1. Options Introduction Call and put option contracts Notation
Definitions
Graphical representations (payoff diagrams)
Finance 30233, Fall 2006
Advanced Investments
S. Mann
The Neeley School at TCU
S.Mann, 2006

2. Options
Right, but not the obligation, to either buy or sell at a fixed price over a time period (t,T)
Call option - right to buy at fixed price
Put option - right to sell at fixed price
fixed price (K) : strike price, exercise price (K = X in BKM)
selling an option: write the option
Notation: call value (stock price, time remaining, strike price)
= c ( S(t) , T-t, K)
at expiration (T):
c (S(T),0,K) = if S(T) < K
S(T) - K if S(T)  K
or: c(S(T),0,K) = max (0,S(T) - K)
S.Mann, 2006

3. "Moneyness" Call “moneyness” Call value K asset price (S) Put
K asset price (S)
Out of the money in the money
(S < K) (S > K)
Put
“moneyness”
Put value
K asset price (S)
in the money out of the money
(S < K) (S >K)
S.Mann, 2006

4. Call value at maturity c (S(T),0,K) = 0 ; S(T) < K5
Call value =
max (0, S(T) - K)
K (K+5) S(T)
S.Mann, 2006

5. Short position in Call: value at maturity
c (S(T),0,K) = ; S(T) < K
S(T) - K ; S(T)  K
short is opposite:
-c(S(T),0,K) = ; S(T) < K [S(T)-K] ; S(T)  K
Value -5
Short call value =
min (0, K -S(T))
K (K+5) S(T)
S.Mann, 2006

6. Call profit at maturity
Call value at T: c(S(T),0,K) =
max(0,S(T)-K)
Value
Call
profit
Profit =
c(S(T),0,K) - c(S(t),T-t,K)
Breakeven point
K S(T)
Profit is value at maturity less initial price paid.
S.Mann, 2006

7. S.Mann, 2006

8. Put value at maturity p(S(T),0,K) = K - S(T) ; S(T)  K5
Put value =
max (0, K - S(T))
(K-5) K S(T)
S.Mann, 2006

9. Short put position: value at maturity
p(S(T),0,K) = K - S(T) ; S(T)  K
0 ; S(T) > K
short is opposite:
-p(S(T),0,K) = S(T) - K ; S(T)  K
Value -5
Short put value =
min (0, S(T)-K)
(K-5) K S(T)
S.Mann, 2006

10. Put profit at maturity Put value at T: p(S(T),0,K) = max(0,K-S(T)) put
Put value at T: p(S(T),0,K) = max(0,K-S(T))
put
profit
Profit =
p(S(T),0,K) - p(S(t),T-t,K)
Breakeven
point
K S(T)
Profit is value at maturity less initial price paid.
S.Mann, 2006

11. Option values at maturity (payoffs)
long put
long call K K
short call
short put K K
S.Mann, 2006

12. European Put-Call parity: Asset plus Put Asset plus European put:K K K
K S(T)
Put
Asset plus European put:
S(0) + p[S(0),T;K] K
S.Mann, 2006

13. European Put-Call parity: Bond plus CallK K K
K S(T)
Call
Bond + European Call:
c[S(0),T;K] + KB(0,T) K
S.Mann, 2006

14. European Put-Call parity: S(0) + p[S(0),T;K] = c[S(0),T;K] + KB(0,T)
Value at expiration
Position cost now S(T)  K S(T) > K
Portfolio A:
Stock S(0) S(T) S(T)
put p[S(0),T;K] K - S(T) 0
total A: S + P K S (T)
Portfolio B:
Call c[S(0),T;K] 0 S(T) - K
Bill KB(0,T) K K
total B: C + KB(0,T) K S(T)
European Put-Call parity:
S(0) + p[S(0),T;K] = c[S(0),T;K] + KB(0,T)
S.Mann, 2006

15. Bull Spread: value at maturity
S(0) = $ value at maturity
position: S(T)  S(T) S(T) > 50
Long call with strike at $45 0 S(T) S(T) -45
Short call w/ strike at $ [ S(T) - 50]
net: 0 S(T) 10 5
Position
value at T
S(T)
S.Mann, 2006

16. Bear Spread: value at maturity
S(0) = $ value at maturity
position: S(T)   S(T)  S(T) >35
Long call with strike at $ S(T) -35
Short call w/ strike at $ [S(T) - 25] - [ S(T) -25]
net: S(T)
- 5
-10
Position
value at T
S(T)
S.Mann, 2006

17. Butterfly Spread: value at maturity
S(0) = $50 value at maturity
position: S(T)  S(T   S(T)  S(T) > 55
Long call , K= $ S(T) S(T) S(T) - 45
Short 2 calls, K= $ [S(T) - 50] -2[S(T) - 50]
Long call , K = $ S(T) - 55
net: S(T) S(T) 10 5
Position
value at T
S(T)
S.Mann, 2006

18. Straddle value at maturity
S(0) = $ value at maturity
position: S(T)  S(T) > 25
Long call, K= $ S(T) - 45
Long put , K= $ S(T)
net: S(T) S(T) - 25 10 5
straddle
Position
value at T
Bottom straddle
S(T)
Bottom straddle: call strike > put strike: put K = 23; call K = 27
S.Mann, 2006