CHAPTER 3: OPTIONS Flashcards
DEF OPTIONS
contracts giving rights to buying or selling underlying asset
DEF EUROPEAN CALL OPTION
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)
DEF EUROPEAN PUT OPTION
give owner the right to SELL it’s underlying asset at the STRIKE PRICE on the EXPIRATION DATE
DEF AMERICAN CALL OPTION
give owner the right to BUY it’s underlying asset at the STRIKE PRICE, BY the EXPIRATION DATE
DEF AMERICAN PUT OPTION
give owner the right to SELL it’s underlying asset at the STRIKE PRICE, BY the EXPIRATION DATE
payoffs of options
European call option
S_T is less than X
S_T = X
S_T is more than X
- 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
payoffs of options
European PUT option
S_T is less than X
S_T = X
S_T is more than X
- 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
Notation: strike price X, expiring in T years.
PRICES TRADED AT:
European Call, European put American call, American put
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 :?
proposition 8 :
relationship between strike price, Spot price, for call and put options and prices traded at
(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
Proof of prop 8:
(a) c, C, p, P ≥ 0.
(b) c ≤ C ≤ S and p ≤ P ≤ X.
- 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)
PROP 9: no dividend paying stock and put and call option prices relating to strike and spot prices
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.
Proof to prop 9 a)
c > S − Xe−rT
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.
Proposition 10 (Put-call parity).
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.
PROOF Proposition 10 (Put-call parity). .
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.
PROP 11:
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.
PROOF PROP 11
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.
proof prop 9
(b) p > Xe−rT − S.
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
PROOF PROP 12:
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
PROP 12:
S − X LESS THAN
C − P
LESS THAN
S − Xe^−rT
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
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.
Parameters affecting the prices of options
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
AMERICAN PUT EXCERCISE WHEN?
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.
