## Introduction

The one-period model is the main building block of a dynamic multi-period model

• It is the easiest way to model uncertainty in financial markets
We introduce concepts such as basis assets, focus assets, portfolio, Arrow-Debreu securities,
hedging and replication
There are only two dates
• Today and tomorrow, this week and next week, this year and next year, now and in 10 min
• The essential feature is that nothing happens between the two dates
We do not know today the security price of tomorrow
•  The state of tomorrow’s world is uncertain
•  We assume there is only a finite number of scenarios, with given probability
It is realistic?
• The number of possible scenarios is much bigger (potentially infinite)
•  There are almost never objective probabilities (subjective conjecture)
•  It is a Model, simple replication of the reality

## Script Codes

%%%%%%%%%%%% Lecture 1 %%%%%%%%%%%%%%%%%%

R = [1; 1; 1]; %Payoff of Riskless Asset
S = [3; 2; 1]; %Payoff of Stock
K1 = 1.5; %Strike of C1
K2 = 1; %Strike of C2

C1 = max(0,S-K1); %Payoff of C1
C2 = max(0,S-K2); %Payoff of C12

A = [R S C1]; %Basis Assets Payoff Matrix
b = C2; %Focus Asset Payoff

det(A);
det(A) == 0; %Is the market complete?

x = inv(A)*b; %Compute the RP

P = [1 2 0.75];%Pricesd Vector for basis assets
P_C2 = P*x; %Compute the no-arbitrage price for C2 (Cost of RP)

%Swap C1 and C2 in the matrix A, and repeat

A = [R S C2];
b = C1;

det(A);
det(A) == 0;

%%%%%%%%%%%% Lecture 2 %%%%%%%%%%%%%%%%%%

R = [1; 1; 1]; %Payoff of Riskless Asset
S = [3; 2; 1]; %Payoff of Stock
K1 = 1.5; %Strike of C1

C1 = max(0,S-K1); %Payoff of C1
e1 = [1; 0; 0]; %Payoff the 1st AD security

A = [R S C1]; %Basis Assets Payoff Matrix
b = e1; %Focus Asset Payoff

det(A);
det(A) == 0; %Is the market complete?

x = inv(A)*b; %Compute the RP

P = [1 2 0.75];%Prices Vector for basis assets
P_e1 = P*x; %Compute the no-arbitrage price for e1 (Cost of RP)

%%%% Check Arbitrage %%%

P = [1 2 0.6]; %Prices Vector for 3 basis assets
A = [R S C1]; %Basis Assets Payoff Matrix

Phi = inv(A’)*P’; %The result must be a positive price vector for the 3 AD securities

Rf = A(1,1)/P(1); %Return for the riskless asset (does not depend on the state of the world, and holds for every A(i,1))
q = Phi * Rf; %Risk-neutral Probabilities

P_q = (A’*q)/Rf; %Risk-neutral pricing

## Security Payoff

We define security any entitlement to receive (obligation to pay) an amount of money

• We know the today’s price, we do not know the tomorrow’s payoff
The payoff is the amount of money given by selling or holding the security
• The price we will sell tomorrow, dividends, coupon
Consider a stock S with today’s price 2, and three possible tomorrow’s payoff: 1, 2, 3
• What I will get tomorrow depends on the State of the World
Option: contract which gives the buyer (the owner) the right to buy or sell an underlying asset
• Specified strike price, on or before a specified date
• The Call Option (Put Option) gives the right to buy (to sell)
Riskless Asset: investment with totally certain payoff
• The tomorrow’s value does not depend on the State of the World
Example
Portfolio with 1 RA, 1 Stock, and 2 Call Options (different Strikes: 1.5, and 1)