principles of option pricing
TRANSCRIPT
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Principles of Option
PricingBy: Ajay Mishra
JSSGIW faculty of Management, Bhopal
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Principles of Option Pricing
Arbitrage opportunitiesare quickly eliminated
by investors.(lets see examples)
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Situation (1) Suppose in a game you draw a ball from a box
known to contain three black and three white
balls. If you draw a black ball you receive nothing.
If you draw a white ball you receive Rs. 10.
Will you Play?
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Situation (1) As no entree fee is mentioned , most people
will play.
You incur no cash outlay up front and havethe opportunity to earn Rs. 10.
Of course this opportunity is too good, but no
one will offer you such option withoutcharging any entry fee.
(lets play another one)
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Situation (2) Now suppose that a fair fee to play Game I is
Rs.4
And for a game II the person offers you to payRs. 20 if you draw a white ball and nothing if
you draw a black ball.
Will the entry fee be higher or lower?
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Situation (2) If u draw a black ball you receive the same
payoff as in game I but if you draw a white
ball you receive a higher payoff.
You should be willing to pay more to play
game II because these payoffs dominate thoseof game I.
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From these simple games and opportunities it
is easy to see some basic principles of how
rational people behave when they faced withrisky situations.
The collective behaviour of rational investors
operates in an identical manner to determinethe fundamental principles ofoption pricing.
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Basic Terminology in Option Pricing
The following symbols are used further:
S = Stock price today
X = Exercise Price. T = Time to expiration
r = Risk free rate
S = Stock price at options expiration; after
the passage of a period of time of legth T
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Basic Terminology in Option Pricing C(S,T,X) = Price of a call option in which the
stock price is S, the time to expiration is T,
and the exercise price is X.
P(S,T,X) = Price of a put option in which the
stock price is S the time to expiration is T andthe exercise price is X.
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Basic Terminology in Option Pricing Ca(S,T,X) = American Call
Ce(S,T,X) = European Call
If there is no a or e subscript, the call canbe either American or a European Call.
In the case where two options differ only by
exercise price, the notation C(S,T,X1) andC(S,T,X2) will identify the prices of the calls
with X1 less than X2.
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Basic Terminology in Option Pricing
Always a the subscript of the lower exercise
price is smaller than the higher exercise
price. In the case where two options differ only by
time to expiration will be T1 and T2, where
T1< T2. Identical adjustments will be made for put
option prices.
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Time to expiration
The time to expiration is expressed as a decimal
friction of a year.
For example if the current date is April 9 and theoptions expiration date is July 18.
We count the number of days between these two
dates.
That would be : April= 21, May =31, June = 30 andJuly = 18 . Total 100 days.
The time to expiration would be 100/365 = 0.274
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Dividend
For most of the examples we shall assume
that the stock pays no dividends.
If during the life of the option, the stock paysa dividends of D1, D2.. And so forth, then
we can make a simple adjustment and obtain a
similar adjustment and obtain similar results. To do so we simply subtract the present value
of the dividends.
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Risk free rate of return r
It is the rate earned on a riskless investment.
An example of such an investment is a treasury bill.
T-bills pay interest not through coupons but byselling at a discount.
The T-bill is purchased at less than face value.
The difference between the purchase price and the
face value is called the discount.
The discount is the profit earned by the bill holder.
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Risk free rate of return r
The rate of return on a T-bill of comparable
maturity would be a proxy for the risk free of
return. All T-bills mature on Thursday because most
exchanged traded option expire on Fridays.
There is always a T-bill maturing the daybefore expiration.
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Risk free rate of return r
Bid and Ask discounts for several T bills for thebusiness day of May 14 of particular year are as
follows.
Maturity Bid Ask
5/20 4.45 4.37
6/17 4.41 4.37
7/15 4.47 4.43
Bid and Ask figures are the discount quoted
by dealers trading in T-bills
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Risk free rate of return r
With first T-bill case if we take expiration
date May 21.
To find the T-bill rate we use the average ofthe bid and ask discount. Which is
(4.45+4.37)/2=4.41
Then we find the discount from par value as-4.41(7/360)= 0.08575, using the fact that the
option has seven days until the maturity
Thus the price is 100-0.08575= 99.91425
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Risk free rate of return r
The yield on our T-bill is based on theassumption of buying it at 99.91425 and
holding for seven days, at which time it will
be worth of 100.
This is a return of
(100=99.91425)/99.91425=1.000858
Suppose we repeat this transaction everyseven days for a full year, the return would
be: ((1.000858)^365/7)-1 = 0.0457
Which can be taken as risk free rate of return.
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Principles of call option pricing
Minimum Value of a call
Maximum Value of Call
Value of a Call at Expiration.
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Minimum Value of a Call
CALL PUT
ExercisePrice
May June July May June July
120 8.75 15.4 20.9 2.75 9.25 13.65
125 5.75 13.5 18.6 4.6 11.5 16.6
130 3.6 11.35 16.4 7.35 14.25 19.65
The below given is one example option data for a
stock for the date may 14. Try to find out the intinsic
value and time values for the call.
