Chapter 3

Question 1

Define Arbitrage

Answer 1

An arbitrage (opportunity) is an investment strategy that yields a positive profit with positive probability and is
not exposed to any downside risk.

Question 2

When do arbitrage opportunities occur?

Answer 2

when the price of the same

asset is different in two different markets

Question 3

What does the portfolio (x0,x1) represent?

Answer 3

xi represents the size of the holding, or no. of units, of asset i.

Question 4

What is an arbitrage-free model?

Answer 4

Models with no possibility of risk-free profits

Question 5

What is the initial value of the total holding corresponding to (x0,x1) in an arbitrage-free model?

Answer 5

x0 + x1 . S0

S0 is the initial price of risky asset 1

Question 6

What is the final value of the total holding corresponding to (x0,x1) in an arbitrage-free model?

Answer 6

x0 . r0 + x1 . S1
(S1 is the price of risky asset 1 at time 1,
r0 is the return of riskless asset 0)

Question 7

Describe 2 cases where there exists an arbitrage

Answer 7

i) Initial value of total holding is negative and final value is certainly positive
ii) Initial value of total holding is non-positive and final value is certainly non-negative and there the probability of the final value being positive is strictly positive

Question 8

What is the binomial asset pricing model?

Answer 8

It is a model for the stock price movement over one time period - it defines the price of the stock at time 1 (S1) as random variable.

Question 9

What is a contingent claim?
From the point of view of the writer?
And of the buyer?

Answer 9

A function of the values of the underlying assets at time 1.
Writer of claim:
payout at time 1 for contract sold to them at time 0. Amount C(w) depends on values of the assets at time 1.
Buyer: profit at time 1

Question 10

What is an option?

Answer 10

the right to buy or sell a particular asset (underlying asset) in the future at a specified price known as the strike price.

Question 11

What might be a benefit of buying an option?

Answer 11

Could make a return on the difference between the strike price and the actual market price of the underlying asset at the time the option is exercised

Question 12

What is a call option?

Answer 12

the right to buy an asset in the future at an agreed strike price - if the asset falls in price to below the strike price, the option is not exercised and gives a zero return.
Bet on upward price movement of underlying asset.

Question 13

What is a put option?

Answer 13

the right to sell an asset in the future at an agreed strike price. Bet on downward price movement of underlying asset.

Question 14

What is a European option? And an American option?

Answer 14

• Options that can only be exercised at an agreed time in the future (expiry time)
• Options that can be exercised at any time up to their expiry time

Question 15

When may a contingent claim be hedged?

Answer 15

If there exists a portfolio (x0,x1) such that
x0 . r0 + x1 . S1 = C
where C is the contingent claim.
This portfolio is called a hedging portfolio.

Question 16

What does it mean to the writer/buyer if a contingent claim can be hedged?

Answer 16

No risk in the claim since the portfolio may be purchased at time 0 at a cost of x0 + x1 . S0 and will replicate the amount C at time 1.