1. Unit 3(a)

2.  Price of option – Fair price and Market price
 Principle of call option pricing:
o Minimum value of a call
o Maximum value of a call
o Value of a call at expiration
o Lower bound of call
o European vs. American Call
 Factors Affecting Call Price
o Effect of Exercise Price
o Time to Maturity
o Interest Rate
o Stock Volatility
 American versus European Style Options
 Early exercise of American call on dividend paying stock
and non-dividend paying stock

3.  A contract between two parties—a buyer
and a seller/writer—in which the buyer
purchases from the seller/writer the right to
buy or sell an asset at a fixed price. The
buyer pays the seller a fee called the
premium, which is the option’s price.
◦ An option to buy an asset at a predetermined
price (also known as exercise price) is known as
the call option

4.  American-style:
An option that can be exercised anytime during its
life. The majority of exchange-traded options are
American.
Since investors have the freedom to exercise their
American options at any point during the life of the
contract, they are more valuable than European
options which can only be exercised at maturity.

5.  European-style:
An option that can only be exercised at the end
of its life, at its maturity. European options tend
to sometimes trade at a discount to its
comparable American option. This is because
American options allow investors more
opportunities to exercise the contract.
European options normally trade over the
counter (OTC), while American options usually
trade on standardized exchanges. A buyer of an
European option that does not want to wait for
maturity to exercise it can sell the option to close
the position.

6.  In-the-Money Option
◦ One that would lead to positive cash flows to the
holder if it were exercised immediately
 At-the-Money Option
◦ One that would lead to zero cash flows to the
holder if it were exercised immediately
 Out-of-Money Option
◦ One that would lead to negative cash flows to the
holder if it were exercised immediately

7. Thank You.

8.  Fair Price:
It is a concept defined as a rational and
unbiased estimate of the potential market price of
Option. There are several options pricing models
that use to determine the fair market value of the
option. Of these, the Black-Scholes model is the
most widely used.
 Market Price:
The current price at which an option can be bought
or sold. Economic theory contends that the market
price converges at a point where the forces of
supply and demand meet. Shocks to either the
supply side and/or demand side can cause the
market price to be re-evaluated.

9. 9
 For the Disney JUN 22.50 Call buyer:
-$0.25
$22.50
$0
Maximum loss
Breakeven Point = $22.75
Maximum profit
is unlimited

10. 10
 For the Disney JUN 22.50 Call writer:
$0.25
$22.50
$0
Maximum profit Breakeven Point = $22.75
Maximum loss
is unlimited

11. 11
 For the Disney JUN 22.50 Put buyer:
-$1.05
$22.50
$0
Maximum loss
Breakeven Point = $21.45
Maximum profit = $21.45

12. Minimum Value of a Call Option
 A call cannot have a negative value
 Thus, Vc  0
 For American Calls, Ca  Max [0,(Vs-E)]
 Minimum value also called intrinsic value
 Intrinsic value positive for in-the-money calls
and zero for out-of-money calls
 Usually, call options trade above their
intrinsic value—Why?

13.  Call derives its value from the underlying
asset/stock on which it is written. Therefore,
it cannot never exceed the value of the
underlying asset
 Thus, Vc  Vs

14.  An option's expiration value is its market value at
expiration. In the case of a call, expiration value
is either:
 zero, or
 the difference between the value of
the underlier and the strike price, whichever is
greater.
 If, at expiration, the underlier value is below the
strike price, the option expires worthless.
Note: The value from which a derivative derives its value is called
its underlier.

15. Market Value of a Call at Expiration

16.  Vc  Max [0,{Vs – E(1+r)-T}] exer
 What if Vce < Max [0,{Vs – E(1+r)-T}] leads to
arbitrage
Buy call and risk-free bonds and sell short the
stock. The portfolio will have positive initial cash
flow, because the call price plus the bond price is
less than the stock price.
At maturity, the payoff is either E – ST if E > ST and
0 otherwise.

17.  The lower bound for any non-dividend paying
call option is: i.e. the current stock price
minus the options strike price multiplied by
the natural e, to the power of negative risk-
free interest rate multiplied by the options
time to expiry.
Example:
Current stock price (Vo) = $20
Strike price (E) = $18
Risk-free rate ( r ) = 10%
Time to expiry (T)= 1 year (T)
Value of (e) = 2.718
Soln. The lower bound for this call option is
$3.71

18. If the market is quoting the European call
option at $3.00, such a price is less than the
lower bound or “theoretical minimum”.
What would happen is that an arbitrageur
would short the stock and then buy the call
option.
This will provided the arbitrageur with a cash
inflow of $20.00 – $3.00 = $17.00.
If this amount is then invested for 1 year at the
market risk-free interest rate of 10% per
annum, then the $17.00 will grow to 17x1.1
= $18.79.