Assuming the current stock price is 125.94 andExpirations : May 21, June 18, July 16
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Minimum value of a Call
A Call is an instrument with limited liability.
If the call holder sees that it is advantageous
to exercise he it, the call will be exercised. If exercising it will decrease the call holders
wealth he will not exercise it.
The option cannot have negative value,because the holder cannot be forced to
exercise it. Therefore, C(S,T,X)>= 0
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Minimum value of a Call
For an American Call it will be
Ca(S,T,X)>= Max(0, S-X)
Max(0, S-X) means take the maximum value of twoarguments, zero or S-X
The minimum value of an option is called its
intrinsic value some time referred to as parity value,
parity or exercise value.
Intrinsic value, which is positive for in the money
calls and zero for out of the money calls.
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Minimum value of a Call
Intrinsic Values and Time Values of a given Call
TimeValue
Exercise
Price
Intrinsic
Value
May June July
120 5.94 2.81 9.46 14.96
125 0.94 4.81 12.56 17.66
130 0.00 3.60 11.35 16.40
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Minimum value of a Call
To check the intrinsic value rule we take the
June 120 Call.
The stock price is 125.94 and the exerciseprice is 120.
Taking Max(0, 125.94-120) = 5.94
Now what would happen if the call werepriced at less than 5.94 , say Rs. 3
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Minimum value of a Call
An option trader could buy the call for Rs.3,
Exercise it
You can purchase the stock for Rs. 120 and then sell
the stock for Rs. 125.94. This arbitrage would provide a risk less profit of
Rs.2.94
All investors would do it, which would drive up the
option price When the price of the option reached Rs.5.94, the
transaction would no longer be profitable.
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Minimum value of a Call
What if the exercise price exceeds the stock
price ? Do it with exercise price = 130
Max(0, 125.94-130) = 0 Then minimum value be zero
Now check at all the given calls.
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Minimum value of a Call
The calls with an exercise price of Rs. 125 have a
minimum values Max(0,125.94-125) = 0.94 and are
priced at no less than 0.94.
The calls with an exercise price of 130 have a
minimum value of Max(0, 125.94-130) = 0
Al those option obviously have nonnegative values
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Minimum Value of a call
The intrinsic value concept applies only to an
American call, because a European call can be
exercised only the expiration day. The price of an American call normally exceeds
its intrinsic value.
The difference between the price and theintrinsic value is called the time value or
speculative value of the call.
Which is defined as Ca(S,T,X)-Max(0,S-X)
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Minimum value of a call
The time value refers what traders are willing
to pay for the uncertainty of the underlying
stock. Time values increase with the time of
expiration
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Minimum value of a European call
(a) European
Call
Price
Stock Price (S)
The call price lies in a shaded area . The European call price lies in the entire area.
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Minimum value of a American call
(b) American
Call
Price
Stock Price (S)
X
Max(0,S-X)
The American call price lies in a smaller area. This does not mean that the American call
price is less than the European call price but only that its range of possible values is narrower.
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Maximum Value of a call
C(S,T,X)S
The most one can expect to gain from the call
is the stocks value less than the exerciseprice. Even if the exercise price were zero,
No one would pay more for the call than for
the stock. However, one call that is worth the stock
price is one with an infinite maturity.
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Minimum and maximum values of a
European call
(a) European
Call
Price
Stock Price (S)
The call price lies in a shaded area . The European call price lies in the entire area.
0
S
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Minimum and maximum values of an
American call
(b) American
Call
Price
Stock Price (S)
The call price lies in a shaded area . The European call price lies in the entire area.
0 X
Max(0,S-X)
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Value of a Call at Expiration
The prospect of future stock price increases isirrelevant to the price of the expiring option,
which will be simply its intrinsic value.
At expiration an American option and aEuropean option are identical Instruments.
Therefore this rule holds for both the options.
C(St,0,X) = Max (0,St-X
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The value of a Call at Expiration.
CallPrice
Stock Price at Expiration (St)
X
Max(0,St-X)
0
*Because of the transaction cost of exercising the option, it could be worthslightly less than the intrinsic value.
C(ST,0,X)
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Effect of time to Expiration.
Consider two American Calls that differ only
in their time to expiration so their price will
be as follows :
(1) Ca(S,T1X)
(2) Ca(S,T2X) (T2 is greater than T1 )
Now think which of these two option will
have a greater value?
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Effect of time to Expiration.
Suppose that today is the expiration day ofthe shorterlived option. The stock price is
the value of the expiring option is :
Max (0, ST1-X).
The second option has a time to expiration of
T2-T1.
Its minimum value is MAX(0,St1-X). Thus
when the shorter lived option expires, its
value is the minimum value of the longer
lived one.
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References:
Don M. Chance, Derivatives and Risk
Management Basics, India edition.
John C. Hull and Sankarshan Basu, Options,Futures, and Other Derivatives seventh
edition.
http://content.icicidirect.com/learning/university.htm