Lecture 3: Option Properties

Question 1

Notation:
 c :	European call option price
 p :	European put option price
 S_0 :	Stock price today
 K :	Strike price
 T :	Life of option 
 σ:	Volatility of stock price
 C :	American Call option price
 P :	American Put option price
 S_T :Stock price at option maturity
 D :	Present value of dividends during option’s life
 r :	Risk-free rate for maturity T with cont comp

Answer 1

Notation:
 c :	European call option price
 p :	European put option price
 S_0 :	Stock price today
 K :	Strike price
 T :	Life of option 
 σ:	Volatility of stock price
 C :	American Call option price
 P :	American Put option price
 S_T :Stock price at option maturity
 D :	Present value of dividends during option’s life
 r :	Risk-free rate for maturity T with cont comp

Question 2

Important factors affecting Option Price:
Variable | c | p | C | P :
S_0        | ? | ?  | ?  |  ? 
K            | ? | +  | -  |  ?
T            | ? |  ?  | +  |  +
σ            | + | +  | ?  |  + 
r             | + | ?  | ?  |  - 
D            | - | ?  | ? |  +

Answer 2

Important factors affecting Option Price:
Variable | c | p | C | P :
S_0        | + | -  | +  |  - 
K            | - | +  | -  |  + 
T            | ? |  ?  | +  |  +
σ            | + | +  | +  |  + 
r             | + | -  | +  |  - 
D            | - | +  | - |  +

Question 3

Stock price drops by value of t’ dividend on ?? date –> dominate ? effect:
Time passes –> c & p not necessarily ?.

Answer 3

Stock price drops by value of t’ dividend on t’ ex-dividend date –> dominate time effect: Time passes –> c & p not necessarily rise.

Question 4

Option Greeks (aka Hedge statistics):
Δ (?), Γ ?, θ (?), ν (?) and ρ (?).
- used by traders to analyse & manage t’ ?of their positions.
- indicate ? & ? of option price to a change in value of a parameter.

Answer 4

Option Greeks (aka Hedge statistics): 
Δ (delta), Γ gamma, θ (theta), ν (vega) and ρ (rho). 
- used by traders to analyse & manage t' riskiness of their positions.
- indicate sensitivity & direction of option price to a change in value of a parameter.

Question 5

Δ (delta):

∂c/∂? >0 , ∂p/∂? >0

Answer 5

Δ (delta):

∂c/∂S>0 , ∂p/∂S>0

Question 6

θ (?):

∂c/∂? >0 , ∂p/∂? >0

Answer 6

θ (theta):

∂c/∂T>0 , ∂p/∂T>0

Question 7

ν (?):

∂c/∂? >0 , ∂p/∂? >0

Answer 7

ν (vega):

∂c/∂σ >0 , ∂p/∂σ >0

Question 8

ρ (?):

∂c/∂?>0 , ∂p/∂?>0

Answer 8

ρ (rho):

∂c/∂r>0 , ∂p/∂r>0

Question 9

Γ (?): Change in ?:

?? >0 , ??>0

Answer 9

Γ (gamma): Change in delta:

∂^2 c/∂S^2>0 , ∂^2 p/∂S^2>0

Question 10

C ? c

P ? p

Answer 10

C >= c

P >= p

Question 11

Call Upper Bound:
C <= ?
c <= ?
If not hold: Buy ? & ??

Answer 11

Call Upper Bound:
C <= So
c <= So
If not hold: Buy Stock & Sell call

Question 12

Put Upper Bound:
P <= ?
p <= ??
If not hold, ? (?) put & ? proceeds at r to yield ?? > K.

Answer 12

Put Upper Bound:
P <= K
p <= Ke^(-rT)
If not hold, write (sell) put & invest proceeds at r to yield pe^(rT) > K.

Question 13

European Lower Call Bound:
c >= ???
If not hold, ? call & ? stock.

Answer 13

European Call Lower Bound:
c >= So - Ke^(-rT)
If not hold, buy call & short stock.

Question 14

European Put Lower Bound:
p >= ???
If not hold, ? at r & ? both put and stock.

Answer 14

European Put Lower Bound:
p >= Ke^(-rT) - So
If not hold, borrow at r & buy both put and stock.

Question 15

Put - Call Parity:

For European options:
       c - p = ???
<=>  c + ?? = p + ?
For American options:
?? <= C - P <= ???

Answer 15

Put - Call Parity:

For European options:
c - p = So - Ke^(-rT)
<=> c + Ke^(-rT) = p + So

For American options:
So - K <= C - P <= So - Ke^(-rT)

Question 16

Call payoff = Max ??

Put payoff = Max ??

Answer 16

Call payoff = Max ( St - K , 0)

Put payoff = Max ( K - St, 0)

Question 17

American-style option: can be exercised at any time ???? maturity date.

Answer 17

American-style option: can be exercised at any time up to and including maturity date

Question 18

Early exercise on American Call on a No-flow paying asset is ? optimal.
Because:
- no ? income is sacrificed.
- ? value’s lost.
- ? is paid early (i.e. cash outflow has ? present value)
- ? upside
- Why not sell it! :
Option Value = Intrinsic Value + Time value
=> C = (S_T - K ) + Time Value > S_T - K
=> C > ???

Answer 18

Early exercise on American Call on a No-flow paying asset is NEVER optimal.
Because:
- no dividend income is sacrificed.
- insurance value’s lost.
- K is paid early (i.e. cash outflow has higher present value)
- unlimited upside
- Why not sell it! :
Option Value = Intrinsic Value + Time value
=> C = (S_T - K ) + Time Value > S_T - K
=> C > So - Ke^(-rT)

Question 19

Early exercise on American Put on a no-flow paying asset: ?? optimal bec:
- immediate cash inflow early (? PV)
- Insurance value ? as S falls
- Limited upside
=> P >= ???

Answer 19

Early exercise on American Put on a no-flow paying asset: may be optimal bec:
- immediate cash inflow early (higher PV)
- Insurance value falls as S falls
- Limited upside
=> P >= (K - So)