19. Therefore at the end of the year, when the option
will expire if the stock price is greater than
$18.00 (the strike price), then the arbitrageur will
be able to exercise the option for $18.00.
By doing this, it will enable them to close out the
short position to make a profit of $18.79 –
$18.00 = $0.79.
This means that the arbitrageur will buy back the
stock at a cheaper rate than what he/she
originally shorted and then invested that money
at the risk-free rate, and then finally met the
shorting obligations by buying the stock back
with the aid of the option at the strike price.

20. What happens if the stock price is less than the strike
price after 1 year (at time of option expiry?)
 Then the arbitrageur will make an even GREATER
profit. This is because he/she will be able to buy
back the stock at the market price which will be
even cheaper than that of the strike price to close
out the shorted stock position.
 For example, if the stock price after 1 year =
$17.00
 The arbitrageur’s profit will then = $18.79 – $17.00
= $1.79
 Which is $1 more than the previous example where
the stock is greater than the strike price.
 The additional difference in profit is the difference
between the strike price and the stock price.

21.  An American call must be worth at least as
much as a European call with the same terms.
◦ Ca  Ce
 An American call on a non-dividend paying
stock will never be exercised early, and we
can treat it as if it is a European call

22.  Time to maturity
 Exercise price
 Interest rate, and
 Stock volatility

23.  Two American call options differ only in their
times to expirations, one with a higher time
to expiration will be worth at least as much as
a shorter-lived American call with the same
terms
 When will the longer-lived call is worth the
same as shorter-lived call?

24.  The price of a European call must be at least as high as the
price of an otherwise identical European call with a higher
exercise price
◦ Ce(Vs, Elow,T)  Ce(Vs, Ehigh, T)
 The price of an American call must be at least as high as the
price of another otherwise identical American call with a higher
exercise price
◦ Ca(Vs, Elow,T)  Ca(Vs, Ehigh, T)

25.  The difference in the price of two American calls that differ
only by their exercise price cannot exceed the difference in
their exercise prices
◦ Ca(Vs, Elow,T) –Ca(Vs, Ehigh, T)  (Ehigh – Elow)
 The difference in the price of two European calls that differ
only by their exercise price cannot exceed the present value
of the difference in their exercise prices
◦ Ce(Vs, Elow,T) –Ce(Vs, Ehigh, T)  (Ehigh – Elow)(1+r)-T
 Or,
 Ce(Vs, Elow,T) –Ce(Vs, Ehigh, T)  (Ehigh – Elow)/(1+r)T

26.  A call option is a deferred substitute for the
purchase of the stock
 If the stock price is expected to rise, the
investor can either choose to buy the stock
or buy the call. Buying the call will cost far
less than purchasing the stock. Invest the
difference in risk-free bonds.
 If rates rise, the combination of calls and
risk-free bonds will be more attractive

27.  Volatility gives rise to risk and need to buy
insurance
 Greater volatility increases the gains on the call if
the stock price rises big time, and
 Zero downside risk if the stock price declines big
time
 Zero downside risk : this is the cushion against
loss, in case of a price decline by the underlying
security, that is afforded by the written call
option. Alternatively, an price amount equal to
the option premium)

28. Early Exercise of a call on a dividend paying
stock
 The exercise of an option contract before its
expiration date.
 Exercise it just before the ex-dividend date if
the DPS exceeds Speculative value of the call
 Do not exercise it if the DPS is less than the
speculative value of the call

29.  If you purchase before the ex-dividend date,
you get the dividend.
Declaration
Date
Ex-Dividend
Date
Record Date Payable Date
7/27 8/6 8/10 9/10

30. Early Exercise of a call on a non-dividend paying
stock
 By exercising a call option early, not only do you
accept full downside risk in the stock but you also
throw away the time value of that call option. To put
it in a more distressing way, you throw money away
in exchange for accepting more risk!
 It does not matter how short of a time period you
intend to hold the stock, either. Even if you plan to
sell the stock the next day, you’re still at risk of
some serious negative news announced before the
opening bell.
 It’s important to remember that, with a call option,
your purchase price is locked in. It doesn’t matter
how high the stock’s price may rise; you will always
be able to purchase it for the strike price, so there
really is no need to take delivery of the stock early.