Welcome to
Our
Presentation

Presentation is on ………..
Properties of
Stock Options
(Put-Call Parity)

Prepared for
Dr. Mahmood Osman Imam
Professor
Department of Finance
University of Dhaka

We are………..
Sl. No. Name Roll No.
1 Saifuddin Ahmed 15-010
2 Sajib Dey 15-034
3 Tanushri Chanda 15-042
4 Tahmina Akter 15-050
5 Hasibul Hasan 15-138

Put call Parity
 Put call parity is a relationship that shows the long run
equilibrium relationship between the value of a
European call with a certain exercise price and exercise
date and the value of a European put with the same
exercise price and same exercise date and vice versa.
c + Ke -rT = p + S0

Continued………
 Consider the following 2 portfolios:
Portfolio A: One European call option on a stock + One
Zero-coupon bond
Portfolio C: One European put option + One share of
the stock
Conclusive Condition: Max(ST , K )

Arbitrage Opportunity…
Arbitrage opportunity is created when put-call parity does not hold. An
Arbitrageur can make profit either buy the call and short both the put and the
stock or short the call and buy both the put and the stock.
Example:
Stock Price = $31; interest rate = 10%; call price = $3. Both put and call have
same strike price of $30 and three months to maturity. When the price of 3
month European put option is $2.25
c + Ke –rT = 3 + 30e –0.1*3/12 = $32.26
p + S0 = 2.25 + 31 = $33.25
Again When the price of 3 month European put option is $1
c + Ke –rT = 3 + 30e –0.1*3/12 = $32.26
p + S0 = 1 + 31 = $32.00

Three month put price = $2.25 Three month put price = $1
Action now:
Buy call for $3
Short put to realize $2.25
Short the stock to realize $31
Invest $30.25 for 3 months
Action in 3 months if ST  $30:
Receive $31.02 from investment
Exercise call to buy stock for $30
Net profit = $1.02
Action in 3 months if ST  $30:
Receive $31.02 from investment
Put exercised: buy stock for $30
Net Profit = $1.02
Action now:
Borrow $29 for 3 months
Short call to realize $3
Buy put for $1
Buy the stock for $31
Action in 3 months if ST  $30:
Call exercised: sell stock for $30
Use $29.75 to repay loan
Net profit = $.27
Action in 3 months if ST  $30:
Exercise put to sell stock for $30
Use $29.75 to repay loan
Net profit = $.27

American options
Put call parity holds for European options. Because
when early exercise is not possible we can argue that
the two portfolios worth the same at time T must be
worth the same at earlier times. When early exercise is
possible the argument falls down. It is said that early
exercise of American call option on a non dividend
paying stock is never optimal, but under some
circumstances early exercise of American put option
on a non dividend paying stock is optimal. On the
other hand, when there are dividends, it can be
optimal to exercise either call or put option early.

Early exercise of a non dividend paying American call
options is not optimal.
American Call Option: (Non-dividend)
=> Delaying exercise delays the payment of the strike price. This means that
the option holder is able to earn interest on the strike price for a longer
period of time.
=> Delaying exercise provides insurance against the stock price falling
below the strike price by the expiration date.
=> No income is sacrificed.

American Call Option: Bounds
As American call options are never exercised early with no
dividends they are equivalent to European call options. So
C=c
max(S0—Ke –rT , 0) ≤ c , C ≤ S0

American Put Option
American put is a trade off between the time value of
money and the insurance value of a put. An American
put when held in conjunctions with the underlying
stock provides insurance. It guarantees that the stock
can be sold for the strike price K. If the put is exercised
early, the insurance ceases. But the option holder
receives the strike price immediately and is able to
earn interest on it between the time value of the early
exercise and the expiration date.

American Put Option: Non-dividend stock
 It can be optimal to exercise an American put option
on a non-dividend-paying stock early. Indeed, at any
given time during its life ,put option should always be
exercised early, if it is sufficiently deep in the money
 Like call option, a put option can be viewed as
providing insurance. A put option when held in
conjunction with stock insures the holder against the
stock price falling below a certain level.

What is the lower bound for the prce of 6 month call
option on a non-dividend paying stock when the stock
price is 80,The strike price is $75. and the risk free
interest rate is 10% per annum???
=80−75e –0.1*0.5
=$8.66

American Put Option: Bounds
Upper and lower bound for American put option is-
max(K–S0, 0) ≤ P ≤ K
max(K–S0, 0)
American Put Price in
this region
S0
Put Price
K
max(Ke –rT –S0, 0)
European Put Price in this
region
S0
Put Price
Ke –rT

Extensions of Put-Call Parity
 American options; D = 0
S0 - K < C - P < S0 - Ke -rT
 European options; D > 0
c + D + Ke -rT = p + S0
 American options; D > 0
S0 - D - K < C - P < S0 - Ke -rT