CHAPTER 3: OPTIONS

Question 1

DEF OPTIONS

Answer 1

contracts giving rights to buying or selling underlying asset

Question 2

DEF EUROPEAN CALL OPTION

Answer 2

give owner the right to BUY it’s underlying asset at a certain price ( STRIKE PRICE) on a certain date the expiration date (the expiration date)

Question 3

DEF EUROPEAN PUT OPTION

Answer 3

give owner the right to SELL it’s underlying asset at the STRIKE PRICE on the EXPIRATION DATE

Question 4

DEF AMERICAN CALL OPTION

Answer 4

give owner the right to BUY it’s underlying asset at the STRIKE PRICE, BY the EXPIRATION DATE

Question 5

DEF AMERICAN PUT OPTION

Answer 5

give owner the right to SELL it’s underlying asset at the STRIKE PRICE, BY the EXPIRATION DATE

Question 6

payoffs of options

European call option

S_T is less than X
S_T = X
S_T is more than X

Answer 6

• European call option with strike price X which has just expired at time t=T. S_T is the SPOT PRICE of the underlying asset @ EXPIRATION
• if S_T is more than X: the holder of the option buys the asset for X and sells it for S_T generating payoff S_T- X
• if S_T ≤ X: option not exercised, no payoff

@ EXPIRATION owner receives payoff, a function of underlying asset price S_T, function = max{S_T-X,0}

PAYOFF| /
| /
|====——-> S_T
X

Question 7

payoffs of options

European PUT option

S_T is less than X
S_T = X
S_T is more than X

Answer 7

• European put option with strike price X which has just expired at time t=T. S_T is the SPOT PRICE of the underlying asset @ EXPIRATION
• if S_T is less than X: the holder of the option sells the asset for X and buys it for S_T generating payoff X-S_T
• if S_T ≥ X: option not exercised, no payoff

@ EXPIRATION owner receives payoff, a function of underlying asset price S_T, function = max{X-S_T,0}

PAYOFF| \
              |   \
              |     \
              |------====-> S_T
                     X

Question 8

Notation: strike price X, expiring in T years.

PRICES TRADED AT:
European Call, European put American call, American put

Answer 8

Notation: strike price X, expiring in T years. option prices ( TRADED AT) :
European Call = c European put= p American call= C, American put=P

we are selling the right to buy from you are selling the right to sell to you

CURRENT PRICE S, STRIKE PRICE X (future price paid), S_T SPOT PRICE IN FUTURE.

ie if WE trade it and its exercised WE have to buy/sell the stock as agreed for X :?

Question 9

proposition 8 :

relationship between strike price, Spot price, for call and put options and prices traded at

Answer 9

(a) c, C, p, P ≥ 0.
(b) c ≤ C ≤ S and p ≤ P ≤ X.

European cheaper than American,
put cheaper than strike price,
call cheaper than spot price

Question 10

Proof of prop 8:

(a) c, C, p, P ≥ 0.
(b) c ≤ C ≤ S and p ≤ P ≤ X.

Answer 10

• if any option has negative value: buy it for the neg -receive option plus v cash and forget about option
• p>P: sell the European put option, buy the American put option. pocket p-P>0. if the European is exercised exercise American and sell the stock
• if c>C: sell the European call option for c, buy the American call option. pocket c-C>0. if the European call option is exercised exercise your American call option and use it to deliver the stock.
• C>S: sell the American call option and buy the stock. if call is exercised deliver the stock otherwise keep it. Pocket C-S>0.
• P>X: sell the American put option for P. if it’s exercised pay X to have stock delivered. Pocketing P-X>0 plus the stock. (if not exercised pocket only P)

Question 11

PROP 9: no dividend paying stock and put and call option prices relating to strike and spot prices

Answer 11

Assume that the stock does not pay dividends and let r be the T -year spot
interest rate.
(a) c > S − Xe−rT
.
(b) p > Xe−rT − S.

Question 12

Proof to prop 9 a)

c > S − Xe−rT

Answer 12

a):
* portfolio A: 1 European call option (traded at c) plus an amount of cash Xe^-rT deposited for T years @ int rate r
* Portfolio B: one share
(ST ≤ X, not exercised, S_t bigger than X exercised)

after T years portfolio A is worth max{S_T, X} ≥ S_t = Portfolio B.

so initial value of A ≥ initial value of B

but sometimes A worth more than B so strict inequality.

