Portfolio Theory

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

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)