3) Discrete-time market models

Question 1

How is a financial market defined in the discrete-time market model

Answer 1

Question 2

What is a dynamic portfolio (or trading strategy)

Answer 2

Question 3

How is the value (wealth) of a portfolio defined

Answer 3

Question 4

How is the gain (cumulative) process of a portfolio defined

Answer 4

Question 5

How is the discounted price process defined

Answer 5

Question 6

How are the discounted value process and the discounted gain process of a portfolio defined

Answer 6

Question 7

When is a portfolio considered self-financing

Answer 7

Investor does not add ot take away wealth when rebalancing the portfolio

Question 8

What is the relationship between the self-financing condition and the discounted price process

Answer 8

Question 9

How does the self-financing condition relate to the portfolio’s discounted value and gains

Answer 9

Question 10

Given (φ1(t),…,φd(t)), under what condition can we construct a self-financing strategy

Answer 10

Question 11

What is an arbitrage opportunity

Answer 11

Question 12

When is a market considered arbitrage-free

Answer 12

A market is arbitrage-free if there are no arbitrage opportunities among all self-financing trading strategies

Question 13

What is a martingale probability measure

Answer 13

Question 14

What happens to the discounted value process of a self-financing strategy under a martingale probability measure

Answer 14

Question 15

What is the First Fundamental Theorem of asset pricing

Answer 15

A market is arbitrage-free if and only if there exists an equivalent martingale probability measure (EMM) P∗ (i.e., S(t) is a martingale under P∗)

Question 16

What happens if two trading strategies have the same terminal wealth in an arbitrage-free market

Answer 16

Question 17

What is a contingent claim

Answer 17

A contingent claim (payoff) X with maturity date T is an arbitrary non-negative FT -measurable random variable representing a cash flow

Question 18

When is a contingent claim is said to be attainable

Answer 18

Question 19

When is a market considered complete

Answer 19

Question 20

What is the Second Fundamental Theorem of asset pricing

Answer 20

An arbitrage-free market is complete if and only if there exists a unique equivalent martingale probability measure P∗

Question 21

What is the payoff of a forward contract

Answer 21

Question 22

What is the payoff of a European call option

Answer 22

Question 23

What is the payoff of a European put option

Answer 23

Question 24

How is trading a claim X at time t for a price Πx(t) represented in the market model

Answer 24

Question 25

When is a number Πx(0) considered an arbitrage-free price for a claim X

Answer 25

A number Πx(0) is an arbitrage-free price of the claim X if, when we extend the market by treating X as a new tradable asset priced at Πx(t), the extended market remains arbitrage-free

Question 26

How is the set of arbitrage-free prices of a claim X at time t defined

Answer 26

Question 27

How are no-arbitrage bounds on the price of a claim X at time 0 defined

Answer 27

Question 28

How is the price of a contingent claim X determined in a complete, arbitrage-free market

Answer 28

Question 29

Describe the proof that if there is an incomplete market where 0 ≤ a < b are both prices for a claim X and there is also a no-arbitrage x with a < x < b for the claim X

Answer 29

Question 30

What are the key properties of the Cox-Ross-Rubinstein (CRR) model

Answer 30

Question 31

How is the arbitrage-free price of a contingent claim given in the CRR model

Answer 31

Question 32

What is the hedging (replicating) strategy for a European call option

Answer 32

Question 33

What happens to the price of an attainable claim across different equivalent martingale measures

Answer 33

Question 34

How are the superhedging and subhedging prices of a claim X at time 0 defined

Answer 34