Question 13

Proposition 10 (Put-call parity).

Answer 13

Proposition 10 (Put-call parity). Assume that the stock does not pay dividends and let r be
the T -year spot interest rate. Then c + Xe−rT = p + S.

Question 14

PROOF Proposition 10 (Put-call parity). .

Answer 14

PORTFOLIOS A AND C as before…

After T years they are both worth max(ST , X) where
S_T is the stock price after T years. These two portfolios must have identical initial values, i.e.,
c + Xe^−rT = p + S.

Question 15

PROP 11:

Answer 15

Assume that the stock does not pay dividends. The optimal exercise time for
the American call option occurs at the expiration of the option and hence c = C.

Question 16

PROOF PROP 11

Answer 16

For any time  0 < τ < T, write S_τ , c_τ , C_τ for the values at time τ years of the stock. EUROPEAN and AMERICAN respectively. Let r be the T-τ spot interest rate at time τ .
We have 
C_τ ≥ c_τ 
> S_τ − Xe^−r(T −τ) 
> S_τ − X

where the second inequality follows from proposition 9.
S_τ − X is the payoff from exercising american, since value of american exceeds this payoff it shouldnt be exercised.

The optimal
exercise of the American option produces the same payoff as a European option , so c=C.

Question 17

proof prop 9

(b) p > Xe−rT − S.

Answer 17

b)
* Portfolio C: 1 European put option and 1 share
* portfolio D: cash Xe^-rT deposited for T years interest rate r.

• After T years D worth X, C worth max{S_T,X} ≥ X(depends if exercised S_T less than X or not exercised ST ≥ X)
• initial value of C must be no less than initial value of D

strict inequality as sometimes C worth more than D

Question 18

PROOF PROP 12:

Answer 18

Since P ≥ p a always,
and since some of the time payoff of the American option will be greater than that of the corresponding European option, P >p.
the second inequality is a consequence of put-call parity

P>p and c=C.
To prove S-X < C-P we consider portfolios:

Portfolio E: one European call option plus cash X deposited for T years at rate r

Portfolio F: one American Put option plus one share.

At expiration, portfolio E worth
 max (S_T - X,0) = max(ST , X) − X + XerT
= max(ST , X) + X(e
rT − 1)
> max(ST , X)

and if American option not exercised before F is worth max(X-S_T,0) + S_T = max(S_T,X). So E expires with higher value than F.

If the American option was exercised earlier time 0 ≤ τ < T , then at that time F worth (X − Sτ ) + Sτ = X. AT that time value of cash in E at least X, since in either case always a time in which E is more valuable than F present value E greater than present F.
C+X>P+S

Question 19

PROP 12:

Answer 19

S − X LESS THAN
C − P
LESS THAN
S − Xe^−rT

Question 20

EXAMPLE:

So we expect that c, C are increasing functions of the volatility of the stock price.
It is reasonable to assume that for longer expiration times T , the value of the stock at time T
will have more variability. But on the other hand, for bigger T , the payoff of the option has to be
discounted by a smaller discount factor. Obviously, C and P are increasing functions of T

Answer 20

Call option on a stock whose present price is £10 expiring in T-years. if the stock price goes up to £15. Payoff at expiration is max(S_T-X,0) where S_T is the price of the stock at expiration. Market expects S
-T to be higher, from rise in stock prices.

Question 21

Parameters affecting the prices of options

Answer 21

Rise in stock leads that market expects S_T to be higher so c and C are increasing functs of S.

p and P are decreasing functs of S

c and C are decreasing functs of X

p and P are increasing functs of X

Question 22

AMERICAN PUT EXCERCISE WHEN?

Answer 22

American put options may have an early optimal exercise date: e.g., suppose that on June
25th, 2002 you held American put options on Worldcom 1
stock with strike price of $65 expiring in
September 2002. Since you bought the stock the company disclosed that it inflated profits for over
a year by improperly accounting for more than $3.9 billion and the stock now sells for $0.20. The
payoff from exercising your option now would be $64.8, almost its theoretical maximum. Things can
only get worse as time progresses and you should exercise your options now.