Review

Question 1

Spot Contract

Answer 1

An agreement to buy an asset S now by paying at the same time. S_o is the price at t=0, the buyer pays seller S₀ at t=0 and the seller gives the asset to the buyer at t=0. If it’s a financial asset (ie. a stock or bond) the buyer will have a long position in the asset.

Question 2

What is a prepaid forward contract?

Answer 2

The buyer will pay the seller F^p_0,T(S) at time 0 and receive the asset at time T.

Question 3

What is a forward contract?

Answer 3

A forward contact is one where the buyer pays the seller F_0,T(S) at time T and receives the asset at time T.

Question 4

Calculating F^P_0,T(S)

Answer 4

For non-dividend-paying stocks:

F^P_t,T(S) = St

For stock indices w/a constant, known dividend yield 𝛿

F^P_t,T(S) = Ste^-𝛿(T-t)

For stocks paying known discrete dividends

F^P_t,T(S) = S_t-PV_t,T(Div)

Question 5

Calculating F_0,T(S)

Answer 5

For non-dividend-paying stocks:

F_t,T(S) = S₀e^r(T-t)

For stock indeces w/a constant, known dividend yield 𝛿:

F_t,T(S) = S₀e^{(r-𝛿)(T-t)}

For stocks paying known discrete dividends:

F_t,T(S) = S₀e^r(T-t)FV_0,T(Div.)

Question 6

Arbitrage Opportunities

Answer 6

When the theoretical price of a derivative is different from the market price, you can create a portfolio that makes no less in the future and costs nothing at time 0 => BUY LOW, SELL HIGH

1. Market Price < Theoretical Price: buy derivative from market at market price and eliminate risk using stocks, the bank account and other derivatives
2. Market Price > Theoretical Price: short sell the derivative at market price and eliminate risk using stocks, the bank account and other derivatives

Question 7

What is a call option?

Answer 7

A call option gives the holder the right to buy the underlying asset by a ceratin date (the expiration date) for a certain price (the strike price)

Payoffs:

• Long Call: payoff = (S_T-K)₊
• Short Call: payoff = -(S_T-K)

Question 8

Call option payoff diagrams

Question 9

What is a put option?

Answer 9

A put option gives the holder the right to sell the underlying aset by a certain date (expiration date) for a certain price (strike price).

Payoff:

• Long put: payoff = (K-S_T)₊
• Short put: payoff = -(K-S_T)₊

Question 10

Put option payoff diagrams

Question 11

Moneyness

Answer 11

Moneyness: whether the payoff of the option would be positive if it were exercised immediately

1. In-the-money (ITM): positive cash flow to holder
2. At-the-money (ATM): zero cash flow to holder
3. Out-the-money (OTM): negative cash flow to holder

Question 12

Put-Call Parity for European Options

Answer 12

c(S_t, t; K, T) - p(S_t, t; K, T) = F^P_t,T(S) - Ke^-r(T-t) = e^-r(T-t)[F_t,T(S) - K]

Question 13

Synthesizing Stock Position

Answer 13

We can synthesize one share at time T (a prepaid forward contract) by making use of the put-call parity

Transaction Cost @ t=0 Payoff @ t=T

S_T<k> <u>S_T&gt;=K</u></k>

Buy a T-yr. K-strike call c(S₀, K, T) 0 S_T-K

Sell a T-yr. K-strike put -p(S₀, K, T) -(K-S_T) 0

Invest Ke^-rT @ risk-free rate Ke^-rT K K

Net pos’n c(S₀, K, T) - p(S₀, K, T) S_T ST

+Ke^-rT

Question 14

What is a dividend forward contract?

Answer 14

A dividend forward contract is a forward contract on the “asset” FV_0,T(Div.). The following portfolio replicates a prepaid forward contract on a stock:

Transaction__Cost @ t=0 Payoff @ t=T

Buy one share and reinvest S₀ S_T + FV_0,T(Div.)

all div.s at a risk-free rate

Short a div. forward contract 0 F_0,T(Div.) - FV_0,T(Div.)

Borrow F_0,T(Div.)e^-rT at risk- -F_0,T(Div.)e^-rT -F_0,T(Div.)

free rate

Net position S₀ - F_0,T(Div.)e^-rT S_T

Question 15

Put-Call Parity in terms of Dividend Forward Price

Answer 15

c(S₀, K, T) - p(S₀, K, T) = S₀ - e^-rTF_0,T(Div.) - Ke^-rT