
G.p;
end;
definition
let Omega be non empty set;
let F be SigmaField of Omega;
let X be non empty set;
let w be Element of Omega;
let G be sequence of X;
let phi be Real_Sequence;
func ElementsOfPortfolioValue_fut(phi,F,w,G) -> Real_Sequence means
:: FINANCE1:def 10
for n being Element of NAT holds
it.n = ElementsOfPortfolioValueProb_fut(F,G.n).w * phi.n;
end;
definition
let d be Nat;
let phi be Real_Sequence;
let Omega be non empty set;
let F be SigmaField of Omega;
let X be non empty set;
let G be sequence of X;
let w be Element of Omega;
func PortfolioValueFutExt(d,phi,F,G,w) -> Real equals
:: FINANCE1:def 11
Partial_Sums(ElementsOfPortfolioValue_fut(phi,F,w,G)).d;
func PortfolioValueFut(d,phi,F,G,w) -> Real equals
:: FINANCE1:def 12
Partial_Sums(ElementsOfPortfolioValue_fut(phi,F,w,G)^\1).(d-1);
end;
registration
cluster non empty for Element of Borel_Sets;
end;
theorem :: FINANCE1:3
for k being Real holds
[.k,+infty.[ is Element of Borel_Sets &
].-infty,k.[ is Element of Borel_Sets;
theorem :: FINANCE1:4
for k1,k2 being Real holds
[.k2,k1.[ is Element of Borel_Sets;
theorem :: FINANCE1:5
for a,b being Real holds
Intersection half_open_sets(a,b) is Element of Borel_Sets;
theorem :: FINANCE1:6
for a,b being Real holds
Intersection half_open_sets(a,b) = [.a,b.];
theorem :: FINANCE1:7
for a,b being Real, n being Nat holds
(Partial_Intersection half_open_sets(a,b)).n is Element of Borel_Sets;
theorem :: FINANCE1:8
for k1,k2 being Real holds [.k2,k1.] is Element of Borel_Sets;
theorem :: FINANCE1:9
for X being Function of Omega,REAL st
X is (Sigma,Borel_Sets)-random_variable-like holds
(for k being Real holds
{w where w is Element of Omega: X.w >=k} is Element of Sigma &
{w where w is Element of Omega: X.w =r}) &
(for r being Real holds
eq_dom(X,r) = {w where w is Element of Omega: X.w =r}) &
(for r being Real holds eq_dom(X,r) is Element of Sigma);
theorem :: FINANCE1:10
for s being Real, f being Function of Omega, REAL st f = Omega --> s holds
f is (Sigma,Borel_Sets)-random_variable-like;
theorem :: FINANCE1:11
for phi being Real_Sequence, jpi being pricefunction,
d being Nat st d>0 holds
BuyPortfolioExt(phi,jpi,d) = phi.0 + BuyPortfolio(phi,jpi,d);
theorem :: FINANCE1:12
for d being Nat st d>0 holds
for r being Real
for phi being Real_Sequence
for G being sequence of set_of_random_variables_on(F,Borel_Sets) st
Element_Of(F,Borel_Sets,G,0) = Omega-->1+r holds
for w being Element of Omega holds
PortfolioValueFutExt(d,phi,F,G,w)
= ((1+r) * phi.0) + PortfolioValueFut(d,phi,F,G,w);
theorem :: FINANCE1:13
for d being Nat st d>0 holds
for r being Real st r>-1 holds
for phi being Real_Sequence,
jpi being pricefunction holds
for G being sequence of set_of_random_variables_on(F,Borel_Sets) st
Element_Of(F,Borel_Sets,G,0) = Omega-->1+r holds
for w being Element of Omega holds
BuyPortfolioExt(phi,jpi,d)<=0 implies
(PortfolioValueFutExt(d,phi,F,G,w) <=
PortfolioValueFut(d,phi,F,G,w) - (1+r)*BuyPortfolio(phi,jpi,d));
theorem :: FINANCE1:14
for d being Nat st d>0 holds
for r being Real st r>-1 holds
for phi being Real_Sequence,
jpi being pricefunction holds
for G being sequence of set_of_random_variables_on (F,Borel_Sets) st
Element_Of(F,Borel_Sets,G,0) = Omega-->1+r holds
(BuyPortfolioExt(phi,jpi,d)<=0 implies
({w where w is Element of Omega:
PortfolioValueFutExt(d,phi,F,G,w) >= 0} c=
{w where w is Element of Omega:
PortfolioValueFut(d,phi,F,G,w)
>= (1+r)*BuyPortfolio(phi,jpi,d)} &
{w where w is Element of Omega: PortfolioValueFutExt(d,phi,F,G,w) > 0} c=
{w where w is Element of Omega: PortfolioValueFut(d,phi,F,G,w)
> (1+r)*BuyPortfolio(phi,jpi,d)}));
theorem :: FINANCE1:15
for f being Function of Omega,REAL st
f is (Sigma,Borel_Sets)-random_variable-like holds
f is ([#]Sigma)-measurable &
f is Real-Valued-Random-Variable of Sigma;
theorem :: FINANCE1:16
set_of_random_variables_on (Sigma,Borel_Sets) c=
Real-Valued-Random-Variables-Set Sigma;
theorem :: FINANCE1:17
Omega --> the Element of Omega2 is random_variable of F,F2;