Lesson 19: Comparing Options

Question 1

Inequalities for call options

Answer 1

+ S ≥ Camer(S,K,T) ≥ Ceur(S,K,T) ≥ max (0, prepaid F(S) - Ke^(-rT)
+ For European call option:
Prepaid F(S) ≥ Ceur(S,K,T) ≥ max (0, prepaid F(S) - Ke^(-rT)

Question 2

Inequalities for put options

Answer 2

+ K ≥ Pamer(S,K,T) ≥ Peur(S,K,T) ≥ max(0,Ke^(-rT) - prepaid F(S)
+ For European put options:
Ke^(-rT) ≥ Peur(S,K,T) ≥ max(0, Ke^(-rT) - prepaid F(S))

Question 3

Every call option has an implicit put option built into it

Answer 3

The implicit put option consists of the option not to exercise the option. This means that at expiry of the call option, you will not exercise the option of the price of the option is less than K -> you “sell” or put the stock at price K back to the writer of the option

Question 4

Compare American options and European options for stocks without dividends

Answer 4

For stocks without div:
- An American call option is worth the same as a European call option
- An American put option may be worth more than a European put option

Question 5

Summary of relationships of option prices and time to expiry, T > t

Answer 5

1. An American option with expiry T must cost ≥ the same one with expiry t
2. A European call option on a nondiv stock with expiry T must cost ≥ the same one with expiry t
3. A European option on nondiv stock with expiry T and strike price Ke^(r(T-t)) must cost ≥ the one with expiry t and strike price K

Question 6

Direction of strike price in options

Answer 6

For a call option, the higher the K, the lower the premium
For a put option, the higher the K, the higher the premium

Question 7

Create arbitrage that maximizes the initial gain (use direction of premiums), suppose K2 > K1 and C(S,K2,T) > C(S,K1,T)

Answer 7

Sell 1 K2- strike call and buy 1 K1-strike call:
+ Immediate gain: C(S,K2,T) - C(S,K1,T)
+ 3 possibilities:
* St ≤ K1 -> neither option pays
* K1 ≤ St ≤ K2 -> the option you bought pays St - K1, the option you sold doesn’t pay
* St > K2 -> the option you bought pays St - K1, the option you sold pays St - K2

Question 8

Create an arbitrage that has minimal cost (direction of premiums) suppose K2 > K1 and C(S,K2,T) > C(S,K1,T)

Answer 8

Sell 1 K2-strike call and buy c = C(S,K2,T) / C(S,K1,T)

Question 9

Slope of strike prices in options

Answer 9

• For a call option, premium decreases more slowly than K increases
• For a put option, the premium increases more slowly than K increases

Question 10

Create arbitrage for call that maximizes the initial gain (use slope of premiums), suppose K2 > K1 and C(S,K1,T) - C(S,K2,T) > K2 - K1

Answer 10

sell 1 K1-strike call and buy 1 K2-strike call

Question 11

Create arbitrage for call that minimizes the initial gain (use slope of premiums), suppose K2 > K1 and C(S,K1,T) - C(S,K2,T) > K2 - K1

Answer 11

Buy 1 K2-strike call and sell cK1-strike call with
c = C(S<K2, T) / [(C(S, K1, T) - (K2 - K1)]

Question 12

Convexity

Answer 12

Option premiums are convex.
- The rate of decrease in call premiums as a function of K decreases.
C(S,K2,T) ≤ [(K2 - K1)C(S,K3,T) + (K3-K2)C(S,K1,T)] / (K3 - K1)
- The rate of increases in put premiums as a function of K increases
P(S,K2,T) ≤ [(K2 - K1)P(S,K3,T) + (K3-K2)P(S,K1,T)] / (K3 - K1)

Question 13

Convexity mispricing

Answer 13

if there’s a mispricing, the option in the middle will be overpriced relative to the other 2 options -> sell the option in the middle and buy the ones at the 2 extremes

Question 14

Arbitrage that maximizes gain when there’s a mispricing in call premium (convexity) (K1 < K2 < K3)

Answer 14

Sell K2-strike call, buy (K2-K1)/(K3-K1) K3-strike call, buy (K3-K2)/(K3-K1) K1-strike call

Question 15

Arbitrage that minimizes loss when there’s a mispricing in call premium (convexity) (K1 < K2 < K3)

Answer 15

Sell one K2-stkie call and buy (u/v)p K1-strike call and (u/v)q K3-strike call
p = (K3-K2)/(K3-K1), q = (K2-K1)/(K3-K1)