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1Derivative Instruments
Lecture Eight
Option Properties
MANA130313 derivative instruments
Lecture Outline
This lecture covers
Option properties
MANA130313 derivative instruments
Factors Affecting Option Price
Six factors
S
0, K, T, σ, r, D
cpCPVariableC:American Call
P :American Put
c:European call
p:European put
3S
0
K
T
σ
r
D++–
+
??++
++++
+–+––
––+
–+–+
MANA130313 derivative instruments
2Factors Affecting Option Price
Current stock price (S
0)and strike price (K)
Payoff on a call:
Max (stock price –strike price, 0 )
Priceofacall:Price of a call:
+ with stock price & -with strike price
Payoff on a put:
Max ( strike-stock, 0)
Price of a put:
+ with strike and –with stock price
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Factors Affecting Option Price
Time to expiration (T)
Longer life => more exercise opportunity
American call & put: + with T
European call & put: pp
usually + with T (there are exceptions)
E.g.: cash dividend is expected to be paid
between the short and long maturity date
-cash div is not adjusted
-long life option is worth less due to stock
price decrease
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Factors Affecting Option Price
Stock price volatility (σ)
Volatility: uncertainty of future price
movements
For stock owner: large volatility => offsetting
MV increase or decrease=> indifference
For option owner: higher is better given
limited downside risk
-call holder benefit from price increase
-put holder benefit from price decrease
-max loss is option price
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3Factors Affecting Option Price
Risk free Rate (r)
Ceteris paribus, when IR increase
-discount rate increase
-PV of CF decreases
Thus, call increases and put decreases
Dynamically, stock price usually changes
-increase in IR => stock price decreases
-net effect of IR increases & S
0decreases
=> can be call decreases & put increases
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Factors Affecting Option Price
PV of dividends during option’s life(D)
Contract is not adjusted for cash div
Div payment decreases stock price on
ex-div day
ThllddtiThus, call decreases and put increases
American vsEuropean Options
An American option is worth at least as
much as the corresponding European
option
Hence, C≥c ; P≥p
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Assumptions & Notation
Similar for deriving forward prices
No transaction costs
Same tax rate
Borrowing and lending at risk free rate
A few key market participants
Non Arbitrage opportunity
Notation
S
0, K, T, S
T, r(nominal, positive & cc to T)
C, P, c, p
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4Upper Bounds for Option Prices
For call options
Right to buy: always cost less than underlying stock
Thus, C <= S
0 and c <= S
0
If call > S
0 , regardless S
0& K
arbitragebysellingcall&buyingstockarbitrage by selling call & buying stock
For put options
Right to sell: always cost less than strike price
Thus, P <= Kand p <= K at maturity
Today, for an European put: P <= Ke–rTand p <= Ke–rT
If p > K, regardless S
0& K
arbitrage by selling put & investing in risk free rate
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Lower Bounds for Option Prices
For European CALL options on non-div-paying stocks
E.g.: An Arbitrage Opportunity?
Suppose that:
c= 3, S
0= 20, T= 1, r= 10% p.a., K= 18
Short stock and buy a call
CF today: $20-$3 = $17
Invest proceeds for 1 year: $17e 10%*1=$18.79
At maturity:
-If S
T> K, exercise @ $18 to buy a share and return:
profit = $18.79-$18 = $ 0.79
-If S
T< K, buy from the market @ S
T,say$17,
profit = $18.79-$17 = $ 1.79
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Lower Bounds for Option Prices
For European CALL options on non-div-paying stocks
If you wish to have a share at T, consider 2 choices:
A: a European call + cash of Ke–rTV.S. B: one share
For A: Invest cash @ r to Tgives K at maturity
-If S
T> K exercise @ K and portfolio is worth S
T
-If S
T< K let expire to buy @ S
T& portfolio is worth K
-Thus, portfolio is worth Max (S
T, K) at maturity
For B: your share will be worth S
Tat maturity
At maturity: if follows that A ≥B and for today:
As the worst of a call is to be out-of-money:
12c+Ke –rT ≥S
0=> c≥S
0-Ke –rT
c≥max(S
0–Ke–rT, 0)
MANA130313 derivative instruments
