Lecture 4: Options

Question 1

Call upper bounds (no dividends)

Answer 1

c≤S0

C≤S0

Question 2

American put upper bound (no divs)

Answer 2

P≤K where K is max profit available at any time

Question 3

Euro put upper bound (no div)

Answer 3

p ≤ K(exp)^-rt

where K(exp)^-rt is max profit at maturity in PV

Question 4

Euro Call lower bounds (no divs)

Answer 4

c≥max(S -K(exp)^-rt,0)

Where a violation is an arbitrage opportunity

We compare this price to obs price, and if there is an imbalance we set up an arb.

Question 5

American call lower bound (no div)

Answer 5

c ≥ max(S0 - K(exp)^-rt, 0)

Question 6

Euro put lower bound (no divs)

Answer 6

p ≥ max(K(exp^-rt - S0 , 0)

Question 7

Two types of arbitrage profits?

Answer 7

1. T = 0 profits
   - This is where we invest the PV of the locked in price to pay at expiry, and reap profits immediately
2. expiry profits
   - Invest all of left-over cash following initial 2 transactions (stock long/short and corresponding option)
   - Reap profits following maturity @ expiry and closing out of position

Question 8

Describe put-call parity (No divs)

Answer 8

Put-call parity is the pricing relationship between a put and call, when they share:

1. Time to expiry
2. Strike price
3. Same S0

S0+p = c+K(exp)^-rT

A violation of this leads to arbitrage opps

Question 9

Put call parity for no div USA options

Answer 9

Early exercise means that arb opps exist beyond a bounds

s0-K≤C-P≤S0-K(exp)^-rT

Beyond these bounds, arb opps exist.

Question 10

How do we evaluate early exercise of American calls?

Answer 10

If the stock is not paying a dividend, and the investor plans to hold the stock for the life of the option

1. No income is being sacrificed
2. Delay exercise price outflow
3. Provides insurance against possibility of price falling below exercise price(early ex. could lead to investor paying >market rate)

Question 11

How should an investor close out an option (in this situation, an American put)

Answer 11

Take the offsetting position (sell the corresponding call), this allows the investor to capture both time value and intrinsic value, whereas early exercise is limited only to intrinsic value

Question 12

What would be a reason for early exercise of an American put?

Answer 12

1. So is zero, pay-off is maxed at K. Take it
2. Interest rates are rising
3. Volatility falling

Question 13

USA put-call parity (with Dividends)

Answer 13

S0 –D–K ≤ C–P≤S0 -K(exp)^–rT

Question 14

Euro put-call parity (with Divs)

Answer 14

c + D + K(exp)^-rT = p + S0

Question 15

How does the view of early exercise of American puts change once dividends are introduced?

Answer 15

It can be optimal to exercise early when divs are coming, just before ex-div date (date at which investors no longer entitled, therefore just before big price drop)

How?

1) As close to t=T before ex-div date (minimise loss of time value)
2) When the dividend will be large (big hit to S)
3) If multiple dividends, exercise just before